paradigms in mathematics, physics, and biology - their philosophical roots(d.strauss-2001)

Paradigms in

Mathematics, Physics,

and Biology:

their Philo soph i cal

Roots

DFM Strauss

(2001)

© TEKSKOR BKP.O. Box 28378

Bloemfontein 9300

2001

ISBN 0-620-31329-3

Preface

The Re nais sance op ti mism of the 14th and 15th cen tu ries soon wit nessed therise and suc cesses of the mod ern nat u ral sci ences. Ga li leo claimed that thelan guage of na ture is writ ten in math e mat i cal sym bols and Al ex an der Popeas sessed the sig nif i cance of New ton's Principia (1686) with his well-knownap pre ci a tion: “Na ture and Na ture's laws hid in night: God said: Let New tonbe! and all was light.” The be lief that the math e mat i cal nat u ral sci ences will be able to un der stand and ex plain all of re al ity cli maxed dur ing the age of rea son, the En light en ment of the 18th cen tury. This clas si cal era of mod ern ism – dom -i nated by an un bri dled trust in the ca pac i ties of hu man rea son – even tu allywas chal lenged by the rise of historicism at the be gin ning of the 19th cen turyand af ter the turn from “thought” to “lan guage” by the end of the 19th cen turythrough the new em pha sis on un der stand ing (her me neu tics) also by the rise of con tem po rary postmodernism.

How ever, re cently Alan Sokal and Jean Bricmont ex posed the un founded way in which prom i nent postmodern think ers abuse the math e mat i cal nat u ral sci -ences in their writ ings (Fash ion able Non sense: Postmodern In tel lec tuals’Abuse of Sci ence, Pica dor, New York, 1998; cf. the Ger man edi tion: Ele -ganter Unsinn: Wie die Denker der Postmoderne die Wissenschaften miß -brauchen, C.H. Beck, München 1999). These au thors did not de velop the pos -i tive side of their cri tique by ex plor ing the in du bi ta ble in ter ac tion be tweenphi los o phy and the nat u ral sci ences. The aim of this book is pre cisely to lookat the con struc tive role of de ci sive philo soph i cal pre-suppositions in the nat u -ral sci ences.

Sci en tific re flec tion is em bed ded in the o ret i cal think ing. The lat ter dif fer en ti -ates into iden ti fy ing and dis tin guish ing on the ba sis of sim i lar i ties and dif fer -ences – which, in the case of schol arly ac tiv i ties, are ar tic u lated in a uniquelan guage [see the sketch on page (iv)] . Yet the re al ity in which we live tran -scends the lim its of log i cal think ing and lin gual ar tic u la tion even though wehave to ac knowl edge that we do not have ac cess to the world out side the do -mains of thought and lan guage. Schol arly re flec tion is con stantly in volved inan an a lyt i cal aware ness of a given “more-than-logical” di ver sity which is al -ways ar tic u lated by a lan guage which is con stantly sub ject to sub tle al ter -ations in mean ing and in ter pre ta tion. The com pli cated in ter ac tion of theseand other even more cen tral and di rec tion-giving fac tors con sti tute the con -flict ing par a digms op er a tive within the dis ci plines.

Of course this rec og ni tion also at once ex plains the his to ric ity of sci en tificendeavours – the fact that we have to rec og nize the in ev i ta ble his tor i cal un -der pin nings of hu man ac tiv i ties. It may there fore of ten be help ful also to pay

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at ten tion to the his tory of the nat u ral sci ences in or der to come to a better un -der stand ing of the di ver gence be tween al ter na tive the o ret i cal par a digmswithin them.

The claim that the nat u ral sci ences are free from philo soph i cal pre-suppo -sitions is it self in deed the ef fect of the in flu en tial philo soph i cal ori en ta tion ofpos i tiv ism and neo-positivism. Against the back ground of de vel op mentswithin mod ern phi los o phy of sci ence this work high lights di verg ing trends ofthought within the dis ci plines of math e mat ics, phys ics and bi ol ogy while fo -cus ing upon the in ev i ta ble philo soph i cal dis tinc tions en tailed in these dif fer -ences. Al though many of the se lected is sues dis cussed delve into some of thefron tiers of nat u ral sci en tific re flec tion the reader with a sec ond ary school ac -quain tance with these dis ci plines will be able to fol low the ar gu men ta tion ex -pressed in the var i ous chap ters.

The Au thor(Jan u ary 2001)

(iv)

Analysis

SimilaritiesIdentification = Synthesisi.e., the bringing together

of the features unitedin a concept

Modal analogies - for examplephysical space/original space;social distance/spatial distance

Entitary analogies - for examplethe elbow of my finger;

the head of the mountain;the modal grid of reality

Similarities-shown-in-the-differences

Differences-evinced-in-the-similarities

Ante- and retrocipations Metaphors

= ANALOGY =

Differences

Abstraction

Social distance

Spatial distance

shown in

Simila

rity

Diff

ere

nc

e

Example

Con tents

Fore word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (iii)

Chap ter IFun da men tal Ques tions in the Phi los o phy of Sci ence

An Ini tial Com ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1(Neo-)pos i tiv ism and re ac tions to it. . . . . . . . . . . . . . . . . . . . . . . . 3Back ground to neo-pos i tiv ism . . . . . . . . . . . . . . . . . . . . . . . . . . 3Re ac tion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4Con tem po rary phi los o phy of sci ence . . . . . . . . . . . . . . . . . . . . . . . 5

The re la tion be tween Pop per, Kuhn and Sneed. . . . . . . . . . . . . . . . 6The Dy nam ics of The ory For ma tion . . . . . . . . . . . . . . . . . . . . . 6Kuhn's crit ics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

Foun da tional prob lems and ba sic dis tinc tions . . . . . . . . . . . . . . . . . . 8The unique na ture of sci ence . . . . . . . . . . . . . . . . . . . . . . . . . . 11

Non-dis tinc tive char ac ter is tics . . . . . . . . . . . . . . . . . . . . . . . 11The dis tinc tive char ac ter is tic of sci en tific(the o ret i cal) thought . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Is Phi los o phy a Sci ence? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13Phi los o phy and the Spe cial Sci ences . . . . . . . . . . . . . . . . . . . . . . 14Philo soph i cal Foun da tional Ques tions in the Spe cial Sci ences . . . . . . . . . 15

Chap ter II

Foun da tional Philo soph i cal prob lems in Math e mat ics

In tro duc tory re marks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Def i ni tions of math e mat ics . . . . . . . . . . . . . . . . . . . . . . . . . . . 19The In fi nite in Greek Thought . . . . . . . . . . . . . . . . . . . . . . . . . . 21A few fur ther con tours from the his tory of the in fi nite . . . . . . . . . . . . . 26In fin i tes i mals and the sec ond foun da tional cri sis of math e mat ics . . . . . . . . 28

Can tor and Ar is totle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32(a) Ar is totle's ob jec tions against the ac tual in fi nite . . . . . . . . . . . . . . . . 32(b) Con ti nu ity in Ar is totle and Can tor-Dedekind . . . . . . . . . . . . . . . . . 34Non-denumerability: Can tor's Di ag o nal Proof. . . . . . . . . . . . . . . . . . . 36

Com ment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37The third foun da tional cri sis in Math e mat ics . . . . . . . . . . . . . . . . . . 38Di ver gence of opin ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Ques tion ing com pleted in fin i tude . . . . . . . . . . . . . . . . . . . . . . . . 42The in flu ence of intuitionism on the ap proach of Dooyeweerd . . . . . . . . . 45Brief sys tem atic as sess ment of the re la tion ship be tween the po ten tial and the ac tual in fi nite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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Chap ter IIIBa sic ques tions in Phys ics

The Prej u dice against Prej u dices . . . . . . . . . . . . . . . . . . . . . . . . 53Dis crep ancy be tween phi los o phers of sci ence and theprac ti tio ners of sci ence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Prop erty terms – the Achil les’ heel of pos i tiv ism . . . . . . . . . . . . . . . . 55The mea sure ment of time and modal time or ders . . . . . . . . . . . . . . . . 56

Time in the as pects of num ber and space . . . . . . . . . . . . . . . . . . 58The kinematical and the phys i cal time or der . . . . . . . . . . . . . . . . 59

The unique ness of Con stancy and Dy nam ics . . . . . . . . . . . . . . . . . . 60Per pet ual mo tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60Closer re flec tions on con stancy and change. . . . . . . . . . . . . . . . . 61

The core of Ein stein’s the ory of rel a tiv ity . . . . . . . . . . . . . . . . . . . . . 62

An al ter na tive for mu la tion of the first main law of ther mo dy nam ics . . . . . . . 63

The the ory of rel a tiv ity and rel a tiv ism . . . . . . . . . . . . . . . . . . . . . . . 63

De ter min ism and indeterminism . . . . . . . . . . . . . . . . . . . . . . 64Or der and de lim i ta tion in phys ics . . . . . . . . . . . . . . . . . . . . . . . . 65

The fi nite and lim ited cos mos in Greek cul ture . . . . . . . . . . . . . . . 66Are there in ac ces si ble lim its in the nat u ral sci ences? . . . . . . . . . . . . 67The un lim ited but fi nite uni verse in Eintein’s the ory of rel a tiv ity . . . . . 67Complementarity – lim its to ex per i men ta tion . . . . . . . . . . . . . . . . 67En ti ties with a phys i cal qual i fi ca tion . . . . . . . . . . . . . . . . . . . . 68

The unity and iden tity of an en tity. . . . . . . . . . . . . . . . . . . . . . . . 73Phys i cally qual i fied en ti ties . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

The wave par ti cle du al ity and the idea of the typ i cal to tal itystruc ture of an en tity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76Phys i cally qual i fied struc tural interlacement . . . . . . . . . . . . . . . . 77

Par a digms in Math e mat ics, Phys ics, and Bi ol ogy:Their Philo soph i cal Roots

Chap ter IVThe Mo saic of philo soph i cal stances in mod ern bi ol ogy

In tro duc tion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83Biotically qual i fied en ti ties . . . . . . . . . . . . . . . . . . . . . . . . . . . 84What guar an tees the iden tity of liv ing things? . . . . . . . . . . . . . . . . . 85The or i gin of liv ing things – a bi o log i cal bound ary ques tion . . . . . . . . . . 87Are vi ruses a tran si tional form be tween ma te rial and liv ing en ti ties? . . . . . . 88Nomi nal ist struc tural un der stand ing in mod ern bi o log i cal lit er a ture . . . . . . 90Structureless con ti nu ity ver sus struc tural dis con ti nu ity . . . . . . . . . . . . . 92Con ti nu ity of de scent? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94The struc ture of a nu clear liv ing cell . . . . . . . . . . . . . . . . . . . . . . 97Phys i cal-chem i cal con stit u ents in the liv ing cell . . . . . . . . . . . . . . . . 98Organelles – the dif fer ent or gans in the cell . . . . . . . . . . . . . . . . . . 100The quest for a ba sic de nom i na tor . . . . . . . . . . . . . . . . . . . . . . . 101Con flict ing views de spite “the same facts”! . . . . . . . . . . . . . . . . . . 103

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Neo-Dar win ism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104Vi tal ism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105Ho lism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107Emer gence evolutionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Pan-psychism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108Me tab o lism as first level of free dom . . . . . . . . . . . . . . . . . . . . . . 109A new mech a nis tic ap proach . . . . . . . . . . . . . . . . . . . . . . . . . . 110Struc tural di ver sity founds structureless fan ta sies . . . . . . . . . . . . . . . 110Struc tural di men sions of the cell – an enkaptic struc tural whole . . . . . . . 111

Chap ter VRe marks about the mys tery of be ing hu man

Con ti nu ity or dis con ti nu ity be tween the var i ous lev els? . . . . . . . . . . . . 117Is the fos sil-re cord con clu sive? . . . . . . . . . . . . . . . . . . . . . . . . 119Is there any thing dis tinc tive to hu man tools? . . . . . . . . . . . . . . . . . 124Do an i mals share the di men sion of (hu man) logi cality? . . . . . . . . . . . . 128The hu man be ing as “Homo symbolicus”?. . . . . . . . . . . . . . . . . . . 132The an a tom i cal con di tions of hu man speech . . . . . . . . . . . . . . . . . . 133Do hu man be ings have ‘speech-or gans’? . . . . . . . . . . . . . . . . . . . 135Does hu man ex pe ri ence of the world dif fer from that of the an i mals? . . . . . 135The un spe cial ized traits of the hu man body . . . . . . . . . . . . . . . . . . 137Is the hu man be ing to be seen as a de fi cient crea ture? . . . . . . . . . . . . . 139The ontogenetic unique ness of be ing hu man. . . . . . . . . . . . . . . . . . 142Hu man free dom – the pre dom i nantly neg a tive ap proachof mod ern phi los o phy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144Au ton o mous free dom ver sus nat u ral cau sal ity . . . . . . . . . . . . . . . . . 144Bridg ing the abyss te leo logi cally. . . . . . . . . . . . . . . . . . . . . . . . 145Entelechie neg a tively de scribed: the in flu ence of Hans Driesch . . . . . . . . 146Re in forced di a lec tics: Ex is ten tial ism and Ex is ten tial Phe nom en ol ogy . . . . 148Free dom at the mo lec u lar level. . . . . . . . . . . . . . . . . . . . . . . . . 148The re jec tion of struc tural con di tions: nomi nal ism . . . . . . . . . . . . . . 149The com mon root of di verg ing trends in mod ern phi los o phy . . . . . . . . . 152Hu man free dom: a sub jec tive re sponse to nor ma tive con di tions. . . . . . . . 154Con clu sion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

Con sulted Works. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

In dex of Tech ni cal Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . 171In dex of Per sons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175

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Paradigms in Mathematics,Physics, and Biology:

Their Philosophical Roots

Chapter I

Fundamental Questions in the Philosophy of Science

An Initial CommentSince the con cept nat u ral sci ence is usu ally iden ti fied with phys ics it is proper to be gin our train of thought with the words of a well known phys i cist of thiscen tury. Carl Friedrich von Weizsäcker once com mented on the pre sup po si -tions of mod ern nat u ral sci en tific thought, say ing that “it is an em pir i cal factthat vir tu ally all lead ing phys i cists of our time phi los o phize” (1972:42).This is no brand new rev e la tion, al though it is care fully hid den be hind themask which a par tic u lar philo soph i cal ap proach forced upon phys ics at thebe gin ning of the 20th cen tury. Pos i tiv ism – also later known as neo-positivism– con vinced nat u ral sci en tists that all good sci ence must be prac tised with outany prej u dices or pre sup po si tions what so ever, and that good sci ence couldonly be prac ticed by al ways re fer ring to “em pir i cal phe nom ena” when mak -ing sci en tific state ments. We can also thank this philo soph i cal trend for a par -tic u lar elab o ra tion of the op po si tion be tween thought and ex pe ri ence. Ein -stein, for in stance, was strongly in flu enced by Ernst Mach – a lead ing nat u ralsci en tist who sup ported pos i tiv ism around the end of the 19th cen tury – butlater de vel oped to wards giv ing pri or ity to the o ret i cal thought. In his Pref aceto the Eng lish trans la tion of Werner Heisenberg's work on the re la tion ship be -tween phys ics and phi los o phy, Northrop com ments on Ein stein's em pha sisthat “...the phys i cal sci en tist only ar rives at his the ory by spec u la tive means.The de duc tion in his method runs not from facts to the as sump tions of the the -ory but from the as sumed the ory to the facts and the ex per i men tal data” (cf.Heisenberg, 1958:3-4).In his Her bert Spencer lec ture at Ox ford on 10 June 1933 Ein stein him self cat -e gor i cally stated that no bridge could be erected be tween pure log i cal thoughtand our ex pe ri ence of re al ity: “Pure log i cal think ing can give us no knowl -edge what so ever of the world of ex pe ri ence; all knowl edge about re al ity be -gins with ex pe ri ence and ter mi nates in it” (con tained in Coley & Hall,1980:144). This state ment touches on the core of epis te mol ogy – i.e. the ques -tion how we ob tain knowl edge of re al ity. This ques tion, how ever, is a typ i cal

1

philo soph i cal ba sic ques tion for ev ery con ceiv able par tic u lar sci en tific dis ci -pline. The an swer we give to it can there fore not be di vorced from par tic u larphilo soph i cal pre sup po si tions and ori en ta tions.

The cor rect ness of this state ment co heres inter alia with the im por tant in flu -ence ex erted his tor i cally – and still to day – by par tic u lar philo soph i cal ap -proaches. Af ter all, the words quoted from Ein stein truly gain depth when un -der stood against the back ground of Im man uel Kant's “Kritik der reinenVernunft” (Cri tique of Pure Rea son) which was orig i nally al ready writ ten in1781 (sec ond im pres sion 17872). Kant was con vinced by his Brit ish pre de ces -sor, Da vid Hume, that ex act nat u ral laws can not be founded in “ex pe ri ence.”Kant could not, how ever, agree with Hume's con vic tion (strik ingly sim i lar toa facet of Ein stein's state ment above) that we ob tain all knowl edge from sen -sory per cep tion or “ex pe ri ence.”1 Kant was im pressed by the abil ity of hu manbe ings to men tally for mu late the laws which the per cep ti ble things in na tureobey. He con se quently fo cused on the (epistemological) ques tion how suchknowl edge is pos si ble.

Kant was par tic u larly im pressed by the con tri bu tion of Ga li leo to the de vel op -ment of the mod ern nat u ral sci ences. Ga li leo's for mu la tion of his law of in er -tia fol lowed ex actly the way of a pure thought ex per i ment. In his fa mous 1638 trea tise on “two new sci ences” Ga li leo used the fol low ing thought ex per i -ment: if a body is put in mo tion on an in def i nitely ex tended track, then thisbody would con tinue its mo tion in fi nitely, i.e. it would not dis con tinue its mo -tion ex cept if some thing ex erts power on it (e.g. grav ity or fric tion).

From this Kant draws the fol low ing con clu sion: if it is pos si ble for Ga li leo tofor mu late a thought ex per i ment out of the spon ta ne ous sub jec tiv ity of his the -o ret i cal thought and to de duce a nat u ral law from it – the kinematical law ofin er tia – this must mean that el e ments of knowl edge are pre vi ously pres ent inthe hu man mind, which in the first place makes our knowl edge of re al ity pos -si ble. With out go ing any deeper into this par tic u lar ques tion, it is al ready clear that we can only re ally un der stand Ein stein's state ment if it is placed in thecon text of this foun da tional philo soph i cal epistemological ap proach. Ein -stein's view of “ex pe ri ence” reaches back to the train of thought of Hume,while his view of the mind (thought) ap pears to be sim i lar to that of Kant,while si mul ta neously dif fer ing from it since he assigns a more in de pend entsta tus to the o ret i cal thought.2

Kant's so-called ‘Co per ni can’ rev o lu tion in epis te mol ogy, in as crib ing thepri macy no lon ger to the ‘ob ject’ but to the (for mal law-giving) sub ject, re in -forced the no tion of things within na ture as ‘ob jects’. Some one in clined to de -

2

1 “To hate, to love, to think, to feel, to see, all this is noth ing but to per ceive.” Hume made thisstate ment in his book: A Trea tise of Hu man Na ture, I,2,6.

2 Al though Ein stein dif fers from Kant's view, he nev er the less did not want to be pre cise as faras his own re la tion ship to Kant is con cerned. In April 1922 the Société Française dePhilosophie ar ranged a dis cus sion about the sig nif i cance of Ein stein's the ory of rel a tiv ity forphi los o phy. An swer ing a ques tion con cern ing his re la tion ship with Kant Ein stein merelystated that ‘each phi los o pher has his own Kant' and that he can not in fer from the ques tionwhat Kant in ter pre ta tion it pre sup poses.

fend the neu tral ity of ob ser va tion nor mally would be will ing to ac cept as themost gen eral ob ser va tion-term the no tion of an ‘ob ject’: all the dif fer entthings in na ture are to be seen as ‘ob jects’. How ever, this ob ser va tion-term init self dis plays the tre men dous subjectivistic as sump tion so deeply in grainedin our West ern no tion of sci ence – as such caus ing the in abil ity to ap praisethings in na ture as gen u ine sub jects, i.e. as be ing sub jected to God'screational or der (law-order) for their ex is tence as ma te rial things, plants or an -i mals. Al though these things could be objectified by humans, this objectifi -cation pre-supposes their pri mary ex is tence as sub jects and not as ob jects!In this brief dis cus sion of the philo soph i cal foun da tions of the nat u ral sci -ences we want to pro vide a brief sur vey of a few per spec tives, us ing a fewprom i nent ap proaches in par tic u lar nat u ral sci ences as ex am ples. The in ten -tion is to si mul ta neously place the dis cus sion in the con text of re cent de vel op -ments in the newer phi los o phy of sci ence which has been greatly stim u lated in par tic u lar by Thomas Kuhn's The Struc ture of Sci en tific Rev o lu tions (1962,19702) – the fo cus of the rest of this in tro duc tory chap ter.

(Neo-)positivism and reactions to it(Neo-)pos i tiv ism can be de scribed as the philo soph i cal idol iza tion of the ex -per i men tal method on the ba sis of sen sory per cep tion. The cen tral con cept ofthe Wiener Kreis, in the twen ties and thir ties of the 20th cen tury, was the “ver -i fi ca tion prin ci ple.”As we briefly in di cated above, the mod ern epistemological her i tage draws adis tinc tion be tween two orig i nal sources of knowl edge: the mind and thesenses (thought and ex pe ri ence – cf. Kant), on oc ca sion amended with in tu -ition as an ul tra-sensory and super-rational or gan of knowl edge. Fol low ingKant we could dis tin guish be tween tran scen den tal ists and empiricists. Thelat ter, par tic u larly well-represented in the Brit ish tra di tion, would even tu allyde velop into (neo-)pos i tiv ism, which de signed a sci en tific meth od ol ogywhich be gins with par tic u lar sen sory data/sense-impressions on the one handand the log i cal con struc tion of en ti ties from these on the other. Thence thepro gres sion of em pir i cal per cep tion-hypothesizing-testing (ver i fi ca tion)-the -ory for ma tion (ver i fied hy poth e sis), a me thod i cal ap proach cer tainly not un -known to the “ex per i men tal” nat u ral sci ences.

Background to neo-positivismRe lated to Kant we find the 19th cen tury pos i tiv ism of Ernst Mach which, onthe grounds of em pir i cal (i.e. sen sory) per cep tion, in cludes only math e mat icsand phys ics in the house of the sci ences. This de lim i ta tion of sci ence led to the po si tion of Wittgenstein, the math e ma ti cian-engineer-philosopher, to thepoint of view that the lim its of my lan guage are the lim its of my world(Tractatus 5.6.). Ac cord ing to Wittgenstein the task of phi los o phy is to de limit the con tro ver sial ter rain of the nat u ral sci ences (= phys ics) (4.113) – and theto tal ity of the nat u ral sci ences con sti tutes the to tal ity of true prop o si tions(4.11). That which tran scends the prop o si tions of phys ics (which is mean ing -ful) and logic (the prop o si tions of which are tau tol o gies and there fore mean -ing less: 4.461) can not be known or lin gually ex pressed – it be longs to the

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sphere of non sense. The ob jec tion that the Tractatus it self would be a vic timof such a de lim i ta tion of sci ence (to logic and nat u ral sci ence) is ob vi ated byWittgenstein with his com ment that his prop o si tions serve an il lu mi nat ingend:

“any one who un der stands me even tu ally rec og nizes them as non sen si cal,when he has used them – as steps – to climb up be yond them. (He must, so topeak, throw away the lad der af ter he has climbed up it” (6.54)!

Reaction

The fa mous phi los o pher of sci ence of the 20th cen tury, sir Karl Pop per, re -acted strongly against this at tempt at de lim i ta tion by Wittgenstein. He in ves ti -gates, for in stance, the fol low ing sen tence by Wittgenstein: “Phi los o phy is nothe ory, but an ac tiv ity” (4.112). This sen tence clearly does not be long to theto tal ity of nat u ral sci en tific prop o si tions, and there fore also not to the to tal ityof true prop o si tions. On the other hand it is not a false prop o si tion ei ther, since if it was, then its ne ga tion would have to be true and there fore be long to thenat u ral sci ences. The only pos si bil ity would then be the men tioned con clu sion of Wittgenstein (6.54): the sen tence is non sen si cal. Al though Wittgensteinad mits with this that the Tractatus is non sen si cal, he de clares in the fi nal para -graph of the Pref ace that the truth of his no tions ap pear to him un as sail ableand de fin i tive. He is even of the opin ion that he has on all car di nal pointsfound the fi nal so lu tion to the prob lems. Pop per re acts sharply to this: “Thisshows that we can com mu ni cate un as sail ably and def i nitely true thoughts byway of prop o si tions which are ad mit tedly non sen si cal, and that we can solveprob lems fi nally by pro pound ing non sense.” The im pli ca tion is that “[i]tmeans that all the meta phys i cal non sense against which Ba con, Hume, Kantand Rus sell have fought for cen tu ries may now com fort ably set tle down, andeven frankly ad mit that it is non sense. For we now have a new kind of non -sense at our dis posal, non sense that com mu ni cates thoughts whose truth is un -as sail able and de fin i tive; in other words, deeply sig nif i cant non sense” (cf.1968, as well as 1966:296ff)!

Pop per asks him self how one can op pose this po si tion of Wittgenstein. Ev erypos si ble ob jec tion against it is af ter all philo soph i cal and there fore non sense!Ac cord ing to Pop per this is sim ply for ti fied dog ma tism, since all that is re -quired, is to delimit the con cept sense (or: mean ing) in an ap pro pri ately nar -row way so as to rid one self of all awk ward ques tions by sim ply say ing thatyou do not find them mean ing ful. Ev ery rea soned ob jec tion to this con cep tionof mean ing is sim ply re jected as non sen si cal: “Once en throned, the dogma ofmean ing is for ever raised above the pos si bil ity of at tack. It is un as sail able and de fin i tive.”

This mean ing-conception of Wittgenstein, with its in cluded de lim i ta tion ofsci ence, is just as un ten a ble as the “ver i fi ca tion prin ci ple” of neo-positivism.The term “log i cal pos i tiv ism” (or log i cal em pir i cism) was brought into be ingto re fer to a group of phi los o phers, lo gi cians and math e ma ti cians who be came known in Vi enna as “der Wiener Kreis.” The move ment orig i nally cen teredaround Moritz Schlick, with philo soph i cally ori ented mem bers such as

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Carnap, Neurath, Feigl, Waismann, Zilsel and Kraft, as well as nat u ral sci en -tific and math e mat i cally ori ented mem bers like Frank, Menger, Gödel andHahn. Carnap, Neurath and Hahn in 1929 pub lished a man i festo en ti tled“Wissenschaftliche Weltauffassung, Der Wiener Kreis.” In this cir cle Witt -genstein's Tractatus was also dis cussed and from it (cf. 4.024) they bor rowedtheir fa mous ver i fi ca tion prin ci ple: the mean ing of a state ment lies in the man -ner by which it is ver i fied.

In his Lan guage, Truth and Logic (1936) A.J. Ayer ex plains that fac tual as ser -tions are sub ject to the fol low ing cri te rion of ver i fi ca tion: a sen tence is mean -ing ful for any spe cific per son if and only if that per son which per cep tionswould lead him (un der cer tain con di tions) to ac cept the prop o si tion as true orto re ject it as false (1967:35). A closer anal y sis causes Ayer to dis tin guish be -tween a strong and a weak sense of ver i fi ca tion. A prop o si tion is ver i fi able inthe first sense, if and only if the truth thereof can be con clu sively de ter minedin ex pe ri ence. A prop o si tion is ver i fi able in the lat ter sense if it is pos si ble toren der the ex pe ri ence prob a ble (1967:36-38). Ayer fully re al izes that gen eralfor mu la tions of laws can not be con clu sively ver i fied – in con se quence he hasto ac cept ver i fi ca tion in the weak sense. In a later pref ace (1946) he is none -the less of the opin ion that there ex ists a class of em pir i cal prop o si tions whichare con clu sively ver i fi able. These are the ba sic prop o si tions which re fers ex -clu sively to the con tent of a sin gle ex pe ri ence and which can be iden ti fied asunique. Ayer is con vinced that he has elim i nated all meta phys ics by means ofthis ver i fi ca tion-criterion.

Contemporary philosophy of science

The newer the ory of sci ence of the past 40 years has re al ized, due to the in flu -ence of Pop per, Toulmin, Polanyi (orig i nally a chem ist) and es pe ciallyThomas Kuhn (phys i cist) that even phys ics is in ev i ta bly gripped by a the o ret i -cal pic ture of re al ity (par a digm) and that it is pos si ble to speak mean ing fullyof an ul ti mate com mit ment in ev ery sci en tific ac tiv ity – a cen tral heart con vic -tion out of which the sci en tist ac counts for the deep est fun da men tal ques tionsof his/her sci en tific prac tice. This re al iza tion came about partly be cause of the non-verifiability of the (neo-positivist) ver i fi ca tion prin ci ple. As an al ter na -tive to (neo-)pos i tiv ism Pop per de fends a crit i cal ra tio nal ism, that is a ra tio -nal ism which rec og nizes that faith in the ra tio nal ity of rea son is it self not ra tio -nal: “we may de scribe it as an ir ra tio nal faith in rea son” (1966:321).1 This cre -

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1 Stegmüller once re marked: “One should not de limit sci ence in or der to cre ate room for faith.Much rather it is the case that one al ready has to be lieve some thing in or der to be able to speak about knowl edge and sci ence as such” (“Man muss nicht das Wissen beseitigen, um denGlauben Platz zu machen. Vielmehr muss mann bereits etwas glauben, um überhaupt vonWissen und Wissenschaft reden zu können” New In tro duc tion, Stegmüller, 1969:33); “some -where an ul ti mate know ing must be given; with out that we could not even start” (“Irgendeinab so lutes Wissen muß es geben; ohne dieses könnten wir überhaupt nicht beginnen”1969:194); “We should al ready ‘dis pose of’ an ab so lute ev i dence. i.e. we have to be lieve in itin ad vance, ...” (“Ab so lute Evidenz müssen wir schon ‘haben’, d.h. wir müssen an sie bereitsglauben, ...” 1969:194); “in sci ence be liev ing is found, in re li gion one knows (or: pre tends to

ated room for the con vic tion that the range of sci ence should not be nar rowlyre duced to the lim its of (the meth ods of) math e mat ics, phys ics and logic (asthe mod ern nat u ral sci ence-ideal claims) – it is as wide as all of cre ation.

The relation between Popper, Kuhn and SneedThe cur rent and dom i nant (even if im plicit) po si tions of nat u ral sci en tists andmany phi los o phers with re gard to the the ory of sci ence claims that the “em pir -i cal sci ences” de vel oped in a lin ear and con tin u ally pro gres sive man ner, thatis that in these sci ences we can speak of a lin ear ac cu mu la tion of knowl edge.Broadly seen this ac cu mu la tive growth me thod i cally gains ac cess to more and better ob ser va tory and mea sur ing in stru ments which give rise to the dis cov ery of new facts. The depth di men sion of this con cerns the the o ret i cal lines ofcon nec tion which can be ac cessed when em pir i cal reg u lar i ties are re placed by reg u lar i ties which can be math e mat i cally for mu lated, and which are then im -bed ded in the o ret i cal de signs. This pro gres sion im plies at the same time thatone can get rid of the ‘un sci en tific el e ments’ in the sci en tific tra di tion.Stegmüller points out that it is ex actly this im age of the cu mu la tive in crease of knowl edge which is false in prin ci ple ac cord ing to Kuhn (cf. 1975, pp.484ff).

The Dynamics of Theory FormationIn his work on The Struc ture and Dy nam ics of The ories for ma tion Stegmüllergives an elab o ra tion of Kuhn's thought as re flected in the spirit of his crit ics(cf. 1976:135ff). His as sump tion, how ever, is that his sketch is an ap prox i ma -tion of the im pres sion which Kuhn's work would make in the mind of im par -tial crit i cal read ers.

The fas ci nat ing his tor i cal back drop which is rel e vant here, is pro vided else -where by Stegmüller when he com pares Hume, Carnap, Pop per and Kuhn.The clas si cal ra tio nal is tic con vic tion that sci ence can pro vide de fin i tive, in -du bi ta ble knowl edge was al ready se ri ously doubted by Da vid Hume, par tic u -larly with ref er ence to “em pir i cal knowl edge.” It was al ready widely ac cepted that the ex pe ri en tial sci ences would pro ceed in duc tively, with out it be ing atall clear what ex actly would char ac ter ise this method.

Hume ad mits that the em pir i cal sci ences could only with aid from the in duc -tive prin ci ple reach gen er al iza tions which bring into view the reg u lar i tieswhich pro vide in sight into fu ture oc cur rences out of past ex pe ri ence. It is re -mark able that ac cord ing to Hume no ra tio nal foun da tion ex ists for this prin ci -ple! Ev ery at tempt to pro vide such a foun da tion leads ei ther to an in fi nite re -gres sion, or a log i cal cir cle. In con se quence the em pir i cal sci ences fol low thein duc tive method, while main tain ing that this method is ir ra tio nal, since thefoun da tional in duc tive prin ci ple can not be founded or le git i mated.

Using the con cept of log i cal prob a bil ity Carnap at tempted to res cue the ra tio -nal sta tus of in duc tion, which con se quently im plies for him that the ex pe ri en -

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know)” (“in der Wissenschaft wird geglaubt, in der Re li gion weiss man (oder: behauptetman, zu wissen)” 1969:212).

tial sci ences pro ceed both in duc tively and ra tio nally.1 Pop per, as is wellknown, re acted with sharp crit i cism to the sup pos edly in duc tive na ture of nat -u ral sci en tific re search. The dis cov ery of sci en tific the o ries is and re mains en -tirely spec u la tive and their can be no foun da tion or le git i ma tion by any means– at most a de duc tive method of test ing. This means that the ory for ma tion pro -ceeds ra tio nally and non-inductively – against ver i fi ca tion Pop per's sets fal si -fi ca tion.2

Ac cord ing to Stegmüller, what would ap pear to be unique and un prec e dentedin Kuhn's work,

“is the fact that he ap pears to im pute ir ra tio nal be hav ior to the prac ti tio ners ofthe ex act nat u ral sci ences (of all peo ple!). And in deed he ap pears to im pute itto both of the forms of the sci en tific prac tice dis tin guished by him. Any one en -gaged in nor mal sci ence is a nar row-minded dog ma tist cling ing un crit i cally tohis the ory. Those en gaged in ex traor di nary re search lead ing to sci en tific rev o -lu tions are re li gious fa nat ics un der the spell of con ver sion, try ing by all meansof per sua sion and pro pa ganda to con vert oth ers to the new par a digm as re -vealed to them selves.” (1976, p.vii).

Not only do the nat u ral sci en tists work in an ir ra tio nal man ner – ac cord ing tothe crit ics of Kuhn it would ap pear as if he is also a pro po nent of thenon-inductive na ture of the nat u ral sci ences.

A com par i son be tween the four men tioned fig ures pro vides the fol low ing pic -ture:(1) Hume: the nat u ral sci ences pro ceed in duc tively and non-rationally;(2) Carnap: the nat u ral sci ences pro ceed in duc tively and ra tio nally;(3) Pop per: the nat u ral sci ences pro ceed non-inductively and ra tio nally;(4) Kuhn: the nat u ral sci ences pro ceed non-inductively and non-rationally;

(cf. Stegmüller, 1975:487-490).

Kuhn's criticsThe first ob vi ous mat ter em pha sized by Kuhn is that no sin gle pro cess thus far ex posed through his tor i cal re search shows even the least sim i lar ity with Pop -per's doc trine of fal si fi ca tion. An ex pe ri ence stand ing in con tra dic tion to athe ory does not, ac cord ing to Kuhn, in di cate a short com ing in the the ory,since it only dis cred its the per son who holds the par tic u lar the ory! Of coursePop per ad mits that his in ten tion was solely to give an ac count of what Kuhnhas called ex cep tional re search and that he con se quently ne glected the phe -nom e non of nor mal sci ence. The nor ma tive meth od ol ogy de signed by Pop -per, still means that he wants to over come the na ture of nor mal sci en tific prac -tice (in Kuhn's sense) in per ma nent rev o lu tion.

Stegmüller none the less points out that it would be mis taken to be lieve that thedif fer ences be tween Kuhn and his crit ics are lim ited to their eval u a tion of the

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1 Keep in mind that the older inductivists were first of all di rected to wards what HansReichenbach called the con text of dis cov ery as dis tinct from the con text of jus ti fi ca tion,whereas mod ern inductivists are fo cussed on the lat ter.

2 Blaise Pascal (1623-1662) al ready de fended a sim i lar view. Ac cord ing to him a hy poth e siscould never be ‘ver i fied’ by ex pe ri ence. Ex pe ri ence may how ever prove it false.

na ture of nor mal sci en tific prac tice. Ac cord ing to Kuhn it af ter all never oc -curs that a new the ory emerges be cause an old the ory could not give ac countof ex pe ri en tial data. The old the ory is rather re placed im me di ately by a newthe ory, with out the me di a tion of any ex pe ri ence (cf. Stegmüller, 1980/I:28).

Re acting to these per spec tives, and es pe cially in re ac tion to the neg a tive (rel a -tiv is tic and irrationalistic) pic ture which crit ics of Kuhn cre ated of Kuhn'scon tri bu tions, Stegmüller aimed rather to de ter mine what Kuhn es tab lished as a com pe tent the o rist of sci ence, and to pro cess this ma te rial log i cally(1980/I:29). Al though it would con tinue our train of thought be yond the pur -poses of the cur rent con text to go into the Stegmüller-Sneed mod i fi ca tion ofKuhn (cf. Strauss, 1987), it is still mean ing ful to men tion its cen tral point. Inthe Stegmüller-Sneed mod i fi ca tion of Kuhn at ten tion is given to the prob lemof the im mu nity of sci en tific the o ries against fal si fi ca tion – with as a cen tralcon cept the non-falsifiable struc tural core of a “the ory.”1

Foundational problems and basic distinctions

In dis tinc tion to Kuhn's em pha sis on the na ture of sci en tific rev o lu tions,Holton gave em pha sis to the per sis tent themes in sci ence – with as cen tralcon cept the role of a par a digm or dis ci plin ary ma trix linked to the idea of aGes talt-switch. The foun da tional prob lem with re gard to this dif fer ence ofopin ion has not only re peat edly played a role in the his tory of the nat u ral sci -ences, but si mul ta neously has dom i nated the his tory of phi los o phy to a greatex tent and can still be found back to day in prac ti cally all ar eas of sci en tificprac tice.

At the be gin ning of this elab o ra tion we briefly noted the thought ex per i mentof Ga li leo with re gard to the na ture of a body in mo tion on a straight course.With this ar gu men ta tion Ga li leo – al beit in flu enced by par tic u lar ear lierthink ers from the tran si tional pe riod from the mid dle ages to the mod ern age –aban doned the clas si cal Ar is to te lian no tion that a mov ing body must be keptin mo tion by some or other causal force. Ga li leo re al ized that mo tion must beac knowl edged as an orig i nal given in its own right. The con cept cause af ter all al ready pre sup poses the given na ture of a con tin u ously mov ing body – sincewe can only mean ing fully con sider changes in mo tion which might oc cur(whether de cel er at ing or ac cel er at ing) on the grounds of this given. In bothcases a cause is nec es sary – but not a cause of mo tion: rather a cause for thechange in mo tion. In pure ki ne mat ics the con cept change strictly speak ing has no place. In a phys i cal sense ac cel er a tion can never oc cur in a dis con tin u ousway. A dis con tin u ous ac cel er a tion would re quire an in fi nite phys i cal force – a phys i cal im pos si bil ity (cp. the re mark by Janich, 1975:69).

Plato was al ready con fronted by the prob lem of con ti nu ity and change. In hisfa mous the ory of ideas he strove to gain knowl edge of all things. Since he also

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1 New ton's law of grav ity is used to sub stan ti ate this ar gu ment – a phys i cal law evinc ing modaluni ver sal ity. Ga li leo's law of in er tia is an other ex am ple of a uni ver sal modal law which couldonly be for mu lated on the ba sis of modal ab strac tion and not on the ba sis of ‘em pir i cal ex per -i men ta tion’. Later on we shall ex plain that modal laws holds uni ver sally in an un spec i fiedway, whereas typ i cal (entitary) laws are only ap pli ca ble to a lim ited class of en ti ties.

stud ied with a stu dent of Heraclitus, namely Cratylus, he was con fronted from the be gin ning with the doc trine that ev ery thing changes. Heraclitus fa mousmaxim is that no-one can en ter the same river twice. Plato asks if this ap -proach does not sus pend the pos si bil ity of knowl edge: if ev ery thing con -stantly changes it would af ter all not be pos si ble to know any thing, since assoon as we at tempt to ap proach knowl edge the thing has al ready es sen tiallychanged and has there fore es caped the grasp of our knowl edge. The only wayout Plato could find was spec u la tive. He as sumes a static be ing – the so-calledaujto; to ei\do" (es sen tial be ing) – which is not sub ject to any change andwhich guar an tees the knowability of ev ery thing. In his di a logue Cratylus hestates it in the fol low ing man ner: if the aujto; to ei\do" of know ing changes into a dif fer ent aujto; to ei\do" no knowl edge (ei ther sub jec tively or ob jec tively)would be pos si ble (440 a-b). Plato pos tu lates a trans-sensory sphere in whichthese sup posed eidè re side. This world of ideas can only be thought of men -tally, it can not be ob served through sen sory per cep tion. Note that we find here an ear lier ex pres sion of the op po si tion be tween thought and ex pe ri ence which we have al ready men tioned with re gard to the man ner in which Ein stein gavepri or ity to the o ret i cal thought above (ex per i men tal) ex pe ri en tial data.

Al though we no lon ger find the spec u la tive so lu tion of fered by Plato ac cept -able, we can not deny that Plato built an ex tremely sig nif i cant in sight into hisar gu ment. He re al ized that it is not pos si ble to de ter mine any change ex cept on the ba sis of du ra bil ity or con stancy. Stated dif fer ently: changes can only bede ter mined on the ba sis of some thing rel a tively con stant.

Al though we shall re turn to this in sight with re gard to Ein stein's the ory of rel -a tiv ity, we want at this stage to in di cate a few fur ther prob lems which haveplayed a role in the his tory of phi los o phy. Time and again think ers have beencon fronted with the ques tion whether re al ity can be il lu mi nated in terms of asin gle point of view, or whether a mul ti plic ity of ex plan a tory ap proachesshould not be re cog nised. The ear li est philo soph i cal de signs al ready at -tempted to find re pose in a sin gle ex plan a tory per spec tive which is el e vated to the foun da tional de nom i na tor in terms of which all of re al ity must be un der -stood. The Py thag o re ans, for in stance, were of the opin ion that ev ery thing isnum ber. Since the me chan ics of Ga li leo mod ern phys ics has been en am oredby the (mech a nis tic) con vic tion that all phys i cal phe nom ena can be de scribedexhaustively in terms of me chan i cal move ment (of mass points, whethercharged or not).

If the way of mul ti ple ex plan a tory ap proaches is cho sen, the next prob lemwould of course be to at tempt to de ter mine which var i ous ex plan a tory ap -proaches can ac tu ally be dis tin guished and to trace what the mu tual co her ence and re la tion among these var i ous pos si ble ap proaches might be.

The dis ci pline of mod ern bi ol ogy neg a tively il lus trates the pos si bil ity of mul -ti ple ex plan a tory ap proaches. We need only con sider the fol low ing trendswhich have yielded prom i nent rep re sen ta tives: the mech a nis tic ap proach,physicalism (in clud ing neo-Darwinism), (neo-)vi tal ism, ho lism, or gan is micbi ol ogy, emer gence evolutionism and pan-psychism (whether mo nis tic in theman ner of Thailhard de Chardin or plu ral is tic in the man ner of Rensch).

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We could also re fer to this as the prob lem of unity and di ver sity, or even unityin di ver sity. A third prob lem is closely linked to both the pre ced ing ques tions,namely the ques tion re gard ing the re la tion ship be tween that which can becon sid ered to be uni ver sal and that which can be con sid ered to be in di vid ual.Peo ple who des ig nate (“ob jec tive”) re al ity to universalia (uni ver sals) havehis tor i cally been known as re al ists. Plato for in stance con fesses that the ideasof his world of ideas pos ses re al ity and uni ver sal ity in de pend ently of hu manknow ing thought. Uni ver sal ity in this case func tions in two senses:

(i) it ex ists in de pend ently of the know ing hu man soul and(ii) it ex erts an ap peal on ev ery thing – wher ever. Va ri eties of this re al is tic

ap proach later re ceived the la bel of pla ton ism.

In op po si tion to this pla tonic re al ism we tra di tion ally en coun ter those who are of the opin ion that no uni ver sal ity ex ists out side the know ing hu man soul.Only the hu man soul is able to ab stract the im mense mul ti plic ity of things out -side it self to the unity of par tic u lar con cepts and/or words. The only form ofuni ver sal ity ac knowl edged by this al ter na tive is that of con cepts or words inthe hu man soul. The lat ter func tion as sub sti tutes, mere nomina (names) forthe things out side the soul. It is for this rea son that this ap proach is known asnomi nal ism. Nomi nal ism not only de nies the re al ity of uni ver sal ity out sidethe hu man soul, but also im plic itly de nies the or der-side of the di ver sitywithin re al ity. At most it of fers as an al ter na tive the no tion that the or der ingin stance in re al ity is hu man thought it self.

Al though these two isms – re al ism and nomi nal ism – are vir tu ally of the sameage as phi los o phy it self, both are still alive and well in con tem po rary sci en -tific thought. To bring their ac tu al ity to your at ten tion it is enough merely tonote that the dom i nant trend in 20th cen tury math e mat ics is pla tonic while thedom i nant trend in con tem po rary bi ol ogy is nominalistic!

With this fleet ing in tro duc tion of three fun da men tal prob lems of sci ence – theprob lem atic re la tion ships be tween con stancy and dy nam ics, unity and di ver -sity, and uni ver sal ity and in di vid u al ity, we have by no means ex hausted themenu! We will how ever suf fice with only point ing out that a fur ther car di nalfoun da tional prob lem is con sti tuted by the re la tion ship be tween laws and thatwhich is fac tu ally sub ject to laws. An swers to the ques tion of the sta tus oflaws com monly di verge in two di rec tions: firstly the di rec tion to wards or derin the va ri ety of ex plan a tory op tions which pro vide ac cess to (the o ret i cal)anal y sis and study of re al ity and sec ondly the di rec tion to wards the typ i calstruc ture of par tic u lar group ings of en ti ties. Be hind this stands the quest fortruth, which, ac cord ing to the phys i cist Stafleu (1980) can dif fer en ti ate inthese two tracks when it is fo cused on the ques tion of or der or struc ture.

One way to form a clearer un der stand ing of this im por tant dis tinc tion can beex plored by be gin ning with a brief in ves ti ga tion into the na ture of sci ence –with the in ten tion of de ter min ing what makes sci ence unique, i.e. what dis tin -guishes sci ence from our ev ery day non-scientific think ing.

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The unique nature of science

To an swer the ques tion “what is sci ence?” we need to iden tify the na ture ofsci ence by not ing those par tic u lar char ac ter is tics which dis tin guish sci encefrom ev ery thing which is not sci ence. Iden ti fi ca tion and dis tinc tion (i.e. anal -y sis) pre sup poses deal ing with sim i lar i ties and dif fer ences. In di cating whatsci ence is has two sides: the sim i lar ity-side and the dif fer ence-side. Sci encehas char ac ter is tics which are not dis tinc tive since in some re gards sci ence issim i lar to non-scientific ac tiv i ties.

As an il lus tra tion of what is meant with iden ti fi ca tion and dis tinc tion, we canask how we can dis tin guish be tween a ma te rial thing and a plant:

(i) There are fun da men tal sim i lar i ties be tween plants and phys i cal things. A plant, for in stance, has a char ac ter is tic mass – a feature shared with ma -te rial things. Sim i larly both plants and ma te rial things have a cer tain spa -tial ex ten sion, a cer tain du ra bil ity and a cer tain unity.

(ii) Only when we pay at ten tion to the fact that plants are alive do we comeinto touch with the dif fer ence-side in the com par i son of mat ter andplants – with the dis tinc tive char ac ter is tic of be ing-a-plant/“plantness.”

A start ing point would ap pear to be given in the ob ser va tion that ev ery sci en -tific ac tiv ity is a thought-activity. This char ac ter is tic still does not dis tin guishsci ence from non-scientific ac tiv i ties, since some one who is not sci en tif i callyen gaged can also think. When we fo cus this ques tion more closely by ask ing:what sort of thought is sci en tific thought? we can be gin with those char ac ter -is tics shared by sci en tific and non-scientific thought.

Non-distinctive characteristics(a) Is sci en tific thought sys tem atic thought? Cer tainly, but this is by no

means a dis tinc tive char ac ter is tic of sci ence. The judge who is pre par inga ver dict in court must sim i larly work sys tem at i cally in his ar gu men ta -tion – but this does not mean that a le gal ver dict changes into a le gal sci -en tific dis ser ta tion.

(b) Is it ver i fi able thought? While the an swer to this must also be in the af fir -ma tive (tak ing into ac count the big controversy in con tem po rary the oryof sci ence over the mean ing of this char ac ter is tic), it is still not a dis tinc -tive char ac ter is tic, since the judge men tioned above must also ver ify that ev ery bit of ev i dence un der consideration is trust wor thy.

(c) Is it me thod i cal thought? Since there are also non-scientific meth ods ofdo ing, it will al ways be nec es sary to dis tin guish be tween sci en tific andnon- sci en tific meth ods. In di cating meth od ol ogy as the dis tinc tive char -ac ter is tic of sci ence only leads to a tau tol ogy (rep e ti tion): sci en tificmethod is sci en tific!

(d) Par tic u larly tired and dis sem i nated is the no tion that sci ence cen ters inthe re la tion ship be tween a sci en tific re searcher (the “know ing sub ject”)and that which is studied, the “study ob ject.” In the first place we need to note that the sub ject-object re la tion is com mon to non-scientific hu manex pe ri ence: we need but con sider the hu man sub ject us ing so cial ob jects(like fur ni ture), or tech ni cal ob jects (tools), eco nomic ob jects (money),

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semiotic ob jects (books), aes thetic ob jects (paint ings), eth i cal ob jects(en gage ment or wed ding rings), le gal ob jects (prop erty), and the like.

All these ob jects in di cate con crete en ti ties which can equally much be stud iedby var i ous spe cial sci ences (each from its own dis tinc tive per spec tive). Let usbriefly look at the many per spec tives which pro vide ac cess to the anal y sis of aso cial ob ject like a lounge chair. The chair has four legs (nu mer i cal as pect: in -ter est of math e mat i cal num ber the ory); it is big or small (spa tial as pect: math -e mat i cal the ory of space); it could be a rock ing chair (ki ne matic as pect: ki ne -mat ics); it is strong or weak (physico-chemical as pect); it could be use ful inhu man life (as bi oti cal ob ject; since the chair it self is not alive – bi ol ogy stud -ies re al ity from the per spec tive of the bi oti cal as pect); it is com fort able (sen si -tive/psy chic as pect: psy chol ogy); some one con ceived it (an a lyt i cal as pect:logic); it is cul tur ally shaped (his tor i cal as pect: his tor i cal sci ence, whichwould be in ter ested in the his tor i cal de vel op ment of var i ous chair styles); ithas a name (a ver bal sign – semiotic as pect: gen eral semiotics and lin guis tics); it is used in the in ter ac tion among peo ple (so cial as pect: so ci ol ogy); it has aprice (eco nomic as pect: eco nom ics); it is beau ti ful or ugly (aes thetic as pect:aes thet ics); it be longs to some one who has a sub jec tive right (the comptenceof en joy ing it and dis pos ing over it) to it (ju rid i cal as pect: le gal sci ence); it issome one's fa vour ite chair (eth i cal as pect: eth ics); and it is trust wor thy – ev -ery body trusts that the chair will bear the weight of a per son sit ting on it (faithas pect: per spec tive of the ol ogy as a sci ence).

From this ex am ple it is clear that the car di nal ques tion is not: with what ob ject(en tity, event or so ci etal re la tion ship) does this or that sci ence en gage it self,but: from what per spec tive (as pect, way of be ing, mode, mo dal ity, func tion,facet) of re al ity are cer tain things, events and so ci etal re la tion ships stud ied bya spe cial sci ence.

Only when we have brought the di ver sity of as pects of re al ity into sight canthese sci en tif i cally dis tin guished as pects serve as gate ways to the study ofdata. How do we bring these as pects into sight? Only once we have asked thisques tion have we stepped be yond the sim i lar i ties of sci ence and non-science.

The distinctive characteristic of scientific(theoretical) thought

Since the as pects of re al ity in di cate the frame work within which all en ti tiesfunc tion con cretely we can also re fer to them as creational ways of be ing. Theman ner in which some thing is ap proached is also known as the mo dus ope -randi, and from this Latin term mo dus we de rive the term mo dal ity (= way ofbe ing, as pect). When we there fore iden tify a par tic u lar mo dal ity (as pect) assuch and dis tin guish it from other mo dal i ties, we must ab stract the con cernedas pect. This pro cess is re ferred to as modal ab strac tion.

Who ever is en gaged in modal ab strac tion, re lin quishes the non-relevant as -pects and fo cuses the o ret i cal-logical at ten tion on one par tic u lar as pect. Thedis tinc tive char ac ter is tic of the o ret i cal-logical (=sci en tific) thought, there -fore, is modal ab strac tion. Ex actly be cause all con crete en ti ties func tion in allthe var i ous as pects of re al ity (cf. the lounge chair ex am ple above), the ab -

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stracted mo dal i ties (as pects) pro vide ac cess to the anal y sis of the struc tures ofsuch en ti ties.Al though non-scientists cer tainly have an an a lyt i cal aware ness of the di verseas pects, this does not mean that in our non-scientific ex pe ri ence of re al ity weever achieve modal ab strac tion. One would not, for in stance, in one's non-sci -en tific ex pe ri ence, re flect on the na ture and struc ture of the nu mer i cal as pectwhen one no tices six peo ple walk ing past one's house, just as lit tle as onewould de velop an eco nomic price the ory when one no tices that a car costsR30 000. With out an an a lyt i cal con scious ness of the di ver sity of as pects in re -al ity, one would not, how ever, have any con cep tion of what was meant whensome one com ments that a cer tain car is so beau ti ful but so ex pen sive. Beauty(aes thetic as pect) and price (eco nomic as pect) are fac ets of one's to tal ex pe ri -ence of cars, al though we would still main tain that these as pects are gen er allyno ticed in a non-abstractive man ner of and in en ti ties.A par tic u lar kind of ab strac tion is part of our non-scientific ex pe ri ence, re -ferred to with the ap par ently con tra dic tory term of con crete ab strac tion (or:entitary ab strac tion). A lit tle child who first no tices a dove and learns its namecan al ready ab stract con cretely, for in stance when s/he shortly there af ter re -fers to a spar row as a “dove.” The child ac tu ally in di cates the con cept birdwith the name (ver bal sign) dove. This is only pos si ble be cause the child haslifted out cer tain bird-char ac ter is tics out of the con crete sen sory per ceived im -age of a dove (e.g. a beak, wings, feath ers) and si mul ta neously re lin quishedthe spe cific char ac ter is tics which dis tin guishes the dove from the spar row.1

This kind of ab strac tion is part of our ev ery day life, since we are con tin u allyiden ti fy ing all sorts of en ti ties, plac ing them in cer tain cat e go ries. Oth er wisehow would one be able to iden tify a par tic u lar horse as a horse (=be long ing to the cat e gory of horses), or a par tic u lar car as a car? With out gen eral con ceptssuch as cars and horses (in which the de tail of par tic u lar cars and horses arere lin quished), this would be im pos si ble.This kind of ab strac tion does not pro vide us with the o ret i cal in sight into thena ture of any as pect, since – as we shall still see – the as pects be long to a sep a -rate di men sion which must be dis tin guished from the di men sion of en -tity-structures.

Is Philosophy a Science?If it is true that modal ab strac tion is the dis tinc tive char ac ter is tic of sci ence,then it should be ob vi ous that we can dis tin guish within the cat e gory of sci -ences be tween the kind of sci ence lim ited to the per spec tive of a par tic u lar as -pect, and the kind of sci ence which pays at ten tion to the foun da tional co her ent interlacement among all the as pects of re al ity – a co her ence which also servesas the foun da tion of a the o ret i cal anal y sis (=an anal y sis via the gate way of ab -stracted as pects) of the interlacement which ex ists be tween the wide di ver sityof con crete en ti ties, events and so ci etal re la tion ships. The lat ter kind of sci -

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1 Note the sym me try be tween anal y sis and ab strac tion: anal y sis rests on the two legs of iden ti -fi ca tion and dis tinc tion which is equiv a lent to the na ture of ab strac tion as it is re vealed in lift -ing out and re lin quish ing (com pare the sketch in the Pref ace).

ence (fo cused on a co her ent to tal view of the di ver sity of re al ity) we call phi -los o phy. Ev ery sci ence lim ited to the per spec tive of a par tic u lar mo dal ity wecall a spe cial sci ence.

Philosophy and the Special Sciences

The fact that the o ret i cal thought en tails modal ab strac tion has im por tant im -pli ca tions for ev ery (mo dally de lim ited) par tic u lar spe cial sci ence. Modal ab -strac tion en tails the o ret i cal anal y sis and we have briefly pointed out that anal -y sis on the ba sis of sim i lar ity and dif fer ence is aimed at the iden ti fi ca tion anddis tinc tion of data. The o ret i cal anal y sis would there fore be aimed at the iden -ti fi ca tion of a par tic u lar as pect in dis tinc tion from other as pects. If re al ity con -tained only one as pect, anal y sis as such would be im pos si ble, be cause we canonly iden tify an as pect by si mul ta neously dis tin guish ing it from all as pectswhich dif fer from the iden ti fied as pect. The o ret i cal anal y sis (modal ab strac -tion) must there fore al ways si mul ta neously con sider at least two dif fer ing as -pects.

If a spe cial sci en tific dis ci pline de lim its its do main of study to the per spec tiveof a sin gle as pect, then it is ob vi ous that the iden ti fi ca tion of the do main ofstudy of a spe cial sci ence can never be seen as an ac tiv ity of that spe cial sci -ence or as an ac tiv ity tak ing place within the per spec tive of a par tic u lar as pect– sim ply be cause more than one as pect is in volved in the iden ti fi ca tion of anyas pect! Since only phi los o phy can en gage more than one par tic u lar as pect inits the o ret i cal pur view, this im plies that no spe cial sci ence can in di cate itsown de lim ited do main of in ves ti ga tion with out pro ceed ing from some orother philo soph i cal view of the co her ence which can ac count for the sim i lar i -ties and dif fer ences among the di verse as pects of re al ity. In other words: thena ture of modal ab strac tion as dis tinc tive char ac ter is tic of sci ence im pliesthat all sci ence has a philo soph i cal base!

This course of rea son ing is closely linked to the ar gu ment which fol lows: Thean swer which a spe cial sci en tist gives to the ques tion: what is sci ence? cannever be a spe cial-scientific an swer, since ev ery such a de scrip tion dis cussesthe spe cial sci ence and thus tran scends its lim its. A de scrip tion such as: math -e mat ics con sists of sub sid iary dis ci plines such as set the ory, al ge bra, to pol -ogy and the like, is no math e mat i cal state ment since the de scrip tion is not inthe least an ax iom/prop o si tion/proof/ar gu ment in set the ory, al ge bra, to pol -ogy and the like! This course of rea son ing is valid for ev ery par tic u lar mo dally perspectival sci ence. Even the ol ogy does not es cape this truth. While ev erystu dent of the ol ogy be comes ac quainted through the study of this dis ci pline of the en cy clo pe dia of the ol ogy (which is re spon si ble for the iden ti fi ca tion andde lim i ta tion of the sub sid iary dis ci plines of the ol ogy), this dis ci pline is neverit self clas si fied as a sub sid iary dis ci pline of the ol ogy (next to e.g. thebibliological, dogmatological and ecclesiological groups). In this way the ol -ogy ad mits that the ques tion: what is the ol ogy? is not a theo log i cal ques tion!

This sit u a tion is re mark able fur ther more since, al though no def i ni tion of anyspe cial sci ence can have a par tic u lar-scientific char ac ter, no such def i ni tioncan be given with out ac count ing for the sci en tific con tent of the spe cial sci -

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ence! Even when it is ar gued that math e ma ti cians, theo lo gians, and the likewould be best equipped to an swer ques tions such as: what is math e mat ics?what is the ol ogy? and the like, this does not abol ish the truth that the an swerwhich they give pre cedes their work in the spe cial sci ence they prac tice. Themea sure is not who gives the def i ni tion, but: what is the na ture of the def i ni -tion! This state of af fairs con firms the in dis sol u ble co her ence which ex ists be -tween ev ery spe cial sci ence and its philo soph i cal foun da tional ques tions.

We could also de scribe this state of af fairs as fol lows: there are ba si cally twokinds of sci ence, (i) the kind of sci ence which, when it dis cusses it self, tran -scends its own lim its, and (ii) the kind of sci ence which, when it dis cusses it -self and the gen eral ques tion of the na ture of sci ence, re mains within its ownlim its. The first op tion in di cates a spe cial sci ence, and the sec ond in di catesphi los o phy. In this sense phi los o phy is the sci ence of sci ences, which is en -gaged inter alia with the philo soph i cal foun da tional ques tions of the spe cialsci ences.

Philosophical Foundational Questions in the Special Sciences

Philo soph i cal pre sup po si tions en tail foun da tional ques tions such as: what isthe co her ence and struc ture of the di verse as pects of re al ity which serve asgate ways for our ex pe ri ence of con crete phe nom ena? Does each of these as -pects have a unique and ir re duc ible own na ture, or can all as pects be re ducedto/ex plained in terms of a few as pects (as sug gested by the many philo soph i -cal and spe cial sci en tific isms: ra tio nal ism, ide al ism, uni ver sal ism, in di vid u al -ism, irrationalism, re al ism, nomi nal ism, physicalism, ma te ri al ism, vi tal ism,historicism, psychologism, aes theti cism, mor al ism, piet ism, and the like)?Can facts and norms (or in the nat u ral sci ences, facts and laws) be di vorced, or are they ir re duc ible cor re lates? What is free dom? Does cau sal ity and free dommu tu ally ex clude each other? What is the na ture of be ing human (and is therea fun da men tal dif fer ence be tween be ing human and be ing an an i mal)? Whatis the re la tion be tween in di vid ual and so ci ety? In what does the creational di -ver sity find its con sum ma tion and cen tral fo cus, and what is its or i gin?

To il lu mi nate the in flu ence of some of these foun da tional philo soph i cal prob -lems in the nat u ral sci ences more closely we will crit i cally ana lyse some ex -am ples of dom i nant (and some times con flict ing) par a digms from the fields ofmath e mat ics, phys ics, and bi ol ogy. In all of them the in ev i ta bil ity of modaldis tinc tions will sur face – thus un der scor ing our claim that modal ab strac tionin deed con sti tutes the dis tinc tive fea ture of schol arly sci en tific ac tiv i ties.

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Chapter II

Foundational Philosophical problems in Mathematics

Introductory remarks

Let us start our re flec tions by con sid er ing one of the stan dard ar gu mentsagainst the pos si bil ity of di ver gent stand points in math e mat ics:

Are you sure 3 + 4 = 7?

This kind of ques tion is more or less a stan dard ar gu ment used by sec u lar hu -man ism in an at tempt to de fend the sup posed neu tral ity of schol arly ac tiv i ties: the state ment 3 + 4 = 7 ap par ently con cerns a fact which is true ir re spec tive ofour be ing Chris tian, Jew ish, athe is tic or com mu nis tic.

Sup pose now that I would ob ject by say ing: 3 + 4 = 5! One way to ‘con vince’me that I am wrong, would be to count some of your fin gers. My re sponse then would be: your ‘fin ger count ing’ op er a tion in deed clar i fies the ini tial state -ment – it con cerns nu mer i cal ad di tion. How ever, I had some thing dif fer ent –but equally le git i mate – in mind: geo met ri cal ad di tion. Just think about a per -son start ing from a spe cific point: He then walks 3 miles to the north and af ter -wards 4 miles east. How far would that per son be away from his point of de -par ture? 5 miles! Clearly we are now con fronted by two dif fer ent kinds offacts: a nu mer i cal fact (3+4=7) and a geo met ri cal fact (3+4=5) – add ing dis -tances is math e mat i cally ac counted for by the the ory of vec tors: a vec tor hasdis tance and di rec tion – which ac tu ally means that we should have used a no -

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ta tion to ac knowl edge this (for ex am ple by us ing pointed ar rows above the‘3’, ‘4’ and ‘5’ of the vec tor-sum).

The point of our two ex am ples is that ‘facts’ are not sim ply ‘facts’ – they areal ways struc tured or qual i fied. In our case they are qual i fied by the as pects ofnum ber and space. Con se quently, in or der to dis tin guish be tween dif fer entkinds of facts an aware ness of the or der-diversity of re al ity is pre sup posed.But ex actly at this point the views of math e ma ti cians de part. Dif fer ent schools of thought in mod ern math e mat ics arose on the ba sis of al ter na tive views withre spect to the na ture of and the co her ence be tween the as pects of num ber andspace. For ex am ple, intuitionistic math e mat ics con structed a whole newmath e mat ics in tro duc ing con cepts and meth ods not found in clas si cal math e -mat ics. Stegmüller strik ingly re marks:

“The spe cial char ac ter of intuitionistic math e mat ics is ex pressed in a se ries ofthe o rems that con tra dict the clas si cal re sults. For in stance, while in clas si calmath e mat ics only a small part of the real func tions are uni formly con tin u ous,in intuitionistic math e mat ics the prin ci ple holds that any func tion that is de fin -able at all is uni formly con tin u ous” (1969:331).

From the seem ingly ‘in no cent’ and ‘neu tral’ state ment of ‘fact’ (3+4=7) weare not only im me di ately en tan gled in foun da tional ques tions of math e mat icsas a spe cial sci ence but also con fronted with a se ri ous chal lenge re gard ing thesup posed ‘ex act’ na ture of math e mat ics as an ac a demic dis ci pline!

A num ber of years ago a well-known math e ma ti cian, Mor ris Kline, wrote awhole book deal ing with the way in which the clas si cal ideal of math e mat icsas an ex act sci ence with cer tainty as its guid ing star was un der mined. He re -marks:

“The de vel op ments in the foun da tions of math e mat ics since 1900 are be wil -der ing, and the pres ent state of math e mat ics is anom a lous and de plor able. Thelight of truth no lon ger il lu mi nates the road to fol low. In place of the unique,uni ver sally ad mired and uni ver sally ac cepted body of math e mat ics whoseproofs, though some times re quir ing emen da tion, were re garded as the acme of sound rea son ing, we now have con flict ing ap proaches to math e mat ics. Be -yond the logicist, intuitionist, and formalist bases, the ap proach through setthe ory alone gives many op tions. Some di ver gent and even con flict ing po si -tions are pos si ble even within the other schools. Thus the constructivist move -ment within the intuitionist phi los o phy has many splin ter groups. Within for -mal ism there are choices to be made about what prin ci ples of metama -thematics may be em ployed. Non-standard anal y sis, though not a doc trine ofany one school, per mits an al ter na tive ap proach to anal y sis which may alsolead to con flict ing views. At the very least what was con sid ered to be il log i caland to be ban ished is now ac cepted by some schools as log i cally sound”(1980:275-276)

It is in deed strange that the his tory of math e mat ics ex plored the dualone-sidedness of an arithmeticistic (founded by Greek math e ma ti cians andagain en throned dur ing the past hun dred years) and a geometricistic ap proach(dom i nant dur ing the in ter me di ate pe riod) and never ven tured to ex plore thefol low ing ob vi ous third pos si bil ity: ac knowl edge both the unique ness and the mu tual co her ence of num ber and space as as pects of the richly var iedcreational or der-diversity.

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We now pro ceed with a sketch of ma jor his toric and sys tem atic anal y ses in or -der to elu ci date the com plex in ter re la tions be tween num ber and space.

First we men tion some mod ern def i ni tions of the na ture of math e mat ics. Fromthese def i ni tions we de duce the im por tance of the no tion of in fin ity for an as -sess ment of the na ture of math e mat ics – sup ported by our his tor i cal over view. This over view, then, is fol lowed by a brief as sess ment of some prom i nent is -sues, and we close our dis cus sion by a suc cinct treat ment of an al ter na tive sys -tem atic per spec tive.

Definitions of mathematics

Though it may seem nat u ral to re late math e mat ics as a spe cial sci ence to theas pects of num ber and space in the first place, the way in which most mod ernmath e ma ti cians de fine their sub ject mat ter does not ex plic itly re fer to theseas pects. Logicism, for in stance Rus sell, wants to stress that math e mat ics is not con cerned with quan tity, but with or der. Al ready W. Ham il ton de fined al ge -bra – in a work from the year 1833 – as the “sci ence of pure time or or der inpro gres sion” (quoted in Cassirer, 1957:85). Cassirer him self con tin ues thisline of thought, ac tu ally dat ing back to Leibniz, in his own way. Smart pointsout that ac cord ing to Cassirer the main pur pose of the crit i cal study of the his -tory of math e mat ics “is to il lus trate and con firm the spe cial the sis that or di nalnum ber is log i cally prior to car di nal num ber, and, more gen er ally, that math e -mat ics may be de fined, in Leibnizian fash ion, as the sci ence of or der”(1958:245).

Works on the foun da tions of set the ory and deal ing with the phi los o phy ofmath e mat ics of ten re fer to math e mat ics as “the sci ence of for mal sys tems”. To those who are in clined to an ax i om atic ap proach this state ment means thesame as “math e mat ics is set the ory” (cf. Meschkowski, 1972:356).

In spite of its re cent or i gin as a math e mat i cal dis ci pline, set the ory has fromthe very start been con fronted with ba sic trends run ning through the his tory of math e mat ics. We only have to re fer to the ten sion of un com pleted in fin i tudeand com pleted in fin i tude – a con trast which has been fa mil iar since Greek phi -los o phy in terms of the op po si tion be tween the po ten tial in fi nite and the ac -tual in fi nite. The un com pleted in fi nite is used to in di cate the con cep tion thatthe in fi nite is lit er ally in-finite, i.e. with out an end. The com pleted in fi nite,again, is seen as a quan tity which is de ter mined in all its parts while it si mul ta -neously ex ceeds ev ery fi nite quan tity.

As founder of set the ory, Can tor was con vinced “that Set The ory deals withthe ac tual in fi nite” (Rob in son, 1967:39). By us ing the com pleted in fi nite Can -tor, in 1874, proved that the set of all real num bers can not be enu mer ated inthe man ner of the set of all nat u ral num bers, i.e. that real num bers arenon-denumerable (we will re turn to this mat ter). But ex actly in this proof,which uses the com pleted in fi nite, H. Meschkowski (1972b:25) sees the“foun da tion of set the ory”.1

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1 Al though one could con sider this state ment also in view of de vel op ments in the field of cat e -gory-the ory and topos-the ory, this falls be yond our cur rent con cerns.

Who ever there fore de fines math e mat ics as set the ory has thereby al readyplaced the prob lem atic re la tion ship be tween un com pleted in fin ity and com -pleted in fin ity at the cen tre of the def i ni tion. In mod ern math e mat ics there arespe cial sci en tific points of view which di verge ex actly in terms of the in fi nite:some in the di rec tion of the un com pleted in fi nite and other in the di rec tion ofthe com pleted in fi nite. Hermann Weyl (1966:89) com ments strik ingly in thisre gard: “If in con clu sion one would want to pro vide a brief slo gan whichwould in di cate the liv ing cen tre of math e mat ics, one would be able to say: it isthe sci ence of the in fi nite”. In view of what we have al ready con sid ered, wewould how ever im me di ately have to add the words of per haps the great estmath e ma ti cian of the 20th cen tury: “The in fi nite has moved the hu man mindlike no other ques tion since the ear li est times; the in fi nite has brought aboutmen tal stim u lus and fruit ful ness like vir tu ally no other idea; the in fi nite how -ever needs clar i fi ca tion like no other con cept” (Hilbert, 1925:163). Clearly,no ac count of math e mat ics can es cape from as sess ing the na ture of the in fi -nite.

How ever, the ques tion of how we gain in sight into the na ture of the in fi niteap peals to the foun da tional ques tion re gard ing the unique do main of in ves ti -ga tion of math e mat ics. It is clear that it is in suf fi cient to de limit math e mat icsas a dis ci pline in terms of “for mal sys tems”, if only be cause other dis ci plinesuse ex ten sive ab strac tions with out be ing math e mat ics (e.g. phi los o phy). Aprom i nent math e ma ti cian such as Gottlob Frege even ques tioned the no tionthat the con cept of ab strac tion is of use in the de ter mi na tion of the na ture ofmath e mat ics. When, for in stance, we be gin with the moon as an en tity, we ar -rive through ab strac tion only at con cepts such as “com pan ion of a planet”,“ce les tial body with out its own light”, “ce les tial body”, “body” and “ob ject” – no where in this row does the num ber 1 ap pear (Frege, 1884:44)!

Frege is of course re act ing against the view that num ber should be seen as a set of units [pure ones] which we gain out of our ex pe ri ence of con crete things via ab strac tion. Angelelli is even of the opin ion that Frege's cri tique in this re gardis dev as tat ing: “by ab stract ing from the par tic u lar dif fer ences and na tures ofthe given ob jects no plu ral ity can be at tained, but only one thing (the con ceptcat, for ex am ple)” (1984:467). In a com ment on Can tor's def i ni tion of a sub set (cf. Can tor, 1962:283), Zermelo also re fers to the ef fort to in tro duce the con -cept car di nal num ber by means of a pro cess of ab strac tion, which would im -ply that a car di nal num ber should be seen as a set of “pure ones”.1

Im man uel Kant al ready per ceived that a purely log i cal syn the sis could neverpro vide a new num ber (Kritik der reinen Vernunft, 1787:15). In a dif fer entway Frege em pha sizes the same point: con crete (or: en tity-directed) ab strac -tion can only con tinue and ar rive at ever more gen eral en tity con cepts – it cannever ar rive at num ber as such. The fun da men tal ques tion, how ever, is: is itpos si ble to dis tin guish dif fer ent char ac ter is tics/prop er ties of one and thesame en tity? In terms of Frege's ex am ple of the moon we can be more spe -cific: does the moon have any nu mer i cal fea tures? This new per spec tive con -

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1 An “aus lau ter Einsen zusammengesetzte Menge” (a “set con sti tuted by pure one's”) – cf.Zermelo's first com ment in Can tor, 1962:351.

cerns the ques tion of how many moons the earth has – and ob vi ously the an -swer in terms of cur rent knowl edge can only be: one. This in tro duces us to adis tinct kind of ab strac tion which dif fers fun da men tally from con crete (en -tity-directed) ab strac tion as this emerges in our ev ery day con cepts (such ashu man, tree, horse, and the like), namely ‘char ac ter is tic-abstraction’. An -other way of in di cat ing this pos si bil ity, is to in di cate it as modal ab strac tion,which should ac tu ally be seen as the dis tinc tive char ac ter is tic of sci en tificthought. We shall pur sue this fur ther later when we give sys tem atic at ten tionto the re la tion be tween the two fun da men tal kinds of in fin ity which have al -ways con fronted math e mat ics.

By first draw ing out a few his tor i cal threads we can at tempt to gain some ini -tial il lu mi na tion on in fin ity and the crit i cal turn ing points in the de vel op mentof math e mat ics. We must at the same time keep in mind that all West ernthought on in fin ity and con ti nu ity has been de ci sively in flu enced by Ar is totle– that is, un til Can tor fun da men tally ques tioned it.1

The Infinite in Greek Thought

In the Milesian phi los o phy of na ture we find a philo soph i cally ex pressed an -swer to their deep est search for the Arche (or i gin, be gin ning) of all tem po ralthings. Anaximander was the first to choose the in fi nite (un lim ited) as Arche:“the Arche of the ex ist ing things is the apeiron (the in fi nite-unlimited)”(Diels-Kranz, 1959-69:B Frag ments 1).2 He adds to this that the “apeiron isage less” (Fr.2) and “the apeiron is with out death and tran sience” (Fr.3).Where the di vine Arche has pre vi ously al ways been iden ti fied with fluid el e -ments (e.g. wa ter by Thales, air by Anaximines and fire by Heraclitus), wefind Anaximander (re mark ably!) see ing the Arche as some thing age less,death less, and intransient – i.e. some thing stand ing in op po si tion to all flu id -ity! It is prob a ble that Anaximander was on the track of an es sen tial el e mentand char ac ter is tic of the in fi nite – but we will re turn to this when we men tionCan tor's de scrip tion of the com pleted in fi nite.

We en coun ter the bi po lar na ture of the in fi nite strik ingly in Zeno's ar gu mentsagainst mo tion. Ar is totle men tions Zeno's four ar gu ments in his Physika (cf.233 a 13ff. and 239 b 5ff.). We will re fer to only two par tic u larly il lus tra tivear gu ments, namely that of Achil les who can never catch up with the tor toise(since the tor toise has al ways again es tab lished a lead by the time Achil les hascaught up with him), and the ar gu ment that it is im pos si ble to move from point A to point B. In or der to do so, af ter all, it is first nec es sary to com plete half the

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1 “Die entscheidende Erkenntnis des Aristoteles war, dass Unendlichkeit wie Kontinuität nurin der Potenz existieren, also keine eigentliche Aktualität besitzen und daher stetsunvollendet bleiben. Bis auf Georg Can tor, der in der 2.Hälfte des 19.Jahrhunderts dieserThese mit seiner Mengenlehre entgegentrat, in der aktual unendliche Mannigfalgtigkeitenbetrachtete, ist die aristotelische Grundkonzeption von Unendlichkeit und Kontinuität dasniemals angefochtene Gemeingut aller Mathematiker (wenn auch nicht aller Philosophen)geblieben” (Becker, 1964:69).

2 Sub se quently we re fer to Diels-Kranz's B Frag ments with the ab bre vi a tion Fr. (=Frag ment).

dis tance, there af ter half of the re main ing dis tance, and there af ter half again ofthe re main ing dis tance – ad in fi ni tum (cf. Dielz-Kranz, 1959-60 B Fr.3). Zeno con cludes: an in fi nite num ber of spa tial sub-intervals must be crossed to move from A to B and this is im pos si ble in a fi nite pe riod of time. It is af ter all im -pos si ble to ac tu ally ex haust the in fi nite. There fore mo tion is im pos si ble. Thisis an in stance of the con tra dic tion be tween the un com pleted in fi nite and thecom pleted in fi nite. We must there fore dif fer from Titze's state ment that the“nu mer i cally-infinite was in con ceiv able in Greek phi los o phy” (1984:141).Anaxagoras with out doubt al ready had a con cep tion of the po ten -tially-infinite, while we ob vi ously find an ini tial con cep tion of the in fi nitedivisibility of a con tin uum in Zeno's thought (cf. his B Fr.3 – ex plained be -low).

Since Ar is totle pro vided the clas si cal for mu la tion of the no tion that a whole ismore than its parts (in his Politeia:1253a19-20), this idea has ex erted an in ex -tin guish able in flu ence in the his tory of phi los o phy and the var i ous spe cial sci -ences – in mod ern times of ten de fended in the form of the state ment that awhole is dif fer ent from the sum of its parts. It would how ever ap pear as if weneed to re turn to Zeno for the first in sight into the divisibility of a spa tial con -tin uum.

Zeno is best known in the lit er a ture for his four ar gu ments against mul ti plic ity and mo tion as this is ren dered in Ar is totle's Phys ics (cf. 233a13ff. and239b5ff.).1 The pe cu liar sense of his third Frag ment lies ex actly therein that itex plic itly ex plores both sides of the whole-part re la tion – ap par ently for thefirst time in the his tory of phi los o phy (and math e mat ics).

Let us note his for mu la tion:

When mul ti plic ity ex ists, then nec es sar ily only as many (things) ex ist as whatare ac tu ally there, no more and no less. When there how ever are as many aswhat ex ist, then it (the num ber thereof) must be lim ited.

In this first half Zeno there fore ar gues from mul ti plic ity to lim i ta tion. Ex actlythe op po site hap pens in the last half:

When mul ti plic ity ex ists, then that which ex ists (the num ber thereof) is un lim -ited. Be cause con tin u ally other ones ex ist in be tween those which ex ist andagain oth ers be tween these. Thus that which is (the num ber thereof) is un lim -ited.

Al though both main parts of this Frag ment be gin with “when mul ti plic ity ex -ists”, Zeno reaches op po site con clu sions in them – in the first in stance the ini -tial com ment im plies that the num ber of ex ist ing things are lim ited, and in thesec ond in stance that it is un lim ited. The static spa tial terms which Parmenidesand his school uses sug gest the pos si bil ity that Zeno is in deed ex plor ing thetwo sides of the spa tial whole-part re la tion (we shall re turn to the na ture of thewhole-part re la tion in a later con text).

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1 It is true that we have of Zeno him self only the fol low ing strik ing for mu la tion in his fourthand last trans mit ted Frag ment: “That which moves, moves nei ther in the space it oc cu pies,nor in the space it does not oc cupy”.

If mul ti plic ity in the ini tial com ment in di cates a mul ti plic ity of parts (of theworld) then their sum to tal must be lim ited (si mul ta neously con sti tut ing theworld-whole). If, al ter na tively, one starts with the world-whole in terms ofwhich to ac count for the parts, then it would in deed be pos si ble to lo cal ize themul ti plic ity of parts in such a way that there would al ways be fur ther partspres ent in be tween – an ar gu ment which of course could be con tin ued in fi -nitely with re gard to all parts.1

The dis cov ery of the whole-part re la tion was there fore in dis sol u bly linked tothe de vel op ment of a no tion of in fin ity in Greek phi los o phy, since it is con -cerned with the in fi nite divisibility of the (world-) whole. To trace the steps ofthe no tion of in fin ity fur ther we would how ever have to re turn to the Py thag o -re ans. One of the foun da tional char ac ter is tics of the ear li est Py thag o reanschool was the arithmeticist state ment: “ev ery thing is num ber” (cf. Thesleff,1970:82).The ap par ent pos si bil ity to arithmetize mu si cal con so nants lead to the gen eralthe o rem: if any two things in their re la tion to each other ap pear as two num -bers, then they them selves ac tu ally are co vert num bers (cf. Scholz and Hasse,1928:6).The in ter est of the Py thag o re ans in the form of fig ures (in clud ingform-congruence/uni-form-ity) ap par entlystim u lated the proof of Py thag o ras’ fa mil iarthe o rem, namely that in any rect an gu lar tri an -gle the square of the di ag o nal side equals thesum of the squares of the two rect an gu lar sides. In Bab y lo nian texts the fol low ing fig ure ap -pears which al ready sug gests the the o rem ofPy thag o ras. Ac cord ing to Euclides’ El e mentsthe orig i nal ar ith met i cal proof of the the o remof Py thag o ras is founded in the con gru ence offig ures:Hippasus of Metapont (450 BC) prob a bly al -ready dis cov ered that this proof is not gen er -ally valid since it pro ceeds from the pre sup po -si tion that the ra tios of all line stretches stand in re la tion to one an other as in te -gers – i.e., can be rep re sented in the form a/b where a and b are nor mal nat u ralwhole num bers. The pentagram2 con vinced Hippasus of the fal sity of this pre -sup po si tion.

Con sider the fol low ing pentagram: If a1 and a0 have a com mon length, a1

could be pre cisely (with out re main der) di vided into a0. If this di vi sion yieldsan in fi nitely con tin ued frac tion, this would mean that the two line-stretches

23

1 Herman Fränkel even ex plic itly uses the whole-part re la tion when he an a lyzes Zeno's Frag -ment 3. Cf. Fränkel 1968, pp.425ff., es pe cially p.430.

2 That is a reg u lar pen ta gon. The Py thag o re ans used a reg u lar pen ta gon of which the sideswhere ex tended to the points of in ter sec tion (cf. Moritz Can tor, 1922:178).

are in com men su ra ble due to the ab sence of acom mon lo gos or ra tio – which nec es sar ilyyields the dis cov ery of ir ra tio nal num bers.

From the ad ja cent fig ure fol lows:ao = a1 + a2

There fore: ao/a1 = 1+ a2/a1and: a1 = =a2 + a3im ply ing: a1/a2 = 1 + a3/a2Sim i larly a2/a3 = 1/a1/a2 and:

a3/a2 = 1/a2/a3 etc.

From the pre ced ing it fol lows:1 + 1

1 + 11 + 1

which con firms the ex is tence of ir ra tio nal num bers

The sim plest ex am ple of an ir ra tio nal num ber is the oblique side of a rect an gu -lar tri an gle where both the rect an gu lar sides mea sure 1.

Ap par ently a proof of the fact that the square root of the num ber 2 is ir ra tio nalwas al ready known to Py thag o ras. In his di a logue Theaitetos (147d) Platomen tions that the Py thag o rean Theodoros had proven fur ther ir ra tio nal i ties.1

The na ture of ir ra tio nal num bers, which can not be in di cated in terms of the re -la tion (ra tio, lo gos) be tween two whole num bers (in te gers), was the core ofthe cri sis in Py thag o rean math e mat ics. Only when Greek math e mat ics is un -der stood in terms of the deep est mo tive ac tive at the root of all Greek thought,does it be come clear why this dis cov ery was ex pe ri enced as so cen tral a cri sis.

The for ma tive and de lim it ing func tion of num ber was how ever un der minedby the men tioned dis cov ery of ir ra tio nal num bers (incommensurability – cf.Von Fritz, 1945:242-264), since it ap peared that the for ma tively-delimitedoblique side, e.g. of a rect an gu lar tri an gle with two rect an gu lar sides with alength of 1, in it self (from an ar ith met i cal per spec tive) con tained an in fi nite(un lim ited) se quence. In other words, in this case the apeiron ab ro gated thede lim it ing func tion of the peras! To avoid this con se quence all al ge braicprob lems2 were trans lated into spa tial terms – hence the geometricization ofmath e mat ics.3 The foun da tional mo tive of Greek thought, namely the mo tiveof the lim ited and un lim ited (tran sient and intransient, or, in Ar is to te lian

24

a1 a1

a1 a1

a1 a1

a2 a2

a3 a3

a2 a2

a0 a0

a a b b

c c d d e e

f f

1 “Our friend Theodoros was prov ing to us some thing about square roots, namely that the sides(or roots) of squares rep re sent ing 3 square feet and 5 square feet are not com men su ra ble inlength with the line rep re sent ing 1 foot; and he con tin ued thus, tak ing each case in turn up tothe root of 17 square feet.”

2 Of course the Greeks did not yet know any al ge bra.

3 Cf. Boyer, 1956:8ff.

terms: the mo tive of form and mat ter), there fore in te grally de ter mined the en -tire di rec tion of Greek math e mat ics!1

The dis cov ery of ir ra tio nal num bers con sti tutes the first foun da tional cri sis ofmath e mat ics. To over come this cri sis Eudoxos de vised a method which ap -proached the mod ern dif fer en tial and in te gral cal cu lus, but due to an ex ces -sive ad her ence to the spa tial per spec tive the most im por tant dis cov ery had towait un til the 17th cen tury A.D.2

Apart from the con tri bu tion of Zeno's B Frag ment 3 to our un der stand ing ofthe na ture of the spa tial whole-parts re la tion ship, we find state ments inAnaxagoras re gard ing the na ture of spa tial con ti nu ity which are still ac tual to -day. He says:

In that which is small there is no small est, since there al ways ex ists some thingsmaller. That which is can never cease to ex ist through fur ther di vi sion, nomat ter how far we con tinue this di vi sion (B Fr.3). And since no small est canex ist, it also can not in su late or con tain it self, but must, as in the be gin ning, ex -ist with ev ery thing else (B Fr.6).

This si mul ta neous ex is tence sug gests the co her ence of spa tial con ti nu itywhich in cludes all (ma te rial) things – a con ti nu ity which is not, how ever, theco-ordination of dis crete (sep a rated) parts, as if sep a rated with an axe (BFr.9).

With these char ac ter iza tions Anaxagoras reaches for ward not only to the view of Ar is totle, but even to the po si tion of intuitionist math e ma ti cians in the 20thcen tury (namely Brouwer and Weyl).3 In Anaxagoras’ course of thought it isclear that the in fi nite should not be un der stood only ex ter nally as spa tial ex -ten sion with out lim its, but also in ter nally (i.e. in wardly) as in fi nitely di vis i blespa tial ex ten sion.

One of the cat e go ries which Ar is totle dis tin guishes is quan tity. As higher ge -nus proximum quan tity en com passes both num ber and space as differentiaspecifica. That is, when one re lin quishes the spe cific dif fer en ti at ing (dis tinc -tive) char ac ter is tics of num ber and space, both be long un der the higher con -cept of kind: quan tity.

“Quan tity is ei ther dis crete, or con tin u ous” (Categoriae, 4 b 20). “Num ber, ...is a dis crete quan tity” (Cat., 4 b 31). The parts of a dis crete quan tity have nocom mon limit, while it is pos si ble in the case of a line (as a con tin u ous quan -tity) to find a com mon limit to its parts time and again (Cat., 4b 25ff., 5 a 1ff.).

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1 P.A. Meijer is of the opin ion that the most ap pro pri ate in di ca tion of the Greek foun da tionalmo tive is to be found in the Greek han ker ing af ter the intransient (cf. 1968:207, and note 15on p.206). Bram Bos re con sid ers form and mat ter to sig nify the foun da tional mo tive(Dooyeweerd) by us ing the al ter na tive des ig na tion of the “ti tanic mean ing per spec tive”(1986). The ten sion be tween be com ing (e.g. of the sea sons) and the un der ly ing quest for con -stancy how ever re mains cen tral to this mo tive. Com pare Bos 1994:220.

2 We shall re turn to this sec ond foun da tional cri sis in the his tory of math e mat ics shortly.

3 H. Weyl points out the sig nif i cance of Greek thought: “Yes, ex actly now we are be ingbrought ev ery where to re turn di rectly to the Greeks in the foun da tions of math e mat ics”(1931:1).

With Zeno Ar is totle de nies the pos si bil ity of the ac tu ally in fi nite (com pletedin fi nite). He be lieves that his par tic u lar view of the po ten tially in fi nite (the un -com pleted in fin ity) over comes the prob lems of Zeno. While a par tic u larline-stretch is in fi nitely di vis i ble, this divisibility is only a po ten ti al ity, a pos -si bil ity, which can never ac tu ally (in re al ity) be car ried through. When some -thing moves it does not, ac cord ing to Ar is totle, move in a count ing man ner,since then Zeno's antinomy (that in the cov er ing of a lim ited spa tial dis tancean ac tu ally in fi nite se quence of num bers would si mul ta neously have to be“counted through”) would be valid. Ar is totle con fronts Zeno's prob lem withthe fol low ing ar gu ment:

“In the act of di vid ing a con tin u ous dis tance into two halves, one point is usedtwice, since we make it the start ing-point and the end-point: ... But if such di vi -sions are made, nei ther the dis tance, nor the mo tion would be con tin u ous: ...and al though that which is con tin u ous con tains an in fi nite num ber of halves,these are not ac tual but only po ten tial halves” (Physica, 263 a 23ff.).

Ar is totle re jects the ex is tence of the ac tu ally in fi nite on two grounds (cf.Physica, 204 a 20ff., Metaphysica, 1066 b 11ff., and Metaphysica, 1084 a1ff.):

(i) if the ac tu ally in fi nite con sists of parts then these parts must them selvesbe ac tu ally in fi nite, which would im ply the ab sur dity that the whole is no lon ger larger than a part; and

(ii) If it con sists of fi nite parts, this would im ply the im pos si bil ity that the in -fi nite can be counted, or there would have to be transfinite (car di nal)num bers which are nei ther even nor un even.

It is un der stand able, there fore, that the for ma tive de ity of Ar is totle (the nousas thought of thought – Metaphysica, 1074 b 34-35) is fi nite. Ac cord ing to Ar -is totle, only that which is lim ited can be known (con ceived), and he con se -quently does not hes i tate to con clude from the un lim ited na ture of mat ter thatmat ter as such can not be known (Metaphysica, 1036 a 8-9).

A few further contours from the history of the infinite

Un der Ar is totle's in flu ence Origines taught in the 3d cen tury A.D. that Godcould not be in fi nite, since if he is in fi nite (un lim ited) no lim its ex ist and hewould not be able to de limit or con ceive him self – which would im ply thatGod could not know him self!

With Plotinus, how ever, we find a re turn of ap pre ci a tion for the in fi nite sincehe char ac ter izes both the One (out of which ev ery thing arises) and the con -trast ing mat ter as in fi nite (cf. Enneads II,4,4; II,4,10; II,4,15; VI,7,32), al -though the term in fi nite is used in a dia lec ti cally op posed man ner with re gardto the One and mat ter: (form less) mat ter re ceives form (as a per ma nent sub -stra tum) – the (form less) One gives form (cf. En. VI,7,17). This re-appre -ciation is re lated to Plotinus’ view of in fin ity as the time less pres ent (cf. thewhole En.III,7), which si mul ta neously ex erted a con sid er able in flu ence onthe con cep tions re gard ing in fin ity of Boethius, Au gus tine (ConfessionesXI,11,13; De Trinitate XII,14), Thomas Aqui nas (Summa Theologica I,10)and Schilder (1948:61).

26

Au gus tine went fur ther than Plotinus and stated ex plic itly that our in abil ity toun der stand the in fi nite should not be used as a mea sure for God, since God inhis om ni science un der stood ev ery in fin ity – also the com pleted in fi nite set ofall num bers – with out any pas sage of thought, at once, with out be fore and af -ter. There fore God can also know his own com pleted in fi nite be ing.1 Cre ation, how ever, is fi nite. At the end of the mid dle ages and the be gin ning of the mod -ern era Cusanus changed this view with his doc trine that God is ac tu ally in fi -nite while re al ity is only end less. Linked to his con vic tion that the in fi nite lineis si mul ta neously a tri an gle, cir cle and sphere (De Docta Ignorantia, I,13-17)Cusanus taught that of God, as the ac tu ally in fi nite, one could in a cer tainsense say ev ery thing and noth ing at all (he is e.g. the big gest and the small est – De Docta Ignorantia, I,5) since all con tra dic tions are re solved in him(coincidentia oppositorum) (De Docta Ignorantia, I,22; De Coniecturis II,1and II,2).2

Des cartes turns the clas si cal view on its head with his view that the in fi nite iscom plete and the fi nite in com plete, so that the fi nite should ac tu ally be re -ferred to as the non-infinite. Since Spinoza iden ti fied God with na ture (Deussive natura), he also saw the uni verse as com pleted in fi nite.

Ga li leo dis cussed the re mark able re la tion be tween square num bers and the se -quence of all num bers in di a logue form in March 1638: All num bers are notsquare num bers (like 1, 4, 9, 16, 25... ). The com bi na tion of all num bers, i.e.square and non-square num bers, are cer tainly more than the square num berson their own. From 0-100 there are only ten squares (100 = 102). That meansonly one tenth are squares; from 0-10000 there are only 100 squares, i.e.100/10000 = one hun dredth; from 0-1000000 there are only a 1000 squares,i.e. one thou sandth, and so forth. If we how ever ask how many square num -bers ex ist, we can an swer: as many as there are square roots, since ev erysquare has a root and ev ery root has a square. Then, how ever, there are asmany squares as the com bi na tion of all num bers!

12 22 32 42..........1 2 3 4...........

Ber nard Bolzano built on this in a post hu mously pub lished work by con sid er -ing an in fi nite set char ac ter ized by the fact that the whole set can be matchedel e ment by el e ment (in the case of Ga li leo's ex am ple: 1 with 12, 2 with 22, 3with 32, and so forth) with a true sub set (the set of squares is a sub set of the setof nat u ral num bers) (Bolzano, 1920, par.20:27ff.). In the case of an in fi nite set the whole is there fore equal (cor rectly stated: equiv a lent) to a part – this in op -po si tion to Ar is totle's men tioned con vic tion that the whole is al ways greaterthan a part (cf. in this re gard Strauss, 1987)!

Be fore we con tinue with Can tor, we re turn for a mo ment to the his tor i cal re la -tion be tween ir ra tio nal num bers and the in fi nite (among the Greeks this re la -tion led to the geometrization of math e mat ics). In 1790 S. Maimon ar gues that

27

1 Cf. Au gus tine: De civitate Dei, Book XII, Chap ter 19; and cf. Heimsoeth, H. (n.d.:68).

2 Ac cord ing to K. Kremer Plotinus and Proclus ac tu ally al ready taught that all con tra dic tionsare re solved in God (1966:354).

an es sen tial dif fer ence ex ists be tween a so-called ir ra tio nal num ber and theapproximating se quence of ra tio nal num bers em ployed to ex presses it.1 In anef fort to han dle the prob lem of in fin ity, Maimon dis tin guishes be tween thelim ited hu man mind2 and the ab so lute un lim ited mind: “an in fi nite num bercan not ap pear oth er wise to us (since our per cep tion is bound to the form oftime) than as an in fi nite suc ces sion in time (which con se quently can not bethought of as com pleted). In the case of an ab so lute mind, on the other hand,the con cept of an in fi nite num ber, with out any pas sage of time, can be thought of at once. Thus that which the mind in its lim ited form con sid ers as a mereidea, is in terms of its ab so lute ex is tence a real ob ject” (Maimon, 1790:228).

Infinitesimals and the second foundational crisis of mathematics

New ton dis cov ered his first “cal cu lus” in 1665-1666. It is also his tor i cally de -ter mined that Leibniz be tween the years 1673-1676 made the same dis cov eryin de pend ently of New ton.3 Yet he only pub lished his dis cov ery in 1684 and1686.

This “cal cu lus” was known as in fin i tes i mal anal y sis and was sub di vided intodif fer en tial cal cu lus and in te gral cal cu lus. In dif fer en tial cal cu lus it is pos si -ble, for in stance, to de ter mine the in cli na tion of a given curve at any point (thein cli na tion is trig o no met ri cally in di cated by means of the tan gent). In te gralcal cu lus, which was prac ti cally de vel oped as the in verse of dif fer en tial cal cu -lus, en ables the math e ma ti cian to de ter mine sur faces and vol umes of fig ures(par tially) de lim ited by curves.

The spa tial rep re sen ta tion of the in cli na tion of a curve at a par tic u lar point isonly the cor re late of prob lems with bod ies in mo tion, which was the ac tualstart ing-point of New ton's dis cov ery. Sup pose the equa tion y = x2 in di catesthe pre scrip tion for the mo tion of a body (where x is the time in sec ondsneeded to move y feet). The prob lem is to de ter mine the speed of the body at agiven mo ment. An anal y sis of this prob lem gives the fol low ing re sult: af ter 1sec ond the body moves 2 feet per sec ond, af ter 2 sec onds 4 feet per sec ond, af -ter 3 sec onds 6 feet per sec ond, and so forth. Thus an equa tion pro vid ing thespeed at any given mo ment (y' = 2x, with y' re ferred to as the de riv a tive) is de -rived from the orig i nal equa tion.

E.T. Bell com ments: “New ton's first cal cu lus, of 1665-6, seems to have beenab stracted from in tu itive ideas of mo tion. A curve was imag ined as traced bythe mo tion of a ‘flow ing’ point. The ‘in fi nitely short’ path traced by the pointin an ‘in fi nitely short’ time was called the ‘mo men tum’ and this mo men tum

28

1 The ir ra tio nal num ber “square root of 2” ( 2) e.g. is ap prox i mated by the se quence of ra tio nal num bers: 1, 1.4, 1.41, 1.414....

2 Ac cord ing to the Kantian con cep tion it is bound to time – as a so-called a pri ori form of in tu -ition.

3 Hawk ing points out that al though it is clear that both dis cov ered this sub sec tion of math e mat -ics in de pend ently, a fu ri ous ar gu ment none the less broke out over who dis cov ered it first.New ton made the dis cov ery be fore Leibniz but pub lished it later. Most of the ar ti cles de fend -ing New ton's po si tion were how ever writ ten by New ton him self and pub lished un der thenames of his friends (1987:182)!

di vided by the in fi nitely short time was the ‘flux ion’” (Bell, 1945:151). This“flux ion” of New ton is noth ing other than the speed at a given mo ment in ourex am ple above.

The de vel op ment of dif fer en tial and in te gral cal cu lus paid vir tu ally no at ten -tion to its foun da tions, un til it be came clear at the be gin ning of the 19th cen -tury that the use of the in fin i tes i mal (the in fi nitely small) causes a great manyprob lems. Slowly but surely this brought about an ar ith met i cally foundedcon sid er ation of lim its.

In 1770 A.G. Kästner de scribed a limit as fol lows: “A mag ni tude ap prox i -mates a value in fi nitely, when the dif fer ence be tween the ap prox i mat ing andap prox i mated value is less than any specifiable mag ni tude. This value is thenre ferred to as the limit”.1

Even Cauchy (1789-1857) still char ac ter izes the de riv a tive of a given mov ingpoint as the “derniére rai son des différences infiniment pitites Dy et Dx” (asthe “ex treme ra tio of the dif fer ence be tween the in fi nitely small Dy and Dx”).Cauchy also uses an un war ranted tran si tion from the ra tio of in fin i tes i mallysmall num bers to the ra tio of nor mal num bers which are suf fi ciently small. Al -though Cauchy still con sid ers Dx and Dy as the vari ables (i.e. mag ni tudeswhich take on suc ces sively chang ing val ues), he none the less pro vides a rel a -tively clear def i ni tion of a limit in his Text book of Anal y sis (1821): “Whenthe suc ces sive val ues as signed to a vari able in def i nitely ap proaches a fixedvalue to the ex tent that it even tu ally dif fers from it with as lit tle as one wishes,then this last (fixed value) can be char ac ter ized as the limit of all the oth ers”.2

Con sider for in stance the se quence 1/n with the num ber 0 as limit when n in -creases in def i nitely through the se quence of nat u ral num bers 1, 2, 3, .... These quence 1/n then fur nishes 1/1, ½, 1/3, 1/4,.... By merely choos ing a suf fi -ciently large n it is there fore pos si ble to bring the value of 1/n as close to thelimit-value 0 as one wishes.

De spite the firmer foun da tion which Cauchy pro vided for anal y sis, he still did not pro vide a suf fi cient foun da tion for the in tro duc tion of real num bers, andex actly real num bers are vi tal for the firm de vel op ment of anal y sis. Cauchybe lieved that ir ra tio nal num bers can be de fined as the lim its of con verg ing se -quences of ra tio nal num bers. Con sider for in stance the fol low ing in ter est ingse quence of frac tions (prob a bly al ready known to the Greeks):

1

1

3

2

7

5

17

12

41

29

99

70

239

169

577

408

1393

985, , , , , , , , , × × × ×

29

1 “Eine Grösse nähert sich einem Werte unendlich, wenn ihr Unterschied von diesem Wertekleiner als jede Grösse werden kann, die sich angeben lässt. Der Wert heisst alsdann ihreGrenze” (1770:1). D'Alembert in par tic u lar ex er cised sharp crit i cism of the no tion of in fi -nitely small mag ni tudes (quantités infiniment petites) and brought to the at ten tion of Eu ro -pean math e mat ics the limit con cept as the cen tral con cept of anal y sis (cf. Rob in son,1966:268-269).

2 “Lorsque les val u ers successivement attribuées à une même vari able s'approchentindéfiniment d'une val uer fixe, de manière à finir par en différer aussi peu que l'on voudra,cette dernière est appelée la limite de toutes autres” – quoted by Rob in son, 1966:269.

The se quence is cal cu lated as fol lows: the de nom i na tor (un der the line) of ev -ery sub se quent frac tion equals the sum of the nu mer a tor (above the line) andde nom i na tor of the pre vi ous frac tion, while the nu mer a tor of ev ery sub se -quent frac tion equals the sum of its own de nom i na tor and that of the pre vi ousfrac tion. The sum of the nu mer a tor and de nom i na tor of 1/1 equals 2 – the de -nom i na tor of the sec ond frac tion – while the sum of the first two de nom i na tors (i.e. 1+2) equals 3 – the nu mer a tor of the sec ond frac tion. In the same way thede nom i na tor of the third frac tion equals the sum of the nu mer a tor and de nom -i na tor of the sec ond frac tion (i.e. 2+3=5) and the nu mer a tor of the third frac -tion equals the sum of the de nom i na tors of the sec ond and third frac tions (i.e.5+2=7). This se quence of frac tions ap proaches 2 alternately from both sides, namely:

1

1

7

5

41

29

239

169

1393

9852

577

408

99

70

17

12< < < < < ×× < < ×× < < < <

3

2

To the left and right of 2 we find two se quences of ra tio nal num bers whichboth ap prox i mate 2 as their limit. Since a limit is it self de fined as a num ber(!) ap prox i mated by the terms of a se quence in such a man ner that the dif fer -ence be tween the terms of the se quence and the limit-value can be made ar bi -trarily small (i.e. smaller than an ar bi trary ra tio nal num ber Î > 0, as it was later for mu lated), it is clear that the nu mer i cal char ac ter of 2 can not be de fined bymeans of the limit con cept, since the limit con cept pre sup poses that what everfunc tions as limit must al ready be a num ber. Even Cauchy was still of theopin ion that ir ra tio nal num bers must be con sid ered as the lim its of con verg ing se quences of ra tio nal num bers. He is there fore caught in the same cir cu lar ar -gu ment since the pres ence of an ir ra tio nal limit pre sup poses its ex is tence as anum ber, which means that the nu mer i cal na ture of ir ra tio nal num bers can notbe de fined in terms of lim its.

In 1883 G. Can tor ex pressly re jected this cir cle in the def i ni tion of ir ra tio nalreal num bers (1962:187). The even tual de scrip tion of a limit still found intext books to day was only given in 1872 by E. Heine, who was a stu dent of K.Weierstrass with G. Can tor (cf. Heine, 1872:178,182). In 1887 Can tor, how -ever, pointed out that the core of the ideas in Heine's ar ti cle were bor rowedfrom him (1962:385). Fur ther more Can tor him self wrote an ar ti cle on trig o -no met ric se ries in 1872 (Mathematische Annalen, Vol ume 5) in which hegave an equiv a lent de scrip tion of a limit with ref er ence to con ver gent se -quences of ra tio nal num bers (1962: 93). A few pages later he also de scribes alimit-point of an in fi nite set of points (i.e. a point each en vi ron ment of whichcon tains an in fi nite num ber of points of the orig i nal set) (1962:98).1 At thesame time his de scrip tion of the en vi ron ment of a point pro vided an im por tantstart ing-point for mod ern to pol ogy.

Al ready in the ap proach of Weierstrass a new pre sup po si tion pen e trated anal -y sis, namely the pre sup po si tion that lim its can be de fined in terms of a static

30

1 In gen eral a num ber l is called the limit of the se quence (xn), when for an ar bi trary 0 < e a nat u -ral num ber no ex ists such that |xn – l| < e for all n ³ n0.

nu mer i cal do main en com pass ing all real num bers. Boyer re fers to this as fol -lows:

“In mak ing the ba sis of the cal cu lus more rig or ously for mal, Weier strass alsoat tacked the ap peal to in tu ition of con tin u ous mo tion which is im plied inCauchy's ex pres sion that a vari able ap proaches a limit. Pre vi ous writ ers gen -er ally had de fined a vari able as a quan tity or mag ni tude which is not con stant;but since the time of Weierstrass it has been rec og nized that the ideas of vari -able and limit are not es sen tially phoronomic, but in volve purely static con sid -er ations. Weierstrass in ter preted a vari able x as sim ply a let ter des ig nat ing any one of a col lec tion of nu mer i cal val ues. A con tin u ous vari able was like wisede fined in terms of static con sid er ations: If for any value x0 of the set and forany se quence of pos i tive num bers d1, d2, ..., dn’ how ever small, there are in thein ter vals x0 – di, x0 + di oth ers of the set, this is called con tin u ous” (1959:286).

The use of the com pleted in fi nite in the math e mat i cal ap proaches ofWeierstrass, Dedekind and Can tor did in deed suc ceed dur ing the last three de -cades of the 19th cen tury in a bril liant way to give shape to the math e mat i calhan dling of the com pleted in fi nite. With ref er ence to Ar is totle's Physica208a6 Can tor (1962:396) dis tin guishes be tween a[peiron dunavmei anda[peiron wJ" ajfwrismevnon: – the for mer is the po ten tial (non-actually) in fi niteand the lat ter is the ac tual in fi nite. Can tor de scribes the po ten tially in fi nite (inour pro vi sional no ta tion: un com pleted in fin ity) as fol lows: “The po ten tial in -fi nite is pref er a bly in di cated where an in def i nite vari able of fi nite mag ni tudeoc curs, which ei ther in creases be yond all fi nite lim its ..., or de creases be yondall fi nite bounds”. Un der the

“ac tu ally in fi nite, though, is un der stood a quan tum, which on the one handdoes not change, but which rather is set and de ter mined in all its parts, a truecon stant, but si mul ta neously on the other hand ex ceeds in mag ni tude ev erysim i lar fi nite mag ni tude”(1962:401).

The un com pleted in fi nite is linked to the na ture of a vari able and the com -pleted in fi nite to the na ture of a con stant. Anaximander how ever al ready de -scribed the in fi nite (apeiron) – which he con sid ered to be the Arche – asintransient (i.e. con stant), and set this against all flu id ity (i.e. tran sience)!Can tor him self ap peals inter alia to Au gus tine's view of the se quence of allnum bers as an ac tu ally in fi nite quan tum (cf. his De civitate Dei, book 12,chap ter 19).

Au gus tine's ex pla na tion that God con ceives the ac tu ally in fi nite se quence ofall num bers at once (with out any pro cess of thought or be fore and af ter), wasin flu enced by the view of Plotinus of eter nity – linked to the time less pres ent.This her i tage we still find (via Cusanus) in the 18th cen tury when Maimonteaches that an “ab so lute mind” thinks the “con cept of an in fi nite num ber atonce with out any pas sage of time”.

Can tor and Weierstrass avoids the cir cle in Cauchy's def i ni tion of ir ra tio nalnum bers through the use of the ac tual in fi nite. Weierstrass sim ply de fines thecom pleted in fi nite set of num bers 1,1.4,1.414, ... as the “square root of two” ( 2). Can tor also de fines ir ra tio nal num bers as com pleted in fi nite sets of ra tio nalnum bers.1 Can tor ex plic itly com ments in this re gard that a real num ber b can -

31

1 Can tor re fers to these as “Fundamentalreihe” (1962:186ff.,410).

not be de fined as the limit of the mem bers of a fun da men tal se quence (an),since this leads to the log i cal er ror of pre sup pos ing the ex is tence of the limit(1962:187). He adds to this “that the ir ra tio nal num ber by means of the na turewhich has per def i ni tion been as cribed to it pos sesses as much of a par tic u larre al ity in our mind as the ra tio nal and even the in te gers and that the ir ra tio nalnum bers are not ob tained only by means of a limit-pro cess, since one is on thecon trary al ready con vinced pre vi ously, due to their pos ses sion, of the prac ti -ca bil ity and ev i dence of limit-pro cesses in gen eral” (1962:187).

Con versely Can tor de fends his doc trine of transfinite num bers (de vel oped onthe ac cep tance of com pleted in fin ity) by means of ir ra tio nal num bers: “Onecan say with out more ado: the transfinite num bers stand or fall with the fi niteir ra tio nal num bers; they con cur with re gard to their in ner most be ing; sinceboth are par tic u lar de lim ited ex pres sions or mod i fi ca tions of the ac tu ally in fi -nite” (1962:395-396). Paul Lorenzen de scribes this mod ern con cep tion of real num bers in terms of the com pleted in fi nite in a way which strik ingly re flectsthe age-old tra di tion on which it rests:

“and thus ev ery real num ber as an in fi nite dec i mal frac tion is al ready rep re -sented as if the in fi nite quan tity of num bers all ex isted at once (auf einmalexistierten)” (1968:100).

Cantor and Aristotle

In the elab o ra tion of his set the ory Can tor takes crit i cal dis tance from Ar is -totle's rad i cal re jec tion of the use of the ac tu ally in fi nite, al though he oth er -wise es sen tially re tains in his de scrip tion of the con tin uum the two stip u la -tions which Ar is totle pos ited for the na ture of con ti nu ity.

(a) Aristotle's objections against the actual infinite

Ar is totle's first ob jec tion, namely that the whole could not in the case of theac tu ally in fi nite ex ceed the part in mag ni tude, had al ready been used byBolzano (anal o gously to Ga li leo) ex actly as cri te rion for in fi nite sets: a set isin fi nite if and only if the whole set can be mapped one-to-one with a true sub -set thereof. In this char ac ter iza tion lies the an swer to the first part of Ar is -totle's sec ond ob jec tion, the ob jec tion namely that it is im pos si ble to count thein fi nite. Can tor how ever only uses the char ac ter iza tion: denumerable. Any set which can be cor re lated one-to-one with the nat u ral num bers (0), 1, 2, 3, 4, 5,... is denumerable. The true mean ing of the in di ca tion “denumerable” how -ever only be came ap par ent when Can tor proved that there does in deed ex istnon-denumerable transfinite num bers.1

When one re lin quishes the na ture and all re la tions of the el e ments of a set(par tic u larly also the or der ing which might ex ist among the var i ous el e ments), then the power (or: car di nal ity) of dif fer ent sets can be com pared. Two sets M and N are re ferred to by Can tor as equiv a lent when their el e ments can bemapped one-to-one (“wenn sie sich gegenseitig eindeutig El e ment für El e -ment einander zuordnen lassen”) (1962:387). It can be in di cated with the term

32

1 Can tor con sid ers it mean ing less to re ject ref er ences to an in fi nite num ber and to speak only of in fi nite sets – both are ac cord ing to him in dis sol u bly linked to each other (1962:394).

or der that for ev ery two el e ments of a set the first pre cedes the sec ond, whilethe sec ond fol lows on the first. If A and B are two or dered sets, and if an or -der-preserving re la tion ex ists among their el e ments, both have ac cord ing toCan tor the same or der-type, while the sets are re ferred to as sim i lar (ähnlich)(1962:297). “A set is called well-ordered, if it meets the re quire ment that ev -ery sub set (Teilvielheit) has a first el e ment”.1 In view of this Can tor de fines an or di nal num ber: “The or der-type of a well-ordered set F we call its due or di nal num ber” (1962:321). In view of the pre ced ing we can state that the transfiniteor di nal num ber (w) must meet the fol low ing four con di tions (strictly speak ing the last con di tion is equiv a lent to the sec ond):(i) It has a first el e ment;(ii) each el e ment has an im me di ate suc ces sor;(iii) each el e ment ex cept the first el e ment has an im me di ate pre de ces sor; and(iv) no last el e ment ex ists.

The re mark able char ac ter is tics of this transfinite or di nal num ber w (w in di -cates the set of nat u ral num bers in their nat u ral or der: 1, 2, 3, 4, 5, 6, ...), en -ables Can tor to an swer Ar is totle's last ob jec tion (namely that ac tu ally in fi nitenum bers would have to be nei ther even nor un even) (cf. Can tor, 1962:178-179).

Within the sys tem of nat u ral num bers, which is closed un der ad di tion andmul ti pli ca tion (i.e. add ing and mul ti ply ing nat u ral num bers in ev ery in stanceren ders nat u ral num bers), the com mu ta tive law is valid with re gard to boththese op er a tions. In other words, with re gard to two nat u ral num bers a and b,a + b = b + a and ab = ba. This com mu ta tive law is not, how ever, gen er allyvalid with re gard to transfinite or di nal num bers. We can il lus trate it as fol lows (note how ever that Can tor in this ex am ple un der stands in the prod uct ba that bis the “mul ti plier” and a the “multiplicandus” – 1962:178):

Take the set A = (1,2) and B = (1, 2, 3, 4, ,5 6, ...) and con sider the (lex i co -graph i cally) or dered prod uct AB; i.e. {(1,1),(1,2),(1,3), ...,(2,1),(2,2),(2,3),..}

or w + w = w.2 (i.e., w ¹ w.2)

The or dered prod uct BA how ever gives us: {(1,1),(1,2),(2,1), (2,2),(3,1), ..}This prod uct clearly meets the four re quire ments, which im plies that 2.w = w.From the prod uct AB we saw that w ¹ w.2:

The num ber w can there fore be rep re sented as 2.w and also as 1 + 2.w – butnever the other way around, since w ¹ w.2 and w ¹ w.2 + 1.

The con clu sion is clear: w is even (namely 2.w), as well as un even (namely1 + 2.w) and si mul ta neously w is nei ther even (namely ¹ w.2), nor un even(namely ¹ w.2 + 1 – cf. Can tor, 1962:178-179).

Com pare this with the no tion of Cusanus that God as the ac tu ally in fi nite is the un ion of all op po sites: the coincidentia oppositorum – he did af ter all rec og -nize some thing es sen tial about the in fi nite!

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1 This de scrip tion is to be found in a let ter to Dedekind (28 July 1899), con tained in Can tor,1962:444 (cf. also Can tor, 1962:312).

(b) Continuity in Aristotle and Cantor-Dedekind

Ar is totle con sid ers it im pos si ble to ex plain the con ti nu ity of a straight line interms of the (in fi nite) num ber of its points. If “that which is in fi nite is con sti -tuted by points, these points must be ei ther con tin u ous or con tin u ously in con -tact with one an other” (Physica, 231a29-31). Points are how ever in di vis i ble(a point has no parts), while “that which ex ists be tween two points is al ways aline” (Physica, 231b8). Ac cord ing to Ar is totle it is clear that “ev ery thingwhich is con tin u ous is di vis i ble into di vis i ble parts which can be di vided in fi -nitely: since if it was di vis i ble into in di vis i ble parts, we would have the di vis i -ble and in di vis i ble in con tact since the lim its of con tin u ous things are one (i.e.the same – DFMS)” (Physica, 231b15ff.).1

In the 19th cen tury a new arithmeticistic ten dency came to the fore in math e -mat ics which would ap pear to be in tent on the ar ith met i cal def i ni tion of spa -tial con ti nu ity. Ber nard Bolzano al ready il lu mi nates this ten dency in par.38 of his (quoted) work on the par a doxes of the in fi nite. He men tions the ob jec tionthat a cir cle would ap pear to be hid den in the at tempt to build ex ten sion out ofparts not them selves ex tended, but is of the opin ion that the prob lem dis ap -pears when it is re al ized that “each whole” ex actly “has nu mer ous prop er tiesab sent in the parts” (1920:72). The ques tion how ever is whether the re la tion -ship: whole-parts is orig i nally ar ith met i cal in na ture?! (We will re turn to thisagain.)

The cri te rion which Bolzano sets for a con tin uum is that “a con tin uum is pres -ent there, but also only there where a set (Inbegriff) of sim ple ob jects (ofpoints ...) finds it self, which is sit u ated in such a way that ev ery sin gle ob jecthas at least an en vi ron ment (Nachbar) (of points – DFMS) in this set for ev erydis tance (Entfernung) how ever small” (1920:73).

Can tor crit i cizes Bolzano's cri te rion as in suf fi cient, since a set con sti tuted e.g. by dis tinct con tinua (and there fore be ing as a whole dis con tin u ous) wouldstill be con tin u ous in terms of Bolzano's def i ni tion (the end points of each dis -tinct con tin uum would af ter all still con tain fur ther points of the par tic u lar setin an ar bi trarily small en vi ron ment – cf. Can tor, 1962:194).

Can tor de clares that he has no other choice but to posit a “gen eral purely ar ith -met i cal con cept of a point-continuum” with the help of the way in which hede fined real num bers (1962:192). A point-continuum he de fines as a per fectlyco her ent set. A set is per fect when ev ery point of the set is a limit-point andwhen all limit-points of the set be long to the set. He calls “T a co her ent pointset, when for ev ery two points t and t’ of this set, at a given ar bi trarily smallnum ber e there are al ways a fi nite num ber of points t1, t2, t3, ..., tn of T’ pres entin mul ti ple ways, so that the dis tances tt1, t1t2, ...tnt’ are all to gether smallerthan e” (1962:194).

34

1 Von Weizsäcker says that the do main in which fig ures are de fined dis plays, when com paredwith nat u ral num bers, the prop erty of con ti nu ity. In this con text he calls upon Ar is totle's viewof con ti nu ity: “Con ti nu ity is de fined by Ar is totle as that which could be di vided end lessly insim i lar parts. The parts of a con tin uum can not be counted; but one can mea sure con tinua”(1993:115).

Comment: Can tor's def i ni tion of co her ence con cerns a met ri cal char ac -ter is tic of the con tin uum. In mod ern to pol ogy, how ever, con ti nu ity isde scribed in terms of open sets (ab stracted from the char ac ter is tics of amet ri cal space – cf. Wil lard, 1970:16-19). P.S. Alexandroff de fines the con tin uum as a non-empty com pactly co her ent set (1956:163ff.,201ff.). A set is com pact if ev ery in fi nite sub set has at least onelimit-point. (A point x is a limit-point of a set A when ev ery en vi ron -ment of x con tains at least one point of A dif fer ent from x). This im -plies that a set in a Eu clid ean space is only com pact if it is de lim ited. Interms of Alexandroff's def i ni tion an in fi nite straight line is there forenot con tin u ous, while it is for Can tor (also cf. Meschkowski, 1967:55).

Dedekind fol lows the con ti nu ity of a straight line ar ith met i cally by time andagain in tro duc ing new num bers:

“If now, as is our de sire, we try to fol low up ar ith met i cally all phe nom ena inthe straight line, the do main of ra tio nal num bers is in suf fi cient and it be comesnec es sary that the in stru ment R con structed by the cre ation of ra tio nal num -bers be es sen tially im proved by the cre ation of new num bers (namely ir ra tio -nal num bers – DFMS) such that the do main of num bers shall gain the samecom plete ness, or as we may say at once, the same con ti nu ity, as the straightline” (Dedekind, 1901:9)

On this foun da tion Dedekind de scribes his well-known cut no tion whichchar ac ter izes con ti nu ity: when

“all real num bers break up into two classes U1, U2, such that ev ery num ber a1

of the class U1 is less than ev ery num ber a2 of the class U2 then there ex ists oneand only one num ber by which this sep a ra tion is pro duced” (1901:20).

Dedekind's no tion of a cut is dealt with in anal y sis text books in such a man nerthat the real num ber which brings about the “split” is greater than or equal toev ery el e ment in the one set and smaller than or equal to all the el e ments of the other set (cf. e.g. Bartle, 1964:51).

Can tor him self re fers to the re la tion which ex ists be tween his view of a per fect set and Dedekind's cut the o rem (1962:194). G. Böhme strik ingly shows in anar ti cle how Can tor's def i ni tion of the con tin uum con tains two stip u la tionswhich both meet the Ar is to te lian def i ni tion of a con tin uum, namely co her ence and a char ac ter is tic which en sures the ex is tence of di vid ing points for in fi nitedi vi sion (1966:309). By means of only al low ing a Dedekind-cut at di vi sions,Böhme jus ti fies his state ment as fol lows:

“when a Cantorian con tin uum as such is di vided in two by means of the in di ca -tion of a point so that the one set con tains those points which are in nu mer i calvalue greater than or equal to the in di cated point, while the other set con tainsthose points of which the nu mer i cal value are smaller than or equal to the nu -mer i cal value of the in di cated point, both parts are again con tin u ous. Such di -vi sions are pos si ble into in fin ity (due to the per fec tion of the con tin uum), andthe parts are still co her ent in the Ar is to te lian sense (i.e. their limit-points arethe same)” (1966:309).

This is a re mark able sit u a tion: the Can tor-Dedekind de scrip tion of the con tin -uum pre sup poses the use of the com pleted in fi nite (in par tic u lar by us ing the

35

com pleted in fi nite set of real num bers), but none the less meets Ar is totle's twore quire ments for a con tin uum – and this while Ar is totle ex actly re jects thecom pleted in fi nite and rec og nizes only un com pleted in fin ity! Did Ar is totleac tu ally use the com pleted in fi nite im plic itly, or is the Can tor-Dedekind def i -ni tion in the last in stance not purely ar ith met i cally founded? We shall shortlyat tempt to find an an swer to this ques tion.

Non-denumerability: Cantor's Diagonal Proof

A set is called (d)enumerable when its el e ments can be cor re lated one-to-onewith those of the set of nat u ral num bers, i.e. any set the el e ments of which canbe ar ranged in a nat u ral se quence of 1, 2, 3, 4, 5, 6, ... It is clear that the in te -gers are denumerable: 0, -1, +1, -2, +2, -3, +3, ... Since all ra tio nal num berscan be de picted by two in te gers in the form of a/b (with b ¹ 0), it is clear thatthey also can be denumerated. No tice the course of the ar rows in the fol low ing de pic tion:

Even all al ge braic num bers are denumerable.1 In a let ter of 29 No vem ber1873 Dedekind men tions to Can tor that he had proven that all al ge braic num -bers are denumerable (cf. Meschkowski, 1972b:23). Dedekind does this byde fin ing the height h of an al ge braic num ber x sat is fy ing a polinomial equa -tion anx

n + an-1xn-1 + ...+ a1x + a0 = 0 as fol lows:

h = n – 1 + |a0| + |a1| + .......... + |an|

Since the co ef fi cients an are in te gers, only a fi nite num ber of al ge braic num -bers be long to each height h. Since ev ery fi nite quan tity is denumerable, theal ge braic num bers as such are also denumerable (cf. Meschkowski,1972b:24).

In 1874 how ever Can tor proved that the real num bers are not denumerable(i.e. are non-denumerable). Only in 1890 does he pro vide his di ag o nal-proof,which we use in our ex pla na tion be low (cf. Can tor, 1962:278-281). Aone-to-one cor re spon dence could be es tab lished be tween all real num bers and the set of real num bers be tween 0 and 1. Fur ther more, ev ery real num ber in

36

1 1 1 1

2 2 2 2

3 3 3 3

4 4 4 4

1 2 3 4

1 2 3 4

1 2 3 4

1 2 3 4

. . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

. . . . . . . . . . . . .

1 Al ge braic num bers are the roots of al ge braic equa tions.

this in ter val can be rep re sented as an in fi nite dec i mal frac tion of the form xn =0.a1a2a3a4 ... (num bers with two dec i mal rep re sen ta tions, e.g. 0.100000... and0.099999 ... are con sis tently rep re sented in the form with nines). Sup pose adenumeration x1, x2, x3, ... ex ists of all the real num bers be tween 0 and 1, i.e. of all the real num bers in the in ter val 0 £ xn £ 1 (i.e. [0,1]), namely:

x1 = 0.a1 a2 a3... ............

x2 = 0.b 1 b 2b3 ............

x3 = 0.c1 c2 c3 ...........

...................................

If an other num ber can be found be tween 0 and 1 which dif fers from ev ery xn,it would mean that ev ery denumeration of the real num bers would leave out atleast one real num ber, which would prove that the real num bers arenon-denumerable. Such a num ber we can con strue as fol lows:

y = y1y2y3y4 ..., with y1 ¹ 0, a1 and 9; y2 ¹ 0, b2 and 9; y3 ¹ 0, c3

and 9; and so forth.

It is clear that y is a real num ber be tween 0 and 1 (i.e. 0 £ y £ 1). The num ber ydoes not have two dec i mal rep re sen ta tions since ev ery dec i mal num ber in itsdec i mal de vel op ment is un equal to 0 and 9. The num ber is also un equal to ev -ery real num ber xn since the dec i mal de vel op ment of y in the first dec i malplace dif fers from the first dec i mal num ber x1, in the sec ond dif fers from thesec ond dec i mal num ber of x2 (namely x2), and in gen eral from the nth dec i malnum ber of xn. It is clear from this that a denumeration of the real num bers willal ways ex clude at least one real num ber (“mis count” it in the denumeration),which con cludes Can tor's proof that real num bers are non-denumerable.

CommentAl though intuitionism ac cepts this proof as valid, it does so in a constructivistsense.1 All constructivist in ter pre ta tions are how ever in ad e quate to reach anon-denumerable con clu sion (cf. Wolff, 1971), sim ply be cause no con struc -tive tran si tion is pos si ble from the po ten tial in fi nite to the ac tual in fi nite. Ourcur rent dis cus sion does not pro vide room for a broader in di ca tion that the ap -proach of Brouwer (al ready in his 1907 dis ser ta tion) and Heyting in deed isam big u ous with re gard to the role of the ac tual in fi nite. The same can af ter allbe said of Poincaré – on the one hand he re jects the ac tual in fi nite (1910) andon the other he at tempts in the same ar ti cle to pro vide an al ter na tive proof forthe non-denumerability of real num bers – with out re al iz ing that non-de -numerability can only be proven if the ac tu ally in fi nite is ac cepted.

There is a fur ther re mark able side to this re sult. In his orig i nal proof of 1874(1962:115-118) Can tor first proved that all real al ge braic num bers are

37

1 Cf. Heyting (1971:40), Fraenkel et al (1973:256,272), and Fraenkel (1928:239 note 1).

denumerable. All real num bers, how ever, are non-denumerable, which proves the ex is tence of a non-denumerable num ber of non-algebraic (also called:tran scen dent) num bers!1

The third foundational crisis in Mathematics

The first two foun da tional cri ses in math e mat ics were the re sult of the dis cov -ery of ir ra tio nal num bers and the found ing of the so-called in fin i tes i mal cal cu -lus.2 By 1895 Can tor dis cov ered that his set the ory con tained anom a lies. Can -tor proved e.g. the prop o si tion that for ev ery set A of or di nal num bers an or di -nal num ber ex ists which is greater than ev ery or di nal num ber con tained in theset. Con sider how ever the set W of all or di nal num bers. Since this set is a setof all or di nal num bers, the fore go ing prop o si tion im plies that an or di nal num -ber ex ists which is greater than ev ery or di nal num ber con tained in W – but this is con tra dic tory, since the set W is sup posed al ready to con tain all or di nalnum bers! A sim i lar antinomy is valid with re gard to Can tor's car di nal num -bers (cf. Meschkowski, 1967:144-145 and Singh, 1985:73).

In 1900 Rus sell made pub lic his antinomy which can be for mu lated in termsof the ABC of set the ory. Con sider the set C with el e ments A and the pre scrip -tion that el e ments of set C may only be those sets A which do not con tainthem selves as el e ments.

Thus C = (A/A Ï A). (The set of ten chairs is e.g. not it self a chair and does notcon tain it self as an el e ment. On the other hand the set of think able thoughts isin it self think able and there fore does con tain it self as an el e ment.) Now sup -pose that C is an el e ment of C (C Î C). Ev ery el e ment of C, how ever, does notcon tain it self as an el e ment – this, af ter all, is the re quire ment for be ing an el e -ment of C. This im plies that if C is an el e ment of C, it must also meet this re -quire ment – but then C Î C im plies C Ï C! Sup pose on the other hand that C Ï C. Then C does meet the re quire ment for be ing an el e ment of C, whichmeans that C Î C! In other words, C is an el e ment of C if and only if C is not anel e ment of C!

C Î C Û C Ï C!

In 1900 the French math e ma ti cian, Poincaré, made the proud claim that math -e mat ics has reached ab so lute rig our. In a stan dard work on the foun da tions ofset the ory, how ever, we read:

“iron i cally enough, at the very same time that Poincaré made his proud claim,it has al ready turned out that the the ory of the in fi nite sys tems of in te gers –

38

1 Just by the way: Can tor de vel oped a whole hi er ar chy of transfinite car di nal num bers on thefoun da tion of his def i ni tion of or di nal num bers.

2 Kline writes: “It was clear to the math e mat i cal world of the late 18th cen tury that proper foun -da tions for the cal cu lus were ur gently needed, and at the sug ges tion of Lagrange the Math e -mat ics sec tion of the Berlin Acad emy of Sci ences, of which he was the di rec tor from 1766 to1787, pro posed in 1784 that a prize be awarded in 1786 for the best so lu tion to the prob lem ofthe in fi nite in math e mat ics” (Kline, 1980:149-150).

noth ing else but part of set the ory – was very far from hav ing ob tained ab so -lute se cu rity of foun da tions. More than the mere ap pear ance of antinomies inthe ba sis of set the ory, and thereby of anal y sis, it is the fact that the var i ous at -tempts to over come these antinomies, ..., re vealed a far-going and sur pris ingdi ver gence of opin ions and con cep tions on the most fun da men tal math e mat i -cal no tions, such as set and num ber them selves, which in duces us to speak ofthe third foun da tional cri sis that math e mat ics is still un der go ing” (Fraenkel,1973:14).

Divergence of opinionAl though the di ver gent opin ions which emerged can not be seen sim ply as amere re ac tion to the antinomies (since their roots reach much fur ther back into his tory), the flow er ing of three schools – logicism, intuitionism, and (ax i om -atic) for mal ism – dur ing the first de cade of the 20th cen tury are none the lessin ex tri ca bly linked to it.Im plicitly or ex plic itly ev ery point of view in math e mat ics must ac count forthe re la tion ships among the var i ous as pects of re al ity, in clud ing num ber,space, move ment, the log i cal and lin gual fac ets. Bertrand Rus sell's logicismwanted to re duce all of math e mat ics to logic – for this rea son he de clares:“math e mat ics and logic are iden ti cal” (1956:v). A. Heyting ex presses the ex -actly op po site intuitionist sen ti ment: “ev ery log i cal the o rem ... is but a math e -mat i cal the o rem of ex treme gen er al ity; that is to say, logic is a part of math e -mat ics, and can by no means serve as a foun da tion for it” (Heyting, 1971:6).Even Hilbert's for mal ism must (un der the in flu ence of Kant) be gin with therec og ni tion of a more than log i cal di ver sity which im plies that no sci ence canbe ex clu sively founded in logic: “Kant al ready taught – and it rep re sents an in -te gral part of his doc trine – that math e mat ics has a guar an teed con tent in de -pend ently of all logic and can there fore never be grounded solely in logic,which im plies that the ef forts of Frege and Dedekind must fail. There is a fur -ther pre req ui site for the ap pli ca tion of log i cal con clu sions and for the per -form ing of log i cal op er a tions, namely that some thing must be given in thecon cep tion: spe cific ex tra-logical con crete ob jects, in tu itively pres ent as im -me di ate ex pe ri ence prior to all think ing” (Hilbert, 1925:170-171).Ac cord ing to the formalistic-axiomatic ap proach math e mat ics should com -pletely re lin quish the truth of the ini tial pos tu lates (ax i oms) and only take ac -count within the re quire ment of con sis tency (non-contradiction) that the validthe o rems can be de duced an a lyt i cally.1 Since, ac cord ing to this point of view,it does not mat ter what we are talk ing about, as long as we are do ing so con sis -tently, Rus sell for mu lated his fa mous ep i gram: “Pure math e mat ics is the sub -ject in which we do not know what we are talk ing about, or whether what weare say ing is true” (quoted by Nagel, 1971:13).Al ready in 1926 P. Finsler showed that in a pure for mal math e mat i cal dis ci -pline, de fined by ax i oms and rules of cal cu lus, there are prop o si tions whichcan be nei ther proven nor con tra dicted (cf. Mathematische Zeitschrift, 25(1926); Finsler, 1975:1-49; as well as Heitler, 1972:50). But in 1931, at the

39

1 The dis tinc tion be tween an a lyt i cal and syn thetic judg ments is used in a Kantian sense by thevar i ous ap proaches in math e mat ics. Cf. Kant, 1787:11.

age of 25, K. Gödel shook the world of math e mat ics with an ar ti cle on the for -mally undecidable prop o si tions in the Principia Mathematica of Rus sell andWhite head, and re lated sys tems. Gödel showed that a proof of the con sis tency of arith me tic can not be re flected in the for mal de duc tions of arith me tic it self – the con sis tency of arith me tic, that is, can not be proven in terms of the ax i omsof arith me tic. In a for mal ax i om atic sys tem Z there al ways ex ists a state ment Awhich can be nei ther proved nor dis proved with the aid of ax i oms of Z. Inother words, to prove that the con clu sions reached from cer tain ax i oms arecon sis tent, it is not pos si ble to use the method in ques tion. In prin ci ple ev eryax i om atic sys tem in math e mat ics is in com plete – it re quires and pre sup posesin sight into its con tent which tran scends its own for mal ism.

H. Weyl com ments strik ingly in this re gard: “It must have been hard onHilbert, the axiomatist, to ac knowl edge that the in sight of con sis tency israther to be at tained by in tu itive rea son ing which is based on ev i dence and not on ax i oms” (1970:269).

Al though this di ver gence in mod ern math e mat ics re lates to the third foun da -tional cri sis of math e mat ics, its philo soph i cal sources are much older. His -torically seen, Brouwer, Gödel, and Hilbert de rive their philo soph i cal pointsof de par ture from the three main parts of the Kritik der reinen Vernunft (1781,2nd edi tion 1787) of Kant: Brouwer from the tran scen den tal aes thet ics,Gödel from the tran scen den tal an a lytic, and Hilbert from the tran scen den taldi a lec tic. This ex plains why even in its sec ond and third foun da tional cri sesmath e mat ics still could not es cape from the fun da men tal philo soph i cal prob -lems al ready pres ent in the foun da tional cri sis of Greek math e mat ics.

Prob lems sur round ing dif fer en tial and in te gral cal cu lus (the so-called in fin i -tes i mals) led to the re for mu la tion of the limit con cept, which in its turnbrought about the use in set the ory of the ac tu ally in fi nite, with the even tualex po sure of the antinomies in the na ive set con cept of Can tor. Hilbert andBernays point out that the fail ure of Frege's logicistic pro ject par tic u larly ex -posed the prob lems in the pre sup po si tion of the to tal ity of nu mer i cal se -quences (Grundlagen, I, 1934:15). Ex actly in this re gard neo-intuitionismchose for the un com pleted infinite, while ax i om atic set the ory (fa thered byZermelo and Fraenkel) at tempted by means of all sorts of lim i ta tions to pro -tect set the ory against antinomies with out sac ri fic ing the ac tual in fi nite.

We shall ar gue be low that the use of the un com pleted in fi nite pre sup poses thenu mer i cal or der of suc ces sion on the law-side of the nu mer i cal as pect and that when this nu mer i cal time-order is dis closed by an tic i pat ing for ward to thespa tial or der of si mul ta ne ity any un com pleted in fi nite set (such as the set ofnat u ral num bers, whole num bers, or ra tio nal num bers) can be viewed as if allthe el e ments are at hand at once as a com plete to tal ity (‘fertige Gesamtheit’ inthe words of Hilbert and Bernays). The re la tion ship be tween these two kindsof in fin ity, which has de manded at ten tion anew since the third foun da tionalcri sis of math e mat ics (it had al ready been at is sue in Greek thought), ul ti -mately ap peals to one of the fun da men tal philo soph i cal is sues of math e mat icsas a spe cial sci ence: what is the re la tion and co her ence be tween the nu mer i caland spa tial as pects of re al ity? The em pha sis in reformational phi los o phy on

40

the cos mic im pact of the creational prin ci ple of sphere-sovereignty there forere tains its rel e vance for math e mat ics with re gard to the mu tual irreducibilityof the nu mer i cal and spa tial as pects. Within the ambit of this study the pur -pose has only been to il lu mi nate the im por tance of philo soph i cal re flec tionalso in math e mat ics, with ref er ence to the foun da tional cri ses in math e mat ics.Par tic u larly against this back ground the fol low ing state ments gain sig nif i -cance:

“from the ear li est times two op pos ing ten den cies, some times help ing one an -other, have gov erned the whole in volved de vel op ment of math e mat ics.Roughly these are the dis crete and the con tin u ous” (Bell, 1965:12);

and

“Bridging the gap be tween the do mains of dis crete ness and of con ti nu ity, orbe tween arith me tic and ge om e try, is a cen tral, pre sum ably even the cen tralprob lem of the foun da tion of math e mat ics” (Fraenkel, A., et al., 1973:211).

Rucker else where re marks:

“The dis crete and con tin u ous rep re sent fun da men tally dif fer ent as pects of themath e mat i cal uni verse” (Rucker, 1982:243).

Also Moore dis cerns two ‘clus ters’ of concetps which dom i nates the his toryof the no tion of in fin ity. In the first clus ter the fol low ing terms are cat e go -rized:

“bound less ness; end less ness; un lim it ed ness; immeasurability; eter nity; thatwhich is such that, given any de ter mi nate part of it, there is al ways more tocome; that which is greater than any as sign able quan tity” (1990:1). Within thesec ond clus ter he men tions: “com plete ness; whole ness; unity; uni ver sal ity;ab so lute ness; per fec tion; self-sufficiency; au ton omy” (Moore, 1990:1-2).

An in ter est ing de vel op ment de serves to be men tioned be fore we con clude this dis cus sion. A. Fraenkel in di cated the in fer til ity of the idea of “in fin i tes i mals”(the in fi nitely small) and its re jec tion by Can tor and the math e mat i cal world at large in the fourth print ing of his Ab stract Set The ory (Am ster dam 1968:120-123). In the mean time, how ever, Abaraham Rob in son de vel oped a newand fer tile use of the in fin i tes i mal ex actly on the ba sis of the use of ac tu ally in -fi nite sets (transfinite cardinalities) by Can tor. A num ber a is called in fin i tes i -mal (or in fi nitely small) if its ab so lute value (that is its value re gard less of theplus or mi nus sign) is less than m for all pos i tive num bers m in Â(Â be ing theset of real num bers). Ac cord ing to this def i ni tion 0 is in fin i tes i mal. The factthat the in fin i tes i mal is merely the cor re late of the transfinite num bers, is ap -par ent in that r (not equal to 0) is in fin i tes i mal if and only if r to the power ofmi nus 1 is in fi nite (cf. Rob in son, 1966:55ff). By means of the in fin i tes i mal itis pos si ble to mean ing fully de fine lim its, de riv a tives, etc.Rob in son points out that Can tor's treat ment of in fi nite sets is nec es sary: “...ab stract set the ory forms a his tor i cal back ground to the free and easy han dlingof in fi nite sets that is re quired in Non-standard Anal y sis” (p. 279). To this headds that “what ever our out look ...., it ap pears to us to day that the in fi nitelysmall and in fi nitely large num bers of a non-standard model of Anal y sis arenei ther more nor less real than, for ex am ple, the stan dard ir ra tio nal num bers”

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(p. 828).1 And rightly so, since both in fin i tes i mal and ir ra tio nal num bers af terall ex ist by the grace of the deep ened sense of num ber which, as we shall pres -ently ar gue, pro vides (at the law-side to the quantiative as pect) math e mat i calmean ing to the com pleted in fi nite.

The con sid er ation of these fun da men tal prob lems will al ways re main rel e -vant, as is con firmed by the state ments of Fraenkel et al. and Kline at the be -gin ning of our ex po si tion.

Herman Weyl pos its against this that “it ap pears from the intuitionist per spec -tive that (com plete) in duc tion2 pro tects math e mat ics from be ing en tirely tau -to log i cal and char ac ter izes its as ser tions as (non-analytically) syn thetic”(1966:86). Fol low ing the pro ject of Kant's cri tique of Pure Rea son (1787:19)Weyl here de fends the ex is tence of syn thetic judg ments a pri ori in math e mat -ics. Ad mit tedly he (and Brouwer) ac cept only Kant's no tion of time. Num ber,ac cord ing to Kant, is due to the sche ma ti za tion of the log i cal cat e gory quan -tity in time as a form of in tu ition (1787:182); it be longs to the syn thetic judg -ments of math e mat ics a pri ori. At his pro mo tion in 1885 D. Hilbert de fendedthe a pri ori char ac ter of nu mer i cal judg ments: The sis II: “the ob jec tionsagainst Kant's the ory of the a pri ori char ac ter of ar ith met i cal judg ments areun founded” (cf. Reid 1970:17). Brouwer bases his math e mat ics on the pri -mor dial in tu ition of con ti nu ity and dis crete ness, a pos si bil ity of think ing to -gether mul ti ple units com bined by an ‘in-between’ which can not be ex hausted through the ‘inter-positioning’ of new units.3

Brouwer's de scrip tion of the pri mor dial in tu ition al ready con tains the re jec -tion of the ac tual in fi nite by intuitionism – our next theme.

Questioning completed infinitude

In intuitionist math e mat ics the in fi nite is taken lit er ally as with out an end,never to be com pleted and con tin u ally be com ing. Al ready in his let ter of July12, 1831 to Schumacher Gauss stated that “in this man ner I pro test against theuse of an in fi nite mag ni tude as some thing com pleted, which is never al lowedin math e mat ics” (re ferred to in Becker 1964:180). The early intuitionist,Leopold Kronecker, a con tem po rary and op po nent of Can tor, rad i cally re -jected the com pleted in fi nite and even at tempted to base all of math e mat ics in

42

1 As men tioned, Can tor also re ferred to the ir ra tio nal num bers with re gard to his transfinitenum bers: “One can sim ply say that the transfinite num bers stand or fall with the fi nite ir ra tio -nal num bers; they are sim i lar in their in ner most be ing” (GA, pp. 395-396).

2 If an as ser tion is true for a num ber 1 and also for n + 1 if it is true for n, then it is gen er ally true(a prin ci ple first dem on strated by Blaise Pascal, ac cord ing to Freundenthal, 1940).

3 “... als het van qualiteit ontdane substraat van alle waarneming en verandering, een eenheidvan continu en dis creet, een mogelijkheid van samendenken van meerdere eenheden,verbonden door een tusschen, dat door inschakeling van nieuwe eenheden zich nooit uitput”(1907:8). Both this math e mat i cal pri mor dial in tu ition of Brouwer and the im me di ate ex tra-and pre-log i cal in tu itive ex pe ri ence of Hilbert is seen in a Kantian sense as a pri ori of sci encein gen eral and math e mat ics in par tic u lar. Brouwer how ever dif fers from Hilbert's dis tinc tionbe tween for mal ized math e mat ics and the in tu itive in ter pre ta tion thereof (the lat ter is also re -ferred to as metamathematics) (cf. Beth, 1965:94). Ad mit tedly an in tu itive math e mat ics isneeded even to de fine a for mal math e mat ics (cf. Kleene, 1952:62)!

the fi nite nat u ral num bers (cf. Scholz 1969:293-294). The early Frenchintuitionist, H. Poincaré (known in life as the great est math e ma ti cian of hisage un til his death in 1912 when Hilbert took over his lau rels), also ex presslyre jected the com pleted in fi nite. The to tal ity of or di nal num bers of the small esttransfinite car di nal num ber (aleph-null: Ào) were used by Can tor to ob tain the next transfinite car di nal num ber, aleph-one (À1): “With re gard to the sec ondtransfinite car di nal num ber aleph-one, I am not en tirely con vinced of its ex is -tence ... (and) whether we can speak of its car di nal ity with out con tra dic tion.The ac tual in fi nite in any case does not ex ist” (Poincaré, 1910:48).

Also Brouwer (who iden ti fies ex is tence and constructibility and de nies theva lid ity of the log i cal prin ci ple of the ex cluded third with re gard to the in fi -nite) re jects the com pleted in fi nite.1 With this Brouwer re jects the transfinitenum ber the ory of Can tor. The no tion of count abil ity af ter all only be comespar tic u larly rel e vant af ter the dem on stra tion of the ex is tence of non-de -numerable cardinalities. Did Can tor not dem on strate that the set of real num -bers is non-denumerable?

In Can tor's di ag o nal proof it is as sumed that all (i.e. the com pleted in fi nite setof) real num bers are cor re lated one-to-one with the set of nat u ral num bers, af -ter which it is dem on strated that a fur ther real num ber can be pre sented whichdif fers from each of the counted real num bers (in at least one dec i mal place),from which the non-denumerability of the real num bers is con cluded. The va -lid ity of this con clu sion de pends, how ever, on the ac cep tance of the com -pleted in fi nite. Some one who rec og nizes only the un com pleted in fi nite cannever ac cept this con clu sion, since the di ag o nal method then only proves thatfor a given constructible se quence of count able se quences (i.e. dec i malexpansions of real num bers) of nat u ral num bers, yet an other dif fer ent count -able se quence of nat u ral num bers can be con strued. Becker states this in thefol low ing way: “The di ag o nal method dem on strates, strictly speak ing, thefol low ing: when one has a counted (law-conformative) se quence of suc ces -sive num bers, a se quence of suc ces sive num bers can be cal cu lated which dif -fers in ev ery place from all the pre vi ous ones” (1973:161 foot note 2). In thisin ter pre ta tion there is no where men tion made of non-denumerability!

A math e mat i cal proof which ap par ently takes an “ex act” course there fore co -mes to con flict ing con clu sions de pend ing on the pre sup po si tions (namelycom pleted in fin ity or un com pleted in fin ity) from which one pro ceeds!Fraenkel points this out em phat i cally:

“Can tor's di ag o nal method does not be come mean ing less from this point ofview, ... the con tin uum (i.e. the real num bers – DFMS) ap pears ac cord ing to itas a set of which only a count able in fi nite sub set can be in di cated, and this bymeans of pre-determinable con struc tions” (1928:239 foot note 1).

Who ever re jects the com pleted in fi nite can not ac cept the de scrip tion of realnum bers given by Dedekind, Weierstrass, and Can tor. Paul Lorenzen ex -plained that a real num ber can be pre sented in terms of the com pleted in fi nite

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1 “immers de intuitionist kan geen andere, dan aftelbare wiskundige verzamelingenconstrueeren” (1919:24).

as an in fi nite dec i mal frac tion of which the in fi nite mul ti plic ity of num bers allex ist at once (i.e. as an in fi nite to tal ity). Cassirer al ready fiercely in di cated in1910 that the con cept of a car di nal num ber de pends on a si mul ta neously givento tal ity.1

Fe lix Kaufmann is how ever of the opin ion that “it is gen er ally rec og nized thatan in fi nite dec i mal frac tion sig ni fies noth ing else but a se quence of nat u ralnum bers, where ... by a se quence is meant not an in fi nite to tal ity, but merelythe do main of a par tic u lar re la tion ” (1968:122-123). With re gard to an ir ra tio -nal num ber Lud wig Fischer writes: “Ev ery rep re sen ta tion of 2, what ever itsna ture might be, can only be taken as an in fi nite and com pletely ‘un fin ish able’se quence of ra tio nally ap prox i mat ing val ues. Only once the in it self con tra -dic tory fic tion of the com pleted in fi nite is added, can the in fi nite dec i mal frac -tion be con sid ered an in stance of 2. With out the con tra dic tory no tion of thecom pleted in fi nite the con cept of an ir ra tio nal num ber can not be formed”(1933:108).

Brouwer and Weyl de fine real num bers in terms of the un com pleted in fi nite,namely as un com pleted in fi nite choice se quences or as a me dium of free re al -iza tion. Weyl de scribes a real num ber as fol lows: “A sin gle real num ber canbe de scribed as an in fi nite se quence of frac tional in ter vals of grow ing mag ni -tude, where each is con tained in the fol low ing one in the se quence”(1966:74-75). Fraenkel, Bar-Hillel, Levy and Van Dalen com ment: “The con -cep tion of the con tin uum as an ag gre gate of ex ist ing points (mem bers), whichis at the bot tom of nine teenth cen tury anal y sis and of Can tor's set the ory, is re -placed by an ag gre gate of parts which are par tially over lap ping and which areso to speak the man i fes ta tions of real num bers still to be gen er ated”(1973:256).

Intuitionism also sees the con tin uum dif fer ently: “In agree ment with in tu itionBrouwer sees the es sence of the con tin uum not in the re la tion of the el e ment to the set, but in that of the part to the whole” (Weyl 1966:74). Fol low ing Ar is -totle Weyl also claims that: “It rather be longs to the es sence of the con tin uumthat each of its parts are in fi nitely di vis i ble” (1921:77). In op po si tion to theno tion of a con tin uum of points Weyl in di cates that the con cept of en vi ron -ment must still be used to sal vage con ti nu ity: “To ren der a con tin u ous co her -ence of points anal y sis has up to to day (since it de com posed the con tin uuminto a set of iso lated points) found ref uge in the con cept of en vi ron ment”(1921:77). It should there fore come as no sur prise that even a non-intuitionistsuch as Paul Bernays – the well-known col league of Da vid Hilbert – sharplyre jects the sup posed suc cesses of arithmeticism in math e mat ics.

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1 “Wenn in der Theorie der Ordnungszahl die Einzelschritte als solche festgestellt und ineindeutiger Folge entwickelt wurden, so tritt jetzt die Forderung ein, die Reihe nicht nurnacheinander in ihrer einzelnen Elementen, sondern als ideelles Ganzes zu erfassen. Dasvorangehende Mo ment soll durch das folgende nicht in ihm aufbehalten bleiben, so dass derletzte Schritt des Verfahrens zugleich alle vorhergehenden und das Gesetz, das diewechselseitig verknupft, in sich fasst. Erst in dieser Synthese vollendet sich die blosse Folgeder Ordnungzahlen zum einheitlichen, in sich geschlossen Sys tem, in welchem jedes Gliednicht nur für sich steht, sondern zugleich den Aufbau und das formale Prinzip derGesamtreihe repräsentiert” – 1910:55.

Else where Weyl de clares: “The se quence of num bers which grows be yondany stage al ready reached by pass ing to the next num ber, is a man i fold of pos -si bil i ties open to wards in fin ity; it re mains for ever in the sta tus of cre ation, but is not a closed realm of things ex ist ing in them selves. ... Brouwer opened oureyes and made us see how far clas si cal math e mat ics, nour ished by a be lief inthe ab so lute that tran scends all hu man pos si bil i ties of re al iza tion, goes be -yond such state ments as can claim real mean ing and truth founded on ev i -dence”. Ear lier on the same page he clearly states: “Brouwer made it clear, as I think be yond any doubt, that there is no ev i dence sup port ing the be lief in theex is ten tial char ac ter of the to tal ity of all nat u ral num bers” (1946:6).

In intuitionist math e mat ics many sub di vi sions of clas si cal math e mat ics nolon ger serve. Hilbert val ues e.g. Can tor's transfinite num ber the ory “as themost won der ful flour ish ing of a math e mat i cal spirit and as such one of thehigh est achieve ments of pure hu man in tel lec tual ac tiv ity” (1925:167), whileA. Heyting con sid ers transfinite num ber the ory as no more than a phan tasm(1949:4). In his work: Das Kontinuum, Weyl worked on the foun da tions ofanal y sis and re con structed large parts in terms of intuitionism, al though hehad to ad mit that he had to sacrifce the the o rem “ev ery de lim ited set of realnum bers has an up per limit” (1932:23-24).

Intuitionism in fact con structed an en tirely new math e mat ics: “Theintuitionists have cre ated a whole new math e mat ics, in clud ing a the ory of thecon tin uum and a set the ory. This math e mat ics em ploys con cepts and makesdis tinc tions not found in the clas si cal math e mat ics” (Kleene 1952:52). E.W.Beth also com ments: “It is clear that intuitionistic math e mat ics is not merelythat part of clas si cal math e mat ics which would re main if one re moved cer tainmeth ods not ac cept able to the intuitionists. On the con trary, intuitionisticmath e mat ics re places those meth ods by other ones that lead to re sults whichfind no coun ter part in clas si cal math e mat ics” (1965:89).

The influence of intuitionism on the approach of Dooyeweerd

Both Brouwer and Dooyeweerd only ac knowl edge the po ten tial in fi nite – as alaw of pro gres sion: “A set is a law on the ba sis of which, when a new value ischo sen re peat edly, a de ter mined se quence of signs is gen er ated for each oneof these choices ... with or with out ter mi nat ing the pro cess” (Brouwer,1925:244). Ac cord ingly, also Dooyeweerd con sid ers an in fi nite se quence ofnum bers as only de ter mined “by the law of ar ith met i cal pro gres sion.” Thismakes it pos si ble “to de ter mine the dis crete ar ith met i cal value in ar ith met i caltime of any fi nite nu mer i cal re la tion in the se ries. For the ra tio nal ist con cep -tion of law this is suf fi cient rea son to at trib ute ac tual com pleted in fin i tude tothe se ries as a to tal ity” (NC, II:92). The ba sic mis take in the idea of the ac tualin fi nite, ac cord ing to Dooyeweerd, is a con fu sion con cern ing the dis tinc tionbe tween law-side and fac tual side: “Num bers and spa tial fig ures are sub ject to their proper laws, and they may not be iden ti fied with or re duced to the lat ter.This dis tinc tion is the sub ject of the fa mous prob lem con cern ing the so-called‘ac tual in fin ity’ in pure math e mat ics. The prin ci ple of pro gres sion is a math e -mat i cal law which holds for an in fi nite se ries of num bers or spa tial fig ures.

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But the in fi nite it self can not be made into an ac tual num ber” (1997-I:98-99note 1).1

Brief systematic assessment of the relationship between the potentialand the actual infiniteIn spite of the fact that Dooyeweerd found the idea of the ac tual in fi nite un ac -cept able, his philo soph i cal the ory of the modal as pects of re al ity does pro videthe start ing-point for a new and non-reductionistic ex pla na tion of the na tureof this kind of in fin ity. Ac cord ing to his gen eral the ory of modal law-spheres,each as pect has a law-side and a cor re lated fac tual side. Within the modalstruc ture of an as pect their are ‘modal mean ing-moments’ that es tab lish theco her ence with ear lier as pects (called retrocipations) and ‘modal mean -ing-moments’ main tain ing the co her ence with later as pects in the cos mic or -der (called an tic i pa tions) (1997-II:75). To this orig i nal con cep tion Dooye -weerd adds his equally unique idea of time. Time is no lon ger iden ti fied withthe phys i cal as pect of re al ity but is seen as a dis tinct di men sion guar an tee ingthe tem po ral or der of suc ces sion be tween the dif fer ent (modal) as pects of cre -ation. Fur ther more, ac cord ing to Dooyeweerd, time ex presses it self within the boundaries of ev ery as pect by ‘tak ing on’ the na ture of the as pect con cerned.It is thus dif fer en ti ated into a modal time-order and a modal time-duration(1997-I:28). Ev ery struc tural el e ment of each modal as pect is qual i fied by its‘mean ing-nucleus’ or ‘prim i tive mean ing’ which also stamps the way inwhich cos mic time evinces it self within the as pect con cerned.The an tic i pa tory anal o gies of an as pect need to be opened up, to be dis closed.Within the not-yet-opened-up mean ing of the nu mer i cal as pect we dis coverthe orig i nal and ba sic mean ing of in fin ity as it man i fests it self at the law-sideof this mo dal ity: in fin ity taken in the lit eral sense of end less ness. This prim i -tive mean ing of the in fi nite is an ex pres sion of the ar ith met i cal time-order ofsuc ces sion which not only lies at the ba sis of the prin ci ple of [math e mat i cal]in duc tion but which also de ter mines ev ery count able (denumerable) end less(i.e., po ten tially in fi nite) suc ces sion of num bers (be it the nat u ral num bers,the in te gers, or the ra tio nal num bers).In stead of us ing the char ac ter iza tion ‘po ten tial in fi nite’ we would pre fer touse an ex pres sion that would re flect the de ter min ing role of the nu mer i caltime-order of suc ces sion at the law-side of the ar ith met i cal as pect: the suc ces -sive in fi nite.Intuitionism only ac knowl edges this prim i tive mean ing of the in fi nite. It isseen as the prod uct of the free and cre ative power of the math e ma ti cian(Brouwer, 1952:140-142). As soon as the suc ces sive in fi nite mean ing of thenu mer i cal time-order is dis closed through the an tic i pa tion from num ber tospace, the mean ing of suc ces sive in fin ity is deep ened by the mean ing of thespa tial time-order of si mul ta ne ity. Any suc ces sive se quence of num berscould then, un der the guid ance of this an tic i pa tory hy poth e sis, be viewed as ifits el e ments are given all at once. The deep ened and dis closed mean ing of thein fi nite en coun tered here could be des ig nated as the at once in fi nite. Un der the

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1 De Swart re marks that for Brouwer's intuitionistically con structed spreads dif fer ent con di -tions hold which are not valid for other sets (1989:41).

guid ance of this hy poth e sis the ini tial suc ces sive in fi nite se quences of nat u ralnum bers, in te gers and ra tio nal num bers could be viewed as ac tual in fin i ties,i.e., as in fi nite to tal i ties given at once.

Ul ti mately, the spa tial or der of at once (si mul ta ne ity) forms the ba sis of theage-old leg acy con cern ing the ac tual (at once) in fi nite as some thing with outbe fore and af ter (Augustin), con nected with the time less at once (Maimon) orwith the ‘all ex ist ing at once’ (Lorenzen). Can tor re ferred to a con stant ‘fixedand de ter mined in all its parts’. An in tu itive ex am ple might help to clar ifywhat is at stake in the idea of the at once in fi nite. Em ploying the idea of the atonce in fi nite one can, in an an tic i pat ing way, cor re late the nat u ral num berswith the points of a straight line – for ex am ple, be tween 1 and 0, with the aidof the ra tio nal num ber se quence 1/n where n ranges over the set of all nat u ralnum bers. This mode of speech only has mean ing un der the guid ance of thereg u la tive hy poth e sis of the orig i nal spa tial time-order of si mul ta ne ity bymeans of which the num ber con cept of the suc ces sive in fi nite is deep ened anddis closed to the num ber idea of the at once in fi nite.1

In Kantian fash ion Da vid Hilbert even re marks: “The role that re mains for thein fi nite to play is solely that of an idea – if one means by an idea, in Kant's ter -mi nol ogy, a con cept of rea son which tran scends all ex pe ri ence and whichcom pletes the con crete as a to tal ity – that of an idea which we may un hes i tat -ingly trust within the frame work erected by our the ory” (1925:190 – this ar ti -cle is trans lated in Benacerraf & Putnam, 1964 – com pare p.151).

Mod ern set the ory claims to de fine the ‘con tin uum’ purely in ar ith met i calterms, i.e., in terms of the ac tu ally in fi nite set of real num bers which is, due toCan tor's well-known di ag o nal proof, non-denumerable. In op po si tion to thisLud wig Fischer de clares: “Be tween dis tinct (non-coalescing) points con ti nu -ity is pres ent ... un der all cir cum stances, there fore, for each in di vid ual point,the law holds: con ti nu ity-point-continuity” (1933:86-87). Since the mea sureof each sin gle point is zero, it ap par ently speaks for it self that ev erydenumerable set of points would have mea sure zero. How ever, in the case ofnon-denumerable sets, ad di tion is not de fined (if one can not enu mer ate the el -e ments of a set, one can not add them). This seem ingly al lows for the arith -meticistic claim that the un count able set of real points could con sti tute a pos i -tive mea sure, larger than zero! Thus, so Can tor and the mod ern math e mat i calmea sure the ory holds, a com plete arithmetization of the ‘con tin uum’ isachieved!

With out ques tion ing the spec tac u lar achieve ments of 20th cen tury math e mat -ics in the do mains of mea sure the ory and in te gra tion, it still must be pointedout that their arithmeticistic claims can not be jus ti fied. There is sim ply nocon struc tive way in which one can bridge the gap be tween denumerable andnon-denumarable in fin ity. Grünbaum, ar dently ad vo cat ing the con cep tion ofthe con tin uum as con sti tuted by an ag gre gate of non-extended el e ments (‘de -gen er ate in ter vals’), re marks: “The con sis tency of the met ri cal anal y sis which

47

1 I have an a lyzed the dis tinc tion be tween con cept and idea in my PhD dis ser ta tion: Con ceptand Idea (Begrip en Idee), Assen: Van Gorcum, 1973.

I have given de pends cru cially on the non-denumerability of the in fi nitepoint-sets con sti tut ing the in ter vals on the line” (1952:302).

It is only pos si ble to dem on strate non-denumerability if one pro ceeds from the as sump tion of the at once in fi nite. This need for the at once in fi nite ex plainswhy one can not, con struc tively, pro ceed from denumerability to non-denu -me ra bility (cf. Wolff, 1971:399-400). In ad di tion to this we have seen that theprim i tive mean ing of num ber does not fur nish any grounds for in tro duc ing the at once in fi nite since it is strictly bound to an end less suc ces sion of num bers(de ter mined by the nu mer i cal time-or der of suc ces sion at the law-side of thear ith met i cal as pect). Only by con sid er ing the an tic i pa tion from num ber tospace do we dis cern the mean ing of the at once in fi nite – com pletely de pend -ent upon the irreducibility of the spa tial or der of si mul ta ne ity.

In other words, with out the na ture of spa tial si mul ta ne ity this sup po si tion ofan at once in fi nite set has no foun da tion. It is an an tic i pa tory spa tial anal ogywithin num ber.

Since the at once in fi nite pre sup poses the ir re duc ible, unique na ture of thespa tial as pect it can not be used sub se quently to re duce space to num ber (a dis -tinct num ber of points) in terms of a non-denumerable set of real points. Thisreductionist at tempt is antinomical and im plies the fol low ing con tra dic tion:space can be re duced to num ber if and only if it can not be re duced to num ber(i.e. if and only if the at once in fi nite is used, which pre sup poses theirreducibility of the spa tial as pect)!

We can there fore fully sub scribe to the words of the well-known math e ma ti -cian, Paul Bernays (co-worker of Da vid Hilbert) where he writes: “Thearithmetizing mo nism in math e mat ics is an ar bi trary the sis. That the field ofre search of math e mat ics ex clu sively pro ceeds from rep re sen ta tions of num ber is no where dem on strated. Much rather con cepts such a con tin u ous curve anda sur face, which are par tic u larly de vel oped in to pol ogy, could not be re ducedto num ber con cepts” (1976a:188).

We should add to our an te ced ent brief ar gu ment that the spa tial as pect (withthe prim i tive mean ing: con tin u ous ex ten sion) not only dif fers from the nu mer -i cal as pect but also can not ex ist with out co her ing with it. It in deed be longs tothe very na ture of a spa tially ex tended con tin uum that any one of its parts al -lows of a suc ces sive in fi nite divisibility. Clearly, this divisibility pre sup posesthe mean ing of the nu mer i cal as pect – ev i dent in the qual i fi ca tion: suc ces sivein fi nite. Re mark ably enough, the set of ra tio nal num bers is called ‘dense’ be -cause the nu mer i cal dif fer ence be tween any two ra tio nal num bers (frac tions)could be ‘di vided’ in def i nitely. This fea ture man i fests an an tic i pa tory co her -ence be tween num ber and space at the fac tual side. Since the in fi nitedivisibility of any spa tial sub ject (func tion ing at the fac tual side of the spa tialas pect) in it self en tails a retrocipation to the nu mer i cal time-order of suc ces -sion (at the law-side of the nu mer i cal as pect), we are jus ti fied to see in the sys -tem of ra tio nal num bers an an tic i pa tion to a retrocipation. The ‘re flect -ing-back’ char ac ter of this an tic i pa tion could best be cap tured by speak ingabout the semi-disclosed na ture of num ber. Af ter all, the divisibility at stake in

48

this con text re mains denumerable – just as the set of ra tio nal num bers in thefirst place could be de scribed in terms of the suc ces sive in fi nite.

This fea ture en abled Brouwer to in tro duce an intuitionistic the ory of the con -tin uum. He ab stracts fully from any mea sure con cept by fo cus sing only on afun da men tal, com pletely or dered and over all dense se quence with a first andlast el e ment: “The sec ond act of intuitionism cre ates the pos si bil ity of in tro -duc ing the intuitionist con tin uum as the spe cies of the more or less freely pro -ceed ing con ver gent in fi nite se quences of ra tio nal num bers” (Brouwer,1952:142).

It is ev i dent that this intuitionistic po si tion is de pend ent upon the men tionednu mer i cal an tic i pa tion to a retrocipation. Thus it re mains within the con finesof a semi-disclosed han dling of the suc ces sive in fi nite in terms of which thereal num bers are ap proached. Un der the pre tense that it pro ceeds purely in ar -ith met i cal terms, Cantorian (and the sub se quent ax i om atic formalistic) setthe ory in fact pro vides us with a deep ened num ber the ory, i.e. with a num berthe ory that is dis closed by the use of the at once in fi nite as an an tic i pa tory hy -poth e sis within the en riched mean ing of num ber. Lack of re al iz ing what isdone caused the math e mat i cal leg acy to re duce the orig i nal mean ing of spaceinto this an tic i pa tory sphere of num ber – ex plain ing why math e ma ti cians arein clined to iden tify the no tion of the ‘con tin uum’ with the real num bers.Intuitionism, on the other hand, re duces the mean ing of space to thesemi-disclosed mean ing of num ber.

In both cases, how ever, and con trary to their true in ten tions, es sen tial el e -ments of the as pect of space are used. In the case of intuitionism, the in fi nitedivisibility of a spa tial con tin uum is ‘bor rowed’, and in the case of ax i om aticfor mal ism the spa tial or der of at once is ‘bor rowed’ in their us age of the atonce in fi nite. This new per spec tive si mul ta neously ex plains why both Ar is -totle and Can tor ad vanced sim i lar cri te ria for con ti nu ity. Since Ar is totle, onthe one hand, ap proached con ti nu ity from the per spec tive of the spa tial as -pect, with its char ac ter is tic end less divisibility, he clearly only had to use thesuc ces sive in fi nite. Can tor, on the other hand, took the av e nue of the nu mer i -cal as pect, and the only way to ‘get at’ con ti nu ity was in terms of the nu mer i -cal an tic i pa tion to space given in the idea of the at once in fi nite.1

Sys tem atically seen, we there fore have to dis tin guish be tween the num bercon cept of denumerability on the one hand, and the num ber idea ofdenumerable and non-denumerable transfinite num bers on the other hand. Afully dis closed treat ment of the real num bers, i.e., a treat ment us ing the at once in fi nite, thus raises math e mat ics to a level of in creas ing thought econ omy (byal low ing, amongst other things, in di rect ex is tence proofs and the free use ofthe log i cal prin ci ple of the ex cluded mid dle) shed ding a new light on the un -nec es sary com plex ity and lim i ta tions of intuitionistic math e mat ics. The prim -i tive term in Zermelo-Frankel set the ory, namely set / el e ment of, re veals the

49

1 Not with out rea son Becker re marks: “Thus the Ar is to te lian the ory of the in fi nite and the con -tin uum, in its pe cu liar prob lem-set ting, is still of ac tual im por tance to a gen u ine ad e quatefoun da tion of higher [math e mat i cal] anal y sis” (1965:xii, note 2).

im plicit de pend ence of set the ory (as a spa tially dis closed num ber the ory, an -tic i pat ing the spa tial whole-parts re la tion and the spa tial or der of at once), onthe ir re duc ible nu clear mean ing of space.1

The acceptance of the in te gral bib li cal ac count of cre ation guided our pre ced -ing anal y sis. On the ba sis of a the o ret i cally ar tic u lated ac count of the or -der-diversity within cre ation (ac knowl edg ing the prin ci ples of ‘sphere- sovereignty’ and ‘sphere-universality’ – en tail ing an tic i pa tions ans retrocipa -tions) our fo cus ought to be on the mu tual co her ence and irreducibility of theas pects of num ber and space. It should also be clear that in pur su ing this av e -nue (as we have re marked in the be gin ning) a third al ter na tive emerges,side-stepping the ex tremes of a geometricized and an arithmetized ap proach.

50

1 Logicism had to con cede that it failed in pro vid ing a suc cess ful re duc tion to logic of the no -tion of in fin ity. Myhill re marks: “the ax i oms of Principia [Mathematica] do not de ter minehow many in di vid u als there are; the ax iom of in fin ity, which is needed as a hy poth e sis for thede vel op ment of math e mat ics in that sys tem, is nei ther prov able nor re fut able therein, i.e., isundecidable” (1952:182). We add the words of Kline, stat ing that Hilbert “did agree withRus sell and White head that in fi nite sets should be in cluded. But this re quired the ax iom of in -fin ity and Hilbert like oth ers ar gued that this is not an ax iom of logic” (Kline 1980:246).

51

Lawside

Numerical time-order of succession

The Successive (potential) Infinite

discrete quantity

Primitivemeaning:

continuous extension

dimension

magnitude

successive infinite divisibility

needs the at once infinite

Primitivemeaning:

(Anticipation from number to the lawside of space)(The primitive meaning of infinityat the lawside of number)

Anticipation

Anticipation

(Semi-disclosed)

Spatial figures(the whole-part relation)

Retrocipation to the

lawside of number

The At Once (actual) Infinite

Spatial time-order of at once

Factual side

NZQ

R

(natural numbers)

(integers)

(fractions)

(real numbers)

retrocipation

retrocipation

(Fractions represent an anti-cipation to a retrocipation)

(The real numbers representthe fully spatially disclosed

structure of number)

NumberNumber SpaceSpace

The mutual coherence and irreducibility of number and space



Chapter III

Basic questions in Physics

The Prejudice against Prejudices

Per haps one of the dis tinc tive fea tures of West ern civ i li za tion is that it ex pe ri -enced an En light en ment dur ing the 18th cen tury. As a re sult of the in creas ingtrust in the ra tio nal ity of the hu man be ing his to ri ans as sess the 18th cen tury as the era of En light en ment. In it self this char ac ter iza tion har bours a rel a tivelypos i tive eval u a tion. As a re sult it would be dif fi cult to have an eye for the tre -men dous blocking out of insight it also brought about.

In or der to dis cern this shadow side of the cen tury of En light en ment we onlyhave to re fer to what Gadamer called the prej u dice against prej u dices! It wasun doubt edly Im man uel Kant, the fa mous phi los o pher of the 18th cen tury,who en throned the o ret i cal rea son. Even law and faith were chal lenged by thisjudge. In the Pref ace to the first edi tion of his Cri tique of Pure Rea son (1781)Kant declares:

Our age is, in ev ery sense of the word, the age of crit i cism, and ev ery thingmust sub mit to it. Re li gion, on the strength of its sanc tity, and law on thestrength of its maj esty, try to with draw them selves from it; but by do ing sothey arouse just sus pi cions, and can not claim that sin cere re spect which rea son pays to those only who have been able to stand its free and open ex am i na tion(A12 – trans la tion F.M. Müller).

It was Kant’s in ten tion to set a limit to the ap pli ca bil ity of (nat u ral) sci encesince he did want to leave open a do main for prac ti cal rea son tran scend ingthe sphere of sense-per cep tion and log i cal un der stand ing. How ever, as wehave seen in Chap ter I, by the end of the nine teenth cen tury and the be gin ningof the twen ti eth cen tury pos i tiv ism emerged as a philo soph i cal trend with theex plicit pur pose to abol ish what ever su per sedes sense per cep tion. We haveseen that it was fore most the neo-pos i tiv ism of the Vi enna Cir cle which, dur -ing the sec ond and third de cades of the 20th cen tury, ad vo cated with great en -thu si asm that the pos i tive (em pir i cal) nat u ral sci ences should at tain the lead -ing role in the fur ther de vel op ment and un fold ing of so ci ety. These pos i tivesciences were (pre-)supposed (!) to operate without any pre-suppositions.

53

In his sharp crit i cism Karl Pop per re al ized that this un der stand ing of sci enceen tails se ri ous dif fi cul ties. He re jected the gen er al ity (uni ver sal ity) of em pir i -cal test ing (so-called ver i fi ca tion) in fa vour of fal si fi ca tion – his newly in tro -duced cri te rion of de mar ca tion. W. Stegmüller un equiv o cally takes the po si -tion that a per son first has to ac cept some thing be fore some thing else could bejus ti fied in terms thereof. In do ing this he im plic itly re jects the prej u diceagainst prej u dices char ac ter is tic of the En light en ment:

A Self-guar an tee of hu man think ing, within which ever do main, does not ex ist. One has to be lieve in some thing in or der to be able to jus tify some thing else interms of it (1969:314).

Discrepancy between philosophers of science and the practitioners ofscience

Al though the de vel op ment of con tem po rary phi los o phy of sci ence dur ing thepast three to four de cades does har bour many points of dif fer ence, there is per -haps one gen eral con sen sus in the unan i mous re jec tion of the posi tiv ist de nialof un avoid able prej u dices in sci ence. At the same time there con tin ues to ex ist a per ti nent dis crep ancy be tween the prac ti tio ners of sci ence in var i ous ac a -demic dis ci plines (who are still con tin u ing this out dated posi tiv ist view of sci -ence) on the one hand, and the current situation in the area of philosophy ofscience.

About a de cade ago I had the op por tu nity to ad dress a num ber of nat u ral sci -en tists who still wished to main tain that only what is ac ces si ble by sen sory ex -pe ri ence qual i fies for nat u ral sci en tific in ves ti ga tion – only that which van beweighed, counted and mea sured falls within the do main of sci ence. This re ac -tion im me di ately re minded me of the mock ing words of the Amer i can so ci ol -o gist, McIver, who re ferred to a sup posed “pre-suppositionless” and “un -biased” positivist attitude as follows:

The fol low ing seems to be the chief ten ets of their creed. First, I be lieve infacts, and to be saved I must dis cover new ones. Sec ond, when I have dis cov -ered them, I must if pos si ble mea sure them, but, fail ing that con sum ma tion, Imust count them. Third, while all facts are sa cred, all the o ries are from thedevil. Hence the next best thing, if one can’t dis cover new facts, is to re fute old the o ries (1967:21).

The prob lem with the posi tiv ist pre oc cu pa tion with facts is given in theunavoidability of do ing sci ence with out em ploy ing the o ret i cal terms. Let uscall these terms pro vi sion ally prop erty terms. The unique power of sci ence ispre cisely given in its abil ity to grasp in a sys tem atic fash ion ex pe ri en tial dataof an ap par ently widely di verg ing na ture in the sys tem atic grip of a spe cificuni ver sal per spec tive. For ex am ple, within the do main of mod ern phys icsmany dif fer ent sorts of en ti ties are en coun tered – from el e men tary par ti clesand at oms up to macro-pro cesses and macro-sys tems. How ever di verse theseen ti ties and pro cesses may be none of them es cape from the in te grat ing anduni ver sal per spec tive of the core phys i cal dis ci pline known as ther mo dy nam -ics. The laws of ther mo dy nam ics – such as the law of en ergy con stancy or thelaw of non-de creas ing en tropy – are af ter all ap pli ca ble to all pos si ble phys i -

54

cal en ti ties and pro cesses there may be, re gard less of the par tic u lar na tures in -volved. The ques tions to be ad dressed to pos i tiv ism at this point are the fol -low ing: how do we ac count for this uni ver sal ity? and: what is the em pir i calsta tus of a property term such as energy-constancy?

By paus ing for a mo ment at the de vel op ment of the con cept of mat ter since an -cient Greece we shall try to an swer these ques tions.

Property terms – the Achilles’ heel of positivism

In the pre vi ous Chap ter we have seen that the Py thag o re ans ad hered to onestate ment above all else: ev ery thing is num ber. Af ter the dis cov ery of ir ra tio -nal num bers – re veal ing within the seem ingly form-giv ing and de lim it ingfunc tion of num ber the form less – Greek math e mat ics as a whole was trans -formed into a spa tial mode (the geometrization af ter the ini tial arithmetiza -tion). As a con se quence ma te rial en ti ties were no lon ger de scribed purely inar ith met i cal terms. Space now pro vided the nec es sary terms used to char ac ter -ize ma te rial en ti ties. This spa tial an gle of ap proach re mained in force un til therise of mod ern phi los o phy, since phi los o phers like Des cartes (1596-1650)and Kant (1724-1804) still saw the ‘es sence’ of ma te rial things in ex ten sion.

It was due to Ga li leo and New ton that the main ten dency of clas si cal phys icseven tu ally caused a shift in modal per spec tive by try ing to de scribe all phys i -cal phe nom ena ex clu sively in terms of (kinematical) move ment.1 Writ ingabout the foun da tions of phys ics, Da vid Hilbert2 re fers to the mech a nis ticideal of unity in phys ics but im me di ately adds the re mark that we now fi nallyhave to free our selves from this untenable ideal.

It is there fore strange that the con tem po rary phys i cal sci en tist from Cam -bridge, Ste phen Hawk ing, still writes: “The even tual goal of sci ence is to pro -vide a sin gle the ory that de scribes the whole uni verse” (1987:10). Since thein tro duc tion of the atom the ory of Niels Bohr in 1913, and ac tu ally since thedis cov ery of ra dio-ac tiv ity in 1896 and the dis cov ery of the en ergy quan tumh, mod ern phys ics re al ized that mat ter is in deed char ac ter ized by phys i cal en -ergy op er a tion – the phys i cal as pect of re al ity must be seen as the qual i fy ingfunc tion of matter.

This brief sketch of the gen e sis and growth of the con cept of mat ter il lus tratesin which way dif fer ent (modal) prop erty-terms served to char ac ter ize mat ter – start ing with the per spec tive of num ber and then pro ceed ing to the as pect ofspace, the kinematical as pect and even tu ally the phys i cal as pect of re al ity.What is im por tant to re al ize is that the de scrip tion of mat ter was de ci sively de -pend ant upon a par tic u lar the o ret i cal view of re al ity (Kuhn would have usedthe ex pres sions par a digm or dis ci plin ary ma trix) which is en tailed in the pref -er ence which is as signed to spe cific prop erty-terms. Is it pos si ble to ac countfor this foun da tional choice in an em pir i cal way? Is it pos si ble to per ceive the

55

1 The Brit ish phi los o pher, Thomas Hobbes (1588-1679), was fa mil iar with the me chan ics ofGa li leo en abling him – as op posed to Des cartes – to em ploy the ba sic con cept mov ing body as de scrip tive tool.

2 Per haps the great est math e ma ti cian of the 20th cen tury.

nu mer i cal as pect? Can we weigh the spa tial as pect? Can we de ter mine the vol -ume of the kinematical as pect? Can we ‘measure’ the ‘distance’ between thespatial aspect and the physical aspect?

The ob vi ous ab sur dity of these ques tions not only il lus trates the untenabilityof the positivistic faith in facts, but at once point at a cru cial dis tinc tion op er a -tive through out the his tory of the spe cial sci ences, namely the dis tinc tion be -tween as pects and en ti ties. Reformational phi los o phy pointed out that theseas pects en able our schol arly re flec tion to es tab lish a uni ver sal co her ence be -tween dif fer ent kinds of en ti ties – just re call the uni ver sal scope of the fun da -men tal laws of ther mo dy nam ics (which hold for all pos si ble phys i cal en ti ties). In gen eral an im plicit choice on this level of sci en tific con vic tions cause a di -ver gence be tween spe cial sci en tists. The ques tion con cern ing the re la tion shipand co her ence be tween the dif fer ent as pects of re al ity (in terms of which wecan de scribe any thing) sim ply can not be set tled with the aid of the positivisticmethod of (empirical) perception and verification.

Pos i tiv ism did re al ize that we can only dis cover the struc tural na ture and thelaws hold ing for phys i cal en ti ties by in ves ti gat ing the law ful ness (law- con for -mity) they evince. How ever, pre cisely the dif fer ence be tween the uni ver sal ityof God’s law and the unique in stances em pir i cally tested in ex per i men tal set -tings once again un veils the untenability of a positivistic po si tion. A lim itednum ber of ex per i men tal in stances could never war rant the claim of uni ver sal -ity contained in law statements.

In its ma te ri al is tic vari ant pos i tiv ism re veals even fur ther in con sis ten cies. Letus look at the typ i cal claim that mat ter is all there is: at oms, mol e cules, andmacro-mol e cules in in ter ac tion. This state ment claims that there is noth ingbe yond mat ter – but what about the state ment mak ing this claim!? Is it true? Ifso, then there is some thing im ma te rial (truth). And what about the nat u rallaws hold ing for ma te rial things? They con di tion be ing ma te rial but are notthem selves ma te rial! Thus both with re spect to the truth-value and the uni ver -sal va lid ity of nat u ral laws the ba sic claim of positivistic materialism isself-defeating!

The measurement of time and modal time orders

Phys i cists nor mally claim that time is an ex clu sively phys i cal phe nom e non.As a con se quence, they also hold that only phys i cists are com pe tent to speakabout the na ture of time. Now sup pose we visit, ac com pa ny ing the His tor i calAs so ci a tion, a his tor i cally sig nif i cant farm. Upon ar rival we no tice that theold cou ple still live as they did al most fifty years ago when they moved to thisfarm – it looks as if noth ing changed dur ing this pe riod, as if time came to astand-still. What does it mean to say dur ing the past fifty years time came to astand-still? If we had phys i cal time in mind ob vi ously this state ment wouldhave been mean ing less, since phys i cal time con tin ues with out any in ter rup -tion! How ever, when we re al ize that the men tioned state ment con cerns ourhis tor i cal aware ness of time, no ab sur dity will be ob served. To phrase this in -sight in terms of the gen eral pat tern of cul tural de vel op ment, tak ing into ac -

56

count the as cend ing line from the stone age, the bronze age, the iron age, andso on, then it is per fectly mean ing ful to say that within the 21st cen tury thereare com mu ni ties still liv ing in the stone age (an era more or less dated between 2 million and 10 000 years ago)!

Let us con sider an other ex am ple. Mod ern gov ern ments are en ti tled to pro mul -gate laws with a ret ro ac tive ef fect! This re al ity would be im pos si ble if le galsci ence had to op er ate with a phys i cal con cept of time. One can not es capefrom the re al ity of such a law by view ing it as a mere le gal fic tion. Surely,phys i cal time is ir re vers ible and there fore flows in one di rec tion only. Even ifwe raise the ar gu ment that his tor i cal and le gal time can only ex ist on the ba sisof phys i cal time, it does not can cel the unique ness (and irreducibility) of theseother modes of time.

The im por tant sys tem atic con clu sion to be drawn from the given ex am ples isthat no sin gle ex pe ri en tial mode of re al ity can ex haust the full mean ing oftime. This novel pro posal was first made by the Dutch phi los o pher, HermanDooyeweerd. Time should be seen as a unique di men sion of re al ity. Withinthe di ver sity of modal as pects time ex presses it self in ac cor dance with theunique na ture of each as pect. In ter est ingly, Ste phen Hawk ing (1987:8), thewell-known con tem po rary phys i cist, cor rectly em pha sizes (with Au gus tine)that time it self is a crea ture and it does not exist from eternity.

By look ing at the his tory of time-mea sure ment sig nif i cant points of con nec -tion are found for dis tin guish ing the first four modes of re al ity in terms of their re spec tive time or ders. It be longs to our gen eral aware ness of time: ear lierand later, si mul ta ne ity, time-flow and ir re vers ibil ity are well-known mo dal i -ties of time. In his work on the foun da tions of phys ics (1980:16) Stafleuremarks:

This is most clearly shown by an anal y sis of the his tor i cal de vel op ment of time mea sure ment. Ini tially, time mea sure ment was sim ply done by count ing(days, months, years, etc.) Later on, time was mea sured by the rel a tive po si -tion of the sun or the stars in the sky, with or with out the help of in stru mentslike the sun dial. In still more ad vanced cul tures, time was mea sured by uti liz -ing the reg u lar mo tion of more or less com pli cated clock works. Fi nally, in re -cent de vel op ments time is mea sured via ir re vers ible processes, for example,in atomic clocks.

What is strik ing in this whole de vel op ment is that dif fer ent time or ders areused, the one af ter the other: the nu mer i cal time or der of suc ces sion,1 the spa -tial or der of si mul ta ne ity,2 the kinematical time or der of con stancy and the ir -re vers ible phys i cal time or der, ex pressed in the re la tion ship of cause andeffect.

On the one hand we al ways en coun ter time as time or der – in which case it ap -pears at the law-side/norm-side of re al ity – and on the other hand as time-du -ra tion (at the fac tual side of re al ity). By vir tue of the cos mic time or der there

57

1 In Chap ter II we re lated this to our most ba sic aware ness of in fin ity, namely the suc ces sive in -fi nite.

2 Com pare the at once in fi nite dis cussed in Chap ter II.

also ex ists a time or der of suc ces sion be tween the various aspects.

Time in the aspects of number and space

Math e ma ti cians who are only ac quainted the dom i nant trend in mod ern math -e mat ics, namely the ax i om atic formalist stand point, will claim straight awaythat time does not have a place in math e mat ics. How ever, those who took no -tice of neo-intuitionist math e mat ics (dis cussed in die pre vi ous Chap ter) ex -em pli fied in the work of L.E.J. Brouwer and his suc ces sors (amongst whomare peo ple like H. Weyl, A. Heyting, D. van Dalen, A. Troelstra and to a cer -tain ex tend also P. Lorenzen), will re al ize that this school ex plic itly pro ceedsfrom the as sump tion of an orig i nal in tu ition of time. In this in tu ition, ac cord -ing to Brouwer, con ti nu ity and dis crete ness co in cide giv ing birth to the pri mal aware ness of one, an other one and son on – a pro cess that, through the end lessad di tion of new units could never be ex hausted. In other words, this pro cess islit er ally in fi nite, with out an end. This intuitionistic con cep tion of time is his -tor i cally de pend ent upon the phi los o phy of Im man uel Kant who saw time asone of the psy chi cal forms of in tu ition of be ing hu man.1

What intuitionism iden ti fies as the in tu ition of one, an other one and so on, re -lates to the ar ith met i cal time or der of suc ces sion at the law-side of the nu mer i -cal as pect. It be longs in deed to the time in tu ition of ev ery per son since with out this nu mer i cal time or der one of the cor ner stones of our mod ern civ i li za tionwill col lapse, in clud ing our mea sure ment and cal cu la tion of (phys i cal) time.Put dif fer ently: our ex pe ri en tial in tu ition of nu mer i cal re la tions pro vides uswith an in sight into the orig i nal (ontically given) numerical time order ofsuccession.

In math e mat ics this time or der lies at the foun da tion of the prin ci ple of (math -e mat i cal) in duc tion – first in tro duced by Pascal. It sim ply says that if a state -ment is valid for the num ber 1 and, sub se quently, if it could be shown thatwhen ever it holds for a num ber n it also holds for the num ber n+1, then it ob -tains uni ver sally. Ac cord ing to Weyl al ready this prin ci ple is suf fi cient tosafe guard math e mat ics against be com ing a mere tau tol ogy, in other words topre vent that a set of for mal ax i oms be the ba sis of math e mat ics in stead of abasic insight that can not be formalized.

The most prim i tive cor re late of the nu mer i cal time or der of suc ces sion isgiven in the se quence of nat u ral num bers: (0), 1, 2, 3, 4, 5, 6, 7, ... (with out anend, end less, in fi nite). Ax i om atic set the ory some times at tempt to de fine or -der. This is done, for ex am ple, by in tro duc ing the con cept of an or dered pair.Even in the stan dard work on Set The ory (sec ond, re vised edi tion, 1973)Fraenkel et al. there sud denly ap pears an un ex pected pe titio principii in thisre gard. With out ex plain ing the tech ni cal de tail it is suf fi cient to take note of

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1 Ac cord ing to Kant the con cept num ber orig i nates through a sche ma ti za tion in time of quan -tity as cat e gory of our un der stand ing.

the re mark added to their ex am ple about an or dered pair (de rived fromKuratowski): “Taken in that or der”!1

Within the as pect of space (cos mic) time ex presses it self in the spa tial time or -der of si mul ta ne ity, cor re lated with fac tual spa tial ex ten sion. Al ready this in -sight can cels the mis con cep tion that time is spaceless and that space is time -less. The aware ness of si mul ta ne ity (that which ex ists at once) be longs to ourbasic intuition of space.

When the ar ith met i cal or der of suc ces sion at the law-side of the as pect ofnum ber is dis closed un der the guid ance of the the o ret i cal in sight into the na -ture of the spa tial or der of si mul ta ne ity we dis cover the reg u la tively dis closedidea of in fin ity, namely the idea of the ac tual or com pleted in fin ity – des ig -nated by us as the idea of the at once in fi nite.

The kinematical and the physical time orderSince the de vel op ment of Ga li leo’s me chan ics clas si cal phys ics at tempted toun der stand all bod ies in terms of the de nom i na tor of me chan i cal move ment.From New ton up to the be gin ning of the 20th cen tury this mech a nis tic ten -dency stamped the main de vel op ment of mod ern phys ics. Max Plank,2 char -ac ter ized this mech a nis tic ori en ta tion as follows in 1910:

The con cep tion of na ture that ren dered the most sig nif i cant ser vice to phys icsup till the pres ent is un doubt edly the me chan i cal. If we con sider that thisstand point pro ceeds from the as sump tion that all qual i ta tive dif fer ences are ul -ti mately ex pli ca ble by mo tions, then we may well de fine the mech a nis tic con -cep tion as the con vic tion that all phys i cal pro cesses could be re duced com -pletely to the mo tions (the ital ics are mine – DFMS) of un change able, sim i larmass-points or mass-elements (1973:53).

In ki ne mat ics all pro cesses are re vers ible in prin ci ple. This re vers ibil ity con -cerns the kinematical time or der. It is anal o gous to the nu mer i cal and the spa -tial time or ders which are also re vers ible. The re vers ibil ity of the nu mer i caltime or der first of all flows from the re vers ibil ity of the + and – di rec tions inthe sys tem of in te gers. Al though con crete events in phys i cal re al ity are uni di -rec tional, the time or der within the nu mer i cal as pect could be ex pe ri encedboth in the pos i tive and the neg a tive di rec tions.3

Al ready in 1824 Car not dis cov ered fun da men tally ir re vers ible phys i cal pro -cesses. The im pli ca tions of this dis cov ery was fur ther de vel oped si mul ta -

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1 The or dered pair (a,b) is “de fined” as the class (a,b) con tain ing the classes (a) and (a,b) “taken in that or der” as el e ments!

2 As men tioned above, he dis cov ered the quan tum of en ergy h (6.62 10-34 joule sec) – por tray -ing the fun da men tal dis con ti nu ity of en ergy. In or der to ac count for the dis crete na ture of theomis sion or absorbtion of en ergy, Planck pos tu lated that ra di ant en ergy is quantized, pro por -tional to the fre quency v in the for mula E = hnv – where n is an in te ger, v the fre quency, and hthe quan tum of ac tion (Wirkungsquantum) with the value 6.624 10-34.

3 It may take 5 min utes (phys i cal time du ra tion) for 100 stu dents to en ter a class suc ces sively.At the end of the class these stu dents may leave the class in a re versed or der, and this may take one min ute only. Al though the phys i cal time du ra tion took place in one di rec tion only, the nu -mer i cal time or der is re versed at the end of the class.

neously by Clausius and Thomp son in their for mu la tion of the sec ond mainlaw of ther mo dy nam ics.1 In 1865 Clausius in tro duced the term en tropy. Thislaw ac counts for the ir re vers ibil ity of phys i cal pro cesses – it de ter mines thedi rec tion of a phys i cal (or chem i cal) pro cess in a closed sys tem.2

Thus the law of non-de creas ing en tropy was es tab lished as the sec ond mainlaw of ther mo dy nam ics. At the same time the clas si cal mech a nis tic re duc tionto pure mo tion was up rooted. Jus ti fi ably there fore Max Planck (in his men -tioned ar ti cle from 1910) re marks that the “ir re vers ibil ity of nat u ral pro cesses” con fronted the “mech a nis tic con cep tion of na ture” with “in sur mount ableprob lems” (1973:55). Con se quently, whereas the time or der in the first threeas pects is re vers ible, it is ir re vers ible in the phys i cal as pect. This is eas ily seenin the a-sym met ri cal re la tion of cau sal ity: it stands to rea son that the causepre cedes the effect!

Since the dis cov ery of ra dio-ac tiv ity it turned out that within mi cro-structursethem selves there are ir re vers ible pro cesses pres ent pro ceed ing spon ta ne ouslyin one di rec tion only. In ad di tion this state of af fairs straight away con firms the irreducibility of the phys i cal as pect to the kinematical as pect (with itsreversible time order).

Al ready in his Isagogè Philosophiae from 1930 Vollenhoven dis tin guishedbe tween the me chan i cal and the phys i cal as pects. How ever, in the edi tion of1936 this dis tinc tion no lon ger ap pears. Dooyeweerd, on the con trary, ini tially main tained the or der nu mer i cal, spa tial, phys i cal – thus iden ti fy ing thekinematical as pect with the phys i cal as pect. Round about 1950 he re al izedthat this dis tinc tion is nec es sary to ac count for the fact that ki ne mat ics(phoronomy) can de fine a uni form mo tion with out any ref er ence to a caus ingforce (compare Galileo’s law of inertia).

The uniqueness of Constancy and Dynamics

Perpetual motionFrom an tiq uity there have been at tempts to make a ma chine which, once set inmo tion, would con tinue this mo tion per pet u ally with out us ing an ex ter nalsource of energy.At the be gin ning of the sev en teenth cen tury Fludd de signed a closed-cir cuitwa ter mill. This ini tially ap peared quite fea si ble, but ev ery ef fort to ac tu allymake it work prac ti cally failed.Al ready in 1775 the French Acad emy for Sci ence and Art de cided to pay nofur ther at ten tion to pur ported de signs of “perpetuum mo bile”. In Eng land also all claims to the pat ent rights on such ma chines were sub jected to the pro vi -

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1 The first law is the law of en ergy con ser va tion.

2 Sup pose an ideal gas in a con tainer is con nected with an other one in which a vac uum is pres -ent. The in ter nal en ergy will not change. How ever, the gas mol e cules will fill the en tire spacespon ta ne ously. This in di cates an in crease in en tropy. Viewed sta tis ti cally an in crease in en -tropy will al ways re flect the oc cur rence of the most prob a ble state. For that rea son, within aclosed sys tem, there will al ways only oc cur an in crease or con stancy of en tropy – but never adecrease of entropy.

sion of a work ing model – to no pos i tive ef fect. The ques tion is: why doesn’t it work?

To un der stand why this sort of per pet ual mo tion ma chine can not work, wemust re fer to the first main law of phys ics. The un der ly ing idea of per pet ualmove ment, af ter all, is that use able en ergy would be pro duced with out us ingany en ergy. Prac ti cally, this means that en ergy would have to be cre ated.What does this first law say?

Stim u lated by Ger man nat u ral phi los o phy at the be gin ning of the 19th cen tury (es pe cially the ideas of the phi los o pher Schelling), Ger man nat u ral sci en tistssearched for a uni fy ing law which would en com pass all phys i cal phe nom enain a sin gle per spec tive. The phys i cists Heimholtz and Mayer and the chem istvon Liebig de fended the no tion of the in de struc ti ble char ac ter of mat ter evenbe fore experimental evidence confirmed their view.

In 1847, at the youth ful age of 26, Helmholtz pre sented a for mu la tion of hisfirst main law of phys ics (ac tu ally ther mo dy nam ics) to the Phys ics So ci ety ofBerlin. He be gan by point ing out that no body had suc ceeded in build ing a suc -cess ful per pet ual mo tion ma chine. This was a log i cal con se quence of the in -de struc ti bil ity of en ergy. Till the pres ent phys i cists rec og nize this law as thelaw of en ergy con ser va tion which means that en ergy can not be cre ated or de -stroyed.1

In view of the law of en ergy con ser va tion it is quite clear to day that the con -struc tion of such a ma chine is im pos si ble in prin ci ple, since it would meanthat use ful (newly cre ated) en ergy would be re leased with out us ing anyenergy!

Com ment: A sec ond such sort of ma chine had also been imag ined – a ma -chine which would draw heat from its en vi ron ment and then con vert this en -tirely into work. The im pos si bil ity of such a ma chine is ev i dent in view of thesec ond law of ther mo dy nam ics, that of non- de creasing en tropy. Sta tis ti callythis means that in any closed sys tem the most likely sit u a tion would oc cur.Ow ing to the dif fer ence in tem per a ture in the en vi ron ment it re quired to con -vert heat into work, the sec ond law im plies the im pos si bil ity of this type of ma -chine.

These two main laws of phys ics are fun da men tal in so far as they are uni ver -sally ap pli ca ble to all phys i cal en ti ties. Laws which in dis crim i nately hold forall en ti ties, must com pletely ig nore the typ i cal dif fer ences be tween such en ti -ties. Such modal laws in di cate the fun da men tal ways of be ing or modi of suchen ti ties. To de duce uni ver sal modal laws re quires that sci en tific ac tiv ity ofanal y sis which we have called modal ab strac tion.

Closer reflections on constancy and change

To grasp the phys i cal mo dal ity (way of be ing) of phys i cal en ti ties, it is nec es -sary to ig nore their non-phys i cal as pects. Amongst other things, this im pliesthat it is es sen tial to dis tin guish clearly be tween the phys i cal as pect of en -ergy-op er a tion and its found ing ki ne matic as pect – that is, the as pect in which

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1 This law does not ex clude the fact that one en ergy form can be trans formed into an other formof en ergy.

we only re fer to uni form move ment with out re fer ring to the cause of mo tion.Move ment – as the mode of con stancy or uni form flow – is an orig i nal given,just as num ber, space, the eco nomic or the eth i cal. For this rea son Ga li leo’slaw of in er tia im plies that we may at most speak of the or i gin of a change inmo tion! All change pre sup poses a con tin u ing ba sis. If you do not re mainyour self (con stancy), you would not be able to age (change)! The im por tanceof our un der stand ing of con stancy and change (dy nam ics), jus ti fies a closerdis cus sion of their na ture and or i gin – which would also enable us todemonstrate further structural characteristics of modal aspects.

The core of Einstein’s theory of relativityEin stein’s the ory of rel a tiv ity is well-known. A phys i cist of his stat ure lendscredit to the pop u lar view linked to his the ory, namely that ev ery thing is rel a -tive and change able. Re mark ably, Ein stein’s the ory rests on a fun da men talpre sup po si tion which is the op po site of all rel a tiv ism. Ein stein had to startwith the idea of an or der which is uni form and con stant – which means that ev -ery thing which he has in di cated to be rel a tive is only rel a tive in re la tion to thiscon stant order.That this is the case is ev i dent from his pos tu late that the speed of light is con -stant in a vac uum. Ein stein worked from the pre sup po si tion that a par tic u larlight sig nal would have the same con stant speed (c) in re la tion to all pos si blemov ing sys tems. It was not even nec es sary for his the ory for such a sig nal toac tu ally ex ist. The fact that later ex per i men ta tion proved ex per i men tally thatthe speed of light does in deed con form to Ein stein’s pos tu late, is – as thephys i cist Stafleu puts it – rel a tively ir rel e vant! One has to keep this in mind incon nec tion with con tem po rary discussions regarding the changing speed oflightThe crux of Ein stein’s the ory of rel a tiv ity is there fore to be found in the na ture of the or der of con stancy which it pre sup poses.1 We are fa mil iar with the nu -mer i cal or der of suc ces sion which founds ev ery count ing ac tiv ity: one, an -other one, an other one, and so on. Just as fa mil iar is the spa tial or der of si mul -ta ne ity. In dis tinc tion from the nu mer i cal or der of suc ces sion and the spa tialor der of si mul ta ne ity, we ex pe ri ence the or der of constancy in the kinematicaspect of movement.

This means that Ein stein’s spe cial the ory of rel a tiv ity of 1905 is a purely ki ne -matic the ory.2 Ein stein’s the ory there fore did not pri mar ily de velop a the oryof rel a tiv ity, but rather one of constancy.

Ga li leo al ready dis cov ered the par tic u lar na ture of the ki ne matic or der oftime, as it was re vealed in his law of in er tia. In terms of this law a body in mo -

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1 Spielberg and Bryon cor rectly em pha size that it is about “invariance” – i.e. con stancy – al -though they un for tu nately thereby con fuse the terms ab so lute and un chang ing: “In deed, Ein -stein orig i nally de vel oped his the ory in or der to find those things that are in vari ant (ab so luteand un chang ing) rather than the rel a tive. He was con cerned with things that are uni ver sal andthe same from all points of view” (1987:6). The term un chang ing is sim ply the de nial (ne ga -tion) of change – a phys i cal term. The term ab so lute can not re ally be ap plied to any thing incre ation, that is, not if one wants to avoid the idolization of created reality.

2 The ir re duc ible na ture of the ki ne matic time or der is im ported with the help of a sub ject which moves at a con stant speed.

tion would con tinue its move ment with out stop ping – un less some thing else (a tion would con tinue its move ment with out stop ping – un less some thing else (a force or fric tion) in flu ences it. That means that our in sight into the na ture ofmove ment does not de pend on a causal power. The term “cause” be longs tothe phys i cal as pect of our ex pe ri ence where we come across the ef fects of en -ergy-op er a tion. It can not be suf fi ciently em pha sized that we can never talk ofa cause of move ment, but rather only of a cause of a change in move ment –thus once again ac knowl edg ing the modal dif fer ence be tween the kinematicaland phys i cal as pects of re al ity.1

The unique na ture of con stancy (that is, the irreducibility of the ki ne matic as -pect) is the foun da tion of all ref er ences to dy nam ics or change. With out a con -stant ba sis all talk of change is sense less. For this rea son phys ics can not linkany mean ing ful con tent to a dis con tin u ous change of move ment – change ofmove ment (ac cel er a tion and de cel er a tion) is al ways con tin u ous, since a dis -con tin u ous change would re quire a phys i cally im pos si ble in fi nite force.2 Con -se quently, we can only es tab lish change on the basis of something continuous.

An alternative formulation of the first main law of thermodynamicsThis foun da tional po si tion of the as pect of move ment en ables us to philo soph -i cally find a for mu la tion of the first main law of ther mo dy nam ics which is true to reality.The phys i cal as pect must not only be dis tin guished from its foun da tional ki ne -matic as pect, since there is also an in dis sol u ble co her ence be tween these twoas pects. For this rea son we shall find in the phys i cal as pect a struc tural mo -ment which re minds us of the foun da tional ki ne matic as pect. Con stancy ap -pears in the phys i cal as pect as a struc tural re minder of the mean ing of mo tion.In philo soph i cal terms we may say that we find an anal ogy of the ki ne maticas pect at the law side of the physical aspect.A for mu la tion of the first main law which in tends to be true to re al ity wouldthere fore have to re fer to en ergy con stancy. Strictly speak ing the use of theterm “con ser va tion” is in ad e quate, since the ac tiv ity of re ten tion it self re -quires an in put of en ergy – as in the case of ther mo dy namic “open sys tems”(or “steady states”). The law of en ergy con stancy il lus trates not only the dis -tinct unique ness of the ki ne matic and phys i cal as pects, but, tak ing into ac -count the dis tinc tion be tween law side and fac tual side, also the in dis sol u bleco her ence be tween them: with out the foun da tional po si tion of the ki ne maticas pect in the or der of the var i ous cos mic as pects we would have no groundsfor dis cern ing an anal ogy of the as pect of move ment in the phys i cal as pect,that is, the anal ogy of en ergy constancy.

The theory of relativity and relativismIn mod ern times there is vir tu ally no sci ence (in clud ing the ol ogy) not be setwith at tempts at his tor i cal rel a tiv ism. Historicism, af ter all, claims that ev ery -

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1 In his men tioned ar ti cle from 1910 treat ing the clas si cal mech a nis tic view of na ture, MaxPlanck sharply and cor rectly dis tin guished be tween a “me chan i cal” and an “energetical”view of na ture (1973:65).

2 Janich stresses a “strict dis tinc tion be tween phoronomic (here af ter named ki ne matic) and dy -namic state ments” (1975:68).

thing changes all the time, that noth ing re mains the same – moral stan dards,re li gious con vic tions, le gal opin ions, eco nomic prac tices – all things continueto change.

The pit fall in this ar gu ment is al ready ev i dent in the fact that ev ery in di ca tionof change is in ev i ta bly ac com pa nied with ki ne matic con stancy terms such as“con tin u ally”, “still”, “al ways”, “incessantly”, etc.

This im plies that we may not iden tify con stancy with some thing static, butthat we should much rather eval u ate it pos i tively as the foun da tion of all dy -nam ics! At the same time, how ever, we should leave aside the one-sided andex ces sive con cern with dy nam ics which is set against all forms of con stancy.1

Such an ap proach only leads to an un jus ti fied di a lec ti cal ten sion: that which is the con di tion and pre req ui site of dy namic change – that is, some thing con -stant – is seen as its opposite pole and enemy.

The re mark able co her ence be tween the terms con stancy and dy nam ics notonly en light ens us re gard ing the nat u ral sci en tific ba sis for the use of theseterms, since it also em pha sizes the in sight that the way in which we talk aboutev ery day oc cur rences can never es cape from the in ev i ta bil ity of hav ing a per -spec tive on particular aspects.

Determinism and indeterminism

Us ing en ergy con stancy as for mu la tion of the first main law of ther mo dy nam -ics im plic itly pre sup poses the dis tinc tion be tween a law (as or der for) andwhat ever is sub jected to and cor re lated with that law. It be longs to the na tureof a law that it de ter mines and de lim its that which is fac tu ally sub jected to it.Con versely, fac tual re al ity is de ter mined and de lim ited by a cor re lat ing law –and in its or der li ness/law-con for mity fac tual re al ity shows this subjectedness. Even with re gard to the sup posed “ini tial state” of the “big bang” Hawk ing ex -plic itly states that one has to as sume “that there are also laws gov ern ing theini tial state” (1987:11). What is strik ing is that Hawk ing does not re flect at allon the or i gin of these laws!

Werner Heisenberg points out that the de vel op ment of quan tum the ory led tothe for mu la tion of phys i cal laws in sta tis ti cal terms. More over, with hiswell-known prin ci ple of un cer tainty, Heisenberg es tab lished that “it willnever be pos si ble to de ter mine both the po si tion and ve loc ity of an atomic par -ti cle with an ar bi trary pre ci sion” (1956:11).2 The then pre vail ing con cep tionof phys i cal cau sal ity was con vinced that phys i cal sub jects ought to be seenmerely as the ex ten sion of phys i cal laws de ter min ing their ex is tenceexhaustively – ex plain ing why this view is also known as de ter min ism. In hismen tioned work Heisenberg ex plains that ac cord ing to this de ter min is tic ap -

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1 Where a few a de cades back one would still re fer with the high est re gard to a res o lute or prin -ci pled per son, to day it is fash ion able to speak of a dy namic person.

2 In April 1927, be fore he made known his re la tion of un cer tainty, Heisenberg (in a per sonalcon ver sa tion) said to Von Weizsäcker: “I be lieve I have dis proved the law of cau sal ity” (VonWeizsäcker, 1993:132, note).

proach ex act knowl edge of na ture or a par tic u lar section of it will suffice todetermine the future. He continues:

If one in ter prets the word cau sal ity in such a strict sense, one also speaks ofde ter min ism and means by that that there ex ist laws of na ture de ter min ing un -equiv o cally from the pres ent the fu ture con di tion of a sys tem (1956:25).1

De ter min ism holds that ev ery ef fect is strictly de ter mined by a cause. In re al -ity this point of view rep re sents a de i fi ca tion of the law-side of the phys i cal as -pect – ex plain ing why it sees phys i cal en ti ties (sub jects) merely as an ex ten -sion of phys i cal laws. De ter min ism re duces the fac tual side to the law-side.Stafleu cor rectly points out, by con trast, that the causal re la tion at the law-side of the phys i cal as pect sim ply states: noth ing hap pens with out a cause – butwhat the ef fect of a spe cific cause may be need not to be fixed in ad vance. Thisfor mu la tion side-steps the one-sid ed ness of both de ter min ism and indeter -minism: it grants de ter min ism that the con cept of a cause is mean ing ful andshould not be dis carded as it is claimed by indeterminism; and it grantsindeterminism that the ef fect need not to be fixed in ad vance (just think aboutthe half-value of ra dio-ac tive el e ments), thus highlighting the untenability ofdeterminism in this regard.

The prin ci ple of un cer tainty of Heisenberg had the ef fect that the paths ofgreat phys i cists of the 20th cen tury parted – con cern ing the ques tion whetheror not the con cept of cau sal ity ought to be main tained in the fur ther de vel op -ment of phys ics. Planck and Ein stein wanted to up hold the claims of de ter -min ism whereas Heisenberg and Bohr (the Co pen ha gen in ter pre ta tion ofquan tum phys ics) opted for the other ex treme: indeterminism. If it is the casethat de ter min ism absolutizes the law-side of the phys i cal as pect, then we haveto say that indeterminism absolutizes the fac tual side of the physial as pect.The al ter na tive ap proach ad vanced here is to view law-side and fac tual side as ir re duc ible cor re lates – an al ter na tive im plic itly sup ported by the ne ces sity toem ploy sta tis ti cal laws in physical theories (cf. Stafleu, 1968:304).

Order and delimitation in physics

In spite of the con tem po rary in ter est in chaos it is re ally not strange that phys i -cists are still look ing for an “un der ly ing or der in the world” (cf. Hawk ing,1987:13). Chaos the ory, in the fi nal anal y sis, wants to un veil a more com plexand over arch ing or der in what ap pears to be dis or derly and cha otic. Thenewly dis cov ered com plex pat terns, how ever, al ways point to a de ter min ingand de lim it ing law, as we have seen. Once this has been re al ized, an other re -lated is sue may gain in prom i nence – the ques tion con cern ing the boundariesof science and reality.

Al ready in the ear li est phases of the West ern sci en tific leg acy schol arlyknowl edge was con fronted with the quest to de ter mine the lim its of sci en tificthink ing. In prac ti cal terms this urge co in cided with de ter min ing the lim its ofthe universe itself.

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1 Max Planck ac tu ally ad hered to this de ter min is tic un der stand ing of cau sal ity. Com pare his1932 ar ti cle on Cau sal ity in Na ture (Planck, 1973:252).

The finite and limited cosmos in Greek culture

It may be sur pris ing to us that Greek thought ap par ently found a point of restin the de lim i ta tion pro vided to their world pic ture by the “large world-sea”,the Okeanos. Ac cord ing to their un der stand ing the earth is a cir cu lar slice de -lim ited and sur rounded by the Okeanos. It seems strange that the Greeks didnot move be yond the bound aries of the Okeanos. Our own mod ern ac quain -tance with the idea of in fin ity al most au to mat i cally forges this question.

In terms of the Greek mind this was im pos si ble. The Okeanos was one of thepri mal forces sub dued when the Olym pic gods started their reign. The or dered cos mos owes its form, mea sure, har mony and de ter mi na tion (con cept) tothese gods. What ever finds it self out side this limit does not dis play anyform-de lim i ta tion and can there fore not be thought or con cep tu al ized. As acon se quence Ar is totle does not ac knowl edge an ab stract or empty space. Helacks our mod ern con cept of space. Ac cord ing to the ma ture Greek un der -stand ing space does not ex ist, only place. Place is a prop erty ex clu sively at -trib uted to a con crete body. In the ab sence of a body there is no sub ject for thepred i cate place. From this it nat u rally fol lows that an “empty place” is theplace of noth ing – in other words, it is no place at all!

The pos si bil ity to un der stand (and en com pass) the or dered cos mos flows from its fi nite and lim ited na ture – for that rea son sci ence is re stricted to this fi niteand lim ited, or dered cosmos.

This ori en ta tion in dif fer ent ways caused ten sions within Greek sci ence andcul ture. In Chap ter II we al ready pointed at the prob lems caused by the dis -cov ery of ir ra tio nal num bers by the Py thag o re ans – even tu ally lead ing to thegeometrization of Greek mathematics.

How ever, the coun ter part of the ques tion con cern ing our knowl edge of thelim its of the cos mos is given in the ques tion whether “space” al lows for a con -tin ued di vi sion or whether the pro cess of di vi sion is blocked by last (small est)in di vis i ble units? The Greek at om ists, Leucippus and Democritus, were con -vinced that there in deed are such last in di vis i ble units, which they called at -oms. Since Des cartes mod ern con cep tions switched to the con vic tion thatphys i cal space is both con tin u ous and in fi nitely di vis i ble.1 By the end of the19th and the be gin ning of the 20th cen tury, how ever, the fol low ing dis tinc tion turned out to be nec es sary: that be tween math e mat i cal space and phys i calspace. Whereas the for mer – in a purely ab stract and func tional per spec tive –is both con tin u ous and in fi nitely di vis i ble (cf. Chap ter 2), phys i cal space isnei ther con tin u ous nor in fi nitely di vis i ble. Since it is bound to the quan tumstruc ture of en ergy it can not be sub di vided ad in fi ni tum. En ergy quanta in -deed rep re sent the limit of the divisibility of energy.

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1 In Chap ter II we have pointed out that this prop erty rep re sents an es sen tial fea ture of spa tialex ten sion. This char ac ter is tic pro vides the ba sis for the (semi-dis closed) intuitionistic math e -mat ics of Brouwer, Weyl and their successors.

Are there inaccessible limits in the natural sciences?One can say that en ergy quanta rep re sent an ac ces si ble (lower) limit. Arethere truly in ac ces si ble phys i cal lim its?The sec ond main law of ther mo dy nam ics, the law of non-de creas ing en tropy(prac ti cally sim ply stat ing that within any closed sys tems there will al ways bean in cli na tion to wards the most prob a ble con di tion), pro vides us with twogood ex am ples in this re gard. This law en tails that no sin gle ma chine can be so ef fi cient that while it pro duces en ergy no en ergy is lost. For this rea son theclas si cal ideal of perpetuum mo bile is un at tain able. Amongst other things italso im plies that the lower limit (ab so lute zero point: -273,16o) is in ac ces si ble. Al though the phys i cist Kurti man aged to reach a tem per a ture as close as onemil lionth from the zero point, it still re mains im pos si ble in prin ci ple to bridgethis last tiny gap – sim ply be cause it will re quire an op ti mally ef fec tive ma -chine, ca pa ble to con vert all the en ergy it needs into us able energy! It isprecisely this that is forbidden by the second main law.

The unlimited but finite universe in Eintein’s theory of relativityEin stein’s the ory of rel a tiv ity ad vanced re mark able per spec tives in this re -gard. Since all ce les tial bod ies are sub jected to the ef fect of grav ity Ein steinin tro duced the no tion of the “curved space of the uni verse” (thus em ploy ingnon-eu clid ean ge om e try). On the one hand it still sug gests that the uni verse isun bounded, i.e., one can move in any di rec tion be yond all lim its. How ever,since world space is curved, on the other hand, it en tails that even tu ally onewill end up where one started – show ing that in spite of be ing un bounded theuni verse is still fi nite! In the fi nal anal y sis his en tire the ory rests on the an other in ac ces si ble limit: the ve loc ity c of light in a vac uum. This ve loc ity is a truecon stant – such that what ever moves is mov ing rel a tively to this el e ment ofcon stancy.1 On the ba sis of these con sid er ations we have al ready pointed outthat strictly speak ing Ein stein’s the ory – in its de pend ance upon this “up perlimit” of mois ac tu ally a the ory of con stancy.

Complementarity – limits to experimentationThere are also re mark able lim its to phys ics in the sense of ex per i men tal ex ac -ti tude and de ter mi na tion. By in tro duc ing his prin ci ple of un cer tainty Heisen -berg showed that it is im pos si ble si mul ta neously to mea sure the im pulse andpo si tion of an elec tron. The Co pen ha gen in ter pre ta tion of quan tum phys icsem ploys the no tion of complementarity in or der to ac count for the im pos si bil -ity to es tab lish both at once - thus al low ing for two ir re duc ible (and com ple -men tary) modes of de scrip tion, in terms of “place” and “im pulse” re spec -tively. In fol low ing some ideas of Mario Bunge the phys i cist Henry Margenau de fends a so-called “mod er ate reductionism.” He takes this the be “the strat -

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1 Some times Ein stein uses the term invariance (cf. Schilpp, 1951:56). Of ten he ex plic itly re fers to the con stancy of light in a vac uum (cf. Schilpp, 1951:54, 56).

egy con sist ing of re duc ing what ever can be re duced with out how ever ig nor -ing emer gence or persisting in reducing the irreducible.1

Entities with a physical qualification

Al though sci ence tends to be oc cu pied pri mar ily with uni ver sal prop er ties orwith the spec i fied uni ver sal ity of types no ac a demic dis ci pline – and there forenei ther phys ics – can es cape from con crete re al ity where we also ex pe ri encethe in di vid ual side of things and events. The his tory of ac a demic re flec tionknows many ex am ples where an at tempt is made to ac count for the in di vid u al -ity of things in terms of a par tic u lar prop erty they may have. Some times in di -vid u al ity is re lated to mat ter – as in the case of Ar is totle and Thomas Aqui nas(mat ter as principium individuationis).

How ever, the brief re marks made ear lier in this Chap ter and the pre vi ousChap ter, al ready showed that it is im pos si ble to re vert to a view which tries tocon sider the as pect of num ber as stamp ing mat ter (the er ror of the Py thag o re -ans). We have also pointed out that both the spa tial and the kinematical as -pects fail to pro vide us with a qual i fy ing as pect of ma te rial things and events.The only can di date left is the phys i cal aspect of ernergy-operation.

In or der to ac count for the type-law of phys i cally qual i fied en ti ties we have toex plain their foun da tional func tion. We have to keep in mind that any en tity of what ever kind func tions typ i cally with all as pects of re al ity, in other words,ev ery en tity in a very con crete and plas tic way dis plays a typ i cal func tionwithin the (uni ver sal) modal struc ture of the var i ous as pects of re al ity. Justcon sider the gen eral fo cus of ther mo dy nam ics – a strictly mo dally de lim iteddis ci pline with a uni ver sal scope in which the typ i cal fea tures of dif fer entkinds of phys i cal en ti ties are dis re garded. In ther mo dy nam ics it does not mat -ter whether we are talk ing about the solid state, the fluid state or the gas eousstate – the spe cific weight and heat re main the same. As soon as we take intoac count the re la tion ship be tween mi cro-struc tures and macro-struc tures (such as within the con fines of sta tis ti cal phys ics) then the for merly ne glected nu -ances do mat ter, be cause the spe cific heat or weight is specified differently ineach of the three mentioned states (solid, fluid, gaseous).

An anal y sis of ma te rial things, fur ther more, can not es cape from the func tional in ter re la tions pres ent be tween the dif fer ent modal as pects in which ma te rialthings func tion in a con crete way. The in ter re la tions are first of all ac countedfor in terms of what in reformational phi los o phy is called modal anal o gies(anti- and retrocipations) – struc tural mo ments within each as pect re flect ingthe co her ence be tween the as pect con cerned and other as pects. But en ti tiesand events al ways have typ i cal func tions within modal as pects. What is typ i -cal about these func tions is that they are evinc ing the ef fect of the qual i fy ingfunc tion of the en tity un der con sid er ation. This im plies that the typ i cal func -tion of ma te rial en ti ties within the first three modal as pects of re al ity will al -ways point to wards (an tic i pat ing) the qual i fy ing phys i cal as pect of mat ter.Uni ver sal modal prop er ties are spec i fied in a typ i cal way. This calls for the

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1 Cf. Margenau, 1982:187, 196-197.

fol low ing ter mi nol ogy: we may speak of an tic i pa tory forms of modal spec i -fic ity. For ex am ple, qual i fied by the phys i cal func tion of en ergy-op er a tion wemeet the modal spec i fic ity of the quan tum of en ergy h which it – for the phys i -cist – a typ i cal num ber. (h = 6.62 x 10-34 joule sec). Some times this kind oftype spec i fic ity is des ig nated with the use of the term: con stants. Sim i lar to the con stant h in quan tum me chan ics, c as the ve loc ity of light (in a vac uum)serves as a con stant in Einstein’s theory of relativity.1

By con trast, the atomic num ber (equal to the num ber of pro tons pres ent in thenu cleus of an atom) char ac ter iz ing a chem i cal el e ment, is an other ex am ple ofa typ i cal num ber.Even a biotically qual i fied en tity – such as a cell with a nu cleus – pres ents it -self with typ i cal spa tial re la tion ships ex pli ca ble in a nu mer i cal con stant K.This con stant K is des ig nated as the “nu clear plas mic in dex” since it re fers tothe ra tio be tween the vol ume of the cell-nu cleus (Vn) di vided by the vol ume of the whole cell (Cc) from which the vol ume of the cell-nucleus is subtracted.Nat u ral and ar ti fi cial crys tals are clas si fied in sys tems on the ba sis of sym me -try prop er ties – where each sys tem of crys tal is char ac ter ized by a pe cu liaraxis sys tem. The ar range ment oc curs ac cord ing to the di min ish ing num ber ofsym me try-el e ments. The seven sys tems are known as cu bic, hex ag o nal,trigonal, tetragonal, orthorombic, monoclinic, and triclinic. When the var i -ous com bi na tions pos si ble be tween these sym me try el e ments are con sid ered,32 pos si ble sym me try classes could be des ig nated – all of them spa tial formtypes en com pass ing all the pos si bil i ties of phys i cally qual i fied crys tal li za -tion.2 In other words, these 32 classes of crys tals rep re sent typ i cal spatialrelationships.We now have to make ex plicit an im por tant dis tinc tion – al though it played ahid den role in our dis cus sions up to this point.Firstly, there are uni ver sal modal laws – such as those treated above (like thelaw of en ergy con stancy and Ga li leo’s law of in er tia). Dis cern ing modal lawsre quires the dis tinc tive fea ture of schol arly ac tiv i ties as de scribed in Chap ter 1 – modal ab strac tion. It is only on the ba sis of our in te gral (mul ti fac eted) ex pe -ri ence of re al ity that we gain the o ret i cal ac cess to the un der ly ing modal struc -ture of it. For this rea son we may call this method of ar tic u lat ing modal prop -er ties tran scen den tal-em pir i cal. Tra di tion ally, es pe cially since Kant’s Cri -tique of Pure Rea son, the word tran scen den tal is em ployed to ac count for that which pro vides the ba sis of all ex pe ri ence in the sense that it makes pos si blewhat we ex pe ri ence. Un like Kant, how ever, we don’t want to as sume that thetran scen den tal con di tions of ex pe ri ence are in ad vance (i.e., a pri ori) con -tained in the for mal struc ture of the know ing per son (Kant’s forms of in tu ition and thought cat e go ries). Much rather, we pro ceed from the con vic tion that the modal con di tion for ex pe ri enc ing phys i cal phe nom ena is given in the uni ver -sal modal struc ture of the phys i cal as pect of re al ity. With this ap proach we in -

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1 In this case, how ever, the speed of light is taken in an un spec i fied uni ver sal modal kine -matical sense.

2 Von Federov and Schönfliess, in de pend ently of each other, came to this clas si fi ca tion al ready in 1890 and 1891.

tend to claim that the phys i cal as pect lies at the foun da tion of what ever we can ex pe ri ence in a phys i cal sense. An anal y sis of the dis tinct modal as pects of re -al ity therefore rests upon a transcendental-empirical approach.

Sec ondly, we have to ac knowl edge that there are entitary laws for dif fer enttypes of en ti ties – suc cinctly des ig nated as type laws. The ex is tence of typelaws en able us to clas sify phys i cal en ti ties and place them in var i ous cat e go -ries. The typ i cal na ture of an en tity spec i fies1 the modal mean ing of the as -pects in which it func tions. These typ i cal na tures of en ti ties pro vide a pe cu liar“colour ing” to their modal func tions. But most im por tantly, type laws do nothold for each and ev ery pos si ble kind of en tity – they ap ply to a lim ited classof en ti ties only. Stafleu ex plains this distinction as follows (1980:11, cf. pp.6ff.):

Hereby we dis tin guish laws which are valid for a lim ited class of sub jects (typ -i cal laws) from those which are valid for all kinds of sub jects (modal laws).Typ i cal laws, in prin ci ple, de lin eate a class of sub jects to which they ap ply,de scrib ing their struc tures and typ i cal prop er ties. Ex am ples of such laws arethe Cou lomb law (ap pli ca ble only to charged sub jects), the Pauli prin ci ple (ap -pli ca ble to fer mions), etc. Of ten the law de scrib ing the struc ture of a par tic u larsub ject (e.g., the cop per atom) can be re duced to some more gen eral laws (e.g., the elec tro mag netic laws in quan tum phys ics). On the other hand, modal lawsare those which have a uni ver sal va lid ity. For ex am ple, the law of grav i ta tionap plies to all phys i cal sub jects, re gard less of their typ i cal struc ture. We callthem modal laws be cause, rather than cir cum scrib ing a cer tain class of sub -jects, they de scribe a mode of be ing, re lat ed ness, ex pe ri ence, or ex pla na tion.2

It is well-known that Im man uel Kant launched his epis te mol ogy by ask ing the ques tion: How are syn thet i cal prop o si tions a pri ori pos si ble? (1787:19). InChap ter 1 we have men tioned his view on the thought cat e go ries of our un der -stand ing and his claim that (in a for mal sense) these cat e go ries are not de rivedfrom na ture but are pre scribed to na ture in an a pri ori way (1783 par.36). Al -though mis di rected by the ra tio nal is tic as sump tions of his epis te mol ogy,Kant, in his search for the syn thetic a pri ori, ac tu ally strug gled with the na tureof modal universality.

To ap pre ci ate Kant’s po si tion better in this re gard we have to re turn to thediffernce be tween modal laws hold ing for what ever there is and type laws ap -pli ca ble to a lim ited class of en ti ties only. Who ever mo dally ab stracts a par tic -u lar as pect in tran scen den tal-em pir i cal man ner gains ac cess to the (un spec i -fied) uni ver sal ity of modal-func tional re la tion ships. Since modal as pects arenot con crete en ti ties or events they can not be treated as if they are entitary inna ture, be cause this would sim ply amount to a rei fi ca tion of modal func tions.If one re ally wants to gain an un der stand ing of the type law of any par tic u lar

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1 Take note that we do not say in di vid u al izes be cause uni ver sal ity does not ex ist on one end ofa con tin uum with in di vid u al ity at its other end.

2 The fact that modal laws – such as those of quan tum phys ics – hold for all pos si ble “ob jects”is clearly seen by Von Weizsäcker: “Quan tum the ory, for mu lated suf fi ciently ab stract, is aunversal the ory for all Gegenstandklassen (classes of ob jects)” (1993:128). When he ex -plains, on the next page, that one can not de duce the kinds of en ti ties of ex pe ri ence from theuni ver sal scope of quan tum the ory, he has in mind what we are calling type laws.

kind of en ti ties one has to in ves ti gate those en ti ties em pir i cally. One can notde rive the typ i cal na ture of dif fer ent kinds of phys i cal en ti ties from modalanal y sis or ab strac tion – what is re quired is em pir i cal test ing throughexperimentation.

This ex plains why even Kant was com pelled to make a dis tinc tion be tween his (sup pos edly uni ver sally valid a pri ori) thought cat e go ries on the one handand so-called em pir i cal laws of na ture on the other hand:

We rather have to dis tin guish em pir i cal laws of na ture, which al ways pre sup -pose par tic u lar per cep tions, from the pure or gen eral nat u ral laws, which,with out hav ing a foun da tion in par tic u lar per cep tions, only con tain the con di -tions of their nec es sary con nec tion in an ex pe ri ence. In re spect of the lat ter na -ture and pos si ble ex pe ri ence are en tirely the same; and since within these thelaw-con for mity of the nec es sary con nec tion of ap pear ances in an ex pe ri ence(with out which we are to tally in ca pa ble of know ing any ob ject of the world ofthe sense), ac tu ally is based upon the orig i nal laws of the un der stand ing, so itini tially does sound strange, but it is none the less cer tain, when I state with re -spect to the lat ter: un der stand ing cre ates its laws (a priori) not out of nature,but prescribes them to nature (1783 par.36:320).

This dis tinc tion runs par al lel with the one which we have drawn be tweenmodal laws and typ i cal laws (type laws). Whereas Kant ought to re ceive credit for wres tling with the di men sion of modal uni ver sal ity, pos i tiv ism andneopositivism ought to be ac knowl edged for their em pha sis on ex per i men taltest ing (not the same as: ver i fy ing!). Only through study ing the or der li ness orlaw-con for mity of en ti ties is it pos si ble to ar rive at an un der stand ing of thetype laws hold ing for that lim ited class of en ti ties con form ing the their pe cu -liar type laws. In the case of phys ics it re quires em pir i cal re search through ex -per i men ta tion. Of course this does not free phys ics from an over arch ing andun der ly ing par a digm (the o ret i cal per spec tive) in which modal prop er ties arealso ac counted for. Some times this di men sion the the ory for ma tion is im plic -itly ac knowl edged when ref er ence is made to the o ret i cal terms which cannotdirectly be tested against actual experiences.

By mak ing an ap peal to Dilthey’s sketch of see ing a nat u ral sci en tific the oryfor ma tion as con struct ing re al ity via log i cal math e mat i cal el e ments of con -scious ness (and thus as sert ing the power over na ture of this sov er eign con -scious ness as an ef fect of the au ton omy of the hu man in tel lect),1 Weyl wantsto fol low the con cep tion of Hugo Dingler re gard ing the prin ci ple of sym bol i -cal con struc tion. Weyl is con vinced that the “con struc tive char ac ter of thenat u ral sci ences, the sit u a tion that their in di vid ual prop o si tions do not have aver i fi able mean ing in in tu ition (Anschauung), but that truth builds a sys temwhich can only as a whole be assessed” (1966:192) has been explained byhim.

Max Planck states a sim i lar per spec tive in a con cise way: “Strictly seen it isto tally im pos si ble to find any phys i cal ques tion which can be as sessed di rectly through mea sure ments with out the aid of a the ory” (1973:341).

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1 Weyl re fers to the sec ond vol ume of the 1923 edi tion of Dilthey's Col lected Works (p.260).Cf. Weyl 1966:192.

Weyl af firms the cor rect ness of Dingler’s def i ni tion of phys ics as that dis ci -pline in which the prin ci ple of sym bol i cal con struc tion is fully car ried through and then adds a state ment once again mak ing an ap peal to the above-men -tioned dis tinc tion be tween modal uni ver sal ity and typ i cal ity: “But what iscon nected with the a pri ori con struc tion is ex pe ri ence and an anal y sis of ex pe -ri ence through the ex per i ment” (1966:192).

Dis cuss ing the na ture of an a pri ori syn thetic el e ment in the “em pir i cal sci -ences,” Stegmüller raises the fol low ing pos si bil ity – also al lud ing to the sameis sue (1969:316):

Surely, this can not im ply that the to tal ity of law-state ments pres ent in a nat u ral sci ence could be of an a pri ori na ture. Much rather, such an apriorism shouldlimit it self to the con struc tion of a lim ited num ber of a pri ori valid law re la -tion ships, while, fur ther more, all more spe cific laws of na ture should be de -pend ent on em pir i cal test ing.1

Keep ing in mind that we must dis tin guish laws in an ontical sense from ourhy po thet i cal law state ments in sci en tific for mu la tions, we also have to notethe sim i lar ity be tween the just-men tioned state ment of Stegmüller and the fol -low ing ex pla na tion of Stafleu (re lated to the dis tinc tion be tween modal lawsand typical laws):

Whereas typ i cal laws can usu ally be found by in duc tion and gen er al iza tion ofem pir i cal facts or lower level law state ments, modal laws are found by ab -strac tion. Eu clid ean ge om e try, Ga li leo’s dis cov ery of the laws of mo tion ...,and ther mo dy namic laws are all ex am ples of laws found by ab strac tion. Thisstate of af fairs is re flected in the use of the term “ra tio nal me chan ics”, in dis -tinc tion from ex per i men tal physics (Stafleu, 1980:11).

It must be clear that what is in tended with the dis tinc tion be tween modal andtyp i cal laws in deed has cap tured the re flec tion of prom i nent think ers. To men -tion one last ex am ple: C.F. von Weizsäcker. He says that al though the ba sicas sump tions of quan tum the ory could be writ ten down on one page (for themath e mat i cally trained reader!), the num ber of known ex pe ri ences con form -ing to this the ory runs into bil lions – and not a sin gle one is found con tra dict -ing quan tum the ory in a con vinc ing way. He then says, al lud ing to the uni ver -sal va lid ity of Kant’s thought forms: “I use an idea of Kant and con jec ture thatquan tum the ory there fore holds uni ver sally in ex pe ri ence, be cause it for mu -lates the con di tions for pos si ble ex pe ri ence.”2

In or der to speak about en ti ties the modal as pects have to be used as points ofen try. Even when we re fer to the to tal ity-struc ture of an en tity the em ployedterms stem from a unique modal as pect: the spa tial mode. The term to tal ity is

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1 Fales en ter tain, in a dif fer ent conext, “the pos si bil ity that there are syn thetic a pri ori truths;truths about ab stract en ti ties may ex press facts which are not merely the re sult of lin guis ticcon ven tion” (1990:148).

2 Von Weizsäcker, 1993:93. “Laws ca pa ble of math e mat i cal for mu la tion fi nally forms the hard core of the nat u ral sci ence: not the im por tant de tail, but the form of uni ver sal va lid ity”(1993:113). In an other con text he writes that the quan ti ta tive re sults of as tron omy are basedupon phys i cal laws and that we pos tu late, as a work ing hy poth e sis, a uni ver sal validity forthese laws (1993:25).

af ter all sim ply syn on y mous with the terms co her ence and whole ness (and itim plies a mul ti plic ity of parts and – at least in the case of spa tial con ti nu ity –the whole is infinitely divisible).

Al though Kant did not fore see that his wres tling with the na ture of the syn -thetic a pri ori ac tu ally bears upon the is sue of modal uni ver sal ity which formsthe coun ter-part of typ i cal ity, the ef fect of this dis tinc tion is still im menselyim por tant for a mean ing ful un der stand ing of phys i cal re al ity and for the dis ci -pline studying it: physics.

We con clude our re flec tion with a brief anal y sis of the struc tural unique nessof an en tity and with a suc cinct state ment of what the idea of a phys i cal qual i -fi ca tion of ma te rial things entail.

The unity and identity of an entity

One of the ba sic prob lems of the o ret i cal re flec tion – oc cu py ing phi los o phersthrough out the cen tu ries – is given in the ques tion how we ac count for the ex -pe ri ence of iden tity which we at tach to dif fer ent things in the world. Whatmakes it pos si ble to rec og nize a chang ing and age ing hu man be ing as thesame hu man be ing over time? Are we jus ti fied in say ing that a tree – with itsdif fer ent ap pear ances in sum mer, au tumn, win ter and spring – is al ways thesame (identical) tree?

Plato wres tled with the prob lem and even tu ally for mu lated his spec u la tivethe ory of static super-sen sory ideal forms, al beit on the ba sis of the last ing in -sight that change re quires con stancy as its ba sis. Per haps it is cor rect to saythat Plato stum bled upon the law for en ti ties in his quest to at tain knowl edgeof change ful things. The type law of and en tity may in deed be seen as the con -di tion for the du ra ble iden tity of an en tity – un der ly ing all the changes and al -ter ations it may ex pe ri ence. The iden tity of an en tity, how ever, can only be ap -proached through the point of en try pro vided by its dif fer ent modal as pects.For that rea son we al ready had to use terms com ing from the kinematical andphys i cal as pects even to for mu late our prob lem – as can clearly be seen whenwe say that changes (phys i cal point of en try) could only be es tab lished on theba sis of some thing rel a tively con stant (kinematical point of en try). In con tin -u ing its iden tity ev ery sin gle en tity fac tu ally dis plays an or der li ness cor re -latively re flect ing the or der for its ex is tence to which it is sub jected. Phraseddif fer ently: in its or der li ness and law-con for mity an en tity in a uni ver sal wayevinces that it is sub jected to the uni ver sally con di tion ing law for its ex is -tence.1

The said or der for, as law for be ing an en tity dif fers in prin ci ple from the static eidos con strued by Plato as the super-sen sory es sence of things. Sim i larly,also Ar is totle did not es cape the one-sid ed ness of his epistemological ap -

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1 Uni ver sal ity and in di vid u al ity are al ways strictly cor re lated at the fac tual side: this tree (in di -vid ual side) is a tree (uni ver sal side).

proach which iden ti fied knowl edge with con cep tual knowl edge.1 As a strictlyin di vid ual en tity Ar is totle’s pri mary sub stance is un know able. This causedAr is totle to in tro duce his sec ond ary sub stance as the uni ver sal sub stan tialform of things – in or der to save the possibility of conceptual knowing!

Our knowl edge of the in di vid u al ity of en ti ties closely co heres with the way inwhich we ex pe ri ence the iden tity of those things. This iden tity is some thinggiven to us in our ex pe ri ence and can there fore never be con strued af ter wardsin terms of the var i ous modal as pects by means of which we gain ex plan a toryaccess to it.

Be ing bound to these points of en try the only al ter na tive is to set apart what isknown as the typ i cal foun da tional and the typ i cal qual i fy ing func tion of an en -tity. But even this ap proach can not re place the given iden tity and unity of anen tity – some thing that we can only ap prox i mate in knowl edge which is of acon cept-tran scend ing na ture, in other words in idea-knowl edge. This en tailsthat our (con sti tu tive) con cept of the or der for and the or der li ness of en ti ties is al ways (reg u la tively) based upon the idea of the tem po ral unity, individualityand identity of an entity.

Physically qualified entities

Al though the his tory of phi los o phy and the nat u ral sci ences have tried forlong to find a qual i fy ing qual i fi ca tion for ma te rial things in one of the firstthree as pects of re al ity, it was only at the be gin ning of the 20th cen tury thatgen eral nat u ral sci en tific con sen sus was reached con cern ing the en er geticqual i fi ca tion of ma te rial things (el e men tary par ti cles, at oms, mol e cules,macro-mole cules, macro-systems).

We have seen that the Py thag o re ans wanted to re duce ev ery thing to num ber.The dis cov ery of ir ra tio nal nu mer i cal re la tion ships led in the school ofParmenides (to which Zeno with his ar gu ments against move ment and mul ti -plic ity also be longed) to the geometrization of Greek math e mat ics and to thecon vic tion that all phys i cal things are spa tially char ac ter ized. This spa tial ori -en ta tion lasted for more than two thou sand years! The fa ther of mod ern phi -los o phy, Des cartes (1596-1650), di vided re al ity into the two spheres of an ex -tended and think ing “sub stance” (res extensa and res cogitans): “the na ture ofbody con sists not in weight, hard ness, col our, and the like, but in ex ten sionalone” (Prin ci ples, Part II, IV). Even un til the 18th cen tury this view ex erts itsin flu ence un changed. As we have seen Kant says that when we re move ev ery -thing which the mind con ceives of in the rep re sen ta tion of the body (like sub -stance, strength, divisibility, etc.) as well as ev ery thing which be longs to our

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1 Con cept-for ma tion al ways oc curs on the ba sis of uni ver sal prop er ties. For this rea son the in -di vid ual side of things is con cep tu ally speak ing un know able! Those who ac knowl edge con -cep tual knowl edge only can not ac count for the knowl edge we have of things (and our selves!) in their (our) in di vid u al ity. We pre fer to call those who iden tify knowl edge with (uni ver sal)con cep tual knowl edge ra tio nal ists and those who re ject con cep tual knowl edge while hold ingon the that kind of knowl edge with which we know things in their individuality asirrationalists.

aware ness of the body (like im pen e tra bil ity, hard ness, col our, etc.), then allthat re mains is ex ten sion and form (Ausdehnung und Ges talt) (CPR, B:35). Incon nec tion with the na ture of con stancy and change we saw that the main ten -dency in clas si cal phys ics (since New ton) was mech a nis tic – in other words, it was be lieved that all phys i cal pro cesses can be re duced to (me chan i cal)move ment. The last great rep re sen ta tive of this mech a nis tic ap proach wasprob a bly Hein rich Hertz – the Ger man phys i cist who did ex per i men tal workabout elec tro mag netic waves more than a hun dred years ago.1 We have men -tioned Planck’s ar ti cle from 1910 where he clearly stated that the“irreversibility of natural processes” confronted the “mechanistic conceptionof nature” with “insurmountable problems” (1973:55).

It is clear that ev ery at tempt to find an arith me tic, spa tial or ki ne matic qual i fi -ca tion for phys i cal en ti ties nec es sar ily runs into the o ret i cal antinomies.

Let us con sider the na ture of an atom for a mo ment. Be sides the arith me ticfunc tion which an atom has (think about the atomic num ber), it also pos sessesa clear spa tial func tion: it is char ac ter ized by a par tic u lar spa tial con fig u ra -tion – the nu cleus of an atom with pe riph eral elec tron sys tems. Ac cord ing towave me chan ics, we find quan ti fied wave move ments around the nu cleus ofthe atom – the ki ne matic func tion of the atom. Al ready in 1911, in Ruther -ford’s atomic the ory, the hy poth e sis was posed that at oms con sist of a pos i -tively charged nu cleus and neg a tively charged par ti cles which moved aroundit (a view which was in spired by the na ture of a plan e tary sys tem). In the fol -low ing year (1912), Niels Bohr set up a new theory which contained twoimportant new ideas:

(i) the elec trons move only in a lim ited num ber of dis crete or bits around thenu cleus and

(ii) when an elec tron moves from an or bit with a high en ergy con tent to onewith a low en ergy con tent, elec tro mag netic ra di a tion occurs.

In 1925, Pauli for mu lated his ex clu sion prin ci ple (Pauli-ex clu sion).2 Ac cord -ing to the di vi sion of charges of elec trons, cor re spond ing elec tron-shells ex ist, and in each peel there is room for a “max i mum” num ber of elec trons. Thismax i mum num ber is given by the sim ple for mula: 2n2. In the first peel (known as the K-peel) there is room for 2 elec trons; in the fol low ing L-peel, there isroom for 8; in the M-sheel for 18; in the N-sheel for 32; and so on. Within asheel with a quan tum num ber n, (where there is room for 2n2 elec trons)

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1 This work not only es tab lished him as the founder of wire less te leg ra phy and the ra dio, butalso im mor tal ized his name in the unit of fre quency (Hertz) named af ter him. Soon af ter hisdeath in Jan u ary 1894 his large the o ret i cal work ap peared: “The Prin ci ples of Me chan ics de -vel oped in a New Con text (Die Prinzipien der Mechanik in neuem Zusammenhangedargestellt (312 pp.).” Re strict ing him self to the first three modal as pects only (rep re sentedby the con cepts time, space, and mass) he re jected the con cept force (a phys i cal con cept) assome thing in her ently antinomic (cf. Katscher, 1970:329). Thus we can see how con sis tentlyhe carried through the mechanistic approach.

2 It ap plies to fer mions, i.e., el e men tary par ti cles with a 1/2 spin (1/2, 3/2, 5/2, etc.) for whichthe sta tis ti cal laws of Fermi-Dirac are for mu lated.

sub-or bits are iden ti fied so that each sub-or bit with a quan tum num ber l hasroom for 2(2l+1) electrons.

It is al ready ob vi ous from these facts that the dis tinct num ber of el e men tarypar ti cles in the in ter nal atom struc ture are joined into a typ i cal spa tial or der of elec tronic or bits which con fig ure the atom as an in di vid ual phys i cal-chem i cal mi cro-to tal ity. The spe cial spa tial con fig u ra tion which is man i fest in the in ter -nal build of an atom, re flects the typ i cal foun da tional func tion of at oms.1

The wave particle duality and the idea of the typical totality structureof an entity

Af ter Ein stein re verted to a par ti cle the ory re gard ing the na ture of light,2 itturned out, on the ba sis of in ter fer ence phe nom ena,3 that it is al ways pos si bleto as cribe a wave-char ac ter to el e men tary par ti cles. Con versely, the Compton- ef fect – re gard ing the in ter ac tion of a pho ton and an elec tron – sup plied ev i -dence to sup port the idea of dis tinct par ti cles. De Broglie broad ened the per -spec tive by show ing that with each and ev ery mov ing par ti cle (at oms, mol e -cules and even macro-struc tures) one can associate a wave (cf. Eisberg,1961:81, 151).

Al though it turned out to be im pos si ble to es tab lish ex per i men tally at the same time both the par ti cle and the wave na ture Bohr claims that these two per spec -tives are com ple men tary (cf. Bohr, 1968:411 ff.).

In the light of the gen er al iza tion pro vided by De Broglie one may ask: if it ispos si ble to de scribe/ex plain en ti ties qual i fied by en ergy in terms of two mu tu -ally ex clu sive ex per i men tal data, namely as par ti cles and as waves, is it thenstill mean ing ful to speak about a uni tary struc ture? This ques tion puts the fin -ger ex actly on that point where the spe cial sci en tific de scrip tion reaches itslim its and needs to fall back upon a per spec tive tran scend ing the con fines ofspe cial sci en tific in quiry. What is here re quired is some or other philo soph i cal ac count tran scend ing the mere com bi na tion of one or more (mo dally de lim -ited) spe cial sci en tific points of view. We have seen that the idea of the unityand iden tity of an en tity could never be pro vided to us by the o ret i cally ex pli -cat ing var i ous modal func tions, sim ply be cause this un der ly ing unity is pre -sup posed in all the o ret i cal ex pla na tions. In a strict and tech ni cal sense thisidea of an en tity in its to tal ity – pre ced ing the anal y sis of its modal as pects –

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1 Dooyeweerd ini tially thought – 1935-1936 – that nat u ral things do not have a typ i cal foun da -tional func tion. In 1950 he re lin quished this po si tion (Cf. 1950:75 note 8).

2 Light quanta are called pho tons and sim i lar to the neu trino they pos sess a zero mass.

3 In ter fer ence phe nom ena were es tab lished af ter Michelson – round 1880 – de signed an in ter -fer om e ter ca pa ble of cut ting light and af ter wards re com bin ing it. Thus one ends up with thesame light beam – with slightly less en ergy. The re mark able re sult was that the sum did notpro duce light but dark ness! How ever, when one of the two halves was blocked with a piece of black pa per the other halve did ap pear. Seem ingly the only way to ex plain what hap penedhere is to as sume that the in ter fer ence of the split light-waves cancel out each other whenreunited.

re fers to an in di vid ual whole em bed ded in the inter-modal and inter-struc turalco her ence of re al ity, to an en tity emerged in the dept-layer of an all-em brac -ing tem po ral ity tran scend ing gen u ine con cept-for ma tion and only to be ap -prox i mated in a con cept-tran scend ing idea.

A deep en ing of this ba sic (tran scen den tal) idea oc curs when – through the o -ret i cal re flec tion and in ves ti ga tion – the di men sion of mi cro-struc tures is un -veiled (the mi cro-world with at oms and sub-atomic par ti cles). It is im por tantin this con text, how ever, to re al ize that con cepts such as par ti cle, field, andwave are not type con cepts but modal func tional con cepts (some times re ferred to as el e men tary ba sic con cepts of phys ics). Con se quently, the terms par ti cleand wave an a log i cally re flect retrocipatory struc tural mo ments within thestruc ture of the kinematical as pect, namely move ment mul ti plic ity (nu mer i calanal ogy) and move ment ex ten sion (spa tial anal ogy). These fac ets are deep -ened in phys i cally qual i fied en ti ties and could be ap prox i mated in phys i calthe ory from the per spec tive of math e mat i cal an tic i pa tions to the phys i calaspect – compare Shrödinger’s wave function formulated in terms ofdifferential equations.

Since num ber, space and move ment re main ir re duc ible as pects re gard less ofthe na ture and type of en ti ties func tion ing in them (their modal uni ver sal ity),it is also from this per spec tive un der stand able why the func tion ally dis tinctcon cepts par ti cle and wave can not be re duced to each other – a state of af fairssup ported by experimental data.

Physically qualified structural interlacement

All en ti ties with a phys i cal qual i fi ca tion be long to the realm of ma te rialthings. At oms rep re sent a rad i cal type within this realm. What is the na ture ofthe re la tion ship be tween atom and mol e cule? Is it pos si ble to see a mol e cule in an atomistic sense as the ex ter nal link ing of at oms ac tu ally con tin u ing to ex -ist? But what then about the ob vi ous to tal ity-prop er ties of mol e cules? Are wenot com pelled, on the ba sis of the lat ter, to con clude (in a ho lis tic sense) thatthe na ture of a mol e cule is such that its con tains and em braces the con sti tu tiveat oms in a trans formed way – as in te gral parts of a new whole?

Van Melsen says that in “most forms of At om ism it is a mat ter of prin ci ple that any com bi na tion of at oms into a greater unity can only be an ag gre gate ofthese at oms.” By con trast, he re fers to ho lis tic ten den cies: “In mod ern the o ries atomic and mo lec u lar struc tures are char ac ter ized as as so ci a tions of many in -ter act ing en ti ties that lose their own iden tity. The re sult ing ag gre gate orig i -nates from the con verg ing con tri bu tions of all is com po nents. Yet, it forms anew en tity, which in its turn con trols the behaviour of its components”(1975:349).

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Chem i cal bond ing is ac counted for in terms of the elec tron shells.1 Nor mallyonly the outer elec tron shells are re spon si ble for chem i cal bond ing. Ions,2

which are nor mally des ig nated with a plus (+) or mi nus (-) sign3 con sti tuteionic bond ing in such a way that the pos i tive ions are bal anced by neg a tiveones.4 Cou lomb forces keep ions in chem i cal bond ing to gether.5

Non-ionic (co va lent) bond ing takes place when cer tain at oms share spe cificelec trons. This type is also known as co va lent bond ing or elec tron pair bond -ing not only be cause the pair of elec trons spend much time in the space be -tween at oms but also be cause they are paired their spins point in dif fer entdirections.

A third type of chem i cal bond ing is found in com mon met als, known as me tal -lic bond ing. Most or ganic crys tals are kept to gether by Van der Waals forces,but be cause they are to weak they should not be seen as true forms of chem i cal bonding.

We may now re turn to the ap par ent ex treme pos si ble po si tions re gard ing there la tion ship be tween at oms and mol e cules in chem i cal bond ing. From the fact that chem i cal bond ing is chem i cally ac counted for in terms of the (outer) elec -tron shells, it is clear that the nu cleus of the atom main tains its in ter nal in teg -rity in the chem i cal bond ing. The nu cleus of the atom is not sim ply an ac ci -den tal fea ture of the atom but in deed that cen tral part of the atom which de ter -mines its place in the pe ri odic sys tem. For ex am ple, in a crys tal lat tice in di vid -ual at oms still serve as sources of ra di a tion when sub jected to Röntgen rays.An other con sid er ation is that the chem i cal bond ing does not af fect the ra -dio-ac tiv ity of atomic nu clei. These kinds of con sid er ations sug gest that at -om ism is cor rect – in the quoted words of Van Melsen: “it is a mat ter of prin ci -ple that any com bi na tion of atoms into a greater unity can only be an aggre -gate of these atoms.”

How ever, this does not tell the whole story, since there are equally force ful ar -gu ments in fa vour of the view that the mol e cule, in the fi nal anal y sis, forms anew unity fully en com pass ing the at oms as parts of an in te gral whole. Bio -chem is try dis cov ered many iso meric forms, that is, they have iden ti fied chem -i cal struc tures which are con sti tuted by the same at oms, viewed purely nu mer -i cally, but that none the less, ow ing to dif fer ent spa tial ar range ments, dif ferchem i cally. The for mula C3H6O may yield the fol low ing (chem i cally dis tinct) struc tures: CH3.CH2,CHO or CH3.CO.CH3. An other ex am ple is C4H4O4.

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1 Weiniger points out that re gard ing the re la tion ship be tween the clas si cal con cept of mo lec u -lar struc ture and quan tum-me chan ics there are still se ri ous un re solved prob lems (1984:940).

2 These are at oms with a pos i tive or neg a tive charge de pend ing on whether or not they have ac -quired or lost an elec tron.

3 For ex am ple Na+ for a ni tro gen ion and Cl- for a chlo rine ion.

4 So dium chlo ride con sists of a lat tice such that each so dium ion (Na+) is sur rounded by sixchlo rine ions (Cl-) and such that each chlo rine ion is sur rounded by six so dium ions.

5 Dis tinct from elec trons with a neg a tive charge the nu cleus of the atom is pos i tively charged.The lat ter is con sti tuted by neu trons (elec tri cally neu tral) and pro tons with a pos i tive charge.

From a neo-Thomistic per spec tive P. Hoenen, for ex am ple, de fends a view ofa mol e cule in line with Ar is totle’s ap proach. He ac cepts Ar is totle’s sub stancecon cept which holds that only the com bi na tion of form and mat ter can yield asub stan tial unity. Any atom main tain ing its ac tual ex is tence with in the mol e -cule would jeop ar dize the sub stan tial unity of the mol e cule. Those fea turesap par ently sug gest ing the ac tual ex is tence of at oms af ter their bond ing in mol -e cules ought to be seen as mere vir tual characteristics.

In con nec tion with the prob lem of the struc tural in ter weav ing of en ti ties,Dooyeweerd de vel oped a the o ret i cal ap proach which ac counts for the con tin -u a tion of the in ter nal na ture of en ti ties which are in ter wo ven (cf.1996-III:627 ff., 694 ff.). When the in ter nal na ture of an in ter wo ven en tity is re tained,Dooyeweerd speaks of enkapsis. When the struc ture of one kind of en tity isfoun da tional for the struc ture of an other kind of en tity, it is re ferred to as aone-sided enkaptic foundational relationship.

With re gard to the in fi nite divisibility of a spa tial whole, there are im por tantlim its in the un qual i fied use of the spa tial whole-parts re la tion. The na ture ofenkaptically in ter wo ven forms il lu mi nate fur ther lim its in this re gard. The in -ter weav ing which ex ists, for ex am ple, be tween the so dium and chlo rine at oms which are found in ta ble salt, is in no way given ac count for with the help of awhole-parts per spec tive. Ev ery di vi sion of ta ble salt must – that is if we stillwant to be work ing with real parts of salt – still pos sess the same chem i calstruc ture (NaCl). The crit i cal ques tion is whether so dium and chlo rine haveeach in di vid u ally got a salt struc ture? Are so dium and chlo rine true parts ofsalt? The an swer is ob vi ous: No, because neither one has a NaCl-structure onits own!

This sim ple ex am ple al ready up roots the un qual i fied way in which, es pe ciallyin mod ern sys tem the ory, lit er ally ev ery thing in re al ity is spo ken of in terms of a whole and parts (sys tems and sub sys tems) (cf. my crit i cism of this inStrauss, 1985).

We have men tioned that within the realm of phys i cally qual i fied en ti ties ween coun ter dif fer ent geno-types. Dif fer ent bonds of the same atom dis play a

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H H

H H

COOH COOH

COOH COOH

C C

C C

Maleic acidcis

Fumaric acidtrans

num ber of vari abil ity types. When an atom en gages in chem i cal bond ing, ween coun ter an enkaptic struc tural to tal ity: be sides an en tity’s in ter nal struc tural work ing sphere there is an ex ter nal enkaptic sphere of op er a tion – a sphere inwhich the enkaptically-bound struc ture stands in ser vice to the enkapticallyencompassing totality.

A wa ter mol e cule, e.g., can ex ist as a struc tural whole on the ba sis of thegeno-type of the bond of the ox y gen and hy dro gen at oms. With out at oms,there can be no men tion of a mol e cule – thus the in di ca tion: uni lat er allyfounded. Does this im ply that the at oms to tally be come part of the chem i calbond which ex ists in the mol e cule? Not at all, be cause the bond ap plies only to the bond ing elec trons and not to the whole atom. Be sides, as we have noted,the atom nu cleus is not just a spe cific char ac ter is tic of the atom, but pre ciselythat nu clear part of an atom which de ter mines its phys i cal-chem i cal geno-type (com pare the atomic num ber = the num ber of pro tons of the nu cleus), as wellas the atom’s place in the periodic table.

The fact that the atom nu cleus re mains struc tur ally un changed in the chem i calbond ing, guar an tees the in ter nal sphere of op er a tion of the atom. Be cause theelec trons can not be dis en gaged from the atom nu cleus, the at oms func tion as a whole in the wa ter mol e cule. Note that we can not say that the at oms func tionin a chem i cal bond. The bond ing does not en com pass the atomic nu clei.None the less the at oms (with their nu clei, elec tron shells and bond ing elec -trons) are pres ent as a whole in the wa ter mol e cule which en com passes themenkaptically. The in di ca tion: enkaptically en com passed, shows that the at -oms, re tain ing their in ter nal na ture, is ex ter nally ser vice able to the wa ter mol -e cule as a whole. The enkaptic in ter weav ing of the at oms in the mol e cule does not make them in trin si cally part of the mol e cule, since this would abrogate the internal sphere of action of the atoms.

The ex ter nal enkaptic func tion of the ox y gen and hy dro gen at oms in the wa ter mol e cule in di cates the func tion ing of the at oms in the mol e cule as to tal ity viathe chem i cal bond. This pres ents us with three facts:(i) First of all, we must dis tin guish the in ter nal sphere of ac tion of the atom.(ii) Sec ondly, we find the chem i cal bond which leaves the atom nu cleus un -

changed be cause it only reaches the outer elec tron shells, so that theatom nu clei can in no way be part of the chem i cal bond ing.1

(iii) Thirdly, we find the enkaptic struc tural whole of the wa ter mol e culewhich enkaptically en com passes the atomic nu clei and bonds and as -cribes to each its struc tural typ i cal place.

This the ory of enkaptic interlacement en ables us to side-step the one-sid ed -ness in an atomistic and a ho lis tic struc tural the ory of chem i cal bond ingwithin a mol e cule – and it also nat u rally rec on ciles ap par ently con tra dic toryex per i men tal data, since it ac counts both for the con tin ued ac tual ex is tence ofat oms in mol e cules (the point of ori en ta tion of at om ism) and for the typ i caluni tary char ac ter of the mol e cule (the em pha sis of ho lism) as a new to tal ityenkaptically founded in the structural nature of atoms.

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1 Ho lism is there fore mis taken in its claim that the atom “loses” its iden tity in the mol e cule. Cf.Van Melsen, 1975:349.

By briefly re turn ing to the orig i nal and an a log i cal mean ing of spa tial ex ten -sion we can dem on strate the ef fect a mis taken con cep tion of space had on theintepretation of data. Whereas the math e mat i cal space – in a purely ab stractand func tional per spec tive – is, as we have ob served, both con tin u ous and in -fi nitely di vis i ble, phys i cal space (by be ing bound to the quatum struc ture ofen ergy) is nei ther con tin u ous nor in fi nitely di vis i ble. For a num ber of years acon tro versy ex isted be tween Millikan and Ehrenhaft. The for mer re ceived in1923 the No bel prize for phys ics for his work which es tab lished that the elec -tron is the fun da men tal and in vis i ble unit for neg a tive elec tri cal charge. Thelat ter be lieved to have ob served elec tri cal chrages smaller than the elec tron.Cush ing remakrs that is ap pears as if Ehrenhaft mis in ter preted the data be -cause he still believed that electrical charge is continuously divisible(Cushing, 2000:10).

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Chapter IV

The Mosaic of philosophical stances in modern biology

Introduction

Of course the de ci sive ba sic philo soph i cal prob lem fac ing bi o log i cal sci en -tific think ing through out its his tory has been that con cern ing the re la tion shipbe tween the bi oti cal as pect and the physico-chem i cal as pect of re al ity inwhich the for mer is founded. Just as math e mat ics is of ten re duced to ei ther the ar ith met i cal or spa tial ex treme, the his tory of bi o log i cal thought is marked bythe ten sion be tween mech a nis tic and vitalistic approaches.

Do biotically qual i fied en ti ties re ally ex ist, or can one ex clu sively and com -pletely de scribe such things in terms of their con sti tu tive physico-chem i calcom po nents? If the lat ter point of view is cor rect, then one has to ask whetherthe dis tinc tion be tween ‘life’ and ‘death’ still makes any sense: if ev ery thingis de ter mined by the in ter ac tion among life less ma te rial con stit u ents then thedif fer ence be tween be ing alive and be ing liveless fades to an il lu sory pe riph -eral phe nom e non of the physical mass of reality.

Hans Jonas once strik ingly typ i fied the mo nis tic forms of vi tal ism andmechanicism. Un like dualists, mo nists do not at tempt to re duce re al ity philo -soph i cally to two fun da men tal prin ci ples, but rather posit a sin gle all-in clu -sive and uni ver sally ex plan a tory prin ci ple. We may there fore just as wellspeak about pan-vi tal ism and pan-mechanicism. Al ready in Greek phi los o phy we come across hulèzoism (zoè = life; hulè = mat ter): one of the in di rectly pre -served aph o risms of Thales sup pos edly was that ev ery thing lives. From thisper spec tive it is un imag in able that ‘life’ may not be the universal rule. Jonascomments:

“In such a world view death is a rid dle con front ing one, a con tra dic tion of thenat u ral, self-ex plan a tory and un der stand able, of the com mon life” (1973:20).

The para graph in which Jonas makes this state ment treats pan-vi tal ism andthe prob lem of death (1973:19ff). On the other hand, peo ple who think pan- mech a nis ti cally em pha size the no tion that liv ing phe nom ena are pe riph eral inan encompassingly ho mo ge neous phys i cal world. Quan ti ta tively neg li gi ble in

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the im mea sur able ex panse of cos mic mat ter, qual i ta tively an ex cep tion to therule of ma te rial char ac ter is tics, sci en tif i cally in ex pli ca ble in an ex pli ca blephys i cal nat u ral re al ity, “life” becomes an insurmountable obstacle forpan-mechanicism:

“Life as prob lem here in di cates rec og ni tion of its strange ness in the me chan i -cal world, which is the real world; to ex plain it means – on this level of the uni -ver sal on tol ogy of death – to deny it, re duc ing it to a vari ant of the pos si bil ityof the life less” (1973:23). This para graph treats pan-mech a nism and the prob -lem of life (1973:22ff).

A first step out of this di lemma is to be found in mak ing a dis tinc tion be tweendif fer ent modal as pects. The fun da men tal modal char ac ter of the phys i cal andbi oti cal as pects re mains only a func tional con di tion for con crete en ti tieswhich con tinue to func tion in these (and other) as pects of re al ity in a typ i calway . What is at is sue here is the ba sic dis tinc tion be tween the as pects of re al -ity and the di men sion of en ti ties – a dis tinc tion con tin u ally dis re garded by thedif fer ent points of view in bi ol ogy which time and again speak of modal func -tions as if they are con crete en ti ties (thus the ha bit ual ref er ence to the or i gin of life, rather than to the or i gin of liv ing things).1 As an as pect of re al ity life hasto do with the how of en ti ties, not their con crete what.

Phe nom ena of life are al ways linked to liv ing en ti ties which – as en ti ties – cannever be en com passed by their bi oti cal as pect. This has been a prob lem es pe -cially in the vitalistic tra di tion – which absolutises life and sees it as va ri et iesof an im ma te rial vi tal force. That it is im pos si ble to con sider the bi oti cal as -pect of liv ing things de tached from the intermodal co her ence within which itfinds it self is con firmed re peat edly by the in her ent anal o gies in the struc tureof the bi oti cal as pect. Even the ex pres sion vi tal force, so of ten used by vi tal -ism (al though of ten re placed with other terms like Gestaltungsfaktor orZentralinstanz), can never in di cate the dis tinc tive ness of the bi oti cal as pect –sim ply be cause it is un mis tak ably a phys i cal anal ogy in the modal struc ture ofthe bi oti cal as pect. The term force re veals the orig i nal (non-analogical) modalmeaning of the physical aspect of energy-operation.

Biotically qualified entities

Biotically qual i fied or char ac ter ized en ti ties be long to the realm of plants. The dis tinc tion be tween the phys i cal and bi oti cal as pects is foun da tional to the en -tity-struc tural dis tinc tion be tween the realm of phys i cal things and the realmof biotically qual i fied en ti ties – i.e. the plant realm. In the ab sence of the nec -es sary modal dis tinc tions it is still com mon for sup port ers of dif fer ing bi o log i -cal points of view to use ex pres sions like: liv ing mat ter and dead mat ter. Ma -te rial things, how ever, are ex clu sively phys i cally qual i fied and can there forenot si mul ta neously have an internal biotical qualifying function.

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1 The latin word for a thing, “res,” ren ders a ser vice to the way in which one can des ig nate theat tempt to treat a modal func tion as if it is an en tity: this fallcy is known as rei fi ca tion.

It is not mean ing ful to re fer to mat ter as be ing “dead”1 – for strictly speak ingonly some thing that once was alive could later on be called “dead.”2

What guarantees the identity of living things?

Al ready the in di ca tion that cer tain things are alive im plies their ac tive func -tion ing in the bi oti cal as pect of re al ity – i.e., their bi oti cal sub ject func tion.The fact that liv ing things must, in a ther mo dy namic sense, be con sid eredopen sys tems, more over in di cates that ev ery liv ing thing in dis tinc tion from its qual i fy ing bi oti cal as pect also has a phys i cal as pect. This is the topic of thewell-known book of Erwin Schrödinger: What is life? The phys i cal as pect ofthe cell (1955). Any liv ing en tity does have sub ject func tions in the first threeas pects of re al ity as well – namely the as pects of num ber, space and move -ment. For ex am ple, linked to the ques tion whether liv ing things can move bythem selves we find an other im por tant dis tinc tion in bi o log i cal sys tem at ics,namely that be tween plants and an i mals.3 The con ti nu ity (en dur ance) of lifeof a plant can be de ter mined in co her ence with the ki ne matic func tion of liv -ing en ti ties. Apart from the pro por tions or spa tial form of liv ing things, theirspa tial func tion is also prom i nently ex hib ited in ex pres sions like bio-mi lieu or Umwelt. The term Umwelt gained prom i nence es pe cially ow ing to the bi o log -i cal think ing of Ja cob von Uexküll (cf. e.g. Von Uexküll & Kriszat, 1970). Aliv ing thing is fur ther more a unity in the di ver sity of its or ganic life pro cesses– if these var i ous pro cesses are not bound to gether as a unity, the liv ing en titydis in te grates and dies.

Since liv ing things main tain, in ther mo dy namic terms, a flow ing equi lib riumin which or der is with drawn from the en vi ron ment (to which Schrödinger re -fers as neg a tive en tropy), it can be said that liv ing things main tain them selvesin a state of high sta tis ti cal im prob a bil ity: in typ i cal growth pro cesses liv ingen ti ties even con tin u ally increases their internal order.

Of course it can not be con sid ered to be the dis tinc tive char ac ter is tic of liv ingthings, since sev eral non-liv ing en ti ties and porcesses – in clud ing flames andgla ciers – are also open sys tems in a ther mo dy namic sense. Only when wetake into ac count the qual i fy ing bi oti cal sub ject func tion of liv ing things canwe un cover their dis tinc tive char ac ter is tic in com par i son with ma te rialthings. This qual i fy ing func tion de ter mines the bi oti cal iden tity of liv ingthings. Ac cord ing to the mech a nis tic point of view in bi ol ogy, how ever, liv ing things are only com plex in ter ac tive sys tems in which, in ac cor dance with thena ture of open sys tems, continuing metabolic processes (anabolism andcatabolism) occur.

Remark: Since Des cartes mod ern phi los o phy and bi ol ogy is ac quaintedwith a ma chine model. Al though we may think that this model con sti -

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1 Oparin, for ex am ple, fol low ing the di a lec ti cal ma te ri al is tic ap proach of Engels, does thatwith out hesitation.

2 Von Weizsäcker ex plic itly high lights this in sight – 1993:32: “Die Steine sind unbelebt. Mansollte aber nicht sagen, sie seien tot. Tot sein kann eigentlich nur etwas das gelebt hat.”

3 This ques tion re gard ing their mo bile ca pac i ties ap peals to the typ i cal func tion of plants or an -i mals within the ki ne matic aspect.

tutes a straight for ward re duc tion – even of the hu man be ing – to “na -ture,” the im plicit technicistic un der tones of this model are lost sightof. The na ture of a ma chine ought te be scru ti nized first, be cause a ma -chine only orig i nated in the course of hu man civ i li za tion. VanWeizsäcker cor rectly says: “Think ing na ture – and with it the hu manbe ing – as a ma chine, sub jects na ture and with it the hu man be ing to aspe cific industiral mode of thought, that of designability. Not the re -duc tion of the hu man be ing to na ture is the mis take here, but the re duc -tion of na ture to the struc tural prop er ties of a very spe cific hu manarte fact.”1

Thus from a mech a nis tic point of view a liv ing thing has a physico-chem i caliden tity con sti tuted by its at oms, mol e cules, and macro-mol e cules. Which ofthese physico-chem i cal com po nents should how ever be con sid ered con sti tu -tive of this sup posed physico-chem i cal iden tity of liv ing things: cur rentlypres ent at oms, mol e cules, and macro-mol e cules, those pres ent years ago, orthose which will be pres ent a few years hence!?2 When liv ing things arephysicalistically re duced to their ma te rial con stit u ents, their bi oti cal iden tityis nec es sar ily lost – since the sup posed el e ments of iden tity con tin u ally vary.

Once the bi oti cal func tion of liv ing things is taken into ac count, it is even pos -si ble to claim that a liv ing thing, biotically con sid ered, is in a sta ble state (re -ferred to as health), while si mul ta neously claim ing – with out any con tra dic -tion – that physico-chem i cally con sid ered (with a view to the flow ing equi lib -rium of its phys i cal-chem i cal con stit u ents) it ex ists in an un sta ble state. If thephys i cal-chem i cal sub stra tum of liv ing things ap proaches a state of highersta tis ti cal prob a bil ity, bi oti cal in sta bil ity in creases as a sign of the fi nal pro -cess of dy ing.

From the per spec tive of his or gan is mic bi ol ogy von Bertalanffy strik ingly in -di cates the cul-de-sacs of the mech a nis tic point of view which elim i nates thebi oti cal func tion of life processes:

“These pro cesses, it is true, are dif fer ent in a liv ing, sick or dead dog; but thelaws of phys ics do not tell a dif fer ence, they are not in ter ested in whether dogsare alive or dead. This re mains the same even if we take into ac count the lat estre sults of mo lec u lar bi ol ogy. One DNA mol e cule, pro tein, en zyme or hor -monal pro cess is as good as an other; each is de ter mined by phys i cal and chem -i cal laws, none is better, health ier or more nor mal than the other” (1973:146).

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1 “Die natur und dann damit den Menschen als Ma chine zu denken, unterwirft die Natur anddamit den Menschen einer spezifisch industriellen Denkweise der Planbarkeit. Nicht dieReduktion des Menschen auf die Natur ist hier die Fehler, sondern die Reduktion der Naturauf the Struktureigenschaften einbes sehr speziellen Menschenwerks” (Van Weizsäcker,1993:38).

2 Jones et al points out that all “the at oms of our body, even of our bones, are ex changed at leastonce ev ery seven years. All the at oms in our face are re newed ev ery six months, all our redblood cells ev ery four months and 98% of the pro tein in the brain in less than a month. Ourwhite blood cells are re placed ev ery ten days and most of the pan creas cells and one-thir -teenth of all our tis sue pro teins are re newed ev ery 24 hours” (1998:40).

The origin of living things – a biological boundary question

Al though the view that liv ing things could spon ta ne ously emerge out of life -less mat ter (generatio spontanea) has been known since Greek an tiq uity, it isno lon ger ac cepted in mod ern times by any nat u ral sci en tist – inter alia. due tothe work of Pas teur. None the less the mech a nis tic (or rather: physicalist) per -spec tive must make at least one ex cep tion: the or i gin of the first liv ing en tityun der cir cum stances en tirely alien to those known to us today.

The old est known fos sils of liv ing en ti ties are those of uni cel lu lar al gae –found near Barberton in South Af rica. By means of the half-life of ra dio- ac -tive sub stances the age of these Archaeosphairoïdes barbertonensis havebeen cal cu lated as ap prox i mately 3 100 mil lion years (cf. Schopf, W. & Barg -hoorn, 1967:508ff).

Since liv ing en ti ties, con sid ered physico-chem i cally, func tion on the ba sis ofboth (en zyme) pro tein and nu cleic acid (DNA), the mech a nis tic point of viewis obliged to pre sume that ini tially there must be a close re la tion ship be tweenpro tein and DNA. Al ready in 1971, how ever, Orgel and Sulston com ment inthis re gard: “This ap proach leads to new dif fi cul ties so se vere that it has neverbeen car ried very far” (1971:91). They con tinue with the strik ing ob ser va tionthat “prog ress” can only be re corded in this re gard when char ac ter is tics are at -trib uted to pro tein and DNA “which have not been dem on strated ex per i men -tally, and which usually seem implausible” (1971:91).

These com ments ac tu ally re fer back to ideas ini tially (and in de pend ently) de -vel oped by Haldane (al ready in 1928) and the Rus sian Oparin (cf. 1953, chap -ters 4-7: pp.64-195). The as sump tions of the Oparin-Haldane ap proach even -tu ally turned out to be ques tion able. That the ini tial at mo sphere of the earthwas mainly com posed of hy dro gen, meth ane, am mo nia and wa ter va por. Inpar tic u lar Oparin holds that car bon “made its first ap pear ance on the Earth’ssur face not in the ox i dized form of car bon di ox ide but, on the con trary, in there duced state, in the form of hydrocarbons” (1953:101-102).

Sil ver points out that there is at pres ent “no ev i dence that the at mo sphere wasre duc ing (meth ane and hy dro gen)” and re marks that “the prev a lent opin ion atthe mo ment is that the Earth’s at mo sphere, at the time that life emerged, wasmainly car bon di ox ide and ni tro gen” (1998:344). The role of meth ane is alsoun ac cept able in the Oparin story since it is one of the com po nents of nat u ralgas which is pro duced by the “ef fect of mil lions of years of pres sure and heatact ing on pre his toric plant ma te rial” (Sil ver, 1998:344). Al though the Hal -dane- Oparin con jec ture was kept alive for a con sid er able time, sup ported bythe ex per i ments done by Stan ley Miller (from Chi cago) in 1953, it does notbring us closer to an un der stand ing of the mys tery of the gen e sis of the liv ingcell. With regard to Miller’s experimentation Silver remarks:

“The Haldane-Oparin hy poth e sis is out of fash ion. Of the forty or so sim plemol e cules that would be needed to form a prim i tive cell, the ex per i ment pro -duces two. It is worth bear ing in mind that glycine con tains only ten at oms andalanine, thir teen. The sim plest nu cle o tide con tains thirty at oms. The prob a bil -ity that a given large mol e cule will be pro duced by chance from small mol e -cules, by sparks, falls dras ti cally as the mo lec u lar size in creases. It has to be

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re al ized that even if heat, ra di a tion, and light ning, on the young Earth, had pro -duced all the amino ac ids and nu cleo tides needed for pres ent forms of life, thegap be tween an aque ous so lu tion of these emolecules and a liv ing cell isstipendous. It’s a ques tion of or ga ni za tion: in the ab sence of a guid ing in tel li -gence, presentday sci en tists are not do ing very well. For the mo ment, let’sshow the Miller ex per i ment to the side door and see who is next in line in thewaiting room” (Silver, 1998:345).

In neo-Dar win ist thought nat u ral se lec tion re ceives much prom i nence.1 Asim i lar story is used to ex plain the or i gin of the first liv ing en ti ties: by meansof se lec tion the ac ci den tal emer gence of or ganic com bi na tions (amino ac ids,nu cleic acid, en zymes, etc.) sup pos edly gave rise to the for ma tion of re pro -duc tive units, vi rus-like forms, proto-or gan isms and even tu ally true liv ingcells. In view of phys i cal laws, Von Bertalanffy, amongst others, alsoquestions this construction:

“In con trast to this it should be pointed out that se lec tion, com pe ti tion and‘sur vival of the fit test’ al ready pre sup pose the ex is tence of self-main tain ingsys tems; they there fore can not be the re sult of se lec tion. At pres ent we knowno phys i cal law which would pre scribe that, in a ‘soup’ of or ganic com pounds, open sys tems, self-main tain ing in a state of high est im prob a bil ity, are formed.And even if such sys tems are ac cepted as be ing ‘given’, there is no law inphys ics stat ing that their evo lu tion, on the whole, would pro ceed in the di rec -tion of in creas ing or ga ni za tion, i.e. im prob a bil ity. Se lec tion of ge no types with max i mum off spring helps lit tle in this re spect. It is hard to un der stand why,ow ing to dif fer en tial re pro duc tion, evo lu tion should have gone be yond rab -bits, her ring or even bacteria, which are unrivalled in their reproduction rate”(1973:160-161).

Those who have re spect for sci en tific mod esty may do well to re flect upon are mark made by Haldane in dis cus sion with Sil ver: “I had a long con ver sa tionwith J.B.S. Haldane, which started off with pol i tics and ended with sci ence.When I ques tioned him about evo lu tion, one of his re marks sparked my in ter -est, and sent me to the li brary that eve ning: ‘Evo lu tion’s not the prob lem. Lifeis’ Then he said, ‘Oparin and I once had an idea about that, but we’ll neverknow the real an swer’ ” (Silver, 1998:353).

Are viruses a transitional form between material and living entities?Vi ruses con sist of nu cleic acid (ei ther RNA or DNA) housed in a man tle ofpro tein and oc ca sional lipids. In 1935 W.M. Stan ley suc ceeded in pu ri fy ingand crystalizing the to bacco mo saic vi rus. Vi ruses are only able to mul ti ply(and in the pro cess act in a deformative way as par a sites) in liv ing cells. Weknow noth ing about the ac tual or i gin of vi ruses, which means that the pos si -bil ity that they could be re duced mi cro-or gan isms, or genes dis man tled fromthe cell-struc ture, or prod ucts of cell me tab o lism, re main spec u la tive. Thusthe sup posed in-be tween-po si tion of viruses remains problematic.What is re mark able about the “in-be tween-po si tion-hy poth e sis” is that itstarts off from the dis tinc tion: ma te rial things and liv ing things. That is why

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1 Dar win al ready de vel oped the the ory that since far more de scen dants are born than could sur -vive, a con tin ual strug gle ex ists in which only the best equipped “or gan ism” makes the grade– out of which grad u ally new kinds emerge.

the ques tion is thus posed: is a vi rus liv ing or is it only a macro-mo lec u lar ma -te rial struc ture? Since it can re pro duce in side a true liv ing cell, it is sug gestedthat it could main tain an in-be tween-po si tion. In re al ity, how ever, we onceagain run into the pre sup po si tions of sci en tific dis tinc tions. All sci en tific dis -tinc tions, af ter all, pre sup pose the whole hu man be ing with all one’s pre-sci -en tific ex pe ri ence of re al ity. The di ver sity im plied in this ex pe ri ence is onlymade ex plicit by means of modal ab strac tion, e.g. when we dis tin guish be -tween the phys i cal and bi oti cal as pects of re al ity. This dis tinc tion, how ever. is em bed ded in our pre- (or rather non-) sci en tific ex pe ri ence of re al ity – andpar tic u larly in the dif fer ence en coun tered be tween ma te rial and plantlikethings (by means of en tity-ori ented ab strac tion). Be fore sci en tists (whetherphi los o phers or spe cial sci en tists) can in ves ti gate the na ture and char ac ter is -tics of plants, they must first, by means of their non-sci en tific ex pe ri ence, gain in sight into the qual i ta tive dif fer ence in kind be tween plants and ma te rialthings. The question: what is botany? after all is a philosophical presup -positional question of botany as a special science.

Al though we would be in clined to claim that bot a nists are in deed able to saywhat a plant re ally is, this priv i lege ac tu ally must ul ti mately be de nied them,since, if they did not al ready in their non-sci en tific con crete ex pe ri ence of re -al ity have had the abil ity to dis tin guish (e.g. by fo cus ing on the dif fer ences be -tween mat ter, plants, and an i mals), then they may well have been study ingma te rial things or an i mals in the mis taken be lief that they were study ingplants! There can be no de nial that sci en tific thought rests on our non- sci en -tific un der stand ing of dis tinct ness. With out this foundation it simply cannotfunction.

The im por tant im pli ca tion of this in sight into the found ing role of our non-sci -en tific un der stand ing of dis tinc tions in this re gard is this: a par tic u lar en titycan only be ei ther (non-liv ing) ma te rial in na ture or (biotically) alive. This ofcourse bears de ci sive im pli ca tions for the ques tion of whether a vi rus is liv ing(i.e. plantlike) or non-liv ing (i.e. ma te rial). Stat ing the ques tion in this man neral ready im plies that the an swer can not be am biv a lent – and in both cases therecan be no ques tion of any in-be tween position.

This “tran si tion prob lem” be tween non-liv ing and liv ing things points to -wards the far greater prob lem caused in mod ern bi ol ogy by the evo lu tion arythe ory of Charles Darwin.

Since Dar win’s no tions be gan to gain bi o log i cal sup port, it has be come a mat -ter of course in mod ern bi o log i cal lit er a ture to talk of evo lu tion in the sense ofan all-en com pass ing de vel op ment across all bor ders and dis tinc tions whichcan sup pos edly still be found to day in the liv ing world. The cat e go ri za tions towhich we can come by only pay ing at ten tion to cur rently liv ing plants and an i -mals is known as the nat u ral sys tem (ab bre vi ated as NS). The cen tral philo -soph i cal ques tion is whether the NS can be used as the foun da tion for somesort of evo lu tion ary the ory or whether one or an other evo lu tion ary the oryshould not rather be used as the foun da tion for a rea soned cat e go ri za tion ofthe NS. Re flec tion on this foun da tional is sue how ever soon co mes in con tact

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with a tra di tional philo soph i cal con flict in which mod ern bi ol o gists havelargely taken sides – in fa vour of nominalism.

Nominalist structural understanding in modern biological literatureWhen G.G. Simpson dis tin guishes in one of his works be tween the phys i calsci ences and bi ol ogy he char ac ter izes the for mer as largely ty po logi cal andide al is tic:

“the phys i cal sci ences are for the most part ty po logi cal and ide al is tic. I meanby that, that they usu ally deal with ob jects and events as in vari ant types, not asin di vid u als with dif fer ing char ac ter is tics” (1969:8).

This ap proach is ac cord ing to him com pletely in ad e quate for the study of phe -nom ena be long ing to the bi oti cal lev els: “for phe nom ena spe cial to the bi o log -i cal lev els” (1969:8). What strikes one in this state ment by Simpson, is that hedis tin guishes be tween two types of phe nom ena, namely phys i cal and bi oti calphe nom ena (al though the lat ter is mis tak enly re ferred to as bi o log i cal). In or -der to ar rive at an iden ti fi ca tion of bi oti cal phe nom ena where a ty po logi cal(and even ide al is tic) method would be of no use, Simpson uses ex actly a ty po -logi cal method – a strik ing in ter nal con tra dic tion: bi ol ogy can func tion non- ty po logically if and only if it is typologically founded!The fun da men tal prin ci ple of intra-bi o log i cal re search is for mu lated by Simp -son as fol lows: “Or gan isms are not types and do not have types” (1969:8-9).The main ground for this state ment is the claim that or gan isms are in di vid u als“and no two are likely ever to be ex actly alike” (1969:9). This ar gu ment al -ready be trays Simpson’s view with re gard to the re la tion ship be tween phys i -cal law and phys i cal in di vid u al ity. Since phys i cal sub jects are ac cord ing tohim stud ied in phys ics only as “ob jects and events” with “in vari ant types”(1969:8), it fol lows that bi ol ogy, which stud ies or gan isms in their in di vid u al -ity only, can not make use of any ty po logi cal method.Simpson’s ap proach typ i cally de scends from clas si cal mech a nis tic phys ics(see the pre vi ous chap ter in this re gard). In clas si cal phys ics phys i cal sub jectshad been con sis tently re duced to the law-side of re al ity (in a ra tio nal ist man -ner). There is none the less no con flict be tween what mod ern phys i cists knowand the at tri bu tion of in di vid u al ity to phys i cal sub jects such as at oms. Shouldone main tain that phys i cally-char ac ter ized en ti ties have in di vid u al ity al -though they are cor re lated with uni ver sally valid phys i cal laws, the ques tionarises why this could not be the case with biotically and psy chic-sen si tivelycharacterized entities (such as plants and animals).Ap par ently with out no tic ing Simpson con tra dicts his own view that or gan -isms do not be long to any type when he re flects on the typ i cal char ac ter is ticsof a hu man be ing. In this re gard he re fers, with out re serve, to mouseness andman-ness (1969:88). Mouseness and man-ness, how ever, al lude to the struc -tural pre con di tions which en ti ties must meet be fore they can be known asmice and hu man be ings. Struc tural pre con di tions are no less real be cause theydo not them selves have a con crete in di vid ual iden tity.1

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1 The con di tions for be ing green are not them selves green – green things sim ply meet thesecon di tions by be ing green.

Since Dar win how ever many bi ol o gists are in clined to dis miss any idea ofstruc ture and uni ver sal ity. This is done since the his tory of the or i gin of theplant and an i mal world must sup pos edly be en com passed in a structurelessevo lu tion ary con tin uum. Even the dis tinc tion be tween plant and an i mal iscon sid ered a mere con ven tion, merely ar bi trary names (nomina) given to anun lim ited num ber of con cretely in di vid ual liv ing en ti ties. The uni ver sal ityim plied by these names have no foun da tion in things out side the hu man mind– this uni ver sal ity is purely a prod uct of hu man thought. The epistemologicalpo si tion of nomi nal ism rests on cos mo log i cal pre sup po si tions found al readyin Greek phi los o phy, but which only became dominant in modern biologicalthought since Darwin.

Over and against the nomi nal ist tech niques of clas si fi ca tion used in many bi -ol ogy texts, men tion is some times made of the older and sup pos edly out datedide al is tic mor phol ogy of Ray, Linnaeus and oth ers. There are none the less still im por tant 20th cen tury rep re sen ta tives of this mor phol ogy, in clud ing E.Dacqué (cf. 1935, 1940, 1948), W. Troll (1949, 1951 and 1973), K. LotharWolf (1951) and W. Leinfeller (1966). Ac cord ing to Troll the foun da tion ofcom par a tive mor phol ogy is to be found in ideas (in the pla tonic sense) whichserve as or der ing “in ner ar tic u la tions of our in tu ition” by means of whichtypes as “Urbildliche Einheiten” (pri mal im ag ery units) be come study sub -jects (cf. Ungerer 1966:232). Troll par tially reaches back to the think ing ofJ.W. Goe the – the Ro man tic poet and nat u ral phi los o pher. In his bi o log i cal in -ves ti ga tions, largely con cerned with mor phol ogy, Goe the em pha sized thechar ac ter of “Ges talt” – form in an al most pla tonic sense – al though he shiftedthe em pha sis to the fac tual side of re al ity, since he did not see the “Gestalt” asrooted in the law, but rather the law in the “Gestalt.”

In ide al ist mor phol ogy a pri mal leaf or pri mal plant is de signed in which cer -tain ba sic ty po logi cal char ac ter is tics are pres ent. Zimmerman en gages in a di -a logue with this ide al ist mor phol ogy in his Evo lu tion und Naturphilosophie(Berlin 1968). He points out that Troll con tin ues to be lieve that mor phol ogyde ter mines the pos si bil ity of the de scent and not the other way around:

“It is not the de scent which is de ci sive in mor phol ogy, but rather the op po site:mor phol ogy has to de cide about the pos si bil ity if de scent.”1

It is pos si ble to be lieve that the prob lems of the NS should serve as the foun da -tion of any pos si ble the ory of de scent with out be ing a sup porter of ide al istmor phol ogy. Portmann com ments for in stance that “few bi ol o gists still con -sider that sys tem at ics is the foun da tion of evo lu tion ary the ory, that this is thecer tain, that which we know, while evo lu tion ary the ory is what we suspect”(1965:10).

It is pos si ble, with out sup port ing ei ther the ar gu ment or point of de par ture ofide al ist mor phol ogy, to agree that the ex ist ing struc tural di ver sity of our con -tem po rary ex pe ri ence of re al ity is of de ci sive im por tance with re gard to the

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1 “Es is nicht die Dezendenz, welche in der morphologie entscheidet, sondern umgekehrt: dieMorphologie hat über die Möglichkeit der Dezendenz zu entscheiden” (Zim mer mann1968:49).

ques tion of de scent of any ex ist ing liv ing en tity. Ex actly in this sense the pres -ent is the key to our un der stand ing of the past. The ge ol o gist J.R. Van de Fliert even goes so far as to say that the doors to the past can only open to the ex tentthat the keys of the pres ent fit them. He men tions the ex am ple of a num ber ofold fos sils of the pre-Cambrium (more than 600 million years ago):

“These fos sils con sisted of im prints of an i mals, which prob a bly had not pos -sessed any hard parts, and which in part could be de ter mined be cause of closere sem blance with the struc ture of liv ing jelly-fish, worms, and other an i mals.Some of these fos sil struc tures, how ever, are so far com pletely un known inliv ing an i mals or plants and as a re sult they are enig mat i cal, ‘problematica’. Inthe ab sence of any struc tural link with the pres ent they could not be at trib utedto any known phy lum.”1

The ex ist ing struc tural di ver sity in the plant and an i mal realms would clearlylimit the con struc tion of fam ily trees in evo lu tion ary the ory. Who everrelativizes the NS in a nomi nal ist sense to a structureless line of de scent, how -ever, chooses in prin ci ple for a cha otic brew in which ev ery tax on omy of liv -ing things be comes ei ther im pos si ble in prin ci ple, or at least en tirely ar bi trary. But the plant and an i mal realms are en tirely mean ing fully di vis i ble, bar ringonly the in stances where our sci en tif i cally de vel oped cri te ria for dis tinc tionare still in suf fi cient. (Think for in stance of the protista, which in cludes al gae,fungi, mu cus fungi, and pro to zoa, of all of which it can not be said with cer -tainty whether they are plants or an i mals. These prob lems do not how everrelativize the fact that each of these protista is either plantlike or animal-like).

Structureless continuity versus structural discontinuity

A core con cept in nomi nal ist evo lu tion ary the ory is sum ma rized in the wordvari abil ity. Think for in stance of the com ment by Simpson, men tioned above,that phys ics is con cerned with in vari ant types while bi ol ogy is con cerned with change able in di vid ual or gan isms. The fun da men tal ques tion con cerns the re -la tion ship be tween con stancy and vari abil ity. The stron ger ten dency in evo lu -tion ary cir cles is to choose for a com plete vari abil ity rather than any struc -tural con stancy. In ide al ist mor phol ogy a choice is ap par ently made for a (pla -ton i cally in flu enced) un der stand ing of con stancy on the op po site ex treme end of the spec trum of op tions. The prob lem, how ever, is that the con cept vari -abil ity only makes sense when de lim ited by some or other typ i cal ity or con -stancy. Here we come across a fur ther anal ogy of the in dis sol u ble co her encebe tween the ki ne matic and phys i cal as pects of re al ity. Bi otic con stancy andbi otic dy nam ics are on to logi cally con sid ered eq ui ta ble – to em pha size the one over the other would en tail dis re gard ing the unique ness of ei ther thekinematic or the physical aspects (cf. the more extensive founding of thisinsight in the previous chapter).

Eisenstein cor rectly points out that the term con stancy en com passes the con -cept of vari abil ity also in bi ol ogy in the sense that vari a tion is only pos si blewithin lim its (1975:278). Van de Fliert com ments in ad di tion that the quan ti -

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1 The Chris tian and Sci ence, un pub lished pre sen ta tion given at Cal vin Col lege, Grand Rap ids,1969, pp.26-27.

ta tive can only be un der stood as some thing de ter mined by the qual i ta tive:“More or less of this re mains this and does not become that” (1969:28).

Who ever em pha sizes vari abil ity has dif fi culty an swer ing the fol low ing ques -tion: If liv ing en ti ties dur ing the past three thou sand mil lion years have beengov erned by a uni ver sal evo lu tion ary law so that they de vel oped in a glob allypro gres sive way to wards the hu man be ing, it can not be ex plained why therestill are, apart from the highly evolved an i mals, such prim i tive en ti ties as bac -te ria, al gae, mosses, amoe bae, worms, etc. – why did the evolved an i mals notalso re main stuck on these orig i nal lev els? Eisenstein writes: “The si mul ta -neous co-ex is tence of the great est va ri ety of life forms, from amoeba to man,any way proves that from the per spec tive of na ture these are all eq ui ta ble andequally vi a ble (existenzfähig = able to ex ist), with out any ne ces sity of fur therde vel op ment” (1975:245). The zo ol o gist W. H. Thorpe com ments: “[i]tseems to me that there is an out stand ing prob lem raised by our dis cus sion –namely the prob lem of fix ity in evo lu tion. What is it that holds so manygroups of an i mals to an as ton ish ingly con stant form over mil lions of years?This seems to me to be the prob lem now – the prob lem of con stancy; ratherthan of change. And here one must re mem ber that the ge netic sys tems whichgov ern ho mol o gous struc tures are con tin u ally chang ing. Thus the con trol sys -tem is con tin u ally chang ing but the sys tem con trolled is con stant, and con -stant over mil lions of years. This prob lem seems to me to stick out like a sorethumb in mod ern evo lu tion ary the ory.”1

Within the frame work of the reformational philo soph i cal tra di tion an at temptis made to avoid the par tial i ties of both nomi nal ist and ide al ist (re al ist) struc -tural ideas. Just as lit tle as a modal phys i cal and typ i cal en tity struc tural lawshould be con fused with any sub ject func tion or con crete phys i cal sub ject,just as lit tle should the struc tural types of plants and an i mals be con fused withpar tic u lar con crete plants or an i mals. All plants and an i mals be long re spec -tively to the realms of struc tur ally ei ther biotically or sen si tive-psy chi cally di -rected or dered types. As true law types these be long to the law side of cre atedre al ity, in which admitedly a rel a tively con stant dy nam ics finds ex pres sion,but which as law types can not be re duced to or equated with tran sient in di vid -ual liv ing plants or an i mals. These rel a tively con stant or der ing types can onlybe re al ized in the course of time in tran sient individual creatures which ascorrelate is subject to the ordering types.

The rec og ni tion of this cor re la tion of the law and fac tual sides of re al ity is astum bling block for the structereless evo lu tion ary con tin uum con strued in anominalistic fashion.

To form a better im age of the na ture of the di ver gence be tween dif fer ent di -rec tions in mod ern bi ol ogy we must con sider the fac tual lim i ta tions set to there li gion of con ti nu ity by paleontology.

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1 A dis cus sion com ment af ter the con tri bu tion of L. von Bertalanffy (Change or Law) in thecol lec tion: Be yond Reductionism, ed ited by A. Koestler and J.R. Smythies, Lon don 1972:77.

Continuity of descent?

Since the first ap pear ance of Dar win’s no to ri ous writ ings much faith has beenplaced in the substantiatory power of paleontological fos sil find ings – a cer -tainty had grown that the nec es sary miss ing links would even tu ally be found.It was a con vic tion that pa le on tol ogy would pro vide di rect ac cess to the keymo ments in the evo lu tion ary his tory of plants, an i mals, and hu man be ings.The pa le on tol o gist D.B. Kitts points out, how ever, that the spa tial dis tri bu tion and tem po ral se quence of or gan isms with which pa le on tol ogy works isfounded in the or der ing prin ci ples of ge ol ogy, and can there fore not be en -com passed in any bi o log i cal the ory: “Thus the pa le on tol o gist can pro videknowl edge that can not be pro vided by bi o log i cal prin ci ples alone. But he can -not pro vide us with evo lu tion. We can leave the fos sil re cord free of a the oryof evo lu tion. An evo lu tion ist, how ever, can not leave the fos sil re cord free ofthe evo lu tion ary hy poth e sis” (1974:466). The dan ger con tin ues to ex ist thatbi ol o gists are con vinced of the ac cept abil ity of the evo lu tion ary hy poth e sis by a the ory which is al ready in her ently evolutionistic: “For most bi ol o gists thestron gest reason for accepting the evolutionary hypothesis is their acceptanceof some theory that entails it” (Kitts 1974:466).

A core paleontological prob lem fac ing the var i ous trends is the pres ence ofstrik ing gaps and dis con ti nu ities in the fos sil re cord. G.G. Simpson openlymen tions this fac tual state of af fairs: “more over, it is a fact that dis con ti nu ities are al most al ways and sys tem at i cally pres ent at the or i gin of re ally high cat e -go ries” (1961:361) and he em pha sizes sev eral pages later “the point that forstill higher cat e go ries dis con ti nu ity of ap pear ance in the re cord is not only fre -quent but also sys tem atic. Some break in con ti nu ity al ways oc curs in cat e go -ries from or ders up wards” (1961:366).1

The quest for miss ing links has al ways been a hope ful glance at pa le on tol ogyon the part of evo lu tion ary the ory – an ex pec ta tion which has ap par ently notre mained en tirely un re quited, since the much sought af ter forms ap pears tohave been found be tween the dif fer ent classes of ver te brates. Four forms areof im por tance in this re gard. The link be tween cer tain fishes (Crossopterygii)and am phib i ans is looked for in the Ichthyostega (first finds in Green land in1931) which be long to the Tetrapoda (quad ru ped ver te brates, in clud ing theam phib i ans, rep tiles, birds, and mam mals) but still have a true fish tail (cf.Kuhn-Schnyder 1967:350-352). Since the Ichthyostega are quad ru peds, theyare placed with the am phib i ans (they are fishlike am phib i ans). Over andagainst this bot tom end of the am phib i ans we find at the top end rep tile-likeam phib i ans, the Seymoria (first finds 1904). D.M.S. Wat son com ments: “Thewhole ef fect of its struc ture is that of a mo saic of sep a rate de tails, some com -pletely am phib ian, some com pletely rep til ian, and very few, if any, show ing apas sage leading from one to the other.”2

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1 The ba sic sys tem atic clas si fi ca tion con sists of the fol low ing: realm, phy lum, class, or der,fam ily, ge nus, and species.

2 Quoted by Kuhn-Schnyder, e.: 1967:357.

A fur ther form, the place ment of which brought about in ter est ing prob lems, isthe Ictidosauria (dis cov ered in 1932) which has both rep til ian and mam ma lian char ac ter is tics. Such an Ictidosaurier can be seen in the Bloemfontein mu -seum (de scribed by A.W. Crompton as Diarthrognathus broomi). Its skull has both a re duced articulare-quadratum-joint and a den tal-squamosum-joint. The pres ence of a den tal-squamosum-joint is a typ i cally mam ma lian char ac ter is tic and caused this form to be thought of as a reptiloid mam mal. The clas si fi ca -tion of Hopson and Kitching re vised this clas si fi ca tion since they grouped theIctidosauria with the Cynodontia (a group de vel oped mammaloid rep tiles ofthe Perm ian and Tri as sic eras). In the class of rep tiles (Reptilia) we there foremeet the or der of mammaloid rep tiles (Therapsida), the suborder Cynodontia,the fam ily of Tritheledontidea (a group of highly de vel oped, car niv o roussmall cynodonts in clud ing the Ictidosauria), and the ge nus Pachygeneluswhich is the same as the Diarthrognathus broomi of Wat son of the late Tri as -sic (red riverbed and cave sand stone strata in S.A.) (cf. Hopson and Kitching1972:76). The pro vi sional re sult is there fore that we are still dealing with areptile, although a mammaloid reptile.

Widely dif fer ing eval u a tions have been the fate of es pe cially the Archaeop -teryx (dis cov ered al ready in 1861), which has both rep til ian and avian char ac -ter is tics. Al though G.G. Simpson and O.H. Schindewolf largely con cur withre gard to the dis cov ered state of af fairs, they ap proach the fac tual in for ma tionfrom rad i cally di ver gent points of de par ture. Schindewolf is of the opin ionthat the tran si tion from the class of rep tiles to the class mam mals found ex -pres sion in the ap pear ance of Archaeopteryx.1 This an i mal was a bird withwings which could fly, the first rep re sen ta tive of a new class – the Aves(birds). In this re gard Simpson com ments: “Schindewolf dis poses of it by say -ing that it is ‘a true bird’ and so can not close the dis con ti nu ity be tween rep tiles and birds. But if we did not know that Archaeopteryx had feath ers, or if wefound its last feath er less an ces tors, then of course we would have ‘a true rep -tile’. The break can be main tained in words even if it is closed by spec i mens”(Simpson 1960:370, cf. p. 342). M. Grene typ i fies Simpson’s ap proach as fol -lows: “Simpson says Archaeopteryx was a spe cies like any other, orig i nat ingby nor mal speciation from other rep til ian spe cies; only when we look backover the whole vista of evo lu tion do we say, this par tic u lar spe cies was thefirst of what turned out to be a new class” (1974:130). Ex actly in view of thisap par ently in ev i ta ble pres ence of gaps in the paleontological re cord Ste phenGould and his fol low ers have be gun dur ing the past two de cades to scram bleback wards with their the ory of punc tu ated equi lib ria.2

All the sup posed tran si tional forms men tioned above there fore do not qual ifyas true tran si tional forms in the sense of an en tirely con tin u ous evo lu tion ary

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1 Schindewold is a great Ger man pa le on tol o gist, whose main work is Grundfragen derPaläontologie 1950, cf. 1969.

2 As an aside we must men tion just how ma jor a prob lem the de vel op ment of rep til ian scalesinto bird feath ers is for evo lu tion ary the ory. In ad di tion no fos sils have been found which canbe con sid ered as an ces tors of cur rently liv ing birds, while the fourth Archaeopteryx was dis -cov ered in 1956, namely Archaeopteryx lithographica.

pro cess. That is why D.B. Kitts can still say that “Evo lu tion re quires in ter me -di ate forms be tween spe cies and pa le on tol ogy does not pro vide them”(1974:467). He points out that Dar win hoped that con tin u ing fos sil findswould fill the gaps and then re marks: “But most of the gaps are still there acen tury later and some pa le on tol o gists were no lon ger will ing to ex plain themaway geo log i cally” (p.467). This fact gives much room for the ac knowl edg -ment of struc tural dis con ti nu ities in the paleontological re cord – the paleon -tological data is en tirely rec on cil able with the hy po thet i cal pre sup po si tion offun da men tal (rel a tively con stant) types of or der within the plant and an i malrealms which as true law types make variability possible in the first place.

The cur rent eval u a tion of the sup posed tran si tional forms is even more com -pli cated by the ex is tence of liv ing “tran si tional forms.” One ex am ple is thewell-known platy pus of east ern Aus tra lia, Tas ma nia, New Guinea, and theSalawati is lands. Mam ma lian char ac ter is tics (Theria) are de ci sive in theplace ment of these an i mals in the sub class Prototheria and the or der ofMonotremata. The platy pus is one of two fam i lies in the or der ofMonotremata and con sists of only one spe cies of platy pus, namelyOrnithorhynchus anaticus. What is so re mark able about this kind of mam malis that apart from mam ma lian char ac ter is tics (such as the den tal-squa -mosum-joint, enucleate red cor pus cles, pres ence of a di a phragm, only leftaorta, hair, milk glands, three ear os si cles), it does not only have rep til ianchar ac ter is tics (eggs with yolk and shell, no ear mus cle, etc.), but also avianchar ac ter is tics (sim i larly to the platy pus birds lay eggs, have a beak and acloaca into which the in tes tine, urine and gen i ta lia dis charge). As is of ten thecase with liv ing fos sils platypi are highly spe cial ized in cer tain par tic u larchar ac ter is tics while re mark ably show ing no fur ther de vel op men tal trend to -wards ei ther birds or more typ i cal mam mals. Ac cord ing to Eisenstein platypitherefore have a right and potential to existence equal to that of birds and other mammals (1975:251).

In this re gard we must point out fur ther more that var i ous pa le on tol o gists arestruck by the fact that the so-called in ter me di ary forms are by no means trulyin ter me di ate since var i ous typ i cal char ac ter is tics are pres ent in tact next to one an other (com pare Wat son’s com ment on the Seymoria above). Schindewolfre fers to these as mixed types (Mischtypen), while G. de Beer in hon our ofD.M.S. Wat son re fers to the mosaïc fig ure of these forms in terms of the Wat -son rule. This rule states that in the tran si tional area be tween two lev els of de -vel op ment mosaïc fig ures ap pear in which each or gan ap pears to have an evo -lu tion ary tempo of its own and in which the rel e vant char ac ter is tics de velopsharply in de pend ently of one an other (cf. Kuhn-Schnyder 1967:362). Thefor mu la tion of this rule how ever pre sup poses evo lu tion ary tran si tions eventhough the tale of con ti nu ity re mains threat ened since true intermediate formssimply do not fit into the picture it paints.

In this con text a dis tinc tion is made be tween lev els of de vel op ment (Stufen)which must be dis tin guished in phylogenetics from true lines of de scent(Ahnenreihe). In the strict est sense of the word it is im pos si ble to con clu sively and ex actly prove any line of de scent. Even if it is taken into ac count that the

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“pri me val bird” Archaeopteryx is sim i lar to mod ern birds in its pos ses sion offeath ers, in other re gards sim i lar to true rep tiles (e.g. in its pos ses sion of areptiloid tail), and that it ap peared paleontologically at a mo ment which pro -vides a link with com pa ra ble rep tiles (Archaeopteryx is about 30 mil lion years youn ger than the com pa ra ble Pseudosuchier) while oc cur ring rel a tivelyshortly be fore the ap pear ance of birds in the Crus ta cean, there is still no con -clu sive proof that con tem po rary birds have de scended from Archaeopteryx.1

Ac cord ing to Wal ter Zimmerman it is more pos si ble rather to show that cer -tain fos sils are not true an ces tors of later forms. This is true even of what ap -pears to be the most sol idly founded lines of de scent (1967:100). The gen eralcon clu sion reached by Zimmerman is for mu lated by him as fol lows: “In short, the an ces tral and typ i cal lines of phylogeneticists are not only on oc ca sion butal ways what O. Abel on oc ca sion re ferred to as ‘level lines’. The fos sil formswhich we en coun ter in the past, as well as cur rent or gan isms rep re sent in thechar ac ter is tics which in ter est us the de vel op men tal level which the par tic u laran ces tor in volved had reached.”2

The im pli ca tion is that the fun da men tal dis ci pline of evo lu tion ary the ory inthis con text is the phylogenetics of dis tinc tive char ac ter is tics (Merkmals -phylogenie), which pro vides the foun da tion of the sup posed lines of de scent(Sippen-phylogenie) (cf. Zimmerman 1967:103). The con tri bu tion of Dar -win’s orig i nal work, which is con cerned ex actly with the ‘or i gin of spe cies’, is clearly threat ened when A. Meyer draws the rad i cal con clu sion that “There isno phy log eny of spe cies, but a phy log eny of the ty po logi cal characteristics ofthe species” (1964:60).

Be fore we con sider the foun da tional philo soph i cal ques tion of the ba sic de -nom i na tor in mod ern bi o log i cal thought, we must first cast a cur sory glance at the re mark able struc tural interlacement be tween the phys i cal-chem i cal con -stit u ent sub stances of liv ing en ti ties and that which we will de scribe be low asthe liv ing or gan ism of e.g. a liv ing cell. Since the cell is the small est vi a ble en -tity known, it is a good starting point for this reflection.

The structure of a nuclear living cellThe un crit i cal sci en tific use of the term ‘life’ – as if it is a con crete quidity –de nies the modal na ture of the bi otic as pect of re al ity. Af ter all, the ear li estfos sil which ap pears on the paleontological ho ri zon is by no means ‘life’,since al gae (or al gae-like liv ing things), have a bi oti cal as pect amongst otheras pects. If the in ar tic u late prac tice of re fer ring to life as an en tity is con sis -tently car ried to its con clu sions, it would im ply that an en tity must be liv ing inall its ar tic u la tions. This is the con sis tent point of view of the neo-ThomistHoenen. The Ar is to te lian-Thomistic sub stance con cept re quires of Hoenen toreach this con clu sion, since the sub stan tial unity of a liv ing thing would be

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1 Wal ter Zimmerman openly ac knowl edges as much – cf. 1967:100.

2 “Kurz, die Ahnen- und Artenreihen der Phylogenetiker sind nicht nur gelegentlich, sondernstets das, was O Abel ‘Stufenreihen’ genant hat. Die fossilen Formen, die wir in derVergangenheit auffinden, sowie die heutigen Organismen Repräsentieren in den unsinteressierenden Merkmalen die entwichlungsstufe, die damals der betreffende Ahn erreichthat” (1967:102).

sus pended if in de pend ent non-liv ing in gre di ents (with a match ing sub stan tialform) were pres ent therein. In Hoenen’s view the sup posed sub stan tial vi talunity of a liv ing thing could never be a mix ture of life and non-life.

Physical-chemical constituents in the living cellOr ganic chem is try, but es pe cially re cent de vel op ments in bio chem is try, con -clu sively de ter mined over the past three de cades that there are all sorts ofmacro-mo lec u lar ma te rial struc tures pres ent in a liv ing cell. A brief syn op sisof the per ti nent in for ma tion which must be taken into ac count in this regardfollows.The chem i cal com po nents of pro to plasm (nu cleus and cy to plasm) are of pri -mary in ter est. Al though we find ex tremely com plex and la bile or ganic com -pounds in both the nu cleus and the cy to plasm, it is strik ing how rel a tively fewel e ments are used. The main in gre di ents are the four so-called or ganic el e -ments: hy dro gen (H), ox y gen (O), ni tro gen (N) and Car bon (C). Apart fromthese the fol low ing in or ganic el e ments can also be found: phos pho rus, mag -ne sium, cal cium, po tas sium, so dium, sul phur, io dine, iron, co balt, man ga -nese, and zinc. A per cent age in di ca tion of the var i ous com pounds in whichthese el e ments are to be found looks as fol lows: wa ter (on its own and in com -pounds) 85-90%, pro tein 7-10% (al bu min, histone, protamine, and nucleo- pro tein), lipids (e.g. fats) 1-2%, other or ganic substances (carbohydrates)1-1,5%, and inorganic substances 1-1,5%.In most cells col loi dal sys tems are found which rep re sent a mix ture of sub -stances with chem i cal char ac ter is tics mid way be tween true so lu tions and sus -pen sions. These sur faces have an enor mous elec tri cal charge which quicklyreg is ters changes in tem per a ture and elec tri cal charge. A more liq uid sit u a tion is re ferred to as a sol state, and a more solid situation as a gel state.The plasma of al most all cells is cov ered with a three-di men sional net work ofpock ets linked with a sys tem of mem branes. These pock ets ap pear most com -monly in the form of cysts or tubes – thence the so-called al ve o lar sys tem(cysts = alveola).1

In 1896 the Buchners dis cov ered al co holic fer ments which serve a cat a lyticfunc tion in cells, ini tially re ferred to as ‘zymase‘, it grad u ally be came ap par -ent that it is a mix ture of en zymes and co-en zymes.2

Pro tein re fers to macro-mol e cules con sist ing of 20 dif fer ent amino ac ids.When an amino group (NH2) of one amino acid is linked with a car -boxyl-group (COOH) of an other amino-acid, a pep tide bond (NH-CO-) isformed – cou pled with the re lease of wa ter (H2O). Mul ti ple amino ac ids arebonded in this way into a macro-mol e cule – a polypeptide.En zymes have a pro tein struc ture built up out of amino ac ids and oc ca sion allyoc curs in their thou sands in a par tic u lar cell. This pro motes chem i cal re ac -tions in the cell, al though each kind of en zyme ca tal y ses only a lim ited num -ber of re ac tions. En zymes are very sen si tive to ab nor mally high tem per a tures– un like in or ganic cat a lysts, who nor mally per form better un der warmer con -

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1 Of course the mem brane func tions as an or gan of the cell.

2 Co-en zymes are or ganic com pounds which play an es sen tial role in re ac tions cat a lyzed by en -zymes, al though it lacks the pro tein struc ture of enzymes.

di tions. The en tire me tab o lism of the cell depends on the functioning ofenzymes.

In the nu cleus of the cell nu cleo tides are formed through the bond ing of asugar and a ni trog e nous base on the one hand and a phos pho rous acid-rem -nant on the other. In this way polinucleotide chains are formed. In the nu cle -onic acid DNA (desoxyribonucleic acid) four nu cleo tides are found, namelyAd e nine (A), Gua nine (G), Cy to sine (C), and Thy mine (T). These spon ta ne -ously as so ci ate in the links A-T and G-C. Out of this mu tual at trac tionemerges two polinucleotide-strings with var i ous pos si bil i ties. A se ries likeATGACGT is com ple mented by a se ries TACTGCA.

The so-called ge netic code con cerns the rule in terms of which a polipeptidese ries is linked to a given polinucleotide se ries. This link age is made pos si bleby RNA – a nu cle onic acid dif fer ing from DNA in that the T is re placed byU(racil). To trans fer the ma trix of DNA to pro tein it ap pears that a com bi na -tion of three let ters is nec es sary in the DNA for ev ery amino acid to beformed.1 This means that some amino ac ids are cor re lated with more than onetrip let of nu cleo tides – i.e. dif fer ent trip lets are oc ca sion ally at tached withonly one amino acid. The trip lets UAA, UAG, and UGA ap pear to be in op er a -tive, since they are not correlated with any amino acids.

The two strings of nu cle onic ac ids are shaped in a dou ble he lix struc ture (ac -cord ing to the model of Wat son and Crick, 1953), and have the abil ity of du -pli cat ing. When du pli cat ing the two strings come apart and each nu cle o tideat tracts its coun ter part out of the free nu cleo tides pres ent in the en vi ron ment,so that the two new DNA spi rals which come into ex is tence are ex act du pli -cates of the sin gle orig i nal. Due to chem i cal in flu ences, Röntgen- or cos micra di a tion it is pos si ble that one or more nu cleo tides can be added or left out,which change the ge netic in for ma tion of the DNA-mol e cule. This ‘mis take’can then be ex actly cop ied – bring ing about a ‘mu ta tion’. Such mu ta tionscould take the form of changes in sin gle genes, in chro mo somes, or even in anum ber of chromosomes, and almost always has negative consequences.

In view of these neg a tive con se quences of mu ta tions, neo-Dar win ism wasforced to still make use of Dar win’s orig i nal no tion of nat u ral se lec tion. When cli mac tic or other nat u ral con di tions change sig nif i cantly, it is con ceiv ablethat the oth er wise dis ad van taged mu tant mem ber of a spe cies could turn out to be ad van taged un der changed cir cum stances. In this man ner na ture se lectsthose liv ing things which has the better chance of suc cess in the strug gle forsur vival, due to mu ta tions. The ge net i cist Th. Dobzhansky sum ma rizes thisthe ory as fol lows: “Mu ta tion alone, un con trolled by nat u ral se lec tion, couldonly result in degeneration, decay and extinction” (1967:41).

Fur ther non-liv ing in gre di ents of the cell in cludes the al ready men tionedgenes (lo cal ized in the chro mo somes).2 Dur ing cell di vi sion a re duc tion in nu -

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1 There are 20 amino ac ids and if we con sider only two pos si ble com bi na tions of 4 DNA nu -cleo tides only 16 amino ac ids can be ex plained: 42 = 16. The men tioned nu cleo tides of A, G,C, and T can how ever be ar ranged in 64 com bi na tions of three: 43 = 64.

2 Chro mo somes are threads in which the colourable sub stance in the nu cleus of the cell vis i blycon tracts dur ing the pro cess of cell di vi sion (chroma = color).

cle onic plasma takes place, twice di vid ing the gen er a tive cells, while the chro -mo somes only di vide once. This pro cess is called mei o sis.1 Sim i lar in gre di -ents are the hor mones and the ‘rest ing nu cleus’, as well as the vac u oles, which con tains cell-sap and is de lim ited by a mem brane. The lat ter does not oc cur inthe cells of bacteria and blue-green algae.

With re gard to the unique man ner in which the liv ing cell func tions in thephys i cal as pect of re al ity, Karl Trincher2 men tions the fol low ing four mac ro -scopic char ac ter is tics (1985:336):1) spa tial macroscopy which de fines the cell as a spa tially de lim ited sur -

face;2) tem po ral macroscopy, which de ter mines the fi nite time in which the en -

ergy cy cle of the cell oc curs;3) the isothermic na ture of the cell, which is re spon si ble for the con stancy

of tem per a ture through out the cell;4) the per sis tent pos i tive dif fer ence be tween the higher in ter nal tem per a -

ture of the cell and the lower ex ter nal tem per a ture of the en vi ron mentad ja cent to the cell surface.

Organelles – the different organs in the cell

The dif fer ent or gans in the cell can be con sid ered true parts of the cell or gan -ism. Al though the whole-parts re la tion ship is a typ i cally spa tial re la tion ship,it re ceives in all liv ing en ti ties a typ i cally bi oti cal qual i fi ca tion. The dis tinc -tive form char ac ter is tics of cells which emerge in the dif fer ent ways in whichthe cel lu lar nu cleus re lates to its sur round ing cy to plasm, were al ready stud iedand clas si fied in the thir ties of the 20th cen tury by the Ger man bi ol o gist R.Woltereck.3

The cel lu lar nu cleus is gen er ally the (ei ther round or ovoid) site of the DNA,and serves, de spite the mu tual de pend ency of nu cleus and cy to plasm, to ini ti -ate the me tab o lism of the cell. The phe nom e non of dual or multinucleic cellsdoes not in the least di min ish the cen tered ness of the struc ture of cell – inmany pro to zoa this is only a pass ing stage re lated to pro cre ation, serv ing thesame func tion as cel lu lar di vi sion among metazoa. Those pro to zoa dis tin -guished by cilia are known as Ciliophora and have a dou ble nu cleus – amacro- (so matic) and a mi cro- (gen er a tive) nu cleus. Bac te ria and blue-greenal gae have a dif fuse nu cle onic sphere rather than a proper nu cleus.4 Bac te riahave no de fin i tive nucleonic membrane between the cyto- and caryoplasm.

The centriole is a cell or gan pres ent in an i mal and a few lower plant cells. Insome cells the centriole tends to be po si tioned at the geo met ric cen tre of thecell. In gen eral it is how ever dis placed by the nu cleus and cy to plas mic prod -ucts. When a cell is not en gaged in mi to sis, centrioles gen er ally ap pear in

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1 The num ber of chro mo somes are for in stance re duced from 46 to 23.

2 Dept. of med i cal phys i ol ogy, Uni ver sity of Vi enna.

3 Com pare his Grundzüge einer allgemeinen Biologie, in which he dis tin guishes amonghylocentric, morphocentric and (in the case of an i mals) kinocentric struc tures in which thetyp i cal cen tral ity of a cell is ex pressed (1932:323-329).

4 Green al gae of the ge nus Cloadophora have multinucleic cells.

pairs. Be fore ini ti at ing cel lu lar di vi sion centrioles must first themselvesdivide.

Ri bo somes, which are mainly to be found in the cy to plasm, are the main siteof pro tein syn the sis. The ge netic mes sage of the chro mo somes is trans ferredto the RNA of the ri bo somes which is ul ti mately re spon si ble for the pro duc -tion of en zyme pro tein. Lysosomes – a term first used in 1955 – are gran u larsubcellular or gans with an en com pass ing mem brane, and con tain par tic u larhydrolitic en zymes. When a cell is for in stance dam aged, this en zyme is re -leased and breaks down the pro tein and nu cle onic acid so that ad ja cent cellscan use these for the re pair of such dam age as oc curred. In any pro cess of dy -ing (autolysis) cells and tis sue are broken down by means of the lysosomicenzymes.

Cer tain fi brous microtubes of ten play an im por tant role in the for ma tion ofcells, while microfilaments are organelles in volved with the mo bil ity of cells.

Mi to chon dria are gran u lar cell or gans of which the in ner mem brane func tionsto “breathe” – mak ing mi to chon dria the power sta tions of the cell. En ergy infoods (cap tured inter alia by means of the cit ric acid and Krebs cy cles) is re -cap tured by the mi to chon dria and trans formed into adenosine triphosphate(ATP) by means of phosphorilization. In this way en ergy is pro duced for var i -ous cellular functions.

Ba cil lary bac te ria are strik ingly sim i lar in shape to mi to chon dria, which hasprompted sug ges tions that mi to chon dria might ini tially have been in de pend -ent prototrophic cells (cf. the com ments of Roodyn and Wilkie 1968:53-57).This sug ges tion is much relativized by the fact that the func tion of mi to chon -dria is de pend ent on the cen tral and di rec tive func tion of the cellular nucleus.

Af ter 1965 di rect bio chem i cal re search con firmed the uni ver sal pres ence ofDNA in mi to chon dria (MDNA). The ba sic com po si tion of MDNA is moreho mo ge neous than that in the cel lu lar nu cleus, and ap pears to be rel a tively ge -netic self-suf fi cient. Iso lated mi to chon dria can syn the size DNA and MDNAis even passed on the daugh ter cells with out be ing bro ken down. De spite thisap par ent in de pend ence, suf fi cient ev i dence re mains for sub stan tial nu cle oniccon trol over the gen er a tion and collection of the ingredients of mitochondria.

The Golgi-com plex, which is rich in lipids, ap par ently con tains the se cret ingfunc tion in the cell or gan ism. Plastides con tain a pig ment and/or food re -serves, and dif fer sig nif i cantly from one kind of cell to an other, al though it isab sent in bac te ria, blue-green al gae, and fungi. Nu cle oli, dis cov ered byFontana in 1781, have a high pro tein con tent – es pe cially phos pho ric pro tein.Other organelles in clude ‘fagozomes’ and ‘peroxyzomes’.

The quest for a basic denominator

Any con sid er ation of the var i ous schools of thought in mod ern bi ol ogy ismade more dif fi cult by the im mense va ri ety of dis ci plines and huge vol ume of in for ma tion rel e vant to such a con sid er ation. None the less, no school ofthought es capes cer tain fun da men tal struc tural re quire ments which de ter -mines and makes pos si ble sci en tific thought in the first place. We dis cov eredal ready in the first chap ter that sci en tific thought is a par tic u lar kind of

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thought, namely deep ened (dis closed or mo dally ab strac tive) log i cal thought.The key mo ment of the log i cal as pect of our ex pe ri ence of re al ity is to befound in the char ac ter of iden ti fi ca tion and dis tinc tion (iden ti fy ing dis tinc -tion/dis tin guish ing iden ti fi ca tion). A sci en tific, sub jec tively log i cal thought- act is by na ture de pend ent on in for ma tion about re al ity which can be iden ti -fied and dis tin guished, and is as such de ter mined and de lim ited by log i calnorms which must be hon oured in all log i cal think ing ac tiv i ties. The di ver sityof re al ity can not be en com passed by the log i cal as pect of re al ity – sci en tificthought is al ways an en gage ment with a cos mic di ver sity which is trans -logical. There are there fore apart from the log i cal norms for sci en tific thoughtalso cos mo log i cal norms, such as the norm of ex cluded antinomy which de -mands that this diversity be honoured if such thought is not to succumb toantinomy (anti-normativities, in distinction from mere logical contradictions).

It ap pears as if the idea of con ti nu ity in the dom i nant bi o log i cal schools ofthought in it self al ready pro vides a de nom i na tor for con sid er ation. Manyphylogeneticists who ar gue with a con sis tent nomi nal ism would ar gue in thefirst place that classi fi ca tory de lim i ta tions are en tirely ar ti fi cial since the ac -tual line of de scent con sists of a structureless con ti nu ity. Ac cord ing to thesethink ers the rec og ni tion of cer tain struc tural ar range ments (Gefügeord -nungen) would not be in con flict with the nomi nal ist con vic tion with re gard to the ar ti fi ci al ity of par tic u lar de lim i ta tions. W. Zim mer mann com ments in thisre gard that “In phylo gen etic de vel op ment the or i gin of a struc tural ar range -ment and the emer gence of lim its to groups of or gan isms do not co in cide in asin gle phase. One may well rec og nize a struc tural ar range ment while none -the less be ing con vinced as a ‘nomi nal ist’ of the ar ti fi ci al ity of lim its. In linesof de scent (Ahnenreihe) it is en tirely un nec es sary for lim its to ap pear. Thepro cesses bring ing about struc tural ar range ments (with re gard to a co her enceof de scent) and those bring ing about the cur rent lim its among liv ing or gan -isms, may well be mil lions of years re moved from each other. Whoever failsto observe the distinction between these two phases, has not yet graspedphilogenetics” (1967:98).

Al though Schindewolf rec og nizes only in di vid u als as re ally ex ist ing en ti ties,ap par ently in line with the nomi nal ist point of de par ture with re gard to thesup po si tion of a structureless con tin uum he is none the less of the opin ion thatthe types sys tem at i cally dis tin guished on all lev els are not mere ar bi trary fic -tions, but are rather gen eral con cepts founded in ob jec tive fac tual data. He ap -peals to the fact that among liv ing things suc ces sive lev els ex ists in ac cor -dance with the de gree of gen er al ity and the de gree of sim i lar ity, in terms ofwhich it is pos si ble to com par a tively co or di nate and sub or di nate the group -ings of these lay ers in terms of the char ac ter is tic com bi na tions pres ent. Apartfrom this he also ap peals to the transitionless dis con ti nu ities among suchstruc tures.1 On oc ca sion he iden ti fies his thought as ide al is tic mor phol ogy,even though his em pha sis on the tem po ral suc ces sion in the emer gence of or -ga ni za tional forms in the his tory of the earth in di cates a dis tance in prin ci plewith re gard to the meta phys i cal pri me val forms of ide al is tic morphology. On

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1 Cf. the ex po si tion by Ungerer 1966:233 ff.

the other hand Schindewolf remains a hardened opponent of the idea of aphylogenetic systematics.

Ex actly in this re gard a com par i son with Simpson is worth while. Simpsoncon sid ers phylogenetics to be the ba sic dis ci pline of bi ol ogy within which hethen places the evo lu tion ary structureless con ti nu ity (with the even tu ally ar ti -fi cial clas si fi ca tion). Ac cord ing to Schindewolf the more gen eral sys tem aticcat e gory ap pears first, and all dif fer en ti a tion and spe cial iza tion can only takeplace within this cat e gory. Schindewolf makes use of the pre sup po si tion ofdis con tin u ous macro-mu ta tion, the no tion that na ture is able to bring forthtruly new types, which he then elab o rates in his the ory of typostrophismwhich ap peals to paleontologically de ter mined trends. The emer gence of newstruc tural types1 Schindewolf calls typogenesis. In the typ i cal de vel op ment ofdif fer ent lev els typogenesis is gen er ally fol lowed by a a pe riod of steady dif -fer en ti a tion and trans for ma tion, which leads to a di rected (orthogenetic) de -vel op ment of the par tic u lar struc tural type which Schindewolf calls typostasis (the flour ish ing of the type). Even tu ally a period of degeneration and eventual extinction follows – typolysis (cf. Ungerer 1966:235-236).

Conflicting views despite “the same facts”!

M. Grene points out that Simpson and Schindewolf ac cuse each other of es -sen tially the same or sim i lar mis takes, mak ing use of un nec es sary and mys ti -fy ing pre sup po si tions. She be lieves that each ac cepts as prem ise the ne ga tionof the other’s con clu sions – while hardly if at all dif fer ing with re gard to thefacts: “Simpson, wed ding pa le on tol ogy to the sta tis ti cal meth ods of pop u la -tion ge net ics, sees a grad ual change in pop u la tions such that the sharp di vi -sions of tra di tional mor phol ogy be come false. Schindewolf, bas ing his the oryon the log i cal pri or ity of mor phol ogy, con cludes that gradu al ist, sta tis ti calpic ture of neo-Dar win ism is false. To put it very sche mat i cally; Simpson ar -gues: the neo-Dar win ian the ory is true; mor phol ogy im plies that neo-Dar win -ism is not true; there fore mor phol ogy is wrong. Schindewolf ar gues: mor -phol ogy must first be ac cepted as true; mor phol ogy im plies that the neo-Dar -win ian the ory is wrong; there fore the neo-Dar win ian the ory is mis taken. Or to put the mat ter an other way, they agree on the major premise: traditionalmorphology and neo-Darwinism are incompatible” (1974:132).

Re fer ring to the the ory of Schindewolf, D.B. Kitts writes: “It per mits an ex -pla na tion of the fos sil re cord as ad e quate as any other” (1974:469).2 From thedis agree ment be tween Simpson and Schindewolf it is there fore clear thatthere are trends in bi ol ogy em pha siz ing ei ther con ti nu ity or dis con ti nu ity. The No ble lau re ate (1973) Konrad Lorenz re jects the mech a nis tic pos tu late ofcon ti nu ity as sharply: “From events in the atom to those in the his tory of hu -man ity in or ganic as well as or ganic de vel op ments oc cur in leaps. Even though

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1 Ac cord ing to Schindewolf the Archaeopteryx is an ex am ple of such a new struc tural type,since he con sid ers it to be the first ex em plar of a new class of ver te brates, namely birds.

2 Kitts re fers to Simpson in this re gard as fol lows: “Simpson did not pro vide com pel ling sup -port for syn thetic the ory against Schindewolfian or Lamarckian, or any num ber of other the o -ries both evo lu tion ary and non-evo lu tion ary” (1974:468).

some quan ti ta tively sum ma rized pro cesses in this course of events might su -per fi cially ap pear con tin u ous, even tu ally it turns out to be as dis con tin u ous asthe ma jor qual i ta tive changes in or ganic evo lu tion, first clearly un der stood byHegel ...” (1973:186). This sit u a tion im plies that the search for a com mon de -nom i na tor needs to con tinue in this direction: under what denominator is boththis continuity and discontinuity discussed?

Neo-Darwinism

The dom i nant neo-Dar win ist syn thetic evo lu tion ary the ory in prin ci plechooses for a phys i cal ba sic de nom i na tor, even though in creas ing ef forts aremade from this per spec tive to ac count for the qual i ta tive dif fer ences whichemerged in the course of the con tin ual evo lu tion ary pro cess. J. Huxley warnsagainst the “noth ing but” trap into which many evo lu tion ary and nat u ral sci -en tific ex plan a tory tech niques fall: .".. if sex ual im pulse is at the base of love,then love is re garded as noth ing but sex; if it can be shown that man orig i nated from an an i mal, then in all es sen tials he is noth ing but an an i mal. This, I re -peat, is a dan ger ous fal lacy. We have tended to mis un der stand the na ture ofthe dif fer ence be tween our selves and an i mals. We have a way of think ing thatif there is a continuity in time there must be a continuity in quality"(1968:137).

Simpson also dis tin guishes be tween non-bi otic and bi otic lev els (of or ga ni za -tion) and is con vinced that it is pre pos ter ous “to base ... a con cept of sci en tificex pla na tion wholly on the non-bi o log i cal lev els of the hi er ar chy and then toat tempt to ap ply it to the bi o log i cal lev els with out mod i fi ca tion” (1969:8).Any treat ment of this prob lem would ac cord ing to Simpson have to avoid theex tremes of both vi tal ism and ‘physicism’ (p.21). Against an ex tremephysicalist reductionism he openly states: “I think it fair to say that in this re -spect, as truly bi o log i cal in ves ti ga tion and an at tempt to ex plain vi tal phe nom -ena, un mod i fied reductionism has failed” (1969:26). Be cause of this he re -mains con vinced that evo lu tion ary organismal bi ol ogy can not be re duced “toa phi los o phy tak ing ac count only of the phys i cal, non-bi o log i cal as pects ofthe uni verse” (1969:7). Simpson re jects an ex treme reductionism (physical -ism), and speaks of the phys i cal and bi o log i cal as pects of re al ity. Does thismean that he im plies with this distinction an irreducibility in principlebetween the physical and biotical aspects?

Ap par ently not, since when he says that the prin ci ples of evo lu tion ary bi ol ogy (which oth er wise do not con tra dict any thing in phys ics) tran scend the prin ci -ples which can be de duced from non-liv ing at oms and mol e cules, he still adds“but with out be com ing any thing other than nat u ral is tic” (p.7). Only the con -cept of or ga ni za tion in the end in di cates that in which liv ing and non-liv ingthings dif fer: “It is the com plex ity and the kind of struc tural and func tional as -sem bly in liv ing or gan isms that dif fer en ti ate them from non-liv ing sys tems”(1969:7). In Simpson’s view the bi oti cal as pect emerges out of the or ga ni za -tional com plex ity of nat u ral sys tems, which ac tu ally im plies that the term “bi -oti cal as pect” can not be un der stood in the sense of ir re duc ible ontic mode. Al -though not stated in ex treme reductionistic, or un mod i fied reductionistic

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terms, Simpson still de fends a form of physicalism, even if a physicalism inwhich it ap pears as if the differences among various levels of organization aretaken into account.

Vitalism

Only in vi tal ism is a prin ci pled choice made for an al ter na tive ba sic de nom i -na tor, namely the bi oti cal – even though not in the sense of what is re ferred toin this text as an as pect or mo dal ity of re al ity. The fa ther of neo-vi tal ism, Hans Driesch, speaks of an im ma te rial vi tal force (to which he re fers as entelechieor psychoide), which would be far more than just the bi oti cal as pect of re al ity.With out sur ren der ing the va lid ity of the mech a nis tic anal y sis of mat ter, andwith out de ny ing the causal claims of the clas si cal hu man is tic sci en tific idealwith re gard to na ture, Driesch tried to ap ply the con cept of nat u ral law (in justas de ter min is tic a sense) to bi oti cal phe nom ena. In agree ment with DrieschRainer Schu bert-Soldern de fended the vitalistic po si tion with a range of bio -chem i cal ar gu ments. As the func tional and for mal unit of life the ex is tence ofthe cell would ac cord ing to Schu bert-Soldern de pend on the ac tu al iza tion of adou ble po ten tial: “(a) the ‘form’ or or der of the cell, and (b) the chem i cal lawsgov ern ing mol e cules. ... This prin ci ple of or der may be called the ‘ac tive po -ten ti al ity’ of the ma te rial parts” (1962:102). His view of the prin ci ple of or derre turns to Ar is totle: “Hence the Ar is to te lian con cept of entelechy cor re sponds ex actly with the prin ci ple of or der, which we see at work mak ing the cell intoa whole. It is a prin ci ple of whole ness which forms a unity from parts whichwould otherwise go their separate ways. Thus a hologenous system is born”(1962:113).

Where Ar is totle, Thomas Aqui nas, and even Driesch still ac count for in di vid -u al ity in terms of the ma te rial com po nents, Schu bert-Soldern chooses an other way: “Since the form brings about the in di vid u al iza tion of some thing whichpre vi ously had been poli-sub stan tial or poli-in di vid ual, it must be the form,which ex presses the in di vid u al ity, which it self must be the in di vid u al ity”(1959:285). In his view the form of a body “brings about a real en tity with anon-ma te rial char ac ter, con cern ing a sub stance which in its es sence possesses its dynamic character” (1959:286).

Simpson chose the term or ga ni za tion to in di cate the es sen tial dis tinc tive char -ac ter is tic of liv ing things. In neo-vi tal ist cir cles or ga ni za tion is un der stood interms of their par tic u lar un der stand ing of form (or der). The bot a nist E.W.Sinnott, for ex am ple, writes “Uexküll and oth ers have em pha sized this ideaand re gard or ganic form as es sen tially an in de pend ent as pect of an or gan ism,par al lel with its mat ter and en ergy. ... In deed, the con cept of or ga ni za tion assome thing in de pend ent of the in ner and outer en vi ron ment im plies that formmust be a ba sic char ac ter is tic of all liv ing things” (1972:51). Against mech a -nis tic at om ism Sinnott em pha sizes in neo-vi tal ist man ner the dy namic-cre -ative and in di vis i bly con tin u ous form of liv ing things: “Form, ... is chang ingand creative. ... It is a category of being very different from matter” (1963:199).

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The neo-vi tal ist bi ol o gist J. Haas em pha sizes the obe di ence of ev ery liv ingthing in the elab o ra tion of the course of its life to an in her ent law orprogramme, which he pre fers to in di cate as its life plan: “The life plan con -tains as com po nents the blue prints of each of its ex pres sions; the ge netic planfor their suc ces sion; the func tional plan for car ry ing out its ac tiv i ties; the be -hav ioral plan for all its ‘acts’” (1974:336). Life plans have (sim i lar to normsand laws in gen eral) an ideal be ing (ideales Sein) in Haas’s view (p.338), andcan not be ex plained phys i cally-chem i cally: “Phys i cal-chem i cal forces andlaws are in them selves un able to bring forth the struc tures of mean ing whichwe iden tify as the life plan, and even less can it pro duce a non-material bearerof life plans” (1974:355).

Fol low ing the (ide al is tic-mor pho log i cal) Aus trian bot a nist Wil helm Troll (cf. his stan dard text Allgemeine Botanik 1973:19 ff.), Wal ter Heitler speaks of aZentralinstanz which must ex ist in ev ery or gan ism (1976:6). Heitler uses thisex pres sion in the con text of the fol low ing hy poth e sis which he would like tode fend (against a con sis tent physicalism): “The or gan ism has its own laws,which partly dis places the laws of phys ics and chem is try with some thingmore gen eral” (1976:3). He be lieves an im por tant point of de par ture for hisar gu ment to be the fact that nei ther phys ics nor chem is try knows or uses a truecon cept of Ges talt or Ganzheit. The an a lyt i cal treat ment of these sci ences dis -turbs the Ges talt. This hap pens be cause phys i cal anal y sis can only be ex -pressed in the sys tem atic mea sure ments of length, time, weight, and tem per a -ture (the so-called c.g.s. sys tem). Due to this “merely an a lyt i cal meth od ol ogythe laws are dif fer en tial, i.e. it makes di rect state ments only about the be hav -iour of ob jects for im me di ately neigh bour ing points in time and space. Bymeans of in te gra tion one is able to ob tain state ments con cern ing the en tire re -la tion ship (e.g. the form of plan e tary or bits), but these must fol low from thedif fer en tial el e ments” (1976:5). The Ges talt of a cell (or of the paw of a cat)tran scends all the de scrip tive pos si bil i ties of the c.g.s. sys tem. For such de -scrip tions it is not rich enough. Af ter all, if one only used dif fer en tial laws,such as those of phys ics, cells would have to di vide ad in fi ni tum with out theemer gence of a cel lu lar com plex. In these terms the ex pres sion of a cat’s pawis un imag in able (1976:5-6). The cen tral in stance di rect ing the even tual te le o -log i cal ac tiv i ties of liv ing things, is re ferred to by Heitler as the biologischenInstanz, who also spec i fies the following sub-instances (Unter-Instanzen):organs, cells, organelles (1976:16).

Re lated to vi tal ism one finds the or gan is mic bi ol ogy founded by L. vonBertalanffy and de vel oped into a gen eral sys tems the ory in which the termswhole and to tal ity are cen tral, with or ga ni za tion sim i larly func tion ing as a key term. Von Bertalanffy con sid ers the or gan is mic world view to be a step be -yond the math e mat i cal more geometrico ideal and also be yond the mech a nis -tic world view: “First came the de vel op ments of math e mat ics, and cor re -spond ingly phi los o phies af ter the pat tern of math e mat ics – more geometricoac cord ing to Spinoza, Des cartes and other con tem po rar ies. This was fol lowed by the rise of phys ics; clas si cal phys ics found its world-view in mech a nis ticphi los o phy, the play of ma te rial units, the world as chaos ... Lately, bi ol ogy

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and the sci ences of man come to the fore. And here or ga ni za tion ap pears asthe ba sic con cept – an or gan is mic world-view tak ing ac count of those as pectsof re al ity ne glected pre vi ously” (1968:66). M. Beckner else where com mentsthat “Even though in fact many biologists agree with the organismic position,they will say they disagree” (1971:60-61).

Holism

Vi tal ism con sists mainly of an at tempt to ex alt life as an im ma te rial sub stancein flu enc ing as an or der ing form a con stel la tion of mat ter, or elab o rat ing a lifeplan within such a con stel la tion. In the ho lis tic bi ol ogy of A. Meyer how ever,an at tempt is made to place the bi oti cal as pect so cen trally that in prin ci ple thephys i cal can be re duced to bi ol ogy. J. Needham sum ma rizes the po si tion ofMeyer: “Thus Meyer, in his in ter est ing dis cus sion of the con cept of whole -ness, main tains that the fun da men tal con cep tions of phys ics ought to be de -duc ible from the fun da men tal con cep tions of bi ol ogy; the lat ter not be ing re -duc ible to the for mer. Thus en tropy would be, as it were, a spe cial case of bi o -log i cal dis or ga ni za tion; the un cer tainty prin ci ple would fol low from the psy -cho-phys i cal re la tion; and the prin ci ple of rel a tiv ity would be de riv able fromthe re la tion be tween or gan ism and en vi ron ment” (1968: 27 note 34). The keyno tion of ho lism (al ready in tro duced in 1926 by genl. J.C. Smuts), is that ofthe whole (Ganzheit, Greek: to holon). Meyer de fines a whole by draw ing asharp dis tinc tion be tween parts (teilen) and ar tic u la tions (Gliedern): “Ganz -heit ist, was nie aus Teilen besteht, sondern stets in Gliedern ensteht und nurgegliedert existiert” (1949:284). With out go ing into the basic principles ofholistic biology we refer only to Meyer’s evaluation of the construction oftrees of descent.

We in di cated above that phy log eny is not ul ti mately a phy log eny of kinds, but rather of ty po logi cal char ac ter is tics. With the aid of ex ten sive em pir i cal in for -ma tion Meyer for mu lates the fol low ing re mark able ‘ba sic ty po logi cal law’:“There is no group of ex ist ing or gan isms be long ing to any tax o nom i cal cat e -gory of the Nat u ral Sys tem, whose mem bers pos sess all group char ac ters intheir most prim i tive or in their most pro gres sive phases only. Rather are prim -i tive, in ter me di ate and pro gres sive char ac ter phases thus com bined with eachother in each real mem ber of a group that an or gan is mic ho lism suited for liv -ing in any real ex ist ing eco log i cal biotope re sults from it. Forms which pos -sess all their mor pho log i cal char ac ters in their prim i tive or in their pro gres sive phases only are nei ther liv ing holisms nor suited for ex is tence in eco log i calbiotopes and are, there fore, but purely ideal con struc tions. ... There fore, theex is tence of all so-called phylo gen etic trees, which make use of such, al wayshy po thet i cal stem-forms, have be come du bi ous” (1964:59-60). On page 113Meyer writes: “But all these phylogenetic trees begin with purely idealisticconstructions.”

In an ear lier work Meyer com mented “that all of the phylo gen etic tree con -struc tion to date is im pos si ble since it de pends on en tirely uto pian pre-sup po -si tions” (1950:8). How can de vel op ment be imag ined with out trees of de -scent? “Only as dis con tin u ous, quan tum-like (quantenhafte) de vel op ment”

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(1950:12), Meyer re plies. It is there fore nec es sary to con tinue to take into ac -count the poliphyletic or i gin of new kinds: “New types are as a mat ter of factnot po ten tially pres ent in one or more kinds of the func tion ing type, but in itscom bined rep re sen ta tives still on hand. Out of this there sud denly breaksthrough a new type with pri me vally sud den force – pa le on tol o gists rightlyspeak of rev o lu tions, and not ini tially in only one kind which must then de -velop at a snail’s pace, but im me di ately in a wholeness of new kinds andforms” (1950:14).

Meyer is in con clu sion of the opin ion that from the start ing point of the ho lis -tic idea, namely that the a-bio sphere must be con sid ered as a sim pli fi ca tion ofthe bio sphere, a su pe rior di a lec ti cal syn the sis (cf. Hegel) is pos si ble be tweenmech a nism and vi tal ism (1964:162). Against the back ground of his idea of aquan tum-like, dis con tin u ous de vel op ment, Meyer con sid ers phy log eny as the his tory of life through emer gence evo lu tion (cf. 1964:147).

Emergence evolutionism

Emer gence evolutionism in gen eral at tempted to take se ri ously the qual i ta tivedif fer ences which in di cate the irreducibility in prin ci ple of var i ous evo lu tion -ary lev els, while si mul ta neously re tain ing the con vic tion that higher evo lu -tion ary lev els emerged out of lower ones. The great emer gence evo lu tion istsopenly ad mit ted that this po si tion con tained an in ner antinomy. R. Woltereckdoes so in his Ontologie des Lebendigen (1940:300ff.), while M. Polanyiwrites:

“We have reached the point at which we must con front the unspecifiability ofhigher lev els in terms of par tic u lars be long ing to lower lev els, with the factthat the higher lev els have in fact come into ex is tence spon ta ne ously from el e -ments of these lower lev els. How can the emer gent have arisen from par tic u -lars that can not con sti tute it” (1969:393).

Th. Dobzhansky calls ar rival at a new level “evo lu tion ary tran scen dence”(1967:44).

“The flow of evo lu tion ary events is, how ever, not al ways smooth and uni form; it also con tains cri ses and turn ing points which, viewed in ret ro spect, may ap -pear to be breaks of the con ti nu ity. The or i gin of life was one such cri sis, rad i -cal enough to de serve the name of tran scen dence. The or i gin of man was an -other” (1967:50). Al though Dobzhansky him self went as far as to ac knowl -edge that dif fer ent lev els are sub ject to typ i cal laws valid for it, he re mained ofthe opin ion that an irreducibility in prin ci ple of these laws is unnecessary:

“The phe nom ena of the in or ganic, or ganic and hu man lev els are sub ject to dif -fer ent laws pe cu liar to those lev els. It is un nec es sary to as sume any in trin sicirreducibility of these laws, but un prof it able to de scribe the phe nom ena of anover ly ing level in terms of those of the underlying ones” (p.43).

Pan-psychism

In the end the no tion of con ti nu ity relativizes Dobzhansky’s rec og ni tion ofdif fer ent kinds of laws. The ac cep tance of the pos tu late of con ti nu ity does nothow ever nec es sar ily im ply the choice of a phys i cal-chem i cal ba sic de nom i na -

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tor. A re mark able po si tion is taken within the de ter min is tic sci en tific ideal bythe German zoologist Bernard Rensch.

Al though Rensch ac cepts the pos tu late of con ti nu ity of the sci ence ideal, heex plic itly dis tances him self from both the mech a nis tic and vitalistic points ofview (the for mer deals with con ti nu ity in terms of a phys i cal-chem i cal de -nom i na tor and the later in terms of a bi oti cal de nom i na tor). Al though he ac -cepts the va lid ity of the nat u ral sci en tific causal an a lyt i cal method, Rensch re -jects ev ery mo nis tic the o ret i cal pic ture of re al ity which at tempts to re duce allof re al ity to a sin gle prin ci ple. Ac cord ing to him world events are gov erned by mul ti ple ba sic laws: “Depsite all ev i dence in fa vour of the mo nis tic prin ci ple,the pri mal ground of world events is plu ral is tic” (1971:33). Rensch re fers inpar tic u lar to .".. the causal law, uni ver sal con stants, the law of conservation,the principles of symmetry, and the logical laws" (1971:33).

Rensch char ac ter izes his own po si tion as ‘panpsychistic’ and ‘identistic’ –that is, all events are founded by some thing which is nei ther psy chic nor ma te -rial, but which has psy chic and ma te rial char ac ter is tics (1971:159). It im pliescon sid er ing the evo lu tion ary con tin uum in terms of a psy chic ba sic de nom i -na tor. If no dis con ti nu ities ex ist in the evo lu tion ary line of de scent, then lower an i mals, plants, and even the in or ganic sphere should ex hibit cer tain cor re -spond ing “psy chic” com po nents – a con se quence drawn by Rensch: “Ac cord -ing to our pre vi ous find ings and dis cus sions we are jus ti fied in as sum ing ...psy chic (par al lel) pro cesses of some kind in all liv ing be ings” (1959:352).‘Psy chic’ con ti nu ity also bridges the tran si tion from liv ing to non-liv ing:“Here again it is dif fi cult to as sume a sud den or i gin of first psy chic el e mentssome where in this grad ual as cent from non liv ing to liv ing sys tems. It wouldnot be im pos si ble to as cribe ‘psy chic’ com po nents to the realm of in or ganicsys tems also, i.e. to credit nonliving matter with some basic and isolated kindof ‘parallel’ processes” (1959:352).

Rensch be lieves that such a panpsychistic ap proach has the ad van tage of nothav ing to as sume that the psy chic, as some thing ba si cally dis tinc tive from thema te rial, ap peared on our planet at some stage af ter the emer gence of liv ingcrea tures. As a sub sti tute for the as sump tion that psy chic phe nom ena ap -peared sud denly af ter an as tro nomic and geo log i cal pre his tory of mil len nia,Rensch con sid ers it far more con ceiv able and ac cept able to link the evo lu tionof the psy chic to the evo lu tion of the ma te rial (anzufügen), i.e. to as cribe aprotopsychic nature to matter (1969:134-135).

Metabolism as first level of freedom

The mod ern an thro po cen tric or hu man is tic sci ence ideal, emerg ing dur ing the time of Des cartes out of the mod ern hu man quest for au ton o mous free dom(the per son al ity or free dom ideal) as an in stru ment of con trol with the aid ofwhich all of re al ity could be brought in the grip of the nat u ral sci ences, hasthreat ened the hu man is tic free dom ideal from its in cep tion, ex actly be cause aclosed caus ally-de ter mined nat u ral or der leaves no room for gen u ine hu manfree dom. Just as Rensch retroprojects psy chic char ac ter is tics to the realm ofma te rial things, H. Jonas is ‘forced’ in the in ter est of the pri macy of the free -

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dom ideal to ‘re cover’ free dom on the level of the ma te rial: “Our po si tion is inac tual fact that it is pos si ble to ob serve free dom al ready at the level of me tab o -lism – yes, even that it is the first form of free dom” (1973:13). Ac cord ing toJonas “life man i fests this po lar ity in a du ra ble fash ion in the fun da men tal an -tith e sis in be tween which it ex is tence weaves it self: the an tith e sis of ex is tenceand non-existence, of self and world, of form and matter, of freedom andnecessity” (1973:15-16).

A new mechanistic approachSup port ers can still be found in mod ern bi ol ogy of the clas si cal mech a nis ticsci ence ideal which wants to con sider all nat u ral phe nom ena in terms of a ki -ne matic (move ment) de nom i na tor. As we saw in Chap ter 3, this was the typ i -fy ing char ac ter is tic of classical physics.Al though Eisenstein ac knowl edges that sen sory ex pe ri ence pro vides us withqual i ta tively dif fer ent things (in terms of which he de vel ops many cut ting ar -gu ments against the evolutionistic con ti nu ity of de scent), he re mains of theopin ion that the in her ent sci en tific ten dency to wards uni for mity (Vereinheit -lichung) car ries ab stract thought to a level tran scend ing the qual i ta tive ex pres -sions of things, where ev ery thing which ap pears to be qual i ta tive, is re ducedas far as pos si ble to dy namic pro cesses of de grees of speed dif fer ing onlyquan ti ta tively (1972:256): “At the high est level of sci en tific ab strac tion we do not there fore think of things as iso lated, es sen tially dif fer ing existences, but,since they have been brought un der a com mon de nom i na tor, we con siderthings in dis sol u bly linked in the co her ence of uni ver sal mo tion” (p.256).1

Structural diversity founds structureless fantasiesFrom this syn op sis it is clear that mod ern bi o log i cal lit er a ture hosts var i ous di -verse schools of thought. The first facet bring ing about a pro vi sional di vi sionis the prob lem of con ti nu ity and dis con ti nu ity – a con cep tual con tra dis tinc tion orig i nally founded only in the spa tial as pect of re al ity (see the pre vi ous chap -ter in this re gard). Even tu ally this spa tial anal ogy finds a closer spec i fi ca tionin the ac tual de nom i na tor un der which the con cerned bi o log i cal po si tion con -sid ers the iden ti fi able and dis tin guish able di ver sity in re al ity: in the case ofEisenstein un der the clas si cal mech a nis tic de nom i na tor of mo tion; amongsup port ers of the gen eral syn thetic the ory of evo lu tion in prin ci ple un der aphys i cal de nom i na tor in which ap par ent (but not prin ci pled) rec og ni tion isgiven to higher struc tural lev els; in vi tal ism, ho lism, and organicism un der abi oti cal de nom i na tor; in the pan-psychistic identism of Rensch un der a sen si -tive-psy chic de nom i na tor, and in the per son al ity ideal-ori ented thought ofJonas un der the de nom i na tor of free dom. Emer gence evolutionism wanted to

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1 Eisenstein links up with the dy namic the ory of Constantin Brun ner (cf. i.a. Brun ner'sMaterialismus und Idealismus, 2nd im pres sion 1962), which ends up in a quasi-He geli an di a -lec ti cal syn the sis in which all fi nite con tra dic tions are rec on ciled in the in fi nite to tal ity ofso-called ab so lute be ing: “From the higher and en com pass ing per spec tive of dy namic the oryall things in the na ture of things have di verse or i gins and merge in the in fi nite to tal ity. In theend ... all types of ex is tence are equal man i fes ta tions of the one ab so lute be ing” (Eisenstein1975:265). Cf. also the obit u ary ar ti cle which Eisenstein wrote in mem ory of the death ofBrunner in Philosophia Naturalis (1987:346-349).

have its cake and eat it by recognizing both a continuity of descent and a(quantitative) discontinuity of being.

The choice of a de nom i na tor im plies (with cos mo log i cal ne ces sity) that allother fac ets of the di ver sity of re al ity must be re duced to the cho sen de nom i -na tor which as an absolutized per spec tive en com passes as as pects all otherdimensions of reality.

What is par tic u larly strik ing, is that all the di verse ap proaches men tioned con -tinue to be con fronted with the di ver sity of re al ity which can be iden ti fied anddis tin guished. No sin gle un der stand ing of con ti nu ity de nies the dif fer encesamong mat ter, plant, an i mal, and hu man, or the dif fer ences among the as pectsof move ment, the phys i cal, bi oti cal, the sen si tive-psy chic, and the post-psy -chi cal – they sim ply de scribe these dif fer ent fac ets and struc tures as non-es -sen tial since it can ap par ently be re duced to one or an other de nom i na tor. Theba sic ques tion re mains whether this di ver sity of choices in de nom i na tor hasany “ob jec tively fac tual” foun da tion. It can not be de nied that the in her ent di -ver sity in re al ity of fers a point of de par ture for this di ver sity in per spec tives,but the be lief that all of this di ver sity can be re duced to one par tic u lar facetwhich would as ba sic de nom i na tor en com pass all oth ers doubt lessly in di catesfun da men tal the o ret i cal pre sup po si tions – the o ret i cal-philo soph i cal pre sup -po si tions which ex ist since the o ret i cal log i cal thought by na ture re quires anidea of the di ver sity in re al ity, while as the o ret i cal pre sup po si tions them selves be ing di rected and de ter mined by ul tra-the o ret i cal con vic tions. No sin gleperspective in modern biology can be released from one or another centralfoundational motive which determines its course as an ultra-theo reticaldunamis.

While most mod ern bi ol o gists in one way or an other sup port nomi nal ism, it isre mark able to note – as we saw in Chap ter II – that most math e ma ti cians ofour day in prin ci ple re ject nomi nal ism in their dis ci pline. Pla ton ism in math e -mat ics is de scribed by P. Benacerraf and H. Putnam as fol lows: “In gen eral,the platonists will be those who con sider math e mat ics as the dis cov ery oftruths about struc tures which ex ists in de pend ently of the ac tiv ity of thought of math e ma ti cians” (1964:15). Paul Bernays is of the opin ion that the use of pla -ton ism is so com mon in math e mat ics “that it would be no ex ag ger a tion to saythat pla ton ism cur rently dom i nates math e mat ics” (1976:65). It is a pe cu liarsit u a tion that the dom i nant di rec tions in mod ern math e mat ics and mod ern bi -ol ogy are di rectly op posed in terms of their the o ret i cal points of de par ture!We con clude this chap ter by once again re turn ing to the struc tural char ac ter of the cell – in terms of the theory of enkaptic structural wholes which wealready considered in chapter II.

Structural dimensions of the cell – an enkaptic structural whole

A re mark able ten dency among the dif fer ent bi o log i cal ap proaches is that vir -tu ally all of them in their own way speak of liv ing and non-liv ing mat ter. Al -though this ex pres sion means some thing dif fer ent in each case, the us agenone the less re flects un re solved prob lems for each of these points of view.

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For the mech a nis tic (-physicalistic) ap proach ev ery thing is in prin ci ple ma te -rial, phys i cally de ter mined – which im plies that any term which ap peals to thebi oti cal as pect of things is ac tu ally prob lem atic. On the other hand it is vi tal -ism which at tempts to find the es sence of “life” in im ma te rial life plans, ges -talt-ex pres sive fac tors or cen tral in stances. This means that it is also prob lem -atic from that per spec tive to speak of liv ing mat ter – a prob lem Haas ob vi -ously per ceived with his em pha sis of the fact that phys i cal sub stances main -tain their “be ing and func tion” also “af ter their as sim i la tion” into liv ingthings. Un der stand ably Haas him self op poses the us age of “liv ing mat ter” –ac cord ing to him bio chem is try and cell phys i ol ogy knows of no “liv ing mat -ter” with “mys te ri ous vi tal char ac ter is tics” (1968:24). He prefers to speak ofthe material substratum of organisms (1968:20-40).

This ap proach of Haas re jects what he con sid ers to be the “mo nis tic vi tal ism”of Ar is totle. At the same time he draws the con clu sions of his own po si tion:“Or gan isms es sen tially con sist of two dis tinc tive re al i ties, a ma te rial and anon-ma te rial com po nent, there fore hav ing in on to log i cal terms a dualisticconstitution” (1968:39).

At oms, mol e cules and macro-mol e cules are not alive – they are phys i callyqual i fied ma te rial struc tures. Just as Heisenberg’s un cer tainty prin ci ple formthe lower limit to phys i cal de ter mi na tion, N. Bohr for mu lates his so-called bi -o log i cal un cer tainty prin ci ple in his ar ti cles from the thir ties (which Heitleruses in 1976 as point of de par ture), which in di cate the up per limit to phys i calde ter mi nacy. This up per limit ac tu ally in di cates that a biotically qual i fied en -tity such as a cell di rects the func tion ing of its ba sic con sti tu tive sub stances to -wards the ex is tence of the liv ing unit as a whole, which im plies that the ac tualma te rial struc tures – apart from this bi oti cal ser vice abil ity – only co mes inview once the liv ing cell dies. The material substances simply do not have abiotical subject function.

Apart from the four (phys i cal) mac ro scopic char ac ter is tics re ferred to byTrincher (cf. 1985:336) the typ i cal bi oti cal qual i fy ing func tion of the cell isstruc tur ally ex pressed in the phys i cal as pect in the typ i cally cen tered (i.e.biotically or ga nized) man ner in which the cell func tions. Driesch had no brieffor the typ i cal in di vid u al ity of liv ing things, since he was of the opin ion thatthe ma te rial com po nents with or with out entelechie did not dif fer. Fur ther -more he failed to de scribe the in flu ence of the im ma te rial entelechie on thema te rial com po nents of liv ing things oth er wise than in terms ap peal ing to thephys i cal as pect. He could not see that sci en tific con cep tu al iza tion nec es sar ilyuti lizes modal points of en try – even pre fer ring to see entelechie as a sys tem of ne ga tions which could not be de ter mined pos i tively: it is non-spa tial,non-mechanical, indivisible (cf. Sinnott and Haas) and non-energetic (1931:297).

By means of a the ory of the enkaptic struc tural whole this ques tion is placed in a dif fer ent con text. In the first place this struc tural the ory pro vides a pe cu liarper spec tive on the di verse points of view of or ganic chem is try and bio chem is -try. In the na ture of things en tity struc tures and their interlacement tran scendsany par tic u lar sci en tific per spec tive. When or ganic chem is try (from a phys i -

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cal-chem i cal per spec tive) in ves ti gates the na ture of mo lec u lar and crys tal linestruc tures, this does not mean that the biotically di rected na ture of the cell canbe re duced to this. Apart from the ex cre tory prod ucts of liv ing cells, manysub stances are pro duced in cells which have e.g. reg u la tory, in duc tive, or ga -niz ing or cat a lytic func tions, with out these sub stances be ing ex ter nally ex -creted. The at tempt to re veal the struc ture of such sub stances is most com -monly at tempted to day by bio chem ists. Such sub stance struc tures ac tu ally,how ever, fall within the re search ambit of or ganic chem is try, since it re tains aphys i cal di rec tive func tion as macro-mo lec u lar struc tures. Con sider for in -stance the study of the chem i cal struc ture of en zymes, which is to day con sid -ered one of the most dis tinc tive fac ets of bio chem is try – which ac tu ally prin ci -pally falls within the ambit of or ganic chem is try. Bio chem ists would prob a bly pro test such a point of view, since they do not clearly distinguish between thestructure of the mentioned material elements in the cell and their bioticallydirected functions.

As phys i cal-chem i cal sub struc ture the liv ing cell or gan ism these ma te rialbuild ing blocks found in the cell are not en tirely self-en closed, since they re -main en tirely open, dy namic, and la bile through be ing dis closed and madeser vice able to the sub jec tive bi oti cal func tion of the liv ing or gan ism. And it isto wards these biotically dis closed and di rected phys i cal func tions of the sub -stances in the cell which bio chem is try should di rect its in ves ti ga tory ef forts.The typ i cal met a bolic func tions of the cell cer tainly oc cur on the foun da tionof its phys i cal-chem i cal con sti tu tive sub stances, but can nev er the less not bede tached from their dis closed directedness towards the qualifying bioticalaspect of the cell.

Since the mo lec u lar and crys tal line struc ture it self al ready ex hib its the form of an enkaptic struc tural whole, we are in this case deal ing with a com plexenkaptic form. The en tity struc ture of the cell rep re sents a uni lat eral enkapticfoun da tional re la tion ship: with out phys i cal-chem i cal con sti tu tive sub stancesthere could be no cell, with out these sub stances there fore par tic i pat ing in thesub jec tive life func tion which qual i fies the cell. (we must re peat that it wouldthere fore be self-con tra dic tory to speak of mo lec u lar bi ol ogy or a bio-mol e -cule, since both ex pres sions sug gest that phys i cally qual i fied en ti ties at thesame time have an internal biotical qualifying function).

The bio chem i cal con stel la tion there fore be gins ex actly where the fo cus shiftsfrom the mo lec u lar or crys tal line struc tures of the or ganic sub stances to theac tual biotically dis closed and di rected func tions of these sub stances. In thebio chem i cal con stel la tion the es sen tial char ac ter of the so-called or ganic sub -stances is nei ther re voked nor ex cluded, since they are only enkaptically (i.e.with the re ten tion of their in ter nal phys i cal-chem i cal struc ture) made ser vice -able to the typical biotical functions of the cell.

The dis clo sure of or ganic chem is try, which placed bio chem is try as an in de -pend ent dis ci pline in the encyclopaedic co her ence of all the sci ences, at thesame time con firmed in a unique way the philo soph i cal de pend ency of thesepar tic u lar sci ences, since only in close mu tual in ter ac tion with or ganic chem -is try can bio chem is try prop erly ful fil its task. In the same way as the phys i -

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cal-chem i cal struc ture of con sti tu tive sub stances is foun da tional to theirenkaptic (i.e. biotically di rected) func tions, or ganic chem is try ought to befoun da tional to bio chem is try, which should fo cus on the dis closed enkapticfunc tions of the sub stance struc tures which or ganic chem is try re veals. Thisfoun da tional re la tion ship con firms the close interlacement of the structureand functions of the constitutive substances of living things.

Within the con text of the or dered (cen tered) struc ture of the cell, we do how -ever (from a bi oti cal per spec tive) come across the var i ous or gans (organelles)which are true parts of a liv ing whole. Since the cell is built up of non-liv ingma te rial com po nents we can not sim ply say that organelles are parts of the cell. To in di cate the bi oti cal sub jec tiv ity of the cell Dooyeweerd uses the term: cellor gan ism. In other words, the var i ous or gans of the cell are all part of the cellor gan ism. The dif fer ent organelles re ferred to above ex ist only on the foun da -tion of phys i cal-chem i cal con sti tu tive sub stances – this is the mean ing of theuni lat er ally enkaptic foundational relationship.

The cell or gan ism there fore is a spe cif i cally biotically qual i fied struc turewhich can only ex ist on the foun da tion of enkaptically bound phys i cal-chem i -cal con sti tu tive sub stances. Since these sub stances are not them selvesbiotically qual i fied, but none the less func tion in the liv ing cell, we are forcedto dis tin guish a struc tural trio if we wish to ac count for the com plex structureof the living cell.

(i) In the first place there are the phys i cal-chem i cally qual i fied con sti tu tivesub stances which them selves al ready rep re sent enkaptic struc turalwholes.

(ii) Sec ondly we en coun ter the liv ing or gan ism of the cell as a bioticallyqual i fied sub struc ture which can only func tion on the foun da tion of theenkaptically bound sub struc tur al substances.

(iii) Fi nally we find the body of the cell as struc tural node which enkaptically en com passes both the pre vi ously men tioned sub struc tures.

Al though the cell or gan ism is liv ing in all its ar tic u la tions, it can not ex ist with -out the enkaptically con tained sub stances, and can only be re al ized in con se -quence in the enkaptic struc tural whole of the body of the cell. Since the bodyof the cell as enkaptic struc tural whole nec es sar ily also enkaptically en com -passes the non liv ing sub stances, the cell can not be en tirely liv ing. In plantstruc tures the liv ing or gan ism is there fore only a qual i fy ing sub struc ture ofthe liv ing body of the cell, which ex ists in a uni lat eral foun da tional re la tion -ship with its molecular substance structure.

At this point we must again clearly dis tin guish the the ory of the enkapticstruc tural whole from a tra di tional uni ver sal ist ic scheme of a whole withparts. Only with re gard to the cell or gan ism can we speak of a true bioticallyqual i fied whole-parts re la tion ship (the whole cell or gan ism which in its typ i -cal struc tural cen tered ness pos sesses dif fer ent sub-or gans). Phys i cal-chem i -cal con sti tu tive sub stances, which as such can never be biotically qual i fied,can there fore nei ther as such be part of the biotically qual i fied cell or gan ism.It re mains only enkaptically bound in gre di ents in the ac tual parts. Macro-mo -le c u lar and quasi-crys tal line sub stance struc tures re main phys i cal-chem i cally

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qual i fied and can not as such be alive. None the less such sub stance struc turesare pres ent in the body of the cell, since without them the cell organism cannot live.

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Chapter V

Remarks about the mystery of being human1

In the pre vi ous chap ter we have dealt with a few as pects of the prob lems pres -ent in the evo lu tion ist ac count of the or i gin of liv ing en ti ties with spe cial ref -er ence to the as sumed evo lu tion ary tran si tion from the non-liv ing to the liv -ing. Surely, this tran si tion is as dif fi cult as it is cru cial for the whole evo lu tion -ist pic ture of the hu man be ing as an ex ten sion of the an i mal realm. At the same time, in or der to pos tu late the or i gin of the first liv ing en tity, a ‘jump’ isneeded just as big as the one from the level of uni-cel lu lar life to that of be inghu man. Dobzhansky says: “The or i gin of life and the or i gin of man are, un der -stand ably, among the most chal leng ing and also most dif fi cult prob lems ofevo lu tion ary his tory” (1967:459). Lately, the con nec tion be tween the mo lec -u lar level and the hu man level is once more em pha sized by de vel op ments inthe study of the re la tion ship between humans and the anthropoids (cf.Chiarelli, 1985; Schwartz, 1985).

Continuity or discontinuity between the various levels?

The tre men dous dif fer ences be tween the var i ous ‘lev els’ of the evo lu tion ary‘path’ seem to be so im pres sive, that var i ous evo lu tion ists tend, in stead of ad -vo cat ing a sim ple gradu al ist per spec tive, to sup port a more ar tic u lated‘emergentistic’ ap proach. Th. Dobzhansky in tro duces a term bor rowed fromPaul Tillich: evo lu tion ary tran scen dence (1967:44): “The or i gin of life andthe or i gin of man were evo lu tion ary cri ses, turn ing points, actualizations ofnovel forms of be ing. These rad i cal in no va tions can be de scribed as emer -gences, or transcendences, in the evo lu tion ary pro cess” (1967:32, cf. 50).Given this at tempt to ac knowl edge some thing ‘novel’ in the di ver sity of re al -ity, it may still be some what sur pris ing to see the fol low ing state ment of

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1 In what fol lows we leave aside the so-called moral is sues fol low ing from the view thatbe ing hu man fun da men tally does not dif fer from any an i mal. A re mark from Azar willsuf fice in this con text to point at the ob vi ous in con sis ten cies pres ent in these cur rentreductionistic neo-Darwinistic views: “In a word, if Ruse sees no fun da men tal dif fer -ence be tween man and the other an i mals, why should he con demn geno cide? We cer -tainly slaugh ter an i mals ev ery day. If we en joy fi let mig non or fried chicken, why objectto killing people?” (1986:233).

Dobzhansky: “Stated most sim ply, the phe nom ena of the in or ganic, or ganic,and hu man lev els are sub ject to dif fer ent laws pe cu liar to those lev els”(1967:43). This could have been said by anyone working within the traditionof reformational philosophy!

We may even go fur ther and quote a state ment from Simpson em pha siz ing the dif fer ence in kind be tween be ing hu man and be ing an an i mal: “Man has cer -tain ba sic di ag nos tic fea tures which set him off most sharply from any otheran i mal and which have in volved other de vel op ments not only in creas ing thissharp dis tinc tion but also mak ing it an ab so lute dif fer ence in kind and not only a rel a tive dif fer ence of de gree” (1971:271). From this state ment, how ever,one can sense the sub tle emer gent evo lu tion ist un der tones: al though hu manbe ings did evolve from other an i mals, this de vel op ment es tab lished “an ab so -lute dif fer ence in kind”. The same sup po si tion is pres ent in Dobzhansky’swords about “dif fer ent laws pe cu liar to those lev els”, men tioned above(1967:43), since, in the very next sen tence, he pro ceeds by say ing: “It is un -nec es sary to as sume any in trin sic irreducibility of these laws, but un prof it able to de scribe the phenomena of an overlying level in terms of those of theunderlying ones”.

This mode of ex pres sion came to the fore when, in the the 19th cen tury, cer -tain bi ol o gists and phi los o phers could no lon ger find peace of mind in theage-old con tro versy be tween mech a nism and vi tal ism. Ac cord ing toPassmore (1966:269) this con cep tion of ‘emer gence’ prob a bly goes back toG.H. Lewis’s work on “Prob lems of Life and Mind” in 1875. Al though LloydMor gan was not a re al ist, he con tin ued this emer gent evolutionism in hisGifford lec tures of 1923 (‘Emer gent Evo lu tion’) and 1926 (‘Life, Mind andSpirit’). In his “Pro cess and Re al ity”, first pub lished in 1920, A.N. White head also chooses for an emer gent evo lu tion ist ap proach. The Ger man bi ol o gist,Rich ard Woltereck, con tin ues the emergentistic trend in his ‘On tol ogy of theLiv ing’ (1940, 300 ff.). The same ap plies to the well-known Ger man Phi los o -pher of the nat u ral sci ences, Ber nard Bavinck (cf. his work of 1954). The con -tem po rary sys tems phi los o pher, E. Laszlo (1971),1 as well as the phi los o -pher-chem ist, M. Polanyi, also ad here to this tra di tion. The Ger man bi ol o gistWal ter Zimmerman once (im plic itly) for mu lated this idea as fol lows: “With -out any doubt or gan isms to day pos sess a typ i cal na ture dis tinct from all other(non-liv ing – D.S.) things in the world. However, this typical nature emergedthrough evolution” (1962:202-203).

On the one hand, the strik ing dif fer ences be tween dis tinct kinds of en ti tiesseems to be so im pres sive that these think ers have to re spect them by dis tin -guish ing dif fer ent lev els/laws of be ing; on the other hand they still want to up -hold their com mit ment to a con tin u ous line of de scent. In other words, theywant to have it both ways: “ge netic con ti nu ity” and “ex is ten tial dis con ti nu -ity”. What is more, as we have seen ear lier, some of them are well aware of the ten sion be tween these two as pects of their views. Just re call what Woltereck

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1 The fact that Laszlo is di rectly in flu enced by the line from Lloyd Mor gan, S. Al ex an derand A.N. White head, is shown by Pretorius (1986:29-37).

and Polanyi say (Woltereck, 1940:300 ff.): “How can the emer gent havearisen from el e ments that can not con sti tute it?” (Polanyi, 1969:393).1

Al though we have shown that there are still dif fer ent trends of thought pres ent in 20th cen tury bi ol ogy, both evo lu tion ist and non-evolutionistic, those ori -ented to the neo-Darwinistic tra di tion are not at all will ing to even con sider anal ter na tive ap proach. In the In tro duc tion to the pub li ca tion of the Pro ceed ingsof an In ter na tional Con fer ence on Hu man Evo lu tion, Tobias, af ter re fer ring to the pres ence of dif fer ences of opin ion be tween some par tic i pants, writes:“This is per haps a good mo ment to re af firm that noth ing in hu man bi ol ogymakes sense ex cept in the light of the evolutionary concept” (1985:iv).Remark: He pro ceeds: “To speak of the con cept or hy poth e sis or the ory

of evo lu tion is, in turn, of ten seized upon by anti-evo lu tion ists as a sign of weak ness in the evo lu tion ary doc trine. Evo lu tion, they are li a ble tode clare, is only a the ory. Thereby, of course, they are be tray ing theirig no rance of the way in which sci ence works – by the cre at ing of hy -poth e ses, the test ing of them and the re fut ing or con firm ing of them.This ap proach, which is the es sence of the sci en tific method, is no signof weak ness; it is surely the very strength of sci ence”. Ev i dently,Tobias is not at all aware of the fact that he still ad heres to the pres ently out-dated positivistic phi los o phy of sci ence – seem ingly with out anyaware ness or knowl edge about the rev o lu tion in the area of the phi los -o phy of sci ence that took place since the 1960s! It is per haps not far- fetched to re quire that a per son ad her ing to the “evo lu tion ary doc -trine”, with its fun da men tal em pha sis on change, should take no tice ofthe far-reach ing changes af fect ing the area of the phi los o phy of sci -ence dur ing the past two to three de cades – re flected in the work ofPopper, Toulmin, Polanyi, Kuhn, Lakatos, Feyerabend, McMullin,Stegmüller and others.

At this stage of our dis cus sion, we must ask the fol low ing ques tion: if the hu -man be ing is evolutionistically con sid ered to be noth ing but an ex ten sion of(and a higher de vel op ment within) the an i mal realm, what should be made ofthe dis tinc tive fea tures of be ing hu man? Or are there no such fea tures? First of all we shall suc cinctly deal with the avail able fos sil data re lated to the sup -posed evo lu tion ary de scent of hu man kind, and af ter wards dis cuss some of the most prom i nent ‘can di dates’ for being uniquely human characteristics.

Is the fossil-record conclusive?With the an nounce ment of the dis cov ery of the Taung child skull by Ray mond Dart in 1925, des ig nated as Australopithecus africanus, a new pic ture of hu -man or i gins took shape. For some time, how ever, the Pilt down hoax com pli -cated the mat ter. Found in a gravel pit on the Sus sex Downs of Eng land be -tween 1908 and 1913, these re mains, in the words of Tobias “showed the as -ton ish ing com bi na tion of a large-brain cra nium, or rather mod ern as pect, with

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1 We must note in this con text, that the anal y sis of Polanyi in 1967 and in 1968 ex hib its the same ‘emergentistic’ am bi gu ity. With ref er ence to the ideas of Polanyi and Laszlo, Hartalso pays at ten tion to this prob lem: “If or ders of kinds are ir re duc ible, can things of cer -tain kinds still arise from things of other kinds?” (1984:121).

an ape-like jaw bone (now known to have be longed to an orang utan –Lowenstein, this vol ume) and lower ca nine tooth. As long as Pilt down was ac -cepted as gen u ine and con sid ered an an cient hu man pre cur sor, it was im pos si -ble to ac cept that Australopithecus was ancestral to man” (Tobias, 1985a:37).

Remark: The story about the Pilt down “man” is not a good one for thesci en tific re li abil ity of evo lu tion ary sci en tists (cf. Weiner, 1955). Dur -ing the twen ties strong claims were made by prom i nent sci en tists as tothe re li abil ity and be long ing to gether of the jaw and the skull of thePilt down “man” (like the anat o mist, Ar thur Keith, and an thro pol o gistGeorge G. MacCurdy from Yale Uni ver sity). With out ac knowl edg ingat all that this forg ery sim ply showed that evo lu tion ary au thor i ties canfan ta size what they want to find (by ig nor ing what they don’t want torec og nize), Tobias sim ply writes: “When the hoax had been per pe -trated more than 40 years ear lier, its fea tures had been in con for mitywith the then fixed ideas about hu man evo lu tion” (1985a:38). If, at acer tain stage, it was pos si ble for a forg ery to ‘fit’ “then fixed ideas”,how cer tain are we that, at an other stage, we are not the vic tims of a“the o ret i cally forgerous” interpretation ‘fitting’ the then known‘facts/fossils’?

By the early fif ties, ac cord ing to Tobias, al most all ob sta cles to the ac cep tance of the Australopithecus dis ap peared, since it gained pretty well uni ver sal ac -cep tance as a mem ber of the homi nids “and as a ge nus, one of more whosespe cies were on the di rect lin eage of mod ern man” (Tobias, 1985a:38).

In the fif ties and six ties this meant that the evo lu tion ary line pro ceeded fromthe Aus tra lo pith ecines and via the Java- and Pe king Ape-men (cur rently clas -si fied as be long ing to Homo erec tus) to Homo neanderthalensis and to Homosa pi ens (cf. Le Gros Clark, 1964:168). Dur ing the six ties and early sev en tiesL.S.B. Leakey (work ing near Lake Rudolph in East Af rica to gether with hisson Rich ard), dis cov ered a new spe cies, called Homo habilis.1 Sim i lar i tieswith mod ern2 hu man be ings caused Leakey to re ject Homo erec tus as a hu man an ces tor (1970:172). At the same time, he ar gues that one can not see the Aus -tra lo pith ecines as an ces tral to Homo habilis since they were for the greaterpart con tem po rar ies!3

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1 This name was pro posed by Leakey, Tobias and Napier in 1964. The term ‘habilis’means that this crea ture not only was better equipped and more suit able than the Aus tra -lo pith ecines for the us age of tools, but that it was also ca pa ble of mak ing stone tools (cf.Gieseler, 1974:486).

2 The nu mer ous dif fer ences sep a rat ing Australopithecus and Homo habilis from Homosa pi ens are de scribed in de tail by Henke & Rothe (1980:80 ff.). In or der to meet the ob -jec tions of Le Gros Clark to the clas si fi ca tion of Homo habilis sep a rately from the ge nusAustralopithecus, Leakey, Napier and Tobias had to in tro duce a new def i ni tion of thege nus Homo, re ject ing cra nial ca pac ity as a de fin ing fea ture of the ge nus Homo (cf.Leakey & Goodall, 1970:161).

3 We should men tion in this con nec tion that T.C. Bromage men tions the fact that “theteeth of Australopithecus re sem ble the great apes more closely than mod ern Homo”(1985:243).

Per haps the most re mark able find ing in this cat e gory is a skull which wasgiven the reg is tra tion num ber 1470 at the Na tional Mu seum of Kenya. Even -tu ally this skull was clas si fied as be long ing to Homo habilis (cf. Henke &Rothe, 1980:95). Leakey re marks: “af ter its care ful re con struc tion, it is themost com plete spec i men of its type: its cra nium and face are vir tu ally in tact,but the lower jaw (the man di ble) is miss ing” (1978:52). Ac cord ing to the de -scrip tion of this spec i men by Rich ard Leakey in the well-known Jour nal “Na -tional Geo graphic” (June 1973), which es ti mated its age at 2,8 mil lion years,1

it “leaves in ru ins the no tion that all early fos sils can be ar ranged in an or derlyse quence of evo lu tion ary change. It ap pears that there were sev eral dif fer entkinds of early man, some of whom de vel oped larger brains than had been sup -posed” (1973:819).2

If we do not take skull 1470 into ac count, it seems rea son able, for a num ber ofevo lu tion ist pa le on tol o gists, to see a line from the Aus tra lo pith ecines, viaHomo habilis to Homo sa pi ens.3

What is rather as ton ish ing to me, is that no one par tic i pat ing in the con fer enceon Hominid Evo lu tion, or ga nized by Tobias in 1985 in South ern Af rica, evenmen tioned the works of Rich ard Leakey or re ferred to skull 1470! Is it be -cause his in ter pre ta tion rules out the pos si bil ity of link ing the Aus tra lo pith -ecines with the hu man lin eage? In terms of Leakey’s in ter pre ta tion they were,af ter all, con tem po rar ies (cf. 1978:52)! Fur ther more, in Leakey’s case, thespec u la tive com mon an ces tor should be pushed back to at least 14 mil lionyears (Kenaypithecus wickeri – found near Fort Kernan in East Af rica), pro -vid ing the start ing-point for two lines of de vel op ment: (i) the one lead ing toHomo sa pi ens while (ii) the other (in clud ing the Aus tra lo pith ecines) becameextinct (Richard Leakey, 1973:829).

Seem ingly in or der to tran scend these prob lems, some schol ars have re centlybeen fo cus sing their at ten tion in more de tail on the (men tioned) pos si bil i tiesof es tab lish ing re la tion ships be tween hu man be ings and their sup posed rel a -

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1 In his work of a few years later, Rich ard Leakey's ref er ences to the age of skull 1470stick to an age of “close to two mil lion years” (cf. 1978:53).

2 It is note wor thy to note that the erec tus forms, which are es ti mated to be 1 mil lion yearsold (Java- and Pe king ‘Ape-men’), have a cra nial ca pac ity sim i lar to that of skull 1470.The lat ter, how ever, is not only much older than these erec tus forms, but its mor pho log i -cal fea tures are also much more sim i lar to that of Homo sa pi ens. The cra nial vol ume ofskull 1470 is 800cc; that of the erec tus forms var ies be tween 700cc-1100cc; the ear li estfinds of Homo habilis were about 650cc; while that of the Aus tra lo pith ecines is 500cc. Itshould also be noted that skull 1470 lacks the prom i nent eye brow-ridge of Homo erec tus – which shows a closer af fin ity to Homo sa pi ens than the lat ter. The Black Skull (com -pare Faul, 1986:10) throws, ac cord ing to Rich ard Leakey, “cold wa ter on the no tion thatas re cently as 3 mil lion years ago there was only one spe cies (of early man) which gaverise to the oth ers”. The brain of this new ex tremely prim i tive hominid skull is about thesize of a mod ern ape's and less than a third of the size of the hu man brain – indeed thesmallest of any hominid measured to date.

3 Cf. McHenry, 1985:222, where he claims, due to 19 traits in which Australopithecusafarensis re sem bles Homo habilis more closely than any other spe cies of Australo -pithecus, that “this spe cies of Australopithecus is the im me di ate an ces tor of Homo” (cp.Fig ure 2 of Clark 1985:75).

tives on the ba sis of mo lec u lar and chro mo somal ev i dence. How ever, also onthis level we can dis cern se ri ous dif fi cul ties. First of all, Schwartz (1985:268)points out that chro mo somal phylogenies and some mo lec u lar and chro mo -somal ev i dence sup port the re la tion ship be tween the hu man be ing and theorang utan – a per spec tive which is, ac cord ing to him, also con sis tent withmor phol ogy. This means that, ac cord ing to this anal y sis, the large Homi nidsdif fer en ti ate into hu man/urangutan and chim pan zee/go rilla sis ter groups(Schwartz, 1985:268). In the same vol ume, how ever, we read the fol low ingcon clu sion from Chiarelli in con nec tion with a fig ure which shows the num -ber and types of chro mo some mu ta tions de tect able in the karyotype of the dif -fer ent apes com pared to the hu man be ing: “The type and num ber of changes,up to now de tected, dem on strate that the orang utan is the most con ser va tiveand the most un re lated to man, among apes, while the Af ri can apes (especially the chimpanzee) share a number of derived changes with the human karyo -type” (1985:400).

With ref er ence to dif fer ent in ves ti ga tions, these two schol ars in deed reach di -rectly op po site con clu sions: the first one re lates hu mans to the orang utan (ex -plic itly re ject ing the chim pan zee as a can di date), and the sec ond one re latesthem to the chimpanzee!

Immuno-bi o log i cal ev i dence (blood an ti gen stud ies) and pro tein homologiespro vide an other in di rect way to re late hu mans and an i mals. Nev er the less,both the di rect and the in di rect meth ods of anal y sis and com par i son only giverise to what Henke & Rothe in di cate as a ‘Sim i lar ity-phenogram’: “Since bio -chem i cal anal y ses do not pro vide the time fac tor nec es sary for any con struc -tion of a phylo gen etic tree” all “at tempts un til now, try ing to es tab lish phylo -gen etic trees on the ba sis of bio chem i cal ev i dence, are not sat is fac tory in view of the nu mer ous and not yet proven pre sup po si tions made in con nec tion withthe tempo of evo lu tion in the mo lec u lar field” (1980:17). What is even moreim por tant, is that they “show im por tant de vi a tions from those phylo gen etictrees which are constructed on the basis of morphological criteria” (Henke &Rothe, 1980:17).

There are even well-known and im por tant sholars which deny the justifia -bility to work at all with a ge netic mode of ex pres sion in pa le on tol ogy and inthe con struc tion of phylogenentic trees. Schindewolf states that the in tro duc -tion of a ge netic rea son ing in phy log eny is not jus ti fied sim ply be cause all thenec es sary pre sup po si tions are ab sent (1969:69). He also re jects Simpson’sno tion of “quan tum evo lu tion” (ex plo sive de vel op ment), since we have nocer tain knowl edge about the adap tive zones or the “ev ery thing-or- noth ing- re -ac tions” (1969:69).

The cru cial point in men tion ing these data and dif fer ences of opin ion, is toshow that there are ex treme dif fi cul ties and prob lems pres ent in the at tempt tocome to a co her ent and ra tio nally jus ti fied pic ture of hu man or i gins even ifone ac cepts the as sump tions of neo-Darwinism.Remark: That the o ret i cal pre sup po si tions are in ev i ta bly part and par cel

of the sci ence of pa le on tol ogy and the con struc tion of phylo gen etictrees (just com pare Grene’s anal y sis of the rad i cal op po si tion be tween

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Simpson and Schindewolf, 1974:130), is ex plic itly con ceded bySchwartz in the fi nal para graph of his men tioned ar ti cle: “So phis ti -cated tech nol ogy does not pro vide more ac cu rate phylogenies thancon ven tional means. Phylo gen etic in ter pre ta tion is ul ti mately a re flec -tion of the the o ret i cal pre dis po si tion of the in ves ti ga tor” (1985:268).The bi ol o gist, P. Overhage, goes even fur ther by em pha siz ing thatsuch an es sen tial and pen e trat ing ques tion as that con cern ing the or i gin of hu man be ings, by its very na ture, reaches into the sphere of ourworld and life view. There fore, also the an swers given to ques tionslike these are nec es sar ily co-de ter mined by pre-sup po si tions andpre-de ci sions which are non-sci en tific in na ture. Es pe cially nat u ralsci en tists mis led many with their sup posed ‘ob jec tiv ity’ and ‘unpre -judiced ness’ by ac cus ing al ter na tive con cep tions of evo lu tion as be ing re stricted by a world and life view. Pre cisely these con vic tions, how -ever, make it very dif fi cult for these sci en tists to re al ize that mostly theop po site is the case. So many di verg ing in ter pre ta tions of fos sil find -ings and so many dif fer ences in the eval u a tion of phylo gen etic co her -ences, evinced fore most in the “trees of de scent”, are not ex plain ablepurely in terms of the cur rent state of af fairs (A. Meyer straight for -wardly dis qual i fies “all these phylo gen etic trees” which pro ceed from“purely ide al is tic con struc tions”, 1964:113, cf.59-60). Much rather, itmakes an ap peal to fun da men tal con vic tions and sup po si tions whichin flu ence the ory con struc tion from the un der ly ing philo soph i cal andworld-and-life-view at ti tude, as well as from the tra di tion withinwhich the sci en tist is work ing (1959:287). To il lus trate this point wemen tion some dif fer ences of opin ion re gard ing Homo habilis.Whereas Clarke (1985:296) em phat i cally claims that “all in di ca tionsare that Homo habilis prob a bly de vel oped into Homo erec tus sometime be fore 1.5 m.y.”, Jelínek ar gues that the dif fer ence from Homosa pi ens to Homo erec tus is not on the spe cies level, but on the sub spe -cies level, im ply ing that the cor rect name should be Homo sa pi enserec tus (1985:345). Aguirre also writes: “The sep a ra tion be tweenHomo sa pi ens and ‘Homo erec tus’ van ishes. The au thors pro pose thatall pop u la tions from the Far East, Af rica and Europe, currentlyreferred to as ‘Homo erectus’, should be considered Homo sapiens”(1985:328).

One of the cru cial ques tions is whether we can re ally rely on an a tom i cal andmor pho log i cal stud ies to ex plain the dif fer ences be tween hu mans and theirsup posed Hominid an ces tors.1 It fre quently hap pens that re course is taken tothe pres ence of tools in or der to de ter mine the hu man na ture of fos sil find ings. But if we con sider ar chae o log i cal ev i dence as an aid to in ter pret fos sil find -ings, are we still work ing within the frame work of paleo-bi ol ogy? Schinde -

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1 In pass ing we may note the strik ing cir cu lar ity pres ent in some of the ar gu ments evo lu -tion ists use to ‘prove’ the ex is tence of a spe cific or gan in terms of its se lec tive value.Gehlen points out (1971:124) that the use ful ness of ev ery ex ist ing hu man abil ity servesas a proof of its se lec tive value: the ex is tence serves as the proof, in stead of show ing (aswas en vis aged), that a spe cific abil ity emerged as a result of its selective value!

wolf warns us that ob vi ously the pa le on tol o gist should ‘dis re gard’ the “tech -ni cal and cul tural achieve ments of man” be cause con sid er ing them wouldtake us “out side a bi o log i cal ap proach” (1969:67). Seem ingly with out be ingaware of the fact that they are tran scend ing the lim its of bi o log i cal re search, as the ar chae ol o gist Narr es tab lishes, even schol ars in clined to fol low a nat u ralsci en tific ap proach now once more start look ing for the line be tween hu mansand an i mals where signs of the typical human spirituality are seen in culturalactivities (1959:393).

The Swiss bi ol o gist, Portmann, warns that, in or der to get a better un der stand -ing of the or i gin of hu man kind, we should dis pense of the un war ranted andun proven as sump tion that hu man spir i tu al ity is a late phe nom e non in the de -vel op ment of the hu man body. If this as sump tion is re jected, how ever, and hu -man na ture is con sid ered in its to tal ity, then the dis tance be tween the hu manbe ing and an i mals will come to the fore in its full mag ni tude (1965:57-58). Tothis we may add his ac knowl edge ment of the fact that his own in ves ti ga tionsinto the ontogenetic unique ness of hu man kind are “guided by the con vic tionthat which can bi o log i cally be grasped is es sen tially co-de ter mined by thoseas pects of hu man kind, which have to be in ves ti gated with meth ods dif fer entfrom those em ployed by the ex per i men tal bi ol o gist” (1969:23-24). The an -thro pol o gist, A. Gehlen, also points out that a to tal view on be ing hu man func -tions as the guid ing philo soph i cal view-point in his re search – and this to -tal-view can not be de duced from the view-point of any spe cial sci ence(1971:13). In one of his ear lier works, P. Overhage dis plays a sim i lar sen si tiv -ity: “To re duce the whole ques tion about the hu man or i gins sim ply to the bi -oti cal-bodily (mor pho log i cal-an a tom i cal) facet, wit nesses an astonishinglyone-sided approach and imply a radical simplification of the total depth of theproblem” (1959a:5).

Is there anything distinctive to human tools?

Orig i nally it was thought that the hu man be ing is the only crea ture that can usetools. When it turned out that an i mals are also ca pa ble of do ing this, Overhage em pha sizes the hu man abil ity to pro duce tools (1973:359). We have men -tioned the fact that the name Homo habilis was in tro duced to in di cate that thisspe cies was able not only to use tools, but also to pro duce them (cf. Gieseler,1984:486). Al though Y. Coppens tries to as cribe some of the old est flakedstone tools of Omo to the Austrolopithecines, Jelínek says that the “whole sit -u a tion is still far from clear” (1985:343, cf. Clarke, 1985:287). Ac cord ing tohim, ar chae ol o gists do ac cept the view that stone tools can be up to two mil -lion years old (1985:343, or even 2.6 mil lion years – cf. Narr, 1974:107).Early Acheulean ar ti facts, pos si bly about 1.6 mil lion years of age, are as so ci -ated with Homo habilis (Clarke, 1985:297).

Jane Goodall ob served that Chim pan zees are able to con struct two kinds oftools:

(i) “they crushed wads of fresh leaves lightly be tween their teeth to in crease their ab sor bent qual ity and then dipped such a wad in wa ter to use as asponge” and

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(ii) “they pre pared slim sticks or the stalks of coarse grasses and used theseob jects to probe into ter mites’ nests. The ter mites would sieze the in trud -ing ob jects in their jaws and then be pulled out to be eaten by the chim -pan zee” (Reed, 1985:90-91).

Some times the chim pan zees would even pre pare their ter mite-sticks be forethey are used at the mound, an abil ity un til then thought to be uniquely hu man. Reed re marks that the cur rent us age of the term ‘tool’ sim ply in di cated an “ob -ject pur posely made and used in a cer tain way” (1985:91).

This definiton caused new prob lems. If a ter mite-stick is a tool, why can’t thesame be said about the nest of a chim pan zee? Beck pro posed that to be a tool,in the words used by Reed in his ex po si tion, “an ob ject must be free of anyfixed con nec tion with the sub strate, must be out side the user’s body at thetime of use, and the user must hold or carry the tool dur ing or just prior to use”(1985:92).

Reed him self opts for a new ap proach. He starts from the phys i cal level whereen ergy and its uti li za tion are the key fac tors. Thus he re verses the eval u a tionof tools. In stead of start ing with a con tem pla tion of hu man kind and its works,the evo lu tion ary per spec tive re quires that one should start with what Reedcalls the “pri mary en ergy-traps”, such as the cel lu lar or pro to plas mic part ofany liv ing be ing (1985:93). Sec ond ary en ergy-traps are A) pro to plas mic se -cre tions – their use as en ergy-traps is ex tra-cel lu lar (ei ther used within or out -side the body); B) hab i tat fea tures; and C) the use of other or gan isms (sym bi o -sis). Tools are de fined as the forth type of sec ond ary en ergy-traps: “A tool is apar tic u lar kind of sec ond ary en ergy-trap, an ob ject or a con trolled chem i calpro cess (fire for ex am ple) in the production of which the environment ismodified” (1985:95).

How wide this def i ni tion ac tu ally is, is seen from the fact that in terms of iteven one-celled an i mals (from the pro to zoan fam ily Difflugiidae) are‘tool-mak ing’. Reed pro ceeds: “Yes, one group of one-celled an i mals maketools, and so do thou sands of other kinds of an i mals; all nests are tools, and soare the bur rows pro duced by phys i cal re moval of sub strate (al ter ation of theen vi ron ment). .... Tool-mak ing is and has been of two kinds, in stinc tive andcul tural. The lat ter, the learn ing anew of tool-mak ing by each in di vid ual ofeach gen er a tion, rep re sents the ac tiv i ties of only a few kinds of pri mates andperhaps elephants” (1985:96-97).

In the fi nal anal y sis, a dis tinc tion is in tro duced be tween in stinc tive and cul -tural, with out pay ing much at ten tion to the way in which the lat ter is de fined.Are there re ally no cri te ria avail able to dis tin guish be tween an i mal tool- mak -ing and human tool-making?

Henk Hart is con vinced that, in this con nec tion, we may dis cern a low levelcat e gory of “for ma tive con trol”: “The sen si tiv ity of lower an i mals found indrive and in stinct is opened up in higher an i mals to con scious and pur pos ively di rected be hav iors which dif fer in prin ci ple from the me chan i cal and au to -mated struc tures of be hav ior” (1984:181). To sup port this view, he men tionsnests, ant hills and bea ver dams, which “re quire skill and con trol in the mold -ing and shap ing of ma te ri als” (1984:179). Ex am ples of “re source ful be hav -

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ior/cre ative be hav ior” are men tioned to sub stan ti ate these mould ing and con -trol ling skills of an i mals fur ther (Hart, 1984:180). Reed holds the op po siteopin ion: “Oth er wise all tool-mak ing is in stinc tive, even the seem ingly pur -pos ive and com plex hive-build ing by some bees and wasps and dam-buildingby weavers” (1985:97).

It seems to me that Hart does not dis tin guish suf fi ciently be tween sen si tive in -tel li gence and ra tio nal in tel li gence on the one hand, and be tween the dif fer -ences pres ent in an i mal and hu man tool-mak ing on the other hand. We shallre turn to the first dis tinc tion pres ently. At the mo ment we want to fo cus our at -ten tion on the unique ness of human tool-making.

In the first phase of the paleolithicum (i.e., the early stone age), Von Königs -wald claims to see ev i dence of a true in ven tion (1968:167). Narr is more ar tic -u late, since he dis tin guishes three cri te ria which de mar cate and qual ify typ i cal hu man tools (cf. Narr, 1973:61-62, 1974:105-107):

a) The form of the tool should not be sug gested: The dis tinc tive fea tures ofthe kind of tool in mak ing should not be ex em pli fied (vorgebildet) by the form of the raw ma te rial – for ex am ple a stick which could only be freedfrom ob struct ing branches and leaves. The fi nal prod uct should still be‘con cealed’ within the raw ma te rial. It amounts to ab stract ing the form“to be brought to the fore” from what is given to the senses.

b) The func tion should not be sug gested: Tools are not pro jec tions of hu -man or gans. They are not to be seen as a strength en ing, el e va tion or ex -ten sion of bodily or gans. Think about a chop ping stone which strength -ens the fist, or about a stick ex tend ing the reach of the arm or the fin gers.When a tool is used to cut, it is per form ing a novel func tion which is notsug gested by the func tion of any of our bodily or gans (this func tion ofcut ting must be dis tin guished from scratch ing with the nails or from tear -ing apart with the teeth). In this sense, tools are the prod uct of gen u inein ven tions in the con text of cre at ing a new prin ci ple of technics and ma -nip u la tion on the ba sis of a true in sight into the nature and relationshipsbetween things.

c) The way of pro duc tion should not be sug gested: Tools should not beman u fac tured sim ply by us ing the nat u ral or gans of the body (hands,teeth). It must be cre ated with the aid of ex ist ing (for in stance, chop ping) tools, al though it is not strictly nec es sary that the lat ter them selvesshould rep re sent artificial products.

Note that these cri te ria deal with ob jects which are qual i fied by the for ma tive(cul tural) mode of re al ity, i.e. tech ni cal tools, which must be dis tin guishedfrom other cul tural ob jects which qual i fied by dif fer ent (non-for ma tive) func -tions (such as mu si cal in stru ments – aes thet i cally qual i fied; money – eco nom -i cally qual i fied; and son on). Clearly, these cri te ria ex plic itly pre sup pose ourtyp i cal and uniquely hu man freely vary ing con trol guided by our for ma tivefan tasy. Kant de fined our fan tasy as the abil ity to rep re sent an ob ject with outits pres ence to the senses (1781:B151). Narr goes fur ther by em pha siz ing that the hu man for ma tive fan tasy must be able to in vent some thing dif fer ent fromwhat is pres ent to the senses. Fur ther more, he also re quires that truly hu man

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tools must be made with the aid of (formed or un formed) tools. Even the mak -ing of sim ple stone tools as such re quires “tool-mak ing tools” (“dasWerkzeug zum Werkzeugherstellen”): “In this we see a trait tran scend ing theknown and ex pected be hav iour of an i mals: It pre sup poses pos si bil i ties andachieve ments which we may view as essentially and specifically human innature” (Narr, 1973:62).

An other way of for mu lat ing this per spec tive is to say that it is typ i cal of themost ba sic hu man tools that their ‘end’ is to be a ‘means’! They are formed inor der to pro duce some thing else. This ap proach is on a par with the sys tem aticchar ac ter iza tion Van Riessen gives for a hu man tool: it is his tor i cally (cul tur -ally) founded and qual i fied (1948:509). Schuurman con tin ues this clas si fi ca -tion in terms of a cul tural foun da tional and qual i fy ing func tion: “All tech ni calob jects are ex cep tional in the sense that both their foun da tional and qual i fy ing func tion are cultural or technical in nature” (1972:16).

In stead of re fer ring to the for ma tive or cul tural as pect of re al ity, we pre fer tospeak about the tech ni cal as pect. The struc tural con di tions of this modal as -pect re quire the ac count able free dom and in ven tive imag i na tion of the hu manbe ing, the only crea ture ca pa ble of act ing re spon si bly within the ma trix ofnor ma tive con di tions. There fore, I can see no rea son to ac cept that both thehu man be ing and an i mals sub jec tively func tion in this as pect of re al ity, as it isclaimed by Hart (1984:179 ff.). In fact, the cri te rion he uses is just as broadand non-dis tinc tive as the def i ni tion which Reed pro vides for tools as sec ond -ary en ergy-traps (Reed, 1985:95). This def i ni tion en ables Reed to in ter pretthe form-prod ucts down to the level of uni cel lu lar an i mals as tools. The onlydis tinc tion which he em ploys on the ba sis of his def i ni tion is that be tween “in -stinc tive and cul tural” tool-mak ing (1985:96-97). How ever, what he con sid -ers to be ‘cul tural’ in cludes both the unique tech ni cal abil i ties of hu man kindand the ex am ples of an i mal be hav iour which show that there are an i malswhich are not to tally de ter mined by their in stincts. How ever, this non-in stinc -tive di men sion of an i mal be hav iour (some times re ferred to as sen si tive in tel li -gence) should not be con fused with the tech ni cal-formative modeencompassing all truly cultural activities of the human being.

Henk Hart does ex plic itly speak about an i mals as “sen tient crea ture(s)”(1984:180), but he does not want to ex plain the in stinc tive for ma tions of an i -mals as for ma tive. In or der to qual ify as ‘for ma tive’ he re quires ‘re source ful’or ‘cre ative’ be hav iour (1985:180). His ex am ples, nev er the less, do not showthat an i mals are ca pa ble of per form ing tasks evinc ing a re spon si ble, nor ma -tively qual i fied tech ni cal in ven tive ness. It merely shows that an i mals can actin sen si tively in tel li gent ways. Else where he cor rectly states that all tools are“tech ni cal ob jects” which are “shaped and designed according to a humanplan” (1984:239).

By con sid er ing the tool-mak ing abil i ties of hu man be ings and an i mals, onemore and more co mes un der the im pres sion that the re la tion ship be tween an a -tom i cal and mor pho log i cal fea tures on the one hand, and par tic u lar be hav -ioural pat terns on the other hand, are not to be seen in a strictly one-to-one cor -re la tion. Narr con sid ers it to be one of the most im por tant re sults of con tem po -

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rary ethol ogy that it showed that close ness in terms of the zoo log i cal sys temdoes not guar an tee sim i lar i ties in be hav iour. The re verse is also true: in Pri -mates which are sys tem at i cally wide apart from each other, correspondingbehaviour does occur (1974: 109)!

If we add this in sight to the un cer tain ties at tached to the fos sil ev i dence con -cern ing hu man or i gins, it does not seem un rea son able to sup port the “doctaignoratia” pro posed by bi ol o gists such as Haas, Overhage and Portmann.1

Just com pare the fol low ing re mark able statement of Overhage:

“Even if, more or less ex actly, the fos sils could be shown which rep re sent thesta dia through which the hu man body passed dur ing a long pro cess of evo lu -tion, the prob lem of the or i gin and the phylo gen etic be com ing of hu man itywould still not be solved. What ever the value of this knowl edge may be, it pro -vides only a par tial per spec tive on the gen e sis of hu man kind, since it can noten com pass the to tal life-form of be ing hu man. Be cause, in the fos sil find ings,the bound aries be tween the bodily mor pho log i cal fea tures of be ing hu man and that of an an i mal fade away, and since the spir i tu al ity of be ing hu man does notex press it self univocally in the mere form, we no lon ger have any cer taintyabout the ques tion whether so matic char ac ter is tics – such as the erect gait, thefree hand and the cra nial ca pac ity (all of which are ex actly de ter mi na ble infos sil find ings) – are stand ing in a strict cor re la tion to the spir i tu ally stampedbe hav iour of the hu man be ing, such that it al lows us to make claims about thena ture and way in which the pri mates ex pe ri enced and viewed the world. ....The rid dle of hu man gen e sis is not solvable simply by transforming andrecombining animal proportions” (1973:374-375).

Do animals share the dimension of (human) logicality?

The fa mous Ger man zo ol o gist, Ber nard Rensch, ex ten sively ar gues, with nu -mer ous ref er ences to ex per i men tal data, that we must as sume the pres ence of“a-ver bal con cepts” (1971:9, 197, 242, 245), for in stance in the an thro poidapes. He does qual ify this mode of re fer ring to an i mal be hav iour im me di atelyby stat ing that the in fer ence of psy chi cal pro cesses of any kind in an i malsmust be for mu lated in an “as if” way: “it is more cor rect to say that chim pan -zees act as if they were ab stract ing and gen er al iz ing, as if they had formeda-ver bal con cepts (i.e., con cepts not con nected with words), and as if theywere pro ceed ing with in ten tion and fore sight” (1971:242; cf. the con trast ingview of Koehler, 1973:199). This should be kept in mind when he dis cerns“con cepts of value”, a “more or less dis tinct con cept of the self” (243) andeven “prim i tive aes thetic feel ings” (244) pres ent in an thro poid apes. Ex per i -ments done with cats show that they are also able to make a choice be tweenwhat Rensch calls “two pat terns cor re spond ing to an ab stract con cept of ‘un -like and like’” (245). These achieve ments are not so much de pend ent on thede vel op ment of the cor tex of the forebrain. With their forebrain only slightlyde vel oped, even fishes can “learn to grasp the sig nif i cance of up to five pairsof pat terns, and they can re tain a pair of patterns for some months andrecognize them even when these are considerably altered” (246).

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1 Portmann also stresses the fact that it is no lon ger ten a ble to up hold the con vic tion of adi rect de scent of hu man be ings from the an thro poids (1977:461).

Clearly, the way in which sen tient crea tures ori ent them selves in the worlddoes show many sim i lar i ties with hu man be hav iour. The log i cal point, how -ever, is that all sim i lar i ties im ply and pre sup pose dif fer ences. In the ab senceof dif fer ences, we meet iden tity, not sim i lar i ties! What are the dif fer ences, ifany, in this con text of be hav ioural sim i lar i ties? Is it enough to say that we “are not just sen si tive to goals and aware of con se quences” since “we fore see themand plan ahead” (cf. Hart, 1984:181)?

If it is pos si ble to show that an i mals can lo cate sim i lar en ti ties and af ter wardsact ac cord ingly, are we then jus ti fied to con clude that they have a-ver bal con -cepts? What do we mean with the term: con cept? This ques tion be comes more ur gent when Rensch stresses that his use of the term ‘a-ver bal’ is meant to em -pha size that “a-ver bal con cepts” in an i mals do not pro ceed from log i cal op er -a tions (1973:118). For this rea son, he claims that the trait de ter min ing the gapbe tween hu mans and the an thro poids is log i cal think ing (1968:147). Renschsays that al though an i mals ap proach causal ‘con cepts’ (con cern ing re la tionsin dif fer ent sit u a tions), hu man kind is tran scend ing it in its unique abil ity tograsp truly log i cal re la tions, ex pressed in concepts like “as a result of”,‘because’, “in case of”, and so on.

Cassirer ar gues that the de ter mi na tion of a con cept as “unity in mul ti plic ity”be longs to the clas si cal leg acy of logic and phi los o phy as such (1929:339).What ever is log i cally grasped can not fully pre scribe in which way the mul ti -plic ity of fea tures should be united in the unity of a con cept (Cassirer,1928:134), be cause it is also a re sult of the real cre ative el e ment in our think -ing: the power to dis cern/ob serve.1 Log i cal con cept for ma tion is al waysaimed at dis cern ing the mul ti plic ity of uni ver sal traits of that which is con cep -tu al ized. As such it is sub ject to uni ver sal (modal) log i cal norms, such as theprin ci ples of iden tity and non-con tra dic tion. Al though the con struc tion ofeach con cept is de pend ent on log i cal sub jec tiv ity - since only a hu man be ingis able to re spond with nor ma tive free dom to these nor ma tive con di tions forlogi cality - no concept is exclusively the product of our subjective logicalfunctioning.

Leakey’s an nounce ment that the “abil ity to see com mon al i ties be tween ob -jects of the same type – classes such as trees, fruit, pred a tors, birds, etc – is acru cial step in cre at ing con cep tual or der in what oth er wise might be an over -whelm ing per cep tual chaos” (1978:204) as cribes, in a typ i cal Kantian fash -ion, a truly for mal cre ativ ity to con cept for ma tion. The ca pac ity of the an thro -poids to rec og nize per ceived ob jects (in an i mals lim ited to a small num ber ofthem) and to as so ci ate them with each other (even in dif fer ent con texts), does

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1 It is dif fi cult to find a suit able Eng lish equiv a lent for the Ger man word used by Cassirer:“...der Aufmerksamkeit als dem eigentlichen schöpferischen Vermögen des Begriffs -bildung” (1910:31). “... the power of ob ser va tion as the truly cre ative abil ity of con ceptformation”.

not pro vide con clu sive ev i dence that they are able to func tion sub jec tively inthe an a lyt i cal as pect of re al ity.1

The per cep tion of a mul ti plic ity of ob jects, the sen si tive de lim i ta tion of par tic -u lar per cep tual ob jects or events (ca pa ble of ex ert ing a con trol ling in flu enceon be hav iour in later sit u a tions – such as avoid ing fire), due to the con ti nu itypro vided by the as so cia tive abil ity of an i mals – all of this are still en closedwithin the do main of sen si tively qual i fied be ings. Pre cisely be cause our hu -man ca pac ity to judge is foun da tional to ev ery act of iden ti fi ca tion and dis tin -guish ing, it dif fers in prin ci ple from a merely sen si tive de lim i ta tion and as so -ci a tion.2

A cor rect log i cal con cept must en tail the mul ti plic ity of iden ti fied char ac ter is -tics united in the con cept in such a way that log i cally jus ti fied judge mentscould be in ferred from it. Any thing log i cally known is, by means of sub jec tive log i cal con cep tu al iza tion, log i cally objectified.3 Objectification is al ways asub jec tive ac tiv ity, con form ing to or con tra dict ing the ap pli ca ble log i cal prin -ci ples. Only this per spec tive can ex plain why we con sider it to be il log i cal(i.e., log i cally in cor rect) when ever judge ments ex pli cate con cep tual el e mentswhich are not an a lyt i cally im plied in the unity of the concept concerned.

Let us take an ex am ple. Does the con cept of a chair only ap peal to its openedup log i cal fea tures, or must we as sume that all the non-log i cal char ac ter is ticsare im plied by open ing up the log i cal ob ject-func tion of a chair? If thesenon-log i cal char ac ter is tics are not im plied, an un bridge able gap, plainly, ex -ists be tween the log i cal sub ject-ob ject re la tion and the non-log i cal as pects ofa chair (in con nec tion with the struc ture of anal y sis, cp. Strauss, 1984). Wehave to con clude, there fore, that in mak ing the log i cal ob ject-func tion of achair pat ent (man i fest), the non-log i cal (modal) char ac ter is tics (spec i fied ac -cord ing to the typ i cal entitary unique ness of the chair) are also log i callyobjectified. There fore, the mul ti plic ity com bined in the unity of the con cept of a chair en ables us to make pred i ca tions like: this chair is beau ti ful (aes theticfea ture); this chair is ex pen sive (eco nomic); this chair is big (spa tial); thischair is heavy (phys i cal); and so on. If these (modal) char ac ter is tics were notan a lyt i cally im plied in the cor rect con cept of a chair, all these men tionedstate ments (ex pli cat ing them in dis tinct judge ments) would, in a log i cal sense, be con tra dic tory. In other words, if the cor rect con cept of the chair does notim ply these char ac ter is tics in an an a lyt i cal way to be gin with, they can not af -

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1 Leakey also re fers to the use of signs by chim pan zees and go ril las. With the aid of dif fer -ent ‘sign-la bels’ these an i mals are sup posed to be able to gen er al ize to cognitively eco -nom i cal con cepts (es sen tial for lan guage): “For in stance, Lucy calls a wa ter melon adrink fruit; Washoe re fers to ducks as wa ter birds, and she in vented the name rock berryfor a brazil nut when she first en coun tered one” (1978:202).

2 O. Koehler ex plic itly claims the con trary: “In the ab sence of ver bal lan guage, we callsuch an op er a tion with rep re sen ta tions, con cepts and judge ments, founded on in tu itionswith out bear ing any names, un named think ing” (1973:119).

3 This objectification is noth ing but the open ing up of the log i cal ob ject-func tion of en ti -ties.

ter wards be pred i cated of the chair, ex cept illogically: From: P is non-Q; onecannot infer: P is Q.

But if it is granted that an i mals can not ar gue and can not in fer in a log i cal or anil log i cal way, shown in judge ments which (do not) con form to log i cal prin ci -ples (such as the principium identitatis and the principium contradictionis),are we not jus ti fied in main tain ing that they do func tion up to the level of(a-ver bal) con cept for ma tion? This ques tion amounts to the fol low ing one:Are there non-log i cal con cepts? and: are an i mals ca pa ble of form ing il log i calcon cepts?! We only have to think about the well-known ex am ple of BertrandRus sell: a square cir cle. Our lin gual abil ity to des ig nate entitary anal o gies, asdis tinct from modal anal o gies (such as the modal dif fer ence be tween math e -mat i cal space – which is con tin u ous and in fi nitely di vis i ble – and phys i calspace – which is not con tin u ous, since it is bound to the quan tum struc ture ofen ergy and there fore is not in fi nitely di vis i ble), is known to us in the form ofmet a phors (viz. “foot of the moun tain”). In our case we only have to thinkabout the na ture of a “box ing ring”. If the sign-mode of re al ity was not dis tinct from the log i cal mode, then this metaphor would have been an assertion that asquare circle exists!

Remark: In spite of the fact that Henk Hart has a clear view on the na tureof modal anal o gies, he does not dis tin guish be tween modal anal o gies(anti- and retrocipations) and entitary anal o gies (des ig nated by met a -phors). His ini tial the sis is that an anal ogy is an interfunctional re la -tion ship (Hart, 1984:87, cf. 153). True ex am ples of interfunctionalanal o gies are men tioned by Hart – such as anal o gies of growth (158)and the modal dif fer ence be tween so cial dis tance and spa tial dis tance(171, cf. 205). How ever, he con stantly equates them with ‘met a phors’(cf. 1984:153, 156, 158, 160). A met a phor can be re placed by an otherone to tally dif fer ent from it. This is not pos si ble with modal anal o gies:ev ery re place ment sim ply turns out to be syn on y mous with the orig i nalone!

This con cept is il log i cal be cause both the iden ti fi ca tion and the dis tin guish ing are not con form ing the rel e vant log i cal prin ci ples, viz. the prin ci ple of iden tity and the prin ci ple of non-con tra dic tion: (in Eu clid ean space) a cir cle is a cir cle(cor rect iden ti fi ca tion), and: a cir cle is not a ‘non-cir cle’ (such as a square –correct distinguishing).

Ber nard Rensch men tions the fact that, in Münster, experimentalists havetried for half a year to teach chim pan zees to copy a given draw ing of a squareor tri an gle, but with out any suc cess (1968:148). My ques tion is: if these an i -mals are not even able to draw these fig ures, how are we go ing to be con -vinced that they truly have the con cept of a square or the con cept of a tri an gle? We must re mem ber that a con cept is some thing dif fer ent from a sen sory pic -ture which can be as so ci ated with some thing else (cf. Overhage, 1972:252) –as is the case in the so-called ‘name-giv ing’ men tioned by Leakey. The de ci -sive point, how ever, in show ing that they do pos sess these con cepts, would beto show that they can think il log i cally by form ing, for ex am ple, the self- con -tra dic tory con cept of a “tri an gu lar square” or a “square tri an gle”! This has

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never been shown by any of the ex per i ments re ferred to by the men tioned au -thors. In other words, an i mals are sim ply not able to func tion sub jec tively inthe an a lyt i cal as pect of re al ity. Con se quently, it should not be sur pris ing thatthey are un able to think – be it in a logically correct way or illogically!

Does this im ply that we can not as cribe any form of in tel li gent be hav iour toan i mals? Buytendijk re fers to etho log i cal re search in or der to sub stan ti ate hiscon clu sion that the an i mal world merely shows grad ual dif fer ences in this re -spect. “Ev ery spe cies has its own prac ti cal in tel li gence, lim ited by dis po si tionand ex pe ri ence” (1970:98). This con clu sion pre sup poses his ba sic dis tinc tionbe tween an i mal and hu man in tel li gence. When, in a given sit u a tion, hu manbe ings and an i mals will pur sue a sim i lar goal, they will ex pe ri ence sim i laremo tional drives. How ever, what is ab sent in the case of the an i mal, is ac tionon the ba sis of judge ments: “There fore, one de fines an i mal in tel li gence as thecon crete ex pe ri en tial and senso-motoric struc tur ing of prac ti cal be hav iour,whereas hu man in tel li gence dis plays it self as a ra tio nal-log i cal, cat e gor i callyjudg ing con cep tu al iza tion of the task-set ting na ture of the con crete sit u a tionand the dis cov ery of a so lu tion which does not fol low from the im me di ate sen -sory ef fect of the sit u a tion” (1970:97). Overhage, re ject ing the an thro po mor -phic mode of speech pres ent in the writ ings of Rensch, Koehler and Lorenz,em pha sizes that an i mal form-per cep tion does not result in genuine conceptformation, since it remains enclosed within the sensory-perceptive sphere(1965:307; cf. Overhage 1972:251-276).

The human being as “Homo symbolicus”?

The unique ness of be ing hu man is some times sought in mo ral ity, in the abil ity to com mit sui cide, in the con scious ness of death (Dobzhansky) or in the lin -gual po ten tial of be ing hu man (Cassirer, Von Bertalanffy). Cassirer (cf.1944)in tro duced the well-known dis tinc tion be tween sig nals and sym bols. The for -mer be longs to the phys i cal world of be ing and the lat ter is a part of the hu manworld of mean ing, the world of hu man cul ture. Von Bertalanffy says thatsym bol ism “if you will, is the di vine spark dis tin guish ing the most per fectlyadapted an i mal from the poor est spec i men of the hu man race” (1968:20). Inor der to iden tify sym bols, he uses three cri te ria: (i) sym bols are rep re sen ta -tive, i.e., the sym bol stands in one way or the other for the thing sym bol ized;(ii) sec ondly, sym bols are trans mit ted by tra di tion, i.e., by learn ing pro cessesof the in di vid ual in con trast to in nate in stincts; (iii) fi nally, sym bols are freelycreated (1968:15, cf. 1968a:134).

Helmut Plessner wants to tran scend the self-con tra dic tory no tion of an‘entelechie’, pre sented to him by his tu tor Hans Driesch. As an al ter na tive, hein tro duces the no tion of positionality. Phys i cal en ti ties are de lim ited by thesur round ing en vi ron ment. In the case of or ganic en ti ties, this de lim i ta tion be -longs to the en tity it self (for ex am ple, the mem brane), and thus evincespositionality (1975:291). This con cept pro vides the pos si bil ity to view hu -man kind as be long ing to the last level of liv ing be ings. An i mals are con sid -ered to be closed and centric, dis tin guished from the hu man be ing as an ec cen -tric (and rel a tively ‘Weltoffen’) liv ing be ing (1975:292). The first an thro po -

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log i cal ‘Grundgesetz’ (fun da men tal law) men tioned at the end of his Book:“Die Stufen des Organischen und des Menschen” (1928, re print 1965) statesthe “vermittelte Unmittelbarkeit” (me di ated immediateness) valid for alleccentric positions (cf. 1975:297).

Lan guage po si tions it self in be tween the grasp of the hand and the view of theeye – the eye as the “or gan of mak ing-some thing-im me di ately-pres ent”.Thus, in var i ous re spects, the hand and the eye be come dis pens able (cf. Hofer, 1972:203). An i mal com mu ni ca tion, ac cord ing to Plessner, does not know a“me di a tion through ob jects” (1975a:380, cf.379). Surely, this phe nom e non ispar tic u larly re mark able, since, in the do main of hu man sen si tiv ity, the senseof see ing and of the sense of touch ing dom i nate that of smelling (cf. Haeffner,1982:16).

Pre cisely by means of the me di ated im me di ate ness of lan guage, hu man be -ings pos sess an aware ness of the past and the fu ture – an aware ness tak ing thelim ited life-span of be ing hu man into con sid er ation. This ex plains theuniquely hu man aware ness of death as well as the pos si bil ity to commitsuicide.

The com mu ni ca tion of an i mals does not re fer to the dis tant past or re mote fu -ture – it is re stricted to the im me di ate needs of the an i mal. As a con se quence,the ‘signs’ used by an i mals (sig nals, in terms of Cassirer’s dis tinc tion), arestrictly univocal. Just com pare the re mark able dance of the bees where the (i)tempo, (ii) the di rec tion and (iii) the tan gent is con stantly as so ci ated with the(i) dis tance, (ii) lo ca tion and (iii) the course to be fol lowed in or der to reachthe de tected source (cf. Overhage, 1972:220 ff.). Lin gual signs, on the con -trary, pre sup pose choice and there fore re quire in ter pre ta tion (cf. Nida,1979:203; De Klerk, 1978:6).

Fur ther more, it is strik ing that the typ i cal hu man lin gual abil ity is de pend enton spe cific an a tom i cal con di tions ab sent in the anthropoids.

The anatomical conditions of human speech

Ever since Des cartes it was be lieved that the unique ness of the hu man brain isre spon si ble for hu man lan guage. The re sult was that anat o mists in sisted thatan thro poids also have the ‘ma chin ery’ avail able to ar tic u late speech. The or -der of pri mates – which in cludes hu man be ings ac cord ing to the prev a lentclas si fi ca tion – is nev er the less, of course with the ex cep tion of hu mans, un -able to vo cal ize. The abil ity to re pro duce hu man speech sounds as it is foundin birds is to tally ab sent in the mam mals. The vo cal po ten tial of the go rilla and urangutan is ex cep tion ally poor. The chim pan zee is some what better off, andthe gib bon can pro duce sounds cov er ing al most an oc tave. All these an thro -poids, how ever, com pletely lack the play ful sounds pro duced by the hu mansuck ling. The un prec e dented pos si bil i ties of hu man sound pro duc tion tran -scend that of the an thro poids by far. In ad di tion since this hu man soundproduction also displays an exceptionally rich modifiability (Overhage, 1972: 242).

Post mor tem stud ies of the up per re spi ra tory tract in mam mals as well as cine -radio graph ic stud ies have shown that the po si tion of the lar ynx is cru cial in

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de ter min ing the way in which an in di vid ual breathe, swal low and vo cal ize(Laitman, 1985:281). This im plies that there are cer tain an a tom i cal pe cu liar i -ties which go hand in hand with the con tri bu tion of brain func tion ing in thepro duc tion of hu man speech, in par tic u lar the grad ual de scent of the lar ynx af -ter the post-natal period (cf. Portmann, 1973:423).

The fail ure of the an thro poids to im i tate hu man sounds fol lows from the to -tally dif fer ent struc ture of their lar ynx. In all an thro poids it is po si tioned ex -tremely high in the neck. Laitman re marks: “This high po si tion per mits theepi glot tis to pass up be hind the soft pal ate to lock the lar ynx into thenasopharynx, pro vid ing a di rect air chan nel from the nose through the naso -pharynx, lar ynx and tra chea to the lungs. ..... In es sence, two sep a rate path -ways are cre ated: a re spi ra tory tract from the nose to the lungs, and a di ges tivetract from the oral cav ity to the esoph a gus. While this ba sic mam ma lian pat -tern – found with vari a tions from dol phins to apes – en ables an in di vid ual tobreathe and swal low si mul ta neously, it se verely lim its the ar ray of sounds anan i mal can pro duce. ... While some an i mals can ap prox i mate some hu manspeech sounds, they are an a tom i cally in ca pa ble of pro duc ing the range ofsounds necessary for complete, articulate speech” (1985:282, cf. Goertler,1972:249).

In or der to pro vide the new born hu man suck ling with a milk tract sep a ratefrom the re spi ra tory tract, the po si tion of the hu man lar ynx at birth is the sameas in that of the mam mals. In the pe riod be tween the first and sec ond year thishighly po si tioned lar ynx starts its de scent in the neck. This down ward move -ment cre ates the phar ynx cav ity, nec es sary for the ar tic u la tion of the richervoice dis po si tion pres ent in hu man be ings. Laitman de clares that the pre cisetime this shift oc curs, as well as the phys i o logic mech a nisms which un der lie it are still poorly un der stood (1985:282). As soon as the lar ynx reaches its des -tined low po si tion, it can no lon ger lock into the nasopharynx. Con se quently,in hu man be ings, the re spi ra tory and di ges tive path ways cross above the lar -ynx. This cre ates the pos si bil ity to suf fo cate, which surely is, eval u ated in it -self, some thing neg a tive. How ever, it is pre cisely this ex panded phar ynxwhich pro vides the hu man be ing with its unique po ten tial to pro duce a rich va -ri ety of speech sounds. The pal ate be tween the mouth and nose cav i ties serveas res o nance ba sis for the pro duced sounds. Goerttler even men tions the factthat in the third month af ter con cep tion a distinctively human structuralelement develops (it is called the vocal chord ‘blastem’ – 1972:250).

It is in ter est ing to note in this con nec tion that Laitman informs us that thebasicranial sim i lar i ties be tween the Aus tra lo pith ecines and ex tant apes sug -gest that their up per re spi ra tory tract was also sim i lar in ap pear ance. Con se -quently, as with the liv ing non hu man pri mates, the phar ynx por tion avail ablefor sound mod i fi ca tion in these early homi nids would have been greatly re -stricted: “As a re sult, these early homi nids prob a bly had a very re stricted vo -cal rep er toire as com pared with mod ern adult hu mans. For ex am ple, the highlar ynx would have made it im pos si ble for them to pro duce some of the uni ver -sal vowel sounds found in hu man speech pat terns” (1985:284). His con jec ture is that the first in stances of full basicranial flexion sim i lar to mod ern hu mans

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do not ap pear un til the ar rival of Homo sa pi ens (es ti mated by him at 300,000to 400,000 years ago): “It may have been at this time that homi nids with upperrespiratory tracts similar to ours first appeared” (1985:286).

Do human beings have ‘speech-organs’?This ques tion points at an other as ton ish ing fea ture of hu man speech pro duc -tion. If we de fine a speech-or gan as that bodily part which ex ists solely in ser -vice of the pro duc tion of speech sounds, then we are in for a sur prise. Let usenu mer ate pos si ble can di dates: the lungs, lar ynx, mouth cav ity, pal ate, teeth,lips and nose cav ity. With out an ex cep tion, all these or gans per form pri maryfunc tions which would nor mally pro ceed even if a per son never ut ters one word (Overhage, 1972:243)! Hu man lan guage sim ply takes hold of all these dif fer -ent or gans in the production of speech sounds.

This highly de vel oped and sub tle co op er a tion, es pe cially of three or gans sohet er o ge neous in char ac ter as the mouth, the lar ynx and the brain in te grated in the pro duc tion of speech sounds, makes it rather dif fi cult, if not hope less, topro vide us with an evolutionistic causal ex pla na tion of this as ton ish ing phe -nom e non. The ques tion arises what num ber of mi rac u lous changes shouldhave oc curred to pro duce the ar tic u la tion con di tions nec es sary for truly hu -man lan guage for ma tion. “Such an un fath om able pro cess of change af fect ingso many dif fer ently struc tured or gans and or gan com plexes, closely cor re -lated with each other, should have pro ceeded har mo ni ously as a to tal change,if it was to come to the unprecedented perfection of human speech” (Over -hage, 1972:250).

Does human experience of the world differ from that of the animals?In the course of our dis cus sion we of ten re ferred to the role of in stinct in an i -mal life. Adolf Portmann is con vinced that an i mals are ac tu ally de ter mined by their in stincts and that they are re stricted to a par tic u lar am bi ent (1969:86).The way in which an i mals ex pe ri ence the world is com pletely de ter mined bytheir nat u ral dis po si tions. They are only con cerned with that which has a di -rect phys i cal, bi otic and sen si tive mean ing to them. Con se quently, they ex pe -ri ence re al ity in terms of places suit able for walk ing or fly ing (phys i cal ac ces -si bil ity), in terms of sex part ners and other an i mals be long ing or not be long ing to the same spe cies, in terms of what can be eaten and what not (bi oti cal in ter -est), and in terms of things or events which are caus ing anx i ety or which maybe comforting (sensitive concern) (cf. Landmann, 1969:162 ff.).

The in stinct de ter mined ness of an i mals func tion in a re mark able way. Due toin her ited coordinations (con cern ing the motoric di men sion) and in born “trig -ger ing-off” mech a nisms (Auslösemechanismus; con cern ing the do main of re -cep tiv ity), par tic u lar an i mals, in given cir cum stances, can act in stinc tivelyand in pre de ter mined ways. These in stinc tive be hav iour pat terns are in her itedand not learned. Eibl-Eibesfeldt gives the ex am ple of a squir rel with its act ofbury ing a nut. Nor mally, these an i mals bury col lected acorns or nuts in di vid u -ally in the ground. In do ing it, they per form cer tain typ i cal ac tions such as run -ning around, lay ing down the nut and bury ing it cov ered with soil. When asquir rel is raised in an ar ti fi cially iso lated cage with out ever be ing given a nut,

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the ma ture squir rel will, when pre sented with one, ex actly ex e cute the typ i cal‘burial cer e mony’ (Eibl-Eibesfeldt, 1972:5). This dem on strates that a‘programmed series of actions’ could be triggered off by a specific stimulus.

The world of dif fer ent an i mals dif fers, and so dif fers the world of an i mals andhu man be ings. Von Bertalanffy ex plains one of the ex am ples de lin eat ing theambients of var i ous an i mals as fol lows: “Take, for in stance, a tick lurk ing inthe bushes for a pass ing mam mal in whose skin it set tles and drinks it self fullof blood. The sig nal is the odour of the bu tyric acid, flow ing from the der malglands of all mam mals. Fol low ing this stim u lus, it plunges down; if it fell on awarm body – as mon i tored off by this sen si tive ther mal sense – it has reachedits prey, a warm-blooded an i mal, and only needs to find, aided by tac tilesense, a hair-free place to pierce in. Thus the rich en vi ron ment of the tickshrinks to metamorphize into a scanty con fig u ra tion out of which only threesig nals, beaconlike, are gleam ing which, how ever, suf fice to lead the animalsurely to its goal” (1973:241).

An other well-known ex am ple, given by Von Uexküll, con cerns an oak tree.Dif fer ent kinds of an i mals ‘dis sect’ their own dif fer ent ambients (Umwelten)from the tree – con stantly en closed within the above men tioned pa ram e terscon cern ing their phys i cal, bi oti cal and sen si tive needs (Von Uexküll,1970:98, 100). He dis cusses the life-worlds of an i mals like the jackal, squir -rel, owl, ant and bee tle. Al though hu man be ings also have ac cess to these di -men sions, one can not say that his ex pe ri ence is closed by or lim ited to theseper spec tives. Hu man func tion al ity en com passes, but also tran scends thephys i cal, bi oti cal and sen si tive as pects of the tree. To the bot a nist the tree maybe an an a lyt i cal ob ject of sci en tific in ves ti ga tion; a per son go ing for a walkmay ex pe ri ence its beauty; a crim i nal can use it as a hid ing place, the car pen ter may use its wood to man u fac ture fur ni ture, and so on. There fore, hu man be -ings are able to experience the tree in a variety of ways which are inaccessibleto animals.

Hu man func tion ing is nei ther com pletely de ter mined by in stincts, nor is itlim ited to only one ‘Umwelt’, sim ply be cause the whole bodily ex is tence ofhu man be ings is di rected to wards and is guided by nor ma tively qual i fiedview-points. The tre men dous flex i bil ity of hu man func tion ing ex e cutedwithin these nor ma tive as pects of re al ity, makes it pos si ble for hu man so ci etyto de velop up to a level with far-reach ing forms of dif fer en ti a tion and spe cial -iza tion, ex pressed in the mutlifarious roles which any per son in such a so ci etycan as sume. Even Simpson stresses this in sight: “Such spe cial iza tion, whichis non-ge netic, re quires in di vid ual flex i bil ity and could not oc cur in a mainlyin stinc tive an i mal” (1969:90). Hart states it with con cise clar ity: “A workerant is just that – and all its func tions are geared to be ing a worker ant. A hu man be ing, on the other hand, has mul ti ple roles to play and is not ex hausted in anyof them” (1984:146). How ever, this hu man-spir i tual spe cial iza tion and flex i -bil ity is de pend ent on a rel a tively un spe cial ized bio-psy chi cal ba sis and foun -da tion. Be cause Van Uexküll ex tended his ‘Umweltlehre’ also to the level ofhu man be ings, Portmann points out that we should take cau tion in this re spect. A com par i son be tween the dif fer ent ambients ex pe ri enced at a tree and the

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dif fer ent do mains of hu man func tion ing, does show the im por tant dif fer encesstill pres ent. The so ci etal struc ture of hu man life en ables com mu nal un der -stand ing be tween all the dif fer en ti ated so ci etal spheres, some thing lack ingbe tween the ambients of different animals (Preface in Von Uexküll,1970:XIV).

The unspecialized traits of the human body

Seen from a mor pho log i cal per spec tive, hu man be ings lack the highly spe -cial ized or gans nec es sary to be per fectly adapted to a par tic u lar hab i tat.Gehlen re fers to the ar chaic (in the sense of prim i tive/un spe cial ized) fea turesdis played by the hu man or gans (1971:86 ff.). Hu man den ture is re mark ablyun spe cial ized if com pared to those of the higher an i mals: it is not spe cial izedsolely for the eat ing of plants or for the eat ing of meat. There are no gaps(diastema) be tween our teeth – some thing typ i cal of the highly spe cial ized na -ture of the more de vel oped mam mals. By con trast we only have to com pare itwith the dis tance be tween the eye-teeth and front mo lars of the an thro poidswhich is closely con nected with the way in which the lat ter are spe cial ized tobe come ca nine teeth (Gehlen, 1971:92). Sim i larly, the hu man hand (Gehlen,1971:98) and foot (Gehlen, 1971:100) rep re sent a prim i tive state in com par i -son with the an thro poids (such as the orang utan, go rilla and chim pan zee).Altner points out that the teeth of the an thro poids are also rel a tively un spe cial -ized. Nev er the less, he does not deny the general tendency present in thephenomena lifted out by Gehlen (1972:199-202).

It should be noted that Gehlen does not use the term prim i tive in the sense oflower but only in the mean ing of un spe cial ized. Ac cord ing to the pre dom i nant neo-Dar win ist evo lu tion ist ap proach there merely ex ists uni di rec tional evo lu -tion ary change: from the less spe cial ized to the more spe cial ized. The Bel giange ol o gist, Dollo, for mu lated this in terms of his law of ir re vers ible spe cial iza -tion. But if this law is uni ver sally ap pli ca ble to all evo lu tion ary change, as it isstill up held by Simpson and other dom i nant neo-Darwinistic think ers, howcan we ‘save’ the an thro poids as can di dates for be ing an ces tral to hu man be -ings? Clearly, they are spe cial ized. How ever, ac cept ing Dollo’s law seem -ingly makes it im pos si ble to de duce the un spe cial ized fea tures of hu man be -ings from the spe cial ized traits of the an thro poids. To sat isfy these strangecon di tions, one has to find (or: hy po thet i cally con strue) a tran si tional formwhich should unite rad i cally op posed traits: ar chaic hu man fea tures and spe -cial ized an i mal char ac ter is tics. Gehlen re marks that this would pro duce such a mon strous and mi rac u lous be ing that it should be awarded a fan tas tic and sep -a rate po si tion in the total realm of animals (1971:87-88)! Notwithstandingthese difficulties, two escape routes have been explored!

(i) Adloff and Klaatsch con structed an hy po thet i cal ‘prim i tive form’ whichis so un spe cial ized that, on the one hand, it can serve as a ba sis and start -ing-point for mod ern hu man be ings, and, on the other hand, may rep re -sent the root en abling the branch which de vel oped into the spe cial iza tion pres ent in the an thro poids. Ob vi ously, this con struc tion is beg ging theques tion, since it ac tu ally starts from a form which is so ‘hu man’ that the

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sup posed or der of de scent is re versed: hu man be ings are an ces tral to theanthropoids (cf. Gehlen, 1971:95)!

(ii) An other pos si bil ity is to ig nore Dollo’s law of ir re vers ible spe cial iza tion by con sid er ing the phe nom e non of neoteny, i.e. the per sis tence of lar valfea tures in the adult or gan ism. Neoteny is found in var i ous kinds of an i -mals, such as worms, in sects and am phib i ans. The Dutch anat o mist,Louis Bolk, used this phe nom e non to ex plain cer tain hu man fea tures.The re mark able sim i lar i ties be tween hu man be ings and the chim pan zees con cern ing the un spe cial ized and ar chaic na ture of hu man or gans, in -spired Bolk to ex plain hu man be ings in terms of the no tion offetalization (sta bi li za tion of pre-na tal traits) and the idea that hu man de -vel op ment un der went a cer tain re tar da tion. In other words, in hu manbe ings the in fan tile char ac ter is tics of the ape be came fixed be cause thema ture ape-form was no lon ger reached, while main tain ing even fe talmarks. Con se quently, in this ap proach, as Landmann re marks, hu manbe ings are not an ces tral to the apes, since they are them selves noth ingbut infantile apes (1969:148)!

Konrad Lorenz, the No bel prize-win ner of 1973 (to gether with Tinbergen and Von Frisch), added to this the ory of Bolk the life-long cu ri os ity of hu man be -ings which cor re spond with the youth ful cu ri os ity of the chim pan zees: “Thecon sti tu tive hall-mark of hu man be ings, their per sis tent, cre atively ac tive en -gage ment with their am bi ent (Umwelt), is a phe nom e non of neoteny”(1973:183-184). Nietz sche once re marked: in ev ery adult hu man be ing a child is con cealed. Lorenz re verses this state ment: in ev ery child a ma ture per sonhides, ea ger to do re search. The cu ri ous child, which dis ap peared com pletelyin the full-grown chim pan zee, is not hid den in adult humanhood, since itcontrols the latter fully (1973:184, cf.242).

To this ap proach Lorenz adds an other per spec tive, fol low ing from the studyof do mes ti cated an i mals, namely that of self-do mes ti ca tion. Do mes ti cated an -i mals dif fer in some typ i cal he red i tary fea tures from wild-liv ing forms. Thesechar ac ter is tics emerged in the pro cess of do mes ti ca tion. For ex am ple, all do -mes tic an i mals show a mea sure of spots, dis play a short en ing of the ex trem i -ties and the basicranial struc ture, are in clined to fat ten eas ily, while an im por -tant in crease in the do main of vari a tion of all pos si ble fea tures of the spe ciesap pear. Al ready E. Fischer pointed out that the pig ment-di vi sion pres ent inhu man eyes which are blue or grey is, though to tally ab sent in all wild-liv ingan i mals, in a cor re spond ing way fully pres ent in al most all do mes tic an i mals.Among the changes caused by the self-do mes ti ca tion of hu man be ings,Lorenz is count ing both the re tar da tion and fetalization men tioned by Bolk.Ac cord ing to Lorenz, cave in hab i ta tion rep re sents the desicive step in the pro -cess of the self-do mes ti ca tion of hu man be ings. This ac count, how ever, is notin ac cor dance with what we know. A num ber of the most typ i cal fea tures of“be com ing do mes tic” are to tally ab sent in the case of hu man be ings. We onlyhave to men tion the early sex ual ma tu rity and the con stant or de creas ing brainde vel op ment of do mes tic an i mals – both phe nom ena which are com pletelyre versed in human beings (cf. Gehlen, 1971:121 and Overhage, 1967:3-4)!

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Fur ther more, an ap peal to phe nom ena of do mes ti ca tion calls upon fac tors notop er a tive in the do main of non-hu man liv ing or gan isms (cf. Overhage,1967:4). Be com ing a do mes ti cated an i mal pre sup poses the cul tural care ofhu man be ings as hu man be ings! The ab surd im pli ca tion of this ar gu mentabout self-do mes ti ca tion is al ready for mu lated by Von Eickstedt more thanfifty years ago: “Cul ture, then, must be older than hu man be ings, since it in flu -enced hu man kind in its bodily de vel op ment. It was not hu man kind whoformed cul ture, be cause cul ture moulded hu man kind” (1934:121)! In otherwords, this view on self-do mes ti ca tion re verses the cul tural sub ject-ob ject re -la tion, by mak ing the hu man being, the cultural subject, an object of culturalcontrol and moulding.

When the in stincts of an i mals de te ri o rate, ow ing to the fact that they be comeac cus tomed to the care of hu man be ings, they do not de velop any com pen sat -ing fac ul ties. Landmann goes a bit too far when he em pha sizes that the pe cu -liar ca pac i ties of hu man be ings do not need any in stincts (1969:148), be causewe must ac knowl edge that hu man be ings still have in stincts, how ever poorlythey are equipped with them if com pared with the ex is tence of an i mals that isse cured by their instancts. How ever, Landmann is fully jus ti fied in say ing that a ‘wild’ hu man form, dom i nated by in stincts, never existed and never couldhave existed.

Even if we want to re ject Dollo’s law, and sup port this re jec tion with Lorenz’s the ory of re tar da tion, fetalization and neoteny, un an swered ques tions keeppop ping up. Ac cord ing to Gehlen it seems ab so lutely im pos si ble to un der -stand in what sense the de vel op ment of thought and lan guage would have en -dowed hu man be ings with a se lec tive ad van tage in com par i son with the an -thro poids. The ques tion should also be asked: in a fight against what wouldthe pro longed and help less youth ful pe riod of hu man be ings pro vide themwith a se lec tive ad van tage in stead of be ing a se ri ous and life-en dan ger ingdis ad van tage (cf. Gehlen, 1971:125)?

Tra di tion ally it was thought that hu man be ings pos sess some thing lack ing inan i mals: in tel li gence (wis dom). This leg acy is re flected in the cur rently stillgen er ally ac cepted (evo lu tion ist) clas si fi ca tion of hu man be ings as Homo sa -pi ens. If we com pare hu man be ings, how ever, with the high level of spe cial -iza tion pres ent in the an thro poids, we must con clude that hu man be ings arelack ing some thing, namely spe cial iza tion. Must we con clude, then, that the“struc tural design” of human beings show shortcomings?

Is the human being to be seen as a deficient creature?

This no tion stems from Gehlen (cf. 1971:20, 30, 80, 354). In deed, if we com -pare the nat u ral pre dis po si tions of hu man be ings with the count less pos si bil i -ties at the dis posal of dif fer ent an i mals, is does look as if hu man be ings aretreated nig gardly by na ture. Hu Man be ings are much slower than many wildan i mals. They do not have nat u ral pro tec tive hair cov er ing his body. The hu -man senses are trun cated in com par i son with the sharp ness and alert ness ofthe senses of the an i mals. Hu man be ings do not have nat u ral and dan ger ousweap ons. They do not pos sess the mus cle power, claws or jaws of any pred a -

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tor. Some an i mals can reg is ter su per sonic waves, some can see ul tra-vi o letrays as light, while there are fishes that can per ceive elec tri cal fields. Birdsori ent them selves, by means of their re mark able nav i gat ing sys tems, to themag netic poles of the earth. And all these ways of experience are withheldfrom human beings (cf. Portmann, 1970:200 ff.).

If we mea sure hu man be ings with the yard stick of an an i mal, we are al mostdoomed to eval u ate them as an i mals that failed! But as soon as we re verse theper spec tive and ac knowl edge the uniquely hu man fea tures which dif fer en ti -ate hu man be ings so clearly from an i mals, a dif fer ent pic ture arises. Portmannex plains this with the following example:

“The nar row lim it ed ness of an i mal in ter est is op posed to flex i ble free dom ofchoice pres ent in hu man be ings. An an i mal can tran scend the bond age to itsdrives only to a lim ited de gree, whereas I am able, in ev ery mo ment and ac -cord ing to my to tal power to dis cern, en com pass ing my full in ner-par tic i pat -ing ded i ca tion, to pay at ten tion to some thing, how ever min ute and un im por -tant it may appear to be” (1974:102).

Why is it that some schol ars em pha size the un spe cial ized na ture of hu man be -ings and even call them de fi cient be ings? No tions like the un spe cial ized na -ture of hu man be ings and the qual i fi ca tion de fi cient be ing only play a role ifwe choose the nat u ral dis po si tion of an i mals as ba sis of com par i son. If weloose sight of this im plicit choice, the ob jec tion of Hans Freyer would havebeen com pletely valid. Ini tially, the fic tion is pos tu lated that hu man be ings are an i mals and then, only af ter wards, it turns out that in this ca pac ity hu man be -ings would rep re sent some thing highly in suf fi cient which, as such, is an im -pos si bil ity! The erect gait, the free hand with the strongly op pos ing thumb(serv ing the for ma tive cul tural fan tasy men tioned ear lier) and the spir i tu allystamped fa cial ex pres sion of hu man be ings - all these fea tures re veal the truena ture of hu man kind - to take over an ex pres sion from Lorenz - as a spe cial istin being unspecialized.

Gehlen is in clined to see the dis tinc tively hu man func tions of hu man be ings as some thing com pen sat ing the lack of be ing in stinc tively se cured. How ever,pre cisely the op po site is the case. The phys i cal, bi otic and sen si tive di men -sions of hu man ex is tence are fully geared to wards the nor ma tively stampedcul tural life of be ing hu man. This cul tural dis po si tion, which should be con -sid ered to be our first na ture (and not our sec ond na ture, as Portmann be -lieves), come to the fore in the abil ity to think, to con cep tu al ize and to ar guein tel li gently. It is also seen in the tech ni cal abil ity to man u fac ture tools af ter afree pro ject man i fest ing a free for ma tive fan tasy, in the lin gual com pe tence todis cern and ar tic u late mean ing ful speech sounds and to interpret thoseproduced by fellow human beings correctly, and so on.

The an a lyt i cal abil ity of a per son, en abling that per son to iden tify (i.e., the lift -ing out of cer tain fea tures) and dis tin guish (i.e., by dis re gard ing other fea -tures), is foun da tional to the tech ni cal func tion ing of hu man be ings: to pro -duce some thing fan ta sized af ter a free pro ject, pre sup poses an an a lyt i cal abil -ity (in clud ing con cep tu al iza tion) in the men tioned sense. Al though Hart ar -gues for a place of the for ma tive/tech ni cal as pect be fore the an a lyt i cal as pect

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(1984:179 ff.), in a dif fer ent con text he does ar gue for the foun da tional role ofcon cept for ma tion: “Much re al ity con tains con cepts as con sti tu tive el e mentsof its na ture. Al most all typ i cally hu man prod ucts, nearly all of what we re ferto as cul ture, can not ex ist ex cept through con cep tu al iza tion. With out our hav -ing con cepts of these re al i ties we can not pro duce them” (1984:411, note 28).This is a good ar gu ment in fa vour of plac ing the log i cal as pect be fore the tech -ni cal-for ma tive as pect in the or der of cos mic mo dal i ties! Al though it is not the place here to give elab o rate ar gu ments for it, I want to men tion the fact that the same ap plies to the sign-mode of re al ity (called the sym bolic as pect by Hart,which he also places be fore the an a lyt i cal as pect – 1984:180 ff.). All typ i calse man tic phe nom ena, such as syno nymi ty, antonymity, re dun dance, meta -phoricity, and so on, not only pre sup pose the an a lyt i cal as pect but only havemean ing if these two as pects are dis tinct and irreducible (cf. Strauss,1981:5-32).

The func tion ing of hu man be ings in these dif fer ent nor ma tive spheres of re al -ity, some times highly dif fer en ti ated in a so cial sense, can not fully be ac -counted for in merely func tional terms. Hart cor rectly states that the in te graliden tity of the hu man per son tran scends hu man func tion al ity. He then pro -ceeds: “Hu man re spon si bil ity and ac count abil ity point to an other di men sionof hu man ex is tence be sides the func tional di men sion, namely, the spir i tual.The spir i tual in hu man ity can not be fully un der stood as func tion al ity, al -though it can be un der stood only if we un der stand it in terms of func tion al ity”(1984:270). Whereas ma te rial things, plants and an i mals are re spec tivelystamped and qual i fied by par tic u lar as pects (viz. the phys i cal, the bi otic andthe sen si tive), the unique ness of be ing hu man is pre cisely seen from the factthat no func tional hu man ac tiv ity can ever en close or en com pass all of our hu -man func tion al ity. Once again Hart puts it well-for mu lated to us: “In the lifeof a per son, there is not a sin gle qual i fy ing func tion that struc tur ally unitesand in te grates all of hu man ex pe ri ence” (1984:276). This also ex plains whyhu man func tion al ity is open, through faith, to being committed to theacceptance of what lies beyond the limits of subjectivity (cf. Hart, 1984:277).

I would pre fer, as Dooyeweerd does in his A New Cri tique (cf. 1997-III:87-89), to deny the pos si bil ity of a hu man realm or king dom (Hart con sis -tently af firms such a king dom – cf. 1984:268 ff.). Of course this is de pend enton the def i ni tion we give for a realm. Hart de fines it by say ing that realms arecat e go ries of ex is tence ac cord ing to prin ci ples of or der (1984:268). This def i -ni tion in tro duces some thing am big u ous be tween Hart’s dif fer ent realms, be -cause the three realms of ma te rial, veg e ta tive and an i mal ex is tence are alluniquely qual i fied by a sin gle modal func tion, whereas the “hu man realm” isnot qual i fied by any single function at all (cf. Hart, 1984:276-7).

Be fore we con sider a num ber of di verg ing per spec tives on the na ture of hu -man free dom, we still have to pay at ten tion to the unique ness of our hu manfunc tion ing within the bi oti cal as pect of re al ity. The rel e vant ma te rial for thisanal y sis is am ply pro vided by Portmann (1969).

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The ontogenetic1 uniqueness of being human

In or der to com pare the ontogenetic de vel op men tal na ture of hu mans with that of the an i mals, Portmann dis tin guishes two dis tinct de vel op men tal types:Nesthocker and Nestflüchter. ‘Nesthocker’, on the one hand, en com pass allthose an i mals which are born with closed eyes, which are na ked and help lessat birth, and which need the care of their par ents in pro vid ing the nec es saryfood as well as a pre pared nest. ‘Nestflüchter’, on the other hand, are rep re -sented by all those an i mals which, at birth, are ca pa ble of mov ing sim i larly totheir full-grown par ents. At birth their eyes are open as well as their au di toryca nals, while their pos ture and bodily pro por tions ex actly cor re spond withthose of the mature members of the species.

Mam mals fall ing within the cat e gory of ‘Nesthocker’ are born af ter a shortpe riod of preg nancy – about 20 to 30 days. Portmann men tions in secti vore(prim i tive in sect eat ing mam mals), gnaw ers and some pred a tors. The num berof new-born an i mals per lit ter is rel a tively large – from 5 to 22. The higher de -vel oped ‘Nestflüchter’ have a long pe riod of preg nancy (more than 50 daysand some times lon ger than 20 weeks), while, in most spe cies, their new-bornoff spring are lim ited to 1 or 2 (sel dom 4). Ex am ples of this cat e gory areungulates, sea lions, wales, horses and mon keys). Mam mals with a less de vel -oped brain are born as ‘Nesthocker’, such as the squir rel, house mouse, redjackal, do mes tic cat and ti ger. Mam mals pos sess ing a more highly de vel opedbrain, ex pe ri enc ing a long growth pe riod within the moth ers womb, en ter theworld as ‘Nestflüchter’, such as the pig, beast, horse, sheep, sea lion and wale) (Portmann, 1969 chapter II).

In or der to sub stan ti ate this clas si fi ca tion, Portmann uses a num ber of dif fer -ent and in de pend ent cri te ria.2 For ex am ple, the growth of the brain is a veryim por tant el e ment in the de vel op ment of an in di vid ual. By com par ing therates of in crease pres ent in the mam mals (with a fac tor of 5 func tion ing as thedi vid ing line), Portmann shows that ‘Nesthocker’ and ‘Nestflüchter’ areuniquely dif fer en ti ated – the lat ter group show ing a fac tor greater than 5 andthe for mer showing one less than 5 (cf. 1969:50)!

The ob vi ous ques tion is: are hu man be ings ‘Nesthocker’ or are they ‘Nest -flüchter’? Hu man be ings are born help less, un able to care for them selves andun able to move like a young or ma ture hu man be ing. Just as the ‘Nesthocker’it is nec es sary to care for them. Is this con di tion suf fi cient to clas sify them asbe long ing to the ‘Nesthocker’? No, the an swer must be neg a tive, be cause hu -man be ings are born with some thing typ i cal of the ‘Nestflüchter’, viz. openeyes and an open au di tory ca nal! But then, surely we have to clas sify them as‘Nestflüchter’. How ever, nor that will do. In sharp con trast to the ‘Nest -

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1 Haeckel's the ory of re ca pit u la tion, stat ing that on tog eny is a suc cinct re ca pit u la tion ofphy log eny, is shown to be sci en tif i cally com pletely un jus ti fi able (cf. Overhage, 1959c).

2 As a re sult of solely con cen trat ing on the head (partly caused by the fact that, in fos silfind ings, the skull is eas ier ac ces si ble), Bolk in tro duced his men tioned fetalization the -ory, pos tu lat ing the per sis tence of pre-na tal fea tures in the hu man form. Portmann re -marks that as soon as the bodily pro por tions of an i mals are taken into ac count, these as -sump tions are con tra dicted by the facts (1969:46).

flüchter’, whose bodily pro por tions are sim i lar to those of the full-grownmem bers of their re spec tive spe cies, at birth hu man be ings are com pletely‘dis pro por tion ate’ in com par i son with their ma ture state, once again shift ingthem back to the al ready un suit able cat e gory of the ‘Nesthocker’! Con se -quently, the only rea son ably jus ti fied con clu sion, based on the de fin ing fea -tures of these two categories of mammals, is that human beings belong toneither of them!

Fur ther more, at birth the mass of the hu man brain ex ceeds that of the an thro -poids at least 2-times (about 370 gram, com pared to the 150 gram of theorang utan or the chim pan zee brain). In com par i son with the an thro poids, hu -man be ings are born too early – al most a year to soon, be cause only af ter anage of one year hu man be ings reach a de vel op ment sim i lar to that of the typ i -cal mam mals at their birth. This means that, for a gen u ine hu man-like mam -mal, i.e. “for a true an i mal-hu man or hu man-an i mal”, an ex tra year is needed(Portmann, 1969:58). Portmann is con vinced that this “too-early-birth-stage”(physiologischen Frühgeburt) can only be un der stood in terms of a broaderper spec tive. He speaks about an “ex tra uter ine” pe riod/year in the bi oticdevelopment of human beings (1969:87 ff.).

Dur ing the sec ond part of the first year the typ i cal hu man fea tures of the de -vel op ing baby start emerg ing, such as the erect pos ture, an a lyt i cal in sight, lan -guage use, free de ci sions, and son on. All these ac tiv i ties are de vel op ingwithin the cul tural mi lieu of hu man so ci ety. The fact that these phe nom enaoc cur within the first post-na tal year, cor re spond ing with the pe riod in whichthe higher mam mals are still in the womb, shows that the “ex tra uter ine year”of the hu man be ing is des tined to be taken up in the typ i cal hu man way oftrans fer ring cul ture. The re tar da tion of the growth rate from the 2nd to the 9thyear also re veals a direct ed ness to wards the com pli cated pro cesses of learn ing and ap pro pri a tion which en able a per son to mas ter the vast cul tural leg acypres ent in the so ci ety within which he is grow ing up. Cor re spond ing to the rel -a tively long youth pe riod, the hu man be ing ex pe ri ences also a rel a tively longpe riod of ma tu rity, pro vid ing time to trans fer, to the next gen er a tion, theheritage of successive generations by means of educational processes andinstitutions.

By and large, all these per spec tives sim ply add weight to the con clu sion that,in spite of the fact that we have to ac knowl edge that both hu man be ings andan i mals func tion within the bi oti cal as pect of re al ity, we must con stantly keepin mind that our hu man bi oti cal func tion ing is to tally unique, clearly shown inhis ex cep tional ontogenetic growth pat tern. Portmann em phat i cally de clares:“In strict cor re la tion to the mea sure in which we see our form of ex is tencemore clearly, we be come more cer tain that the ques tion about the or i gin of hu -man kind, as well as the equally dif fi cult ques tion about the rise of the big‘form-spheres’ (Gestaltenkreise) of the liv ing, are not an swer able with theresearch tools at our disposal” (1969:163).

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Human freedom – the predominantly negative approach of modernphilosophy

As is shown ex ten sively and con vinc ingly in the writ ings of Dooyeweerd,mod ern phi los o phy re ceived its ul ti mate di rec tional im pulse from the di a lec ti -cal op po si tion which pre vailed be tween the ideal of an all-en com pass ingcausal (nat u ral sci en tific) ex pla na tion on the one hand, and the ideal of the hu -man be ing as an au ton o mously free per son al ity on the other hand – i.e., themo tive of na ture and freedom (cf. 1997-I:207 ff., 216 ff.).

Autonomous freedom versus natural causality

At the birth of mod ern phi los o phy, dur ing and af ter the Re nais sance, a newideal of an au ton o mously free per son al ity came to the fore, al though its firstaim was to mas ter na ture ra tio nally with the aid of the newly de vel op ing nat u -ral sci ences. Des cartes’ em pha sis on the maxim that our ideas should be clearand dis tinct (con sid er ing clear ness to be more fun da men tal than dis tinct ness – Prin ci ples, XLVI), is ori en tated to wards math e mat ics as model of thought.Even the cer tainty that God ex ists is only ac com plished by clear and dis tinctun der stand ing – show ing – in the fi nal anal y sis, that he uses the idea of God inor der to fur nish his de i fied math e mat i cal thought with the fea ture of cer tainty, thus stamp ing the in fal li bil ity of the new math e mat i cal method of anal y sis.Hav ing men tioned Galilei’s mathematization of na ture and mod ern phy -sicalistic ra tio nal ism, Edmund Husserl char ac ter izes this new phase in mod -ern phi los o phy as hav ing given birth to a rationalistic ideal of science(rationalistischen Wissenschaftsideal – 1954:119).

How ever, the mod ern free dom mo tive which, al most with an in ner ne ces sity,gave birth to the dom i na tion mo tive in the sci ence-ideal (na ture mo tive), fi -nally came into con flict with it self. If the whole of re al ity, by means of “re con -struct ing cre ative thought” could be framed in terms of ex act and in ex o ra blenat u ral laws of cause and ef fect (uni ver sal de ter min ism), it stands to rea sonthat the free dom of the sup pos edly au ton o mous per son al ity is re duced to, andde ter mined by, in vari able causal laws of na ture with out any free dom at all!The sci ence-ideal turned out to be a real Fran ken stein – dem on strat ing the in -her ent di a lec tic be tween the freedom-pole and the nature-pole in modernphilosophy.

The sub tle but ba sic dis tinc tion be tween ‘Erscheinung’ (ap pear ance/phe nom -e non) and “Ding an sich” (thing in it self), which Kant uses, is com pletely inser vice of his fun da men tal aim to safe guard a sep a rate (and super-sen sory)realm for be ing hu man as an au ton o mous eth i cal be ing (Zelbstzweck). Thecat e gory of cause and ef fect (to gether with all the other cat e go ries) is only ap -pli ca ble to ap pear ances and not to things in them selves (such as the free willof the human soul).

Kant re al izes that an un lim ited em ploy ment of the cat e gory of cau sal ity (un -der stood in the de ter min is tic and mech a nis tic sense of clas si cal phys ics) in ev -i ta bly im plies the ab o li tion of all free dom. Kant ex plains his basic problem asfollows:

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“Now let us sup pose that the dis tinc tion, which our Cri tique has shown to benec es sary, be tween things as ob jects of ex pe ri ence and those same things asthings in them selves, had not been made. In that case all things in gen eral, asfar as they are ef fi cient causes, would be de ter mined by the prin ci ple of cau sal -ity, and con se quently by the mech a nism of na ture. I could not, there fore, with -out pal pa ble con tra dic tion, say of one and the same be ing, for in stance the hu -man soul, that its will is free and yet is sub jected to nat u ral ne ces sity, that is,not free. For I have taken the soul in both prop o si tions in one and the samesense, namely as a thing in gen eral, that is, as a thing in it self; and save bymeans of a pre ced ing cri tique, could not have done oth er wise. But if our Cri -tique is not in er ror in teach ing that the ob ject is to be taken in a two fold sense,namely as ap pear ance and as thing in it self; if the de duc tion of the con cepts ofun der stand ing is valid, and the prin ci ple of cau sal ity there fore ap plies only tothings taken in the for mer sense, namely, in so far as they are ob jects of ex pe ri -ence – these same ob jects, taken in the other sense, not be ing sub ject to theprin ci ple – then there is no con tra dic tion in sup pos ing that one and the samewill is, in the ap pear ance, that is, in its vis i ble acts, nec es sar ily sub ject to thelaw of na ture, and so far not free, while yet, as be long ing to a thing in it self, itis not sub ject to that law, and is there fore free” (1967-B:XVII-XVIII).

It is clear that Kant’s ul ti mate con cern to safe guard the (au ton o mous) free domof hu man be ings ne ces si tated this dis tinc tion be tween ap pear ance and thing in it self. This is most ev i dent from the en tire Tran scen den tal Di a lec tic. In hisdis cus sion of the so lu tion of the third cos mo log i cal idea he once more ex -plains that we are not al lowed to as cribe any ab so lute re al ity to ap pear ances:“The com mon but fal la cious pre sup po si tion of the ab so lute re al ity of ap pear -ances here man i fests its in ju ri ous in flu ence, to the con found ing of rea son. For if ap pear ances are things in them selves, free dom cannot be upheld” (1967-B:564).

The fi nal re mark in this sub sec tion re veals the ba sic mo tive of Kant’s wholeCri tique of Pure Rea son (1967-B:565):

“My pur pose has only been to point out that since the thor ough-go ing con nec -tion of all ap pear ances, in a con text of na ture, is an in ex o ra ble law, the in ev i ta -ble con se quence of ob sti nately in sist ing on the re al ity of ap pear ances is to de -stroy all free dom. Those who thus fol low the com mon view have never beenable to rec on cile na ture and free dom” (I am emphasizing – DS).

Bridging the abyss teleologically

This in her ent di a lec tic, en closed in the ba sic mo tive of na ture and free dom, al -ready in his Cri tique of Pure Rea son brought Kant to a neg a tive in ter pre ta tionof hu man free dom: free dom is seen as be ing free from nat u ral ne ces sity(1967-B:651-652). In his Cri tique of Judge ment Kant de vel ops a most in flu -en tial for mu la tion of the way in which na ture and free dom pre sup poses eachother dia lec ti cally. Al though the hu man un der stand ing a pri ori ap plies the cat -e gory of cau sal ity, as an in ex o ra ble law, to na ture, Kant ap proaches or ganicna ture te leo logi cally. It means that na ture is thus rep re sented as if the mul ti -plic ity of laws pres ent in it is con tained in the uni fy ing ba sis of an un der stand -ing (1968-B:VIII). The con cept of a nat u ral te le ol ogy is pro posed by the ca -pac ity to judge, in or der to func tion as a me di at ing con cept be tween the con -cepts of na ture and the con cepts of free dom. How ever, the pur pos ive ness of

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na ture only func tions as a reg u la tive prin ci ple to the (re flect ing) ca pac ity tojudge (1968-B:LVI). As guid ing prin ci ple, this nat u ral pur pos ive ness is never to be used in a con sti tu tive way, since then our re flect ing abil ity be comes a de -ter min ing fac ulty of judge ment, im ply ing that once again we are in tro duc ing a new cau sal ity (a fi nal cause; nexus finalis; cf. 1968-B:269) into nat u ral sci -ence (1968-B:270).1

The te le o log i cal prin ci ple merely func tions as a sub jec tive maxim in judg ingna ture. There fore, it can not be ap plied to the ob jec tive re al ity of things in na -ture. Con se quently, the rec on cil i a tion be tween the caus ally de ter min ing andthe te leo logi cally re flect ing view of na ture is sought in the unity of a su -pra-sen sory prin ci ple which is sup posed to be valid for the to tal ity of na ture as a sys tem (1968-B:304). This ‘so lu tion’ did not re ally rec on cile the op pos ingpoles of na ture and free dom, since it sim ply re in forces the ba sic du al ism be -tween nat u ral ne ces sity and super-sen sory freedom – each with its ownlaw-giver (cf. 1968-B:LIII-LIV).

Fr. Schelling at tempted a syn the sis be tween na ture and free dom. Ac cord ing to him, in the ab sence of the antinomy (Widerspruch) be tween ne ces sity andfree dom, not only phi los o phy, but also ev ery higher will of the spirit willshrink into in sig nif i cance (1968:282). As a re sult of this com mit ment he be -lieves that in na ture it self a prin ci ple of free dom is con cealed, while his tory isfounded on a hid den prin ci ple of ne ces sity. Clearly, the re sult is not a real syn -the sis or rec on cil i a tion, since it amounts to noth ing but a du pli ca tion of theorig i nal di a lec tic: ne ces sity is pres ent in the do main of free dom, and freedomis present in the domain of necessity!

Entelechie negatively described: the influence of Hans Driesch

With out re ject ing the clas si cal mech a nis tic anal y sis of mat ter, Driesch, in hisneo-vitalistic bi ol ogy, ex tends the ap pli ca tion of the de ter min is tic con cept oflaw to bi otic phe nom ena. The tra di tional mech a nis tic ap proach is lim ited byhim to the ma te rial ba sis of liv ing things. We have seen that Driesch in ter -preted the re gen er a tive phe nom ena discernable in liv ing things in terms of histhe ory of liv ing en ti ties as “equi-po ten tial har mo ni ous sys tems” and in termsof his no tion of an entelechie op er at ing as a “to tal ity-causal fac tor” (Ganz -heitskausalität). The im por tant con tri bu tion which Driesch made to the prob -lem of free dom, is given in his no tion of the ‘entelechie’ as some thing thatcan not be de ter mined in any pos i tive sense. As such, he con sid ers it to be a“sys tem of ne ga tions” (1920:513; 459 ff.), i.e., it can not be pos i tively de ter -mined: ‘entelechie’ is some thing non-me chan i cal, it is not en ergy, not force,not a con stant (1920:460) and non-spa tial (1920:513). The dif fer ence be -tween the atomistic ‘Einselkausalität’ and the ho lis tic ‘Ganzheitskausalität’ isalso framed in terms of the op po si tion ‘Ganzheit’ and ‘Zufall’ (to tal ity andchance). In the thought of Driesch de ter mi na tion is op posed to gen u ine free -dom. He de clares that the ques tion about free dom is to be con sid ered as ameta phys i cal ques tion of faith which can not be an swered by the sci ence of

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1 Excatly this was done in the neo-vitalistic bi ol ogy of Hans Driesch. Cf. his no tion of‘Ganzheitskausalität’ (1920:416 ff., 542 ff.).

phi los o phy (cf. 1931:93-122). Al though Kant and Driesch dif fer in their viewon the na ture of phi los o phy, they agree that free dom is not a question ofscientific proof, but one of (practical) faith.In his the ory of the free dom of the will, Ar nold Gehlen con tin ues Driesch’sneg a tive de scrip tion of the ‘entelechie’. How ever, with an ex plicit ap peal tothe free dom ide al ism of Schelling, he im me di ately trans forms it in or der topro vide a point of en try for free dom. At the same time he re al izes that Drieschac tu ally brought bi oti cal phe nom ena un der the reign of the de ter min is tic clas -si cal ideal of sci ence. There fore, once again he wants to re strict cau sal ity tome chan i cal cau sal ity: “Since cau sal ity is only think able as me chan i cal cau sal -ity, the entelechie is neg a tively free, i.e. spon ta ne ous and pri mary in a sensewhich cannot be subjected to a closer determination” (1965:60).The ten sion be tween na ture and free dom brought Max Scheler to hiswell-known char ac ter iza tion in terms of what he calls the ‘Weltoffenheit’ ofhu man be ings (1962:38, 40).1 Against this back ground Plessner de vel opedhis own per spec tive on the hu man be ing as an ec cen tric crea ture, while bi ol o -gists and an thro pol o gists such as Portmann, Overhage and Gehlen gave theno tion of ‘Weltoffenheit’ a prom i nent place in their writ ings. Even the ol ogytook ad van tage of this no tion. Wolfhart Pannenberg, for ex am ple, in ter prets it in terms of what he calls “der grenzenlosen Angewiesenheit des Menschen”(the un lim ited de pend ency of the hu man be ing) while re lat ing it to the “fun da -men tal bi o log i cal struc ture of be ing hu man” (1968:11; cf. also Scherer’streat ment of the ‘Weltoffenheit’ of the hu man be ing, 1980:79 ff.). Ul ti mately,this term ‘Weltoffenheit’ is used to em body the re ac tion against the claims ofthe sci ence ideal, namely that the hu man be ing is de ter mined in all re spects. In the fi nal anal y sis, the in ten tion of these au thors is to show that the humanbeing is free from being determined by natural causality.In his Ph.D-the sis, deal ing with philo soph i cal as pects in the bi ol ogy ofPortmann, R. Kugler states that Portmann es sen tially un der stands the hu manbe ing in terms of free dom (1967:75). At the same time, Portmann is wellaware of the fact that, as a “philo soph i cal idea”, free dom with draws it selffrom a sci en tific grasp. Kugler places this ap proach within the “large tra di -tion” of a “philo soph i cal de ter mi na tion of the hu man be ing,” dat ing back toIm man uel Kant: “The in ner most es sence of the hu man be ing is free dom, it isthe pos si bil ity of the hu man be ing to trans form it self into that what it is”(1967:81). Com pare this an nounce ment with the fol low ing words of Plessner: “As ec cen tri cally or ga nized crea ture the hu man be ing must make itself intothat what it already is” (1965:309).Gehlen points out that this mode of ex pres sion man i fests the log i cal schemepres ent in a nor mal te le ol ogy. This tra di tion is in flu enced by Fichte: “I want to be free ... means: I want to make my self into that what I shall be be fore I am it,in or der to be able to per form it” (cf. Gehlen, 1965:103-104). And we haveseen that Fichte him self is de pend ent on Kant, who in tro duced te le ol ogy as a

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1 In this work, Scheler sketches ab so lute be ing as an end less, re cip ro cal interpenetrationof spirit (Geist) and drive (Drang) – the for mer has to guide and di rect the lat ter, but onlyre ceives its power from this equally orig i nal life-drive.

bridge to hu man free dom. The philo soph i cal tra di tion in which “me chan i calcau sal ity” and “te le ol ogy” (na ture and free dom) is al ways dia lec ti cally re -lated, in spires Ed. von Hartmann to re mind nat u ral sci en tists in the fol low ingway: “If our nat u ral sci en tists were philo soph i cally better trained, they wouldhave been aware of the fact that the whole Ger man spec u la tion, from Leibnizto Kant and up to the pres ent, equally de ci sively re jects a te le ol ogy sep a ratedfrom me chan i cal cau sal ity, as it does with a mechanical causality divorcedfrom teleology” (quoted by Haas, 1959:456).

Reinforced dialectics: Existentialism and Existential PhenomenologyNot with stand ing the fact that var i ous philo soph i cal trends of the 20th cen turyde parted from the ra tio nal is tic phi los o phy of Kant, the un der ly ing mo ti vat ingpower pres ent in the “leit mo tif” of na ture and free dom re mained in force. Theex is ten tial phenomenological thinker, Merleau-Ponty, for a great part re ly ingon the re sults of psy cho log i cal and psy cho-patological stud ies, un der standsthe hu man be ing dia lec ti cally in terms of two ba sic de nom i na tors: be ing abody (taken in a bi oti cal sense as an or gan ism) and ex is tence (in ter preted asbe ing his tor i cal in na ture). On the one hand, to gether with Sartre, he ac ceptsthe the sis: “I am my body”. On the other hand, how ever, he also holds theopin ion that one’s his tor i cal ex is tence must re press the bodily organism downto the pre-personal level of an anonymous complex.

In spired by the na ture-pole of the ba sic mo tive (ground-mo tive) of hu man ism, Merleau-Ponty writes: “I can not un der stand the func tion of the liv ing bodyex cept by en act ing it my self, and ex cept in so far as I am a body which rises to -wards the world” (1970:75). From the op po site mo ti va tion he states: “... so itcan be said that my or gan ism, as a pre-per sonal cleav ing to the gen eral form of the world, as an anon y mous and gen eral ex is tence, plays, be neath my per -sonal life, the part of an in born com plex” (1970:84). On the one hand I am mybody, and on the other hand my body is seen as a pre-re flex ive, pre-per sonal,anon y mous com plex by vir tue of its be ing-in-the-world (1970:79, 80, 82, 83,86). Na ture and free dom re cip ro cally en dan ger and pre sup pose each other:“... for most of the time per sonal ex is tence re presses the or gan ism with out be -ing able ei ther to go be yond it or to re nounce it self; with out, in other words,be ing able ei ther to re duce the or gan ism to its ex is ten tial self, or it self to theor gan ism” (1970:84). The di a lec ti cal move ment, to and fro, be tween thesepoles is best il lus trated in his fol low ing words: “Man taken as a con crete be ing is not a psy che joined to an or gan ism, but the move ment to and fro of ex is tence which at one time al lows it self to take corporeal form and at others movestowards personal acts (I am emphasizing – DS)” (1970:88).

Per haps Karl Jaspers saw the im passe of this whole di a lec ti cal leg acy mostclearly. His con fes sion reads: “Since free dom is only through and against na -ture, as free dom it must fail. Free dom is only when na ture is” (1948:871).

Freedom at the molecular levelSome times it is strik ing to see what the ef fect is of the pre sumed con tin u ousand un in ter rupted line of as cent from mol e cules to the hu man be ing. Clearly,if one wants to as cribe free dom to the hu man be ing, the con ti nu ity of the pos -

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tu lated ge netic pro cess de mands that noth ing truly novel can arise some whereon the line. Con se quently, Hans Jonas, pro ceed ing from the pri macy of thefree dom mo tive, is ‘forced’ to rec og nize free dom al ready at the mo lec u larlevel! “Our po si tion is in deed that al ready me tab o lism, the bot tom layer of allor ganic ex is tence, re veals free dom, yes, that, in it self, it is the first form offree dom” (1973:13). ‘Life’, ac cord ing to Jonas, “con stantly man i fests it self in the di a lec ti cal an tith e ses be tween which its ex is tence is stretched: the an tith e -sis be tween be ing and non-be ing, be tween self and world, form and mat ter,free dom and ne ces sity (I am em pha siz ing – DS)” (1973:15-16). Ber nardRensch is com mit ted to ex actly the op po site con clu sion, al though he sharesthe con vic tion about the con ti nu ity of the line from mol e cules to the hu manbe ing: “Ac cord ing to our pre vi ous find ings and dis cus sions we are jus ti fied in as sum ing .... psy chic (par al lel) pro cesses of some kind in all liv ing be ings”(1959:352). This ‘psy chic’ con ti nu ity must also bridge the gap be tween theliv ing and the non-liv ing: “Here again it is dif fi cult to as sume a sud den or i ginof first psy chic el e ments. It would not be im pos si ble to as cribe ‘psy chic’ com -po nents to the realm of in or ganic sys tems also, i.e. to credit non liv ing mat terwith some ba sic and iso lated kind of ‘par al lel’ pro cesses” (1959:342). Thusmat ter re ceives a “proto-psy chi cal na ture” (1969:134-135). And since theuni verse is ruled by eter nal ba sic laws, Rensch can not ac cept any free dom ofthe will: “If ‘free will’ really existed it would have emerged in the head of thehuman being, thereby disrupting the causal law which governs the processesof the brain” (1971:211).

The rejection of structural conditions: nominalismIt is tre men dously dif fi cult for mod ern phi los o phy to ac cept con stant and uni -ver sal con di tions un der ly ing our hu man free dom. As a re sult of the over -whelm ing in flu ence of mod ern nomi nal ism, the uni ver sal creational or der forand the (uni ver sal) or der li ness of en ti ties sub ject to the for mer con di tions, aremostly re jected. Some of the most prom i nent trends in mod ern phi los o phystress the ever-chang ing and con tin gent na ture of the world in which we live.Rauche, for ex am ple, is con vinced that a per son’s “ba sic con tin gent ex pe ri -ence of the world”, its be ing in ter wo ven with life’s “chang ing con di tions”,im plies that the the o ries a per son ad vo cates “can never be con clu sive”, andcon se quently, they “should be re garded as blue prints which at tempt to give”one “mean ing ful guid ance in the per ma nent flux of be com ing” that one findsone self “in and of which they are an in te gral part” (1985:11; cf.20, 75, 87, 96,137). It is not nec es sary to re late the idea of con stancy to ‘God’, ‘Be ing’ or ameta phys i cal ‘Ab so lute’ (cf. Rauche, 1985:12), since the first appeal shouldbe to the conditioning order for creaturely existence.Dur ing a visit to South Af rica Rich ard Rorty was even an nounced as a spe cial -ist on con tin gency! I do not want to deny the unique ness, in di vid u al ity andcon tin gency in deed pres ent in our ex pe ri ence of the world. How ever, I dothink we have to be very cau tious in or der to avoid the pit falls of mod ernnomi nal ism. In this con text, it will suf fice to point out that all con tin gency and changes can only take place within the bound aries of struc tural con di tionswhich are not only uni ver sal, but which are also con stant. In an other con text, I

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have an a lyzed some as pects of this prob lem (cf. Strauss, 1985:133 ff, 138 ff.;cf. also Strauss, 1984:36-37 and Hart, 1984:65 ff.). The fun da men tal point incon nec tion with the prob lem of hu man free dom is to re al ize that hu man free -dom should be eval u ated in terms of the strict cor re la tion be tween uni ver salnor ma tive con di tions and sub jec tive re sponse of hu man kind to them. Hart isfully jus ti fied in say ing that he “will de fend the view that be ing free is not op -posed to be ing de ter mined” since “only what is de ter mined can be free andonly what is free can be de ter mined” (1984:298). Here ‘de ter mined’ can mean noth ing but “be ing sub ject to a uni ver sal con di tion ing or der”. How ever, the‘or der-di ver sity’ in cre ation con fronts us with dif fer ently struc tured sub jec -tive re sponses by crea tures, and it is in terms of this per spec tive that we seehu man free dom as an out come of the unique ac count able abil ity of hu man be -ings to re spond (‘re sponse-abil ity’). What is unique about hu man be ings isnot that they are free from con di tions, but that they in their sub jec tion to them,ac tu ally are free to obey them in uniquely vary ing ways and even, ul ti matelyas an ef fect of sin, has the temp ta tion to dis obey them. The his tory of ar gu ingfor a spe cial place of hu man be ings on earth is of ten dia lec ti cally mo ti vated by the urge to see the human being, not as something conditioned, but, inopposition to nature, as the (autonomous) origin of conditions (cf. Hart,1985:295).

The no tion of a uni formly mov ing body, un der ly ing Galilei’s law of in er tia,was for mu lated by him in terms of a thought-ex per i ment, with out tak ing ac -count of any real sense-ex pe ri ence. This in spired Kant’s whole epis te mol ogy(cf. Holz, 1975:345-358). Von Weiszäcker frames Kant’s epistemologicalprob lem in terms of the ques tion: What is na ture, that it must obey laws whicha hu man be ing could for mu late with his/her un der stand ing (1971:128)? Kantim plic itly in ter prets the Gal i lean pro ce dure as fol lows: Since the law of in er tia is de rived and pre scribed to mov ing en ti ties out of the pure un der stand ing ofthe hu man be ing in its spon ta ne ous sub jec tiv ity, Kant brings about the (his tor -i cally cru cial) Co per ni can turn in epis te mol ogy, in as crib ing the pri mary nolon ger to the ob ject, but to the sub ject. Kant draws the rad i cal (ra tio nal is tic)hu man is tic con clu sion – the laws of na ture are a pri ori con tained in the sub -jec tive un der stand ing of the hu man be ing: “the cat e go ries are con di tions ofthe pos si bil ity of ex pe ri ence, and are there fore valid a pri ori for all ob jects ofex pe ri ence” (1967-B:161); “Cat e go ries are con cepts which pre scribe laws apri ori to appearances, and therefore to nature, the sum of all appearances”(B:163).

This ra tio nal is tic in cli na tion of Kant was even tu ally historicized. The man nerin which Rauche frames the prob lem of truth clearly por trays Kantian un der -tones: “Truth is a mat ter of the mind. It is the trans la tion of our sense-ex pe ri -ence into ra tio nal terms or con cepts, while the real is per ceived through thesenses and is yet still cha otic and un or ga nized by the mind” (1971:9). He nolon ger ac cepts the Kantian no tion of un der stand ing which is ca pa ble of a uni -ver sally valid act of form-giv ing (or der ing) – ev ery per son can only ac countfor his/her own par tic u lar con sti tut ing ac tiv ity. Our rel a tive hu man per spec -tives are rooted in a finite and contingent world verified by our experience.

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“It is the world of be com ing and change, namely, the ever chang ing con creteob jects, which rep re sent the en vi ron ment of hu man be ings, caus ing them tofeel un cer tain and in se cure, so that they are im pelled to or der it ra tio nally. It isthus not a vague ab stract ”some thing“ to ward which we di rect our in tent, but itis a con crete sit u a tion that causes us to build our world, which in its stage ofcon sti tu tion must be pe cu liar, dif fer ent and in this sense con tra dic tory to thecon sti tuted world of our fellow human beings” (Rauche, 1966:99).

The Kantian no tion of uni ver sal va lid ity is fun da men tally historicized – leav -ing us with noth ing but com pet ing and con tra dic tory or dered worlds of dif fer -ent peo ple in dif fer ent his tor i cal sit u a tions (cf. Rauche, 1971:34).

Al though the ra tio nal is tic and irrationalistic trends in mod ern phi los o physeem to di vert rad i cally, their com mon root in nomi nal ism tran scends this su -per fi cial di ver gence. Ra tio nal ism con sid ers uni ver sals to be the only source of knowl edge, thus leav ing no room for knowl edge of things in their in di vid u al -ity. Surely, con cept-for ma tion is al ways bound up with the uni ver sal or derfor, and the uni ver sal or der li ness of things. This im plies, as al ready dis cov -ered by Ar is totle, that one can not con cep tu ally com pre hend the in di vid ualside of an en tity. Un for tu nately, in a typ i cal ra tio nal is tic way, he iden ti fiesknowl edge with con cep tual knowl edge, im ply ing that some thing in di vid ualcan not be known (cf. Metaf. l040 a 5 ff.). Con trary to this ra tio nal is tic po si -tion, we must em pha size that in fact we do have knowl edge of things in theirin di vid u al ity, al though this kind of knowl edge is not con cep tual. Much rather, it is of a con cept tran scend ing and ap prox i mat ing na ture, re fer ring to the in di -vid ual side of things in terms of uni ver sal fea tures. But this is pre cisely whatidea-knowl edge is all about – an idea con cen trates a con cep tual di ver sity upon (resp. re fers it to) that which tran scends the lim its of all con cept-for ma tion.There fore, ra tio nal ism leaves no room for idea-knowl edge. Irrationalism, onthe other hand, al ways wants to em pha size the con tin gent unique ness of thein di vid ual side of en ti ties or events tran scend ing the lim its of con cept-for ma -tion. Consequently, irrationalism leaves no room for real conceptualknowledge.

In re spect of the typ i cal struc ture of en ti ties, nomi nal ism does not ac cept anycon di tion ing or der (uni ver sal struc tures for), or any or der li ness (uni ver salstructuredness of) such en ti ties. Ev ery en tity is strictly in di vid ual. In terms ofour dis tinc tion be tween ra tio nal ism and irrationalism, nomi nal ism surely rep -re sents an irrationalistic view in con nec tion with the na ture of en ti ties, sinceev ery in di vid ual en tity is com pletely stripped from its uni ver sal or der li ness(law-con for mity) and con di tion ing or der. This char ac ter is tic ap plies to bothmod er ate nomi nal ism, viz. con cep tu al ism (Locke, Ockham, Leibniz and oth -ers), and to ex treme nomi nal ism, that re jects all gen eral and ab stract ideas andac cepts only gen eral names (Berke ley and Brentano). This irrationalistic sideof nomi nal ism, how ever, does not ex haust the mul ti fac eted na ture of nomi nal -ism, be cause uni ver sals are fully ac knowl edged in the hu man mind, at least asgen eral words in the case of Berke ley’s and Brentano’s ex treme nomi nal ism.This re stric tion of knowl edge to uni ver sals is typ i cal of ra tio nal ism in thesense de fined by us. There fore, it is pos si ble to see nomi nal ism as be ing si -mul ta neously ra tio nal is tic (in terms of the universals – concepts and words –

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in one’s mind), and irrationalistic (in terms of the strict individuality ofentities).

The common root of diverging trends in modern philosophy

This dual na ture of nomi nal ism forms the start ing-point of two di verg ingphilo soph i cal de vel op ments in mod ern philosophy.

(i) On the one hand, it pro vided ra tio nal ism with the pos si bil ity to el e vatehu man rea son to the level of the cre ator of a ra tio nal or der in re al ity. This fol lows from the fact that nomi nal ism in fact trans poses the uni ver salside of en ti ties into the hu man mind. But the uni ver sal side of en ti ties isnoth ing but the man i fes ta tion of the conditionedness of en ti ties by therel e vant uni ver sal or der for their ex is tence. Con se quently, if an en tity isstripped of its or der li ness (its uni ver sal side), it is si mul ta neouslystripped of its be ing sub jected to a uni ver sal creational or der. What isleft is fac tual re al ity in its un struc tured, cha otic in di vid u al ity and par tic -u lar ity (con tin gency) (cf. Rauche, 1966:97). Driven by the new mo tiveof log i cal cre ation, this very fea ture of nomi nal ism en abled mod ern phi -los o phy from Des cartes on wards to re con struct all of re al ity in terms ofnat u ral sci en tific thought. Only the ex treme con se quences of this nat u ral sci ence-ideal, cancelling in prin ci ple also hu man free dom, were ques -tioned by Kant. Within the (lim ited) do main of the sci ence-ideal, how -ever, Kant draws the ul ti mate ra tio nal is tic con clu sion of nomi nal ism. In -deed, Kant tries to con sol i date and strengthen the pre ced ing nat u ral sci -ence-ideal, be it in the re stricted form of the ra tio nal is ti cally el e vated un -der stand ing which (though lim ited to sen si bil ity in or der to save a sep a -rate super-sen sory do main for the prac ti cal-eth i cal free dom of au ton o -mous hu man ity), is con sid ered to be the a pri ori (for mal) law-giver ofna ture! Nomi nal ism cre ated a vac uum by leav ing fac tual re al ity in its in -di vid u al ity un struc tured. In or der to fill up the lack of de ter mi na tion thus cre ated, Kant in tro duces hu man un der stand ing to take hold of this va -cant po si tion. To be sure, Kant not merely trans poses the uni ver sal sideof en ti ties into human understanding, since in fact he elevates humanunderstanding to the level of the conditioning order for things.

(ii) On the other hand, nomi nal ism pro vided a start ing-point for all thosetrends in mod ern phi los o phy which, in an irrationalistic fash ion, want totake the unique and con tin gent char ac ter of (mostly des ig nated as: his -tor i cal) re al ity se ri ous. This av e nue opened up by nomi nal ism was fol -lowed up by a va ri ety of historicistic de signs in mod ern phi los o phy, forex am ple from the forth phase of Fichte’s thought up to prag ma tism, ex -is ten tial ism and con tem po rary neo-Marx ism. If re al ity is trip ped both ofits or der li ness and of its be ing sub jected to a con di tion ing uni ver salcreational or der, it seems to be a “self-ev i dent historicistic truth” that, ul -ti mately, ev ery thing is his tor i cal and there fore taken up in the dynamicand ever-changing contingent flow of historical events.

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How ever, this does not nec es sar ily mean that the or der ing func tion of un der -stand ing is cancelled, as is ev i dent in Rauche’s con cep tion of the hu man taskof self-con sti tu tion in or der to tran scend pure contingency.

At this point we can link up the in flu ence of nomi nal ism with the pre dom i nant neo-Darwinistic evolutionism. The re mark of Simpson, re ferred to in an ear -lier con text, namely that plants and an i mals are not types and do not havetypes, since ev ery one of them is unique (1969:8-9), is a fully-fledgednominalistic con vic tion. The gen e sis of plants, an i mals and hu man be ings aretaken up in a structureless con tin uum. Sys tem atic dis tinc tions, ex em pli fied indif fer ent tax on o mies, are noth ing but ar bi trary names (nomina) given to anim mense num ber of in di vid u ally dif fer ent liv ing en ti ties. The uni ver sal ity im -plied in these names is a prod uct of our con sti tu tive hu man un der stand ingwith out any foun da tion in the “things out side the mind”. Al ready CharlesDar win ad hered ex plic itly to this view in his “Or i gin of Spe cies”. He says“that no line of de mar ca tion can be drawn be tween spe cies” (1968:443) andpro ceeds: “In short, we shall have to treat spe cies in the same man ner as thosenat u ral ists treat gen era, who ad mit that genera are merely artificialcombinations made for convenience” (1968:456).

Within the con text of an evo lu tion ary epis te mol ogy Van Huyssteen re centlydem on strates a strange mix ture of dif fer ent po si tions at once. He is in searchaf ter a new (in ter dis ci plin ary) space for the in ter ac tion be tween the ol ogy andsci ence. He pro ceeds from the as sump tion that evo lu tion is a fact (1998:143)and that we have to “take very se ri ously the gen eral con clu sions and find ingsof gen eral cos mol ogy ” – “that is that this uni verse is evolv ing, that all that iswithin it has had a com mon phys i cal or i gin in time, and that all it con tains is inprin ci ple ex pli ca ble by the nat u ral sci ences” (1998:75). How ever, he does notes cape from fun da men tal am bi gu ities. The neo-Dar win ian pre sup po si tion ofcon ti nu ity (cf. 1998:111) and chance are slowly but surely sub sti tuted with amix ture of emer gent evolutionistic (cf. 1998:134, 151) and vitalistic (cf.1998:37, 121, 125, 127) over tones – with out evinc ing an aware ness that thesepo si tions are al ter na tive to neo-Dar win ian the ory and that they con tra dict itsba sic as sump tions.1

An amaz ing re turn to the ra tio nal is tic po si tion of Kant and mo der nity is seenin his iden ti fi ca tion of the struc ture of the uni verse with hu man ra tio nal ity and math e mat ics: “What is as tound ing, how ever, is to what ex tent our world istruly ra tio nal, i.e., in con for mity with hu man rea son” (1998:68). While men -tion ing Davies he re fers to the “fact that the ra tio nal na ture of our uni verse isre flected in its ba sic math e mat i cal struc ture” (1998:71). Van Huyssteen andthe mod ern ist (ra tio nal is tic) tra di tion on this point do not dis tin guish be tween

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1 The fol low ing state ments show his emer gent evolutionistic po si tion, con ti nu ity in as -cent, dis con ti nu ity in ex is tence: “Cul ture in deed has evolved, but the prin ci ples of cul -ture are not the same as the prin ci ples we know from or ganic evo lu tion” (1998:146); andin af firm ing the ap proach of Wuketits he says: “cul ture is not re duc ible to bi o log i cal en -ti ties” (1998:157). On page 130 he ex plic itly em ploys the phrase “emergent evolution.”

ontically given uni ver sal fea tures of re al ity and the na ture of con cept- for ma -tion. From the fact that con cepts are formed on the ba sis of uni ver sal traits itdoes not fol low at all that these ontic porperties them selves are ra tio nal in na -ture! This po si tion is taken while at the same time an equally force ful attemptis made throughout the work to hold on to a postmodern perspective!

Human freedom: a subjective response to normative conditions

What is at stake here, is a con fron ta tion with historicism on the ba sis of ac -cept ing norms or prin ci ples in the fol low ing sense: a prin ci ple or norm is auni ver sal and con stant unit that can only be made valid (en forced) in dif fer entsit u a tions by a com pe tent or gan pos sess ing an ac count able will which pro -vides the free dom of choice to es tab lish a nor ma tively cor rect or anti norm -ative positivization (form-giv ing) of the pos si bil i ties con tained in such a start -ing-point. Only a positivized prin ci ple pos sesses va lid ity. It is there fore con -trary to the very na ture of a pre-pos i tive prin ci ple, pro vid ing the start ing-point for form-giv ing ac tiv i ties in all dif fer ent sit u a tions, to char ac ter ize such apre-pos i tive start ing-point as uni ver sally valid. The va lid ity of any positivized prin ci ple is fun da men tally re stricted to the unique set ting of a spe cific place ata par tic u lar time. Con se quently, the nat u ral law view is un ten a ble, since it as -cribes a va lid ity to norms that hold for all times and places. Hart uses the ex -am ple of ex press ing re spect, which was instantiated in greet ing rit u als of var i -ous dif fer ent kinds – from tak ing off one’s hat up to sim ply rais ing the hand.In spite of all that var ies, he says, “some thing ‘in prin ci ple’ re mains in vari antthrough all this his tor i cal de vel op ment” (1984:59), viz. show ing re spect. Thisprin ci ple should not be tied down to only one kind of re sponse (lift ing the hator rais ing the hand): “The legalist who claims that those who just tip their hatsare in prin ci ple not show ing the proper re spect is mak ing the same mis take asthe nomi nal ist: he is fail ing to dis tin guish un der ly ing prin ci ples in theirinvariance from the ob serv able pat terns of vari ant be hav ior” (1984:59). It is apity that Kugel, who ex plic itly fol lows Groenman’s reformational model ofbe ing hu man (cf. 1982:135), only ac knowl edges four types of norms – viz. the eco nom i cal, the ju rid i cal, the eth i cal, and the aes thet i cal (1982:280-283). It isnot even enough to re fer to the normativity of all the post-sen si tive as pects,since, at the norm-side of each one of these mo dal i ties, ev ery retrocipation and an tic i pa tion un veils a fundamental modal norm (I have treated thisperspective elsewhere in more detail – Strauss, 1979: 254-264).

The pos si bil i ties con tained in any uni ver sal and con stant start ing-point func -tion as the ba sis for spe cific acts of form-giv ing (positivization) in di verg ingunique his tor i cal sit u a tions. The ra tio nal is tic trait of nat u ral law con cep tionscan not ac count for this free dom to positivize con tained in a prin ci ple. Theirrationalistic na ture of historicism, on the other hand, can not do jus tice to theuni ver sal ity and con stancy of such a start ing-point which ac tu ally form theba sis of ever-chang ing positivizations. This one-sid ed ness of both nat u ral law and historicism is a di rect con se quence of the au ton omy-theme in mod ern phi -

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los o phy men tioned ear lier. The au ton omy-ideal hypostatized the free dom topositivize – thus try ing to elim i nate the very na ture of a prin ci ple as a uni ver sal and con stant start ing-point for hu man ac tion. When positivizations are el e -vated to the level of be ing uni ver sally valid, we en coun ter ra tio nal is tic ca su -istry. And when the free dom to positivize is one-sidedly accentuated, weencounter an irrationalistic situational ethics.

The way in which the ma jor ity of con tem po rary so cial sci en tists use terms like val ues, norms, be liefs (cf. Sorokin, Par sons, Znaniecki, and oth ers), some -times called the cul tural sys tem, does not al low for prin ci ples as uni ver sal andcon stant start ing-points that ul ti mately con di tion hu man ac tion in a task-set -ting way, since they iden tify these terms with the re sult of free and for ma tivehu man ac tions – typical of historicism.

The long-stand ing in flu ence of nomi nal ism in our mod ern West ern cul turehas ul ti mately suc ceeded in rul ing out the bib li cal view on the creational or der for the ex is tence of creaturely sub jects. The rel a tiv is tic and self-con tra dic toryna ture of historicism is sim ply a symp tom of the con tem po rary world view.Any con fron ta tion with historicism that does not pen e trate into this pre-sci en -tific root has not suc ceeded in un veil ing its deepest motivation and impasse.

What, in the fi nal anal y sis, is there fore ul ti mately de ci sive, is the ba sic (pre- the o retic) com mit ment to the mod ern historicistic world view with its au ton -omy-ideal1 and nominalistic the o ret i cal ar tic u la tions on the one hand, or, onthe other hand, the com mit ment to a dif fer ent world- and life-view, namelythat of bib li cal Chris tian ity, which does al low for the ac cep tance of uni ver saland con stant prin ci ples which (as creational or der for) con di tion hu man sub -jec tiv ity in a truly nor ma tive way and at the same time leave hu man ity withthe ac count able au then tic free dom to positivize responsibly in changing histo -rical situations.

The or der of cre ation in deed shows us the good di rec tion to wards obe di enceto the will of God. How ever, due to the rad i cal na ture of the fall into sin, thisGod-obe di ent di rec tion was re di rected in ser vice of some or other idol bornefrom the apos tate heart of hu man kind. The creational or der still ex er cises itsnor ma tive ap peal to obey the will of God, but in or der to ac com plish this wemust be freed from the ef fects of sin by the re demp tive work of Christ. Only in Him and through the work of the Holy Spirit are we, in prin ci ple, freed fromthe apos tate in cli na tion of our hearts and re di rected to wards obe di ent ser viceto God within the world-wide, all-en com pass ing King dom of God in Christ.Obe di ence to God-given creational pos si bil i ties is a pos i tive task, not some -thing struc tur ally neg a tive which we have to transcend.

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1 The def i ni tion which Rous seau gives for free dom, dem on strates the hu man is tic ideal ofself-de ter mi na tion (au ton omy) ex plic itly: “obe di ence to a law which we pre scribe toour selves is liberty” (1966:16).

Conclusion

The hu man be ing is not sim ply an ex ten sion of the an i mal realm. By means ofthe pre ced ing ex po si tion, this con vic tion is sub stan ti ated with the aid of var i -ous ar gu ments and in terms of di verse per spec tives. The cru cial ‘turn -ing-points’ in the (neo-Darwinistic) ac count of the all-en com pass ing pro cessof evo lu tion, namely the or i gin of ‘life’ and the emer gence of hu man be ings,in a marked way high light the in ad e quacy of this pre dom i nant mode of think -ing. First of all we dis cerned a dif fer ence of opin ion in con nec tion with the no -tions of con ti nu ity and dis con ti nu ity. Sec ondly, mod ern bi o log i cal think ingtries to sub sume this prob lem of con ti nu ity/dis con ti nu ity un der dif fer ent ba sic de nom i na tors, such as the me chan i cal (Eisenstein), the phys i cal (neo-Dar win -ism), the bi oti cal – in dif fer ent ways (neo-vi tal ism, ho lism, or gan is mic bi ol -ogy), the psy chi cal (be it mo nis tic: Teilhard de Chardin – or plu ral is tic: Ber -nard Rensch’s pan-psychistic identism), while even free dom is cho sen (HansJonas). Some times, the ob vi ous struc tural dis con ti nu ities discernable be -tween ma te rial things, plants, an i mals and hu man be ings, caused an am biv a -lent (emergentistic) at ti tude, try ing to have it both ways: ge netic con ti nu ityand ex is ten tial dis con ti nu ity (Lloyd Morgan, Whitehead, Woltereck, Bavink,Polanyi, Laszlo, Dobzhansky and even certain statements of Simpson andJulian Huxley).

None of the treated schol ars, how ever, asked the ques tion why, in spite of thesup posed con tin u ous (and: structureless) change con stantly oc cur ring, thethe o ries about these evo lu tion ary changes re main con fined within the men -tioned modal di ver sity. Much rather, this state of af fairs con fronts mod ern bi -o log i cal thought with the in es cap able con di tion ing role of this modal di ver sity for the o riz ing as such. The di a lec ti cal es cape route which these trends try topur sue is to ig nore this given di ver sity by ar gu ing as if it does not ex ist. Con -trary to this in ten tion, nev er the less, ev ery the ory pre sented to us sim ply cameup with an over-es ti ma tion of one of these modal per spec tives, not re al iz ingthat it is only while work ing and think ing within the con di tion ing ‘or der-di -ver sity’ of re al ity, that we even can at tempt ig nor ing this con stant ‘or der-di -ver sity’! Ul ti mately, the choice of any spe cific ba sic de nom i na tor is fully inthe grip of the un der ly ing ba sic com mit ment of the thinker in question – acommitment transcending the realm of theoretical thinking as such.

Un cer tain ties and even con tra dict ing in ter pre ta tions of these fun da men talques tions warn us to be mod est in our of ten pre ma ture con clu sions. Muchrather, it is im por tant to em pha size what one can know about the unique nessof be ing hu man – both in terms of the ex cep tional abil ity to re spond in nor ma -tive free dom and in re spect of the way in which hu man be ings func tion dis -tinctly in those as pects of re al ity which they share with other creatures.

We have seen that the prev a lent di a lec ti cal ap pre ci a tion of the mean ing of hu -man free dom is the out come of an un der ly ing mo tive power op er a tive in thisphilo soph i cal leg acy, which is, ul ti mately, apos tate in char ac ter. The unsol va -ble ten sion be tween the poles of na ture and free dom in ev i ta bly leads to a neg -

156

a tive idea of free dom, dia lec ti cally op posed to na ture. Fur ther more, our hu -man ‘nat u ral’ fea tures, such as the bodily con fig u ra tion of the hu man be ing,our unique bi oti cal de vel op men tal sta tus, and the rel a tively un spe cial ized or -gans, are, to gether with the erect gait and spir i tual ex pres sion of the face, all in ser vice of our nor ma tively qual i fied truly hu man re spon si bil ity to obey theuni ver sal con di tions of God’s creational or der. Though, in the pres ent sin fuldis pen sa tion, we shall al ways be tempted to dis obey these nor ma tive con di -tions, in Christ we are in prin ci ple saved from this sin ful in cli na tion and freedto con stantly act in more norm-con form ing ways, show ing, in an tic i pa tion toGod’s coming Kingdom, that already now we share in the restored paradiseorder of obedience and peace.

157

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170

In dex of Tech ni cal TermsA

Ad e nine 99adenosine triphosphate 101aes theti cism 15al bu min 98al ge bra 14amino ac ids 99am phib i ans 94-95, 138Anaxagoras 22, 25Anaximander 21, 31Anaximines 21Angelelli 20an thro poids 117, 128-130,

134, 137-138, 140, 143an thro po mor phic 132an tic i pa tions 77antinomy 102, 108, 147antonymity 141Archaeopteryx 96, 103articulare-quadratum-joint 95as tron omy 73at once in fi nite 58-59atomic nu clei 78, 80-81atomic num ber 69, 75, 80atomistic 77, 81, 147Aus tra lo pith ecines 120-122,

135Australopithecus 121, 163,

165, 169Australopithecus africanus 120,

165

Bbac te ria 88, 93, 100-102bio chem i cal con stel la tion 114bi o log i cal

– dis or ga ni za tion 107– the ory 94

bio-milieu 85bi oti cal in sta bil ity 86biotope 108blue-green al gae 100-102bond ing elec trons 80

Ccal cium 98car bo hy drates 98car bon di ox ide 87cell or gan ism 100, 102,

113-115cell-structure 89cen tral in stance 112

chem i cal bond ing 78, 80-81chro mo somes 100-101cine radio graph ic stud ies 134classes of ver te brates 94Cloadophora 101co balt 98col loi dal sys tems 98com par a tive mor phol ogy 91Compton-effect 76con cept for ma tion 130-132,

141con cept-transcending 74, 77con cep tual knowl edge 74, 152con sis tent physicalism 106con stancy and change 62, 75con text of dis cov ery 6con text of jus ti fi ca tion 6Cou lomb forces 78co va lent bond ing 78crit i cal ra tio nal ism 5crys tal lat tice 78crys tal line struc ture 113-114Cynodontia 95cy to plasm 98, 100-101cy to sine 99

Dde fi cient crea ture 140den tal-squamosum-joint 95desoxyribonucleic acid 99Diarthrognathus broomi 95dif fer en tial re pro duc tion 88Difflugiidae 126dis ci plin ary ma trix 8, 56dis con ti nu ity 59, 92, 94-95,

104, 111, 117, 154, 157divisibility 67, 75, 79dog ma tism 4dou ble he lix struc ture 99du ra tion 59

Eelec tri cal fields 140elec tro mag netic waves 75elec tron pair bond ing 78elec tron-shells 76el e men tary ba sic con cepts 77el e men tary par ti cles 55, 76emer gence evolutionism 9enkapsis 79enkaptic interlacement 81

enkaptic in ter weav ing 80enkaptic struc tural whole 81,

112-115En light en ment 53-54entelechie 105, 113, 133, 147en tropy 55, 60-61, 67, 85, 107epis te mol ogy 2, 70-71, 151,

154equi lib rium 85-86erect gait 128, 141, 157ev ery thing is num ber 9, 55Evo lu tion ary

– con tin uum 91, 94, 109– hy poth e sis 94– the ory 96-97, 104

ex pe ri ence 1-3, 7, 9, 11, 13,15, 53-54, 70-74, 89, 92,102, 110, 128, 132,136-137, 140, 144-145,150-151

ex per i men tal phys ics 72ex per i men ta tion 8, 62, 67, 71,

87ex ter nal enkaptic func tion 80

Ffac tual side 74, 91, 94fac ulty of judge ment 146faith in rea son 5fal si fi ca tion 7-8, 54fetalization 143fos sil re cord 94, 104free will 145

Ggen er al iza tions 6generatio spontanea 87ge netic sys tems 93geno-types 80grav ity 2, 8, 67Greek phi los o phy 83, 91gua nine 99

Hhistone 98historicism 15, 155-156ho lis tic bi ol ogy 107-108homi nids 120, 135Homo erec tus 121, 123Homo

– habilis 121, 123, 125,160

171

– neanderthalensis 120– sa pi ens 121-124, 135,

140– symbolicus 132

ho mol o gous struc tures 93hu man free dom 110, 142, 146,

148, 150-151, 153, 157hy dro gen 80, 87, 98

IIchthyostega 94-95Ictidosauria 95ide al ism 15, 147ide al is tic mor phol ogy 91, 103in di vid u al ism 15in di vid u al ity 70, 74, 90, 106,

113, 150, 152-153in er tia 2, 8, 60, 62-63, 69, 151in fi nite divisibility 79in fi nite re gres sion 6in fi nitely di vis i ble 66-67, 73,

131in ter fer ence phe nom ena 76in ter fer om e ter 76in ter me di ate forms 96-97intuitionism 58io dine 98iron 57, 61, 85, 98-100,

106-107, 125-126, 133,136, 151

ir ra tio nal num bers 55, 66irrationalism 15, 152ir re vers ibil ity 57, 60, 75ir re vers ible spe cial iza tion 138iso meric forms 79

Kkinematical time or der 59

Llar ynx 134-135law-conformity 56, 64, 71, 74,

152law-side 58-59, 65, 90lipids 88, 98, 102log i cal prob a bil ity 6

Mmacro-molecules 56, 74, 86,

99, 112macro-mutation 103mag ne sium 98mag netic poles 140mam mals 94-96, 134,

136-137, 142-144

man ga nese 98mass points 9ma te ri al ism 15, 56me chan i cal move ment 9, 59me tab o lism 89, 99, 101, 110,

149me tal lic bond ing 78metaphoricity 141meta phys i cal 4, 103, 147, 150metazoa 101meth ane 87mi cro-organisms 89mi cro-structures 68, 77mid dle ages 8miss ing links 94mi to chon dria 101-102modal ab strac tion 8, 12-14, 61,

69, 89modal uni ver sal ity 8, 71-73, 77mo lec u lar bi ol ogy 87, 114Monotremata 96mor al ism 15more geometrico 107

Nnat u ral num bers 58nat u ral sys tem 90, 105neoteny 138-139Nesthocker 142-143neu tral ity 3neu trino 76ni tro gen 78, 87, 98-99nomi nal ism 10, 15, 90-91,

102, 112, 150, 152-153,156

non-contradiction 132non-decreasing en tropy 55,

60-61, 67nucleo-protein 98nu cleo tides 99

Oobjectification 3, 130open sys tems 63, 85-86, 88or der li ness 64, 71, 74, 150,

152-153or ganic

– chem is try 113-114– com pounds 99

or gan is mic bi ol ogy 9, 86, 107,157

Ornithorhynchus anaticus 96orthogenetic 103

PPachygenelus 95paleo-biology 124pa le on tol ogy 94, 96, 104, 123pan-psychism 9panpsychistic 109-110pan-psychistic identism 111,

157par a digm 5, 7-8, 56, 71Pauli-exclusion 76pep tide bond 99per pet ual mo tion 61per sis tent themes 8phar ynx cav ity 134philo soph i cal foun da tions 3phos pho rus 98phylo gen etic trees 108,

122-123phys i cal en ti ties 55-56, 61-62,

65, 68, 70-71, 75phys i cal laws 73, 88, 90phys i cal time or der 59physicalism 9, 15, 105-106piet ism 15Pilt down hoax 120pla ton ism 10, 112platy pus 96Plotinus 27, 31polinucleotide 99polypeptide 99pos i tiv ism 1, 3-5, 53-56, 71pos tu late of con ti nu ity 104, 109po tas sium 98pre-natal traits 138pre sup po si tions 1-2, 15, 89,

91, 104, 111-112, 122-123pri mary sub stance 74prin ci ple of the ex cluded mid dle

169prop erty terms 54protamine 98protista 92pro to plasm 98, 125Prototheria 96pro to zoa 92, 101, 126psychoide 105psychologism 15punc tu ated equi lib ria 96Py thag o ras 23-24

Qquad ru peds 95quan tum the ory 70, 72-73

172

Rra di a tion 76, 78, 99ra dio-activity 55, 60, 78ra tio nal ism 5, 15, 145,

152-153ra tio nal ity 5, 53, 154re al ism 10, 15re dun dance 141Reformational phi los o phy 56reg u la tive prin ci ple 146rei fi ca tion 71rep tiles 94-95, 97re spi ra tory tract 134-135, 164re tar da tion 138-139, 144retrocipations 68, 131Ri bo somes 101

Ssci ence-ideal 5, 145, 153sec ond law of ther mo dy nam ics

61sec ond ary en ergy-traps 125,

127self-domestication 139set the ory 14, 58Seymoria 95-96so dium 79-80, 98struc tural con stancy 92structureless con tin uum 103

sub jec tiv ity 2, 114, 151, 156suc ces sive in fi nite 57sul phur 98su per sonic waves 140sur vival of the fit test 88syno nymi ty 141

Tthe o ret i cal terms 54, 71the o ret i cal thought 1-2, 9, 14the ory of rel a tiv ity 9, 62-64, 67,

69Therapsida 95ther mo dy nam ics 55-56, 60-61,

63-64, 67-68thought cat e go ries 70-71Thy mine 99time-duration 58to pol ogy 14to tal ity-structure 73tran scen den tal-empirical 69, 71Tritheledontidea 95type laws 70-71typ i cal

– foun da tional func tion 76– laws 70-72, 109

typogenesis 103typolysis 103typostasis 103typostrophism 103

Uul ti mate com mit ment 5ul tra-violet rays 140uni ver sal sub stan tial form 74uni ver sal ism 15Uni ver sal ity 10, 74, 168un spe cial ized 137-138,

140-141, 157

Vvac u oles 100Van der Waals forces 78vari abil ity types 80ver i fi ca tion prin ci ple 3-5vi tal ism 9, 15, 83-84, 105,

107-108, 111-112, 118,157

Wwave

– func tion 77– par ti cle du al ity 76

wave-character 76Weltoffenheit 148whole-parts re la tion ship 100,

115Wiener Kreis 3

Zzinc 98

173

In dex of Per sonsA

Adloff 138Aguirre 159Alexandroff 35, 159Allesch 159Altner 137, 159, 165Anaxagoras 22, 25Anaximander 21, 31Anaximines 21Angelelli 20, 159Apolin 159Ar is totle 21, 34, 36, 45, 50, 66,

68, 74, 79, 105-106, 112,152

Au gus tine 27, 31, 57Azar 117, 159

BBar-Hillel 45, 161Bartle 35, 159Bavink 157, 159Becker 21, 50, 159Bell 29, 41, 159Bendall 159, 161, 163Bernays 41, 45, 49, 112, 159Beth 43, 46, 159Boethius 27Bohr 55, 65, 75-76, 113, 159Bolk 143, 159Bolzano 27, 32, 34-35, 160Born 160Bos 25, 160, 162, 165, 167Boyer 24, 31, 160Bricmont (vi), 168Bromage 121, 160Brouwer 43, 46-47, 49, 66,

160, 169Bryon 62, 160, 168Buytendijk 132, 160

CCan tor 19-21, 23, 32-33, 38,

42-45, 47-50, 160, 168Car not 60Cassirer 19, 44, 130, 133, 160,

167-168Cauchy 29-32Chiarelli 117, 122, 160Clark 121, 123, 125, 160, 164,

169Clarke 123, 125, 160, 169Clausius 60

Coley 2, 161Crompton 95Cusanus 27, 32, 34Cush ing 81, 161

DDacque 161Dar win 9, 88-89, 91, 94,

96-97, 100, 104, 117, 119,123, 138, 153-154,156-157, 159, 161, 163

De Broglie 76De Klerk 134, 161De Swart 46, 161Dedekind 33-37, 39, 44, 161Des cartes 27, 55, 66, 75, 86,

107, 110, 134, 144, 153Diels-Kranz 21, 161Dingler 72Dobzhansky 100, 109,

117-118, 132, 157, 161Dollo 138-139Dooyeweerd 25, 46, 76, 79,

114, 142, 144, 160-161Driesch 105-106, 113,

146-147, 161Duley 161

EEhrenhaft 81Eibl-Eibesfeldt 136, 161Eigen 161Ein stein 1-2, 9, 62-63, 65, 67,

69, 76, 161, 163, 167Eisberg 76, 161Eisenstein 111, 157, 161Euclides 23Eudoxos 25

FFales 72, 161Faul 121, 161Feyerabend 119Fischer 44, 48, 139, 161Fontana 102Fraenkel 38-39, 41-42, 44-45,

59, 161Frege 20, 39, 41, 159, 162-163Freudenthal 162Friedrich 1, 162, 164

GGadamer 53, 161-162Ga li leo (vi), 2, 8-9, 27, 32, 55,

59-60, 62-63, 69, 72, 162Gehlen 124, 137-141,

147-148, 162-163Gieseler 120, 125, 162Goerttler 135, 162Gould 96Goulian 162Greenberg 162Greene 162Green field 162Grene 95, 103, 123, 162

HHaas 106, 112-113, 128, 148,

162Haeffner 133, 162Haldane 87-88Hallonquist 162Har ri son 162Hart 119, 126-129, 131, 137,

141-142, 148, 150-151,155, 162, 169

Hasse 23, 162, 168Hawking 28, 55, 57, 64-65,

162Hebeda 163Heberer 162-163, 170Heimholtz 61Heimsoeth 27, 163Heine 30, 163Heisenberg 1, 65, 67, 112, 163Heitler 40, 106, 113, 161, 163Henke 121-122, 163Hentschel 163Heraclitus 8, 21Hertz 75Heyting 38-39, 45, 58, 163Hilbert 43, 50, 55, 163, 167,

170Hippasus 23, 169Hobbes 55Hoenen 79, 98Holz 151, 163Howells 163Husserl 145, 163Huxley 104, 157, 163

JJammer 163

175

Janich 8, 63, 163Jansen 163Jaspers 149, 163Jevons 163Jonas 83-84, 110-111, 149,

157, 163Jones 86, 163

K

Kant 2-4, 20, 40, 43, 47, 53,55, 58, 69-71, 73, 75, 127,130, 145-149, 151-154,160, 162-164, 168

Katscher 75, 164Kaufmann 44, 164Keith 120Kerkut 164Kitts 104, 164Klaatsch 138Kleene 43, 46, 164Kline 18, 38, 50, 164Koehler 130, 132, 164Kremer 27, 164Kronecker 43Kugel 155, 164Kugler 148, 164Kuratowski 59

LLagrange 38Laitman 134-135, 164Lakatos 119Landmann 136, 138-139, 164Laszlo 118-119, 157, 164, 167Le Gros 121, 164Leakey 120-121, 130, 132,

164Leinfeller 91, 164Lenk 164Levy 45, 161Liebig 61Linnaeus 91Lloyd Mor gan 118, 157Lorenz 32, 44, 47, 58, 104,

132, 138-139, 141, 164Lorenzen 32, 44, 47, 58, 164Lowenstein 120

MMacCurdy 120Maimon 28, 32, 47, 164Mal thus 164Margenau 68, 165McHenry 121, 165

McMullin 119, 165Meijer 25, 165Merleau-Ponty 149, 165Meschkowski 19, 35, 37-38,

165Meyer 97, 107-108, 123, 165Miller 87, 165Millin 81Monod 165Moore 41, 165Munson 165, 168Myhill 50, 165

NNagel 40, 165Narr 124-128, 165Needham 107New ton (vi), 8, 28-29, 55, 59,

75Nida 134, 165Nietz sche 138

OOparin 85, 87-88, 165Orgel 87, 165Origines 26Overhage 142, 148, 165-166

PPannenberg 148, 166Passmore 118, 166Planck 63, 65, 72, 75, 166Plato 8-10, 24, 73-74, 112Plessner 133, 148, 166Plotinus 27, 31Polanyi 5, 109, 119, 157, 166Pongratz 166Pope (vi)Pop per 4-7, 54, 119, 166Portmann 92, 128, 143-144,

148, 164, 166, 170Pretorius 118, 167Py thag o ras 23-24

RRahner 165Rauche 150-153, 167Ray 91, 120Reed 125-127, 167Reid 42, 167-168, 170Rensch 9, 109-111, 129, 132,

157, 167, 169Rickert 167Rob in son 19, 29, 42, 167Rombach 167

Rosas 159Rucker 41, 167Rus sell 4, 19, 50, 131, 167

SScheler 148, 167Schelling 61, 147, 167Schilder 27, 167Schilpp 67, 167-168Schindewolf 103-104,

123-124, 167Scholz 23, 43, 162, 168Schopf 87, 168Schu bert-Soldern 105-106,

168Schuurman 127, 168Schwartz 117, 122-123, 168Sil ver 87-88, 168Simpson 104-106, 118, 123,

137-138, 153, 157, 168Singh 38Sinnott 106, 113, 168Smart 19, 168Smuts 107Sokal (vi),168Spielberg 62, 160, 168Spinoza 27, 107Stafleu 57, 62, 65, 70, 72, 168Stan ley 87-88Strauss 8, 80, 131, 141, 150,

155, 161, 168-169

TThesleff 23, 169Thomas Aqui nas 27, 68, 106Thomp son 60Thorpe 93, 169Titze 22, 169Tobias 120-122, 159-160,

162-165, 167-169Trincher 100, 113, 169Troll 91, 106, 169

VVan Dalen 45, 161Van Huyssteen 154, 169Van Melsen 81, 169Van Peursen 169Van Stigt 169Vollenhoven 60Von Bertalanffy 88, 107, 136,

167, 169Von Eickstedt 139, 169Von Fritz 24, 169

176

WWat son 95-97, 99Weierstrass 30-32, 44Weiner 120, 162, 170Weiniger 78, 170Weyl 25, 40, 42, 45, 66, 72,

170

White head 50, 118, 157

Wil lard 35, 170

Wolf 38, 48, 91, 148, 170

Woltereck 101, 108, 118-119,157, 170

Wundt 170

ZZeno 21-23, 25-26, 75, 162Zermelo 20, 41, 50Zimmerman 91, 97, 102, 118,

170

177