kreisel, 1998 - second thoughts around some of göde's writings

G. KREISEL

SECOND THOUGHTS AROUND SOME OF GDELS WRITINGS:A Non-Academic Option?

TABLE OF CONTENTS

Preamble in the light of TENOS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :p. 99Miscellaneous reading around Gdels piece : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : p. 114Sundry items from Vol. III : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : p. 121Diversity remembered by whom it may concern : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :p. 133Appendix: Logical complements in the light of TENOS : : : : : : : : : : : : : : : : : : : : : : : : :p. 134Notes: Casual conversations; Views taken by many a (minds) naked eye;

Non-academic aspects : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : p. 149

Preamble in the light of TENOS, short for: tested experience, not onlyspeculation (about possibilities). As has been stressed before, for example,in [Nl(c)],1 Gdels work has been widely publicized. This is repeated herein line with a property of knowledge that will be prominent throughout: Itis one thing to have knowledge, another to remember it when an occasionarises (tacitly, in the part of the universe encountered and taken in). NBThe erudites may remember this property more vividly by contrast withFreges ideal(ization of another aspect of knowledge): behauptende Kraft.In terms of a pun this is the power of a proposition (Behauptung) to im-pose its(elf, in particular, its) meaning; tacitly, on the attention of Fregesideal(ization of the knowing) subject. That power is present, but may beweak in any particular situation. More generally, the diversity of properties(of knowledge) will be recalled below by contrast with the idea(l) in thephilosophical literature of (its) mere truth.

As explained in the editorial note, particular attention will be givento material in Vol. III (Gdel 1995), which was previously unpublished,? Editorial Note. This article was originally commissioned as a Review Essay (fo-

cussing on Volume III of Gdels Collected Works). But it became clear that the broaderthemes discussed here have general interest beyond that original telos. The article shouldthus be read as a review in the sense described by the author, i.e. as second thoughts on andaround foundational issues, with special reference to some of Gdels writings (not only inVolume III).

Synthese 114: 99160, 1998. 1998 Kluwer Academic Publishers. Printed in the Netherlands.

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but, broadly speaking, matches the publications reprinted in Vols. I (Gdel1986) and II (Gdel 1990) in chronological order. (Material written duringGdels final illness, which became evident (to me) in the latish 60s, is hereleft aside.) The match is even closer if published reports of conversationswith Gdel are added to Vols. I and II. There are endless opportunitiesfor meticulous comparisons, even of wordings, and as elsewhere in thebroad academic tradition, subsidized academic trades that reward thosewho seize those opportunities. For them the aspects stressed below are, byTENOS, not rewarding.

1. One piece (pp. 376382 of Vol. III) stands out; it dates from the early60s, and has the title: The Modern Development of Foundations in the Lightof Philosophy. It is the last piece in Vol. III written before Gdels finalillness. It is phrased in a way of which Gdel was fond (and for which Iacquired a passive taste, tacitly, when practiced by him in conversations).Details aside for the moment, in plain English it is above all a reminder offamiliar facts.

Logical Foundations had ambitious (cl)aims about mathematics andmathematical knowledge, still enshrined in such terminology as logical(in)dependence and (in)completeness; tacitly, as logical idea(lization)s ofexperience meant by household words like proof; with emphasis on prin-ciples, a.k.a. (mere) understanding-in-principle. Contrary to those (cl)aimssuch logical aspects, which are of course present in mathematical knowl-edge, generally do not require (or reward) close attention. Both internalsquabbles in so-called foundational debates in the 20s, and such pastimesas belabouring paradoxes distract from this.

As an example of what is lacking in those foundations, Gdel mentionscertain non-scientific meditation exercises for the specific purpose of dis-covering suitable new axioms of infinity, an idea to which he remainedattached since the 40s. Those exercises, proposed in the early 60s, are tobe read in(to) Husserls writings, a kind of Wesensschau; in contrast to thekind of meditation popular by the end of the 60s such as the transcendentalvariety or by better living through chemistry. (Gdel explicitly warnedme that I should find Husserl boring, and, being interested in other things,I had no occasion to test his impression.) But, (these) specifics aside theexample is a reminder that some aspects are lacking. It is a separate ques-tion which of them, if any, lend themselves to rewarding study, let alone ofa theoretical variety.

By the way, the only specific item in the modern development of foun-dations used in Gdels piece (apart from those axioms of infinity) is hisincompleteness theorem. This serves again, by contrast and by TENOS,

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not only me as a salutary reminder; of such areas as real algebra forwhich a complete set of principles has long been familiar. But progressrequired giving principal attention to some suitable aspects neglected inthose logical idea(lization)s of mathematics; principal since as readerswith a little logical education will know occasionally that completenessis used in so-called transfer theorems; cf. 3(a) below for more.

2. Given the malaise about logical foundations sampled in (1), one ideathat has struck many a minds eye, is to ignore the whole enterprise (whilemany another minds not much less naked eye was fascinated).2

(a) As so often with such ideas the option has to be tempered by suit-able understandings; after all, as mentioned, knowledge has some logicalaspects. Foundations for the Working Mathematician (Bourbaki 1949) or rather what I read in(to) it now, not 35 years ago provides one suchunderstanding. It selects suitable items of logical knowledge, here meant incontrast to mere understanding-in-principle, that is, of set-theoretic princi-ples. It could be compared to (arithmetic) foundations for the working car-penter, learnt before ones teens as basic arithmetic (in contrast to Peanosaxioms). It will come up below, viewed as an example of a general asym-metry in knowledge: By themselves such foundations do not go far, butnot knowing them can be a disaster (in a sense of this word suitable to theacademic situation involved).

(b) Equally broadly, but at another extreme, that idea has, by TENOS,not served well at least one reader of those who wanted to pontificateabout their malaise. By TENOS many cases of prejudice (by gifted peoplesteeped in the subject involved) are confirmed by more thorough knowl-edge; but without those details it is harder to say what is known. In the nextsection some earlier critical literature on logical foundations will be takenup; complementary to Gdels piece in Vol. III.

(c) Patently, the option of simply ignoring logical foundations wouldbe hopeless for a review around Vol. III (or, by [N1], Vols. I and II),where review is meant in the sense of second thoughts; in contrast toa schoolboys prcis (required to satisfy the school masters regulations).Though often perfunctory, references to logical foundations and even tofoundational debates introduce or conclude material in Vol. I (and Vol.II from before the mid 40s, or appear in footnotes with afterthoughts).Sometimes the references are implicit; famously in the words relativeconsistency in the title of work which would otherwise come under theheading: constructible sets. In Vol. II, in an essay on Russells mathemati-cal logic, there is a fanfare about logic being a science prior to all others,emphasis on paradoxes as related to an amazing breakdown of various

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logical intuitions in the middle, and a peroration about promises of logicfor mathematics. So unless Gdels piece from the 60s is to be ignored,too, a rereading in the sense of rethinking, not of a prcis of Vols. IIIIrecommends itself.

This was actually done in [Nl], suitably for some, but, by TENOS, notfor all (with a logical education). As a matter of personal TENOS, with thereminders in Gdel more vividly before me, I have spotted since then someitems that are not reviewed critically enough in [N1]. (Some are taken upbelow in the section on Sundry items from Vol. III under the heading hardcore foundations.)

(d) Given the intellectual vitality, both of Gdels own work and ofsome (not all) familiar logical literature that continued it, it was a foregoneconclusion that a review would not leave one empty-handed. In eruditeterms, going back to Aristotle, the enterprise of logical foundations pre-sented itself (to many a minds naked eye) as a privileged telos for math-ematical logic. But contrary to a familiar view of Aristotles sound bite,by TENOS, it can be rewarding to shift the emphasis to different targets,among infinitely many candidates in our infinitely diverse world.

Suitable TENOS for second thoughts on such shifts is available to read-ers with a little experience in applied mathematics; specifically, for re-viewing the application of mathematical logic to that enterprise. It is easyenough to pay pious lip service to the principle (of rewarding shifts); byTENOS, its practice is demanding. Thus it is not merely the words asso-ciated with the original telos that remain enshrined in the mathematicalscheme, but the ideas or, if preferred, objects (categories in the colloquialsense or whatever): Are they, let alone, their descriptions, suitable for thenew purpose and, more generally, for extended knowledge?

In short, there is no free lunch; (suitable) shifts of telos have a price,which, depending on ones resources, may be a bargain or a dead loss. Anecdotes. By a fluke, as an undergraduate I still encountered those Tri-pos Exercises (at Cambridge, UK) in Applied Mathematics before WorldWar II, notorious for elaborating painstakingly primitive idea(lization)sof physical phenomena, especially, in rational mechanics, mentioned al-ready in [N1], for example, on p. 602 of (c). Some elementary items haveremained among the foundations for the working (theoretical) physicist,others had been developed with mathematical vitality in the theory of func-tions (of a complex variable). But those elaborations were, by and large,sterile; except for the fact that the dons concerned made a, to them, by andlarge, satisfactory living. From what I remember they had no illusions ofcontributing to physics or mathematics, but were doing something differ-ent (tacitly, from such contributions). For the readers above my experience


in applied mathematics may be good as a salutary reminder: to be preparedfor similar elaborations in the case of primitive logical idea(lization)s ofknowledge introduced in logical foundations.

As to TENOS on the extent to which the likes of Tripos exercisesactually occur in the logical literature, this has a (social) price as follows.Over the last few years Zbl. Math. has sent me a stream of articles, whichcontain, sometimes extreme, examples; including occasionally the kind oflip service mentioned earlier and thus providing TENOS that the prac-tice of the principle is more demanding than appears to the authors; cf.Zbl. 815.03036 together with its editorial note for a convenient record. Itincludes reports of (the authors) and speculations about (the reviewers)intentions.3 The authors indignation aside, some malaise about the reviewis fitting3 inasmuch as it is in conflict with academic conventions. It isunconventional to use (academic) publications for the kind of TENOSabove, which has been compared to the use of bacterial cultures in bac-teriology. Also it is unconventional to use Zbl. as a convenient record ofsuch TENOS, in particular, in the case above concerning the diversityof knowledge perceived as understanding among those of us with alogical education; cf. 3(c) and 3(d) below for more under the heading: anon-academic option.

So much for parochial rewards for attention to logical foundations;parochial in contrast to the following:

3. A broader view looks at the likes of logical foundations and at alternatives to formal (mathematical) refutations such as Gdels incom-pleteness theorem. A familiar look alike is >2500 years old; with arith-metic replacing logic: Number is the measure of all things. The refutationuses the irrationality of

p2, the measure, a.k.a. length, of the diagonal of a

unit square. It can also be reworded in terms of Dedekind (in)completeness:Q is incomplete, and incompletable by any countable set (of real adjunc-tions). Remark A specific parallel will be used below: all finite decimalsand all finite sequences of integers are rational and recursive respectively.

A common gut feeling focuses on the lack of proportion between themind-bogglingly crass (cl)aims of the look alikes above and the elegantformal refutations. (NB. This feeling does not seem to be shared by thetrade of so-called exact philosophy, with scholastic logic chopping in for-mal dress as its principal commodity.) On the other hand, while I haveshared the feeling since my teens, especially then I also fancied it, and,by TENOS, so have others; famously Andr Gide: Les extrmes me (notonly, se) touchent. Other attractions of the formal refutations are familiar,so need not be dwelt on, but must not be forgotten here either. For those

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in the market for entertainment or uplift (&), they are jolly orbeautiful. The lasting market for the arithmetic item is known, for Gdelsexpected. Given the intellectual vitality of the ideas used (for refutations),the potential for shifts of telos has been present from the start; at the kind ofprice documented in 2(d) above. Readers with a suitable background willfind more in [A2(c)], short for item (c) in the second part of the appendix;in particular, about the rocky evolution of basins of attraction within thearea of diophantine problems (evolved from the insolubility of n2 = 2m2 inpositive integers).

Such alternatives as (a) and (b) below are reservations regarding thosecrass (cl)aims rather than formal refutations, and not nearly as flashy. Bya refrain of this article, it is a separate matter where either alternative issuitable; more so than, for example, the option of ignoring those claims.This is taken up in (c) and (d) below; as mentioned, as a non-academicoption.

(a) In contrast to the irrationality ofp2, the alternative meant empha-sizes diversity: generally, of the world, more specifically, of numbers, of aspects of any one thing and of suitable measures. This is by no meansad hoc (tacitly, after footnote 3), inasmuch as it is in contrast with therepository of (not only logical) foundational ideals. They all demand theessence, a.k.a. nature, of things, and unity of, say, a body of knowledgearound them. When taken in-the-raw, these ideals are not satisfied by thealternative of relatively few measures suitable for relatively many situa-tions encountered. (As always, they may be suitably tempered by suitable,possibly tacit, understandings).

Anecdote. Already in my teens I was taken by the views and personalityof J.E. Littlewood whose introduction describes A mathematicans mis-cellany as suitable except for those irreconcilable persons who demand anappearance of unity and a uniform level. I dont remember if, in my teens,I associated this demand with ill will. Since then I have learnt by TENOSthat some simply either do not take in knowledge presented in a miscellanyor, if they do, do not remember it suitably. Some of [N1] has the characterof a miscellany, but less so than the world at large.4

(b) In the case of logical foundations, one alternative to Gdels formalrefutation combines the last paragraph in (1) on real algebra with the re-minder at the outset: (tacitly, true) knowledge has other properties besides(mere) truth. In the background is the master reminder (in [N2]): rewardingknowledge, here, around real algebra, may not long continue to concernaspects that strike many a minds naked eye. The choice of real algebra fits


the idea earlier on that a critique (here, of logical foundations) will besnappier if it uses knowledge of the subject.

Logical foundations emphasize the fact that exactly the true formulaein the (elementary) logical theory of R are (logical) consequences of afamiliar axiom schema, say, RCF. As corollaries, RCF is complete and(recursively) decidable; with some refinements of the latter, going back>60 years. Now, by TENOS, within mathematics another view among,by (a), infinitely many! has evolved, which shares in the first place onlya literally superficial, a.k.a. syntactic, element: the formal axioms RCF.

RCF is interpreted as describing the abstract idea, to the erudites a.k.a.structure, of real closed fields, on top of a few particularly well knownbasic structures (or structures mres) such as groups or fields. The unde-cidability of their elementary theory has been known for nearly 50 years(and its incompleteness much longer).

A disclaimer and a reminder of a conflict. I have no idea to what extent,if any, spotting abstract structures is an exercise in Wesensschau; at least,in effect, not generally by intention of most people in abstract mathematics(inasmuch as they, too, have not read Husserl; cf. (1) above and [Nl(e)]). A view at an opposite extreme is not merely implicit in the logical tradition,but quite explicit in the (recent) literature on the allegedly only way offinding theorems of a structure; cf. 1.-5 to 1.-3 on p. 159 of (Pohlers 1996).In plain English, in familiar mathematical reasoning formal deduction gen-erally isnt used even if it could be. Instead without premature precision such logical consequences are seen in ways not obviously different fromseeing (old or new) axioms; reducing the likelihood of errors resulting, forexample, from boredom (with logical deduction). This point will come uprepeatedly below, with labels like Wesensschau or intuition serving aspegs strong enough to hang ideas on, which are rooted in TENOS.

Those of us who have paid attention to abstract mathematics, as it hasevolved in the last few decades, have by now plenty of TENOS for secondthoughts on the matter brought up in 2 (d): Where are logical categories, inparticular, those of completeness and decidability, suitable for understand-ing knowledge about and around the abstract structures above? Reminderof a different kind of TENOS: Logical categories have appeal for many aminds naked eye; in the simple sense that they are related on purposeand in effect to epistemological notions, which are of venerable vintage,and thus enshrine views taken by many such eyes. Logical categories aregeneral, being defined across the board. But by a(nother) refrain of thisarticle, this is a separate matter from being suitable across the board.

The mathematical TENOS reported in the following sections is elemen-tary, but in terms introduced earlier good enough for reservations about

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those logical categories; with corrollaries for (the suitability, not meretruth, of) Gdels incompleteness theorem, which refutes precise, some-what extravagant formulations of certain foundational (cl)aims. In contrast,reservations are suitable not only w.r.t. to those logical formulations, butalso when the admittedly crass (cl)aims are stated in terms not specificto any academic education. This is an example of what was called a non-academic option, here, for using ones (mathematical, including)logical education. It is not offered as a free lunch; cf. 2(d). When the time isripe (at the end of the article), it may be considered (by individual readers)whether the price is right (for them). But enough is known already for afew words on the label non-academic.

(c) As mentioned, a principal alternative to the option is simply toignore those (and similar) crass (cl)aims together with refutations andreservations. As a matter of personal TENOS, some of us perceive speech-lessness here, in the face of those (cl)aims as a kind of helplessness, andthus find the alternative unsatisfactory. Evidently by academic traditions,this (cl)aim for the option is a mere luxury. NB The defect is parochial,inasmuch as not all effective thought is academic (let alone, scientific; cf.(d) below).

But this is not all. Closer inspection allows one to specify actual con-flicts between the option and diverse idea(l)s of various academictraditions; first, from academic philosophy. The examples below quote2 memorable items in the philosophical literature, which implicitly oreven explicitly are in conflict with TENOS used by the option. The firstquotation comes from C. S. Peirces (Coll. Papers: 4, 237): It is . . . easyto be certain. One has only to be sufficiently vague.

Implicitly, this is assumed to be too easy to be rewarding. But abstractmathematics provides much TENOS around situations where, what is suf-ficiently vague for certainty, is also sufficiently precise for the matter inhand. NB Sufficient in the sense of a suitable kind, not only degree,of precision; in plain English, what one is precise about (in contrast toscholastic logic chopping earlier on). Here abstraction is meant as in ab-stract mathematics (in (2) of the next section) and in contrast to logic,where it means higher types. For example, a problem which is not easy toanswer for a particular number field, say Q, may be discovered to be easywhen (made) suitably vague (in the sense of non-specific, for example, forall number fields). It is a logical convention that this ceases to be vaguewhen it is put in formal dress: the totality of fields considered is splendidlyvague, and precise enough for most situations actually encountered.

A second quotation is Quines quip on an idea(l) of ideas that sustainintensive analysis, stated (at the end of the piece on creation), but not


followed in his Quiddities (Quine 1989). Many ideas satisfy the ideal, forexample, when (infinitely) many logical consequences are derived (relent-lessly; by TENOS, for some in our diverse world such intensive analysis isits own reward). TENOS shifts the emphasis to other well tested propertiesof ideas besides their being subjects for analysis.

Thus they may be (suitably) selected or combined; with due regard tocompatibility, prominent in (b) above in TENOS on combining logical cat-egories and knowledge around abstract structures (in Zbl. 815.03036, andagain in 2(a) of the next section). In [A 713] Kant mentions that they maybe constructed; true to form, given his general preoccupation. Last, but notleast, ideas may be good enough as they stand, provided they are remem-bered suitably. In this case they are liable to be spoilt by elaboration; asmentioned in connection with foundations for the working mathematician(or carpenter), underlined by the warning provided by Tripos Exercises.In terms of a hackneyed metaphor, here a first step restricted to an inch,may approach a target by, say half an inch, but, pursued without suitablechange of direction, may miss any sensible target by a mile.

W.r.t. the label non-academic the broad conclusions above are de-scribed in colloquial terms without academic erudition. The terms ac-quire additional meaning in the senses of being memorable, of othersocalled uses, and presumably of some neurological aspects when re-lated to the academic education involved in the TENOS adumbrated. Inshort, the option consolidates the (non-academic) knowledge expressedcolloquially.

(d) As to conflicts between the option and traditions evolved within the sciences and mathematics, there is above all the following contrast inconcerns of the common-or-garden varieties among those traditions. (Thesocalled fundamental sciences are discussed separately in (4) below.) Interms used above, those traditions respect, at least tacitly, such matters asthe diversity of the world and of kinds and degrees of suitable precision(prominent in uses of the option); foundational ideals, e.g. of unity, aretempered accordingly.

By and large, the relative contribution of a scientific or mathematicaleducation generally outweighs that of consolidated non-academic knowl-edge. For example, one does not expect to get far without a scientific edu-cation in a study of phenomena in astronomy and spectroscopy measuredwith an accuracy to 9 or 11 significant digits. Of course not all thought isscientific. But then not all objects of (rewarding) thought are paid attentionto in those traditions either. By TENOS, within those disciplines, whathave been called sound intellectual reflexes serve academics quite wellto get on with their jobs (laid down by their traditions). By a refrain of

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this article this is a separate matter from describing those jobs (withoutleaving the rest of us speechless). By and large, and not, for example,in enterprises of discovering suitable shifts of emphasis as in 2(d) above;away from the likes of Tripos exercises.

For the present non-academic option the article mentioned in 2(d) andreviewed in Zbl. 815.03036, is of particular interest just because it is notsome kind of isolated, socalled personal aberration. It enshrines conve-niently in one place several idea(l)s that lend themselves to correction byuse of the option.

To conclude this general orientation, a couple of observations are worthnoting. First, the option is genuine, even w.r.t presenting (memorably) thekind of non-academic knowledge in question: it is not suitable for all. ByTENOS, for some, colloquial formulations are not weighty enough (to besuitably remembered). Such people strike the rest of us as close relativesof those irreconcilable persons in endnote 4. They have a different option,called so wahrhaft philosophisch by G. Lichtenberg; cf. footnote on p.143 of [N1(b)] for a quotation. Roughly, his colloquial paraphrase recallshow in practice thoughtful scientists have always taken into account thatboth events and their observation have a part in knowledge of nature; butthen Lichtenberg stresses that Kants way of saying this was exceptionalin being s.w.p.

Secondly, there is a vivid relation between the option and Gdels (dif-ferent) words in his piece from the 60s to the effect that not all thought isscientific. This is certain even if to be realistic the word scientific isleft vague; even because Peirces Law in (c) requires a little care in thecase of negative propositions. Gdel takes such popular extensions of non-academic knowledge, a.k.a. gesunder Menschenverstand, as meditation orreligion; cf. Einsteins credo about science without religion being lame;cf. Nature 146 (1941) p. 605. Now, I personally find science with religiousfervour a drag. So the (low-brow) non-academic option suits me better: ithelps me remember (available) non-academic knowledge better (withoutagony over the kind of knowledge provided by those other extensions).This evidently leaves open the matter of TENOS where the option is goodenough.

4. Throughout [N1] there are (brief) references to another side of logi-cal foundations, but without mention of the label in (3) of any non-academic option. Foundational (cl)aims are viewed as parallel to those of(successful) fundamental or even, in contemporary jargon, final theories;the parallel being available for comparisons and contrasts. In particular,logical foundations are viewed as a final theory of mathematics; cf. top


of p. 5 in Russells Principles of Mathematics (in the first decade of thiscentury): The fact that all Mathematics is Symbolic Logic is one of thegreatest discoveries of our age. He adds the ideal of an understanding-in-principle, and, in the introduction to the second edition of Principia, theideal of unity (achieved by the formal reduction to just one kind of objectand relation, namely, sets and membership, one propositional operator andone quantifier).

So there is nothing disturbingly original, in particular, untested, letalone unheard-of, in those references. The references were brief, becauseat the time I had not come across (or had not recognized) enough TENOSon actual rewards for attention to this side of logical foundations.

(a) By the restriction to successful fundamental theories it is a foregoneconclusion that the parallel will turn out to be a parody. The view takenhere is:

Even if it is a parody, glaring limitations of logical foundations areliable to have counterparts, which are much less visible (in successfultheories), but as always, in certain situations present a comparableobstruction; above all, to successful combinations with other knowledge.

Inasmuch as it is a parody (that is, presents a striking relation) at all,glaring successes in other theories are liable to draw attention to elements whose counterparts are lacking in logical foundations; and thus help saywhat they are (not necessarily supply any, and not only see that somethingis lacking).Reminder (cf. [N2]) This view is for many a minds naked eye no lessperverse than the heliocentric view for many an ordinary eye. In our diverseworld the view is not advocated generally; no more than the heliocentricview to those making sun dials. (By TENOS, a vivid description of thisview or of any other option is liable to be (mis)understood as advocacy.)

(b) As in 3(d) on the non-academic character of the option, in par-ticular, on not expecting some direct contributions to academic business the parallel is not presented for any scientific interest to successful theoret-ical scientists. By way of a manifesto I dissociate myself from popularin(s)anities about (such thoughts as) the parallel providing clarification,and about taking ones impressions of some socalled heuristic value (ofsuch thoughts, for example, to oneself) at face value. Of course, the im-pressions may be as clear as the geocentric view of the suns motion; cf.the reminder at the end of (a). More generally, blithe talk about heuristicvalues, in which the idea(l) of causes is prominent, recalls TENOS in end-note 3 on items from the repository around knowledge of causes, and ofvalues associated with (unsuitable) orderings.

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(c) In the 90s, after the material in [N1] was published, several semi-popular books around fundamental science appeared; by authors who, bycommon consent, have been very successful in the broad areas of thosebooks (here meant in contrast to the kind of speculation on brain physi-ology in Gdels Gibbs Lecture touched below under the heading: Brainsand faster computers). As in (b), at least for me, there is no question ofhelping the authors to be (even) more successful in their science or (even)more popular, tacitly, in suitable sectors of the market (in the commerce ofideas). A main concern here is decidedly non-academic: to derive fromthose books some knowledge outside the areas of ones academic educa-tion, without (hopeless) illusions of understanding, but also without being(hopelessly) put off by passages on matters of the kind touched in 3(c), andcalled general by one of the authors.5

Now, the normal course of nature provides opportunities for spot checkson ones impressions of understanding by specialists with (academic)knowledge around those books. It is not (academically) conventional tohave ones knowledge tested by colleagues; but then there is no free lunch.As to those passages, some of them recall (my) teenage emotions when(I was) left speechless both by (cl)aims and by objections in the caseof logical foundations. One option is to ignore the passages in question;tacitly, for the concern above, not for the authors agenda in (d) below. Thenon-academic option provides the (non-academic) luxury of an alternative.

On pp. 4344 of Dreams of a Final Theory S. Weinberg (1993) ago-nises over an objection by another successful scientist, P. Anderson: Arethe discoveries of the (double) helical structure of DNA and of Turingsidea(lization) of computability not comparably fundamental (to the dreamt-of theory)? A moments attention to the diversity of the world, prominentin (3) as a principal element of the non-academic option, is enough as awarning; of the need for remembering tacit understandings, which temperidea(l)s of some fundamental ordering of knowledge, theoretical or not; cf.endnote 2 (on terrestrial mechanics) extended to such terrestrial affairs asliving organisms or software engineering, and on (metaphorically) celes-tial matters of high energy physics (with astronomical costs). Remark forreference in (d) below. Inasmuch as practically, universal monetaryorders, a.k.a. values, do apply to (the costs of) acquiring a wide rangeof knowledge, those tacit understandings are liable to be incompatiblewith monetary orders; as with orders (of prominence) of the same itemof logical knowledge in foundations for the working mathematician and inelaborations satirized above by reminders of Tripos Exercises.

Weinbergs kind of objection to the comparison with Turings ideawould also apply to the present parallel between logical foundations and


fundamental scientific theories. In effect, though not in these words, Wein-berg appeals to the doctrine of category mistakes; here, the alleged mistakeof mixing the categories of knowledge about natural science and aboutmathematics (or engineering). As a general doctrine it is hardly compellinginasmuch as, by TENOS, it is often a good idea to count objects in differentcategories (which would count as a category mistake).

The irony is that, with better knowledge, Anderson(s case) would belaughed out of court, at least by readers with suitable background. As tocontributions of Turings idea to contemporary mathematics or to down-to-earth computation, the items on hard core foundations provide secondthoughts; cf. the section (below): Sundry items from Vol. III. Briefly,those contributions are relatively limited; compared to the prominence ofTurings idea in connection with the idea(l) of absolute definitions for epis-temological notions (on p. 150 of Vol. II). But then these second thoughtstemper the ideal itself.

As to broad knowledge about and around DNA here meant in con-trast to the specific geometric form of the molecule there is, realistically,no chance of competition for general interest by any dreamt-of final the-ory. (I dont know the money expended by pharmaceutical companies onknowledge around even single genes.) W.r.t. that specific geometric as-pect of DNA, the matter is more delicate: It is good enough to refutecertain precise formulations of traditional vitalism, with claims which areno more coarse-minded than those refuted by the irrationality of

p2 or

by Gdels incompleteness theorem. (Realistically, the knowledge of bio-chemistry used is more demanding than that of arithmetic and logic inthe other cases.) However, neither Weinbergs book loc. cit. nor presum-ably Andersons case mentions what even I (as an outsider) know about limitations on the understanding-in-principle provided by that helicalstructure w.r.t. not just Life Itself, but specifically self-replication;cf. Is DNA really a helix? (Crick et al. 1979). For the present option it isworth noting that the background is typical. A mathematician, Pohl, madea song and dance about a (perfectly valid) point many (of us) had noticed inthe 50s: the advertisement about self-replication by just un-zipping DNAdid not bear (even my) second thoughts; (it was known) there were just toomany turns of the helix to be unwound in the time available during mitosis.But apparently Herr Pohl was not satisfied with this observation, but wanted to have an opinion on what does happen. He collected some chumsto propose a different (false) geometrical structure to which the title of thepaper above refers. (More interestingly (to me), the paper also shows, byuse of technology evolved since the 50s, that a battery of enzymes cut andglue fragments of DNA and their replicates respectively, and thus adds to

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those elementary second thoughts more demanding knowledge.) So muchfor those pages in Weinbergs book.

Cricks words (1994, cf. endnote 5) about all our thoughts and feelingsbeing a matter of neurons etc. are reminiscent of Russells at the begin-ning of this subsection; just add a matter of after is in: mathematics issymbolic logic. The hypothesis in question, that the scientific search (byCrick) for the soul might be unsuccessful, does not astonish me; in fact,w.r.t. this matter of astonishment, there is enough of it (for my taste) inTENOS around common-or-garden varieties of psychological and psychi-atric phenomena not to seek it in the mere (non-)existence of the soul.But also, by reference to the parallel with Russells sound bite I am notleft speechless nor put off from reading the material offered in Cricksbook (and having my understanding tested whenever an occasion presentsitself). Perhaps, it would have been fitting (for my taste in view of the soundbite on neurons) to have more on brain-damaged patients; not instead, butin addition to straight experimental psychology. Fittingly (for applicationsof the non-academic option), time and again the text recalls (endnote 3 witha sound bite of) Aristotle on the business of understanding and knowledgeof causes; here, neurological causes of psychological phenomena.

Even now enough is known about the parallel between logical and neu-rological (would-be) fundamental theories for TENOS on the idea of aparody adumbrated above by comparison between logical foundations forthe working mathematician and neurological foundations for the workingpsychologist and especially psychiatrist. The little I know of neurology hasrelatively more weight for the common-or-garden varieties of psychologi-cal phenomena than all of logic in mathematics. Reminder. This conclusionis already discounted (in the stockbrokers sense) by shifts to discoveredaims for a logical education.

On the other hand the parallel above, between those scientifically verydifferent foundations, highlights similarities if the emphasis is shifted backto such foundational ideals as the essence or nature of this or that X; sat-isfied by definitions of X in answer to the nursery question: What is X?Reminder (of Platos enthusiasm about definitions in Euclids geometry,for example, of the circle, an idea that has struck many a naked eye). Aselsewhere by general TENOS, with suitable tacit understandings on the X(and on the kind of definitions) considered, the ideal becomes impecca-ble. Now, in conversations with Gdel, some 40 years ago, about diverseepistemological notions there was like-mindedness (cf. [N1(e)]) regardingthe view that such notions have struck many (a minds naked eye; in plainEnglish) before or without scientific experience. In terms of Gdels piecein (1) such notions are defined by a kind of Wesensschau, which I pursued


as what I called at the time a calculated risk (without his conviction);now (cf. (4) of the next section) I prefer the term milksop foundations.These are rewarding to me, in retrospect, not foreseen for informationon points of diminishing returns for attention to those notions.

Viewed this way some claims, popular at the time (in the 50s), aboutDNA and life, another idea that strikes many a minds eye, and about DNAbeing the secret or at least the molecule of life, no longer leave me speech-less. For one thing, in the light of TENOS around epistemological notions,this is (now) simply no big deal for me, if my what else? idea of life(at that time) was meant. That idea did not include much about viruses,nothing about retroviruses etc. (nor such matters as protein synthesis). Bynow those popular claims are among the less interesting sides of what isknown (even by me) around DNA: I want a better idea (of life). Anec-dote. In my teens the people whom I heard pontificate about these matters,struck me as pretentious. Now, with a little more TENOS, they are seen tobe clueless; in particular, about the general diversity (even) in the (small)world of human affairs. Reminder. This diversity includes of course alsoareas of uniformity, as it were basins of attraction, especially within thoseacademic trades that have evolved traditions aptly called disciplines. Herethe non-academic option adumbrated is relatively rarely suitable.

(d) The emphasis throughout this Preamble has been on personalTENOS; evidently in the last paragraph, but also in the fanfare about theproperty of knowledge being suitably remembered, tacitly, by oneself.Plainly, there are other uses of knowledge, including communication, per-suasion or even inspiration. This applies to Weinbergs and Cricks differ-ent agenda (reflected) in their books. The former set out to communicatehis enthusiasm for a final theory to a bunch of people, and to persuadethem to spend $12 billion dollars (over 5 years) on that theory (rather than,perhaps, on metaphorical black holes such as national or social security).6

Cricks message (of scientific salvation) was to inspire with luck,gifted people to work on (tacitly, neurological aspects of) conscious-ness now. Without exaggeration, once the soul(ful tradition) is discarded,inspiration is seen as a matter of statistical TENOS, which I do not have(and so, by temperament, I do not have any opinion on it; cf. endnote 5for contrast). Remark on personal TENOS, in particular, reported in (c).For me the broad idea of the message is not at all controversial (no morethan the broad idea of determining the molecular structure of DNA, tacitlywithin the tradition of biochemistry in, say, the 40s). This leaves openthe chance of success with the resources available (to an individual). Thewords of the message do not suit me at all; I am no more interested inmy idea of consciousness than of life above. There is no word on the (to

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me) most encouraging, if not inspiring, aspect of TENOS around DNA:when progress did not quite live up to expectations (and/or rhetoric) aroundself-replication, and the geometric (helical) structure remained a mem-orable, but isolated bit of information, (also Cricks) attention turned toother, almost immediately rewarding aspects such as protein synthesis andthe linear order of bases. (Flexibility of mind and energy can be usefulresources, compensating a taste for flat generalities.)

Reminder of relations between this Preamble and Gdels piece on pp.376382 of Vol. III (at the end of (3) above): Both have qualms about log-ical foundations, but differ w.r.t. the kind of attention given to them and tooptions for using the mathematical logic originally developed for them. The Appendix is for readers with a taste for the jolly side of (mathematical)logic.

Miscellaneous reading around Gdels piece. The words in its title arehere understood broadly, as in the Preamble. Thus attention is given tophilosophy as a repository of memorable reminders; both of views taken bymany a minds naked eye, of feelings about them and of thoughts generallypractised, but not elsewhere described so wahrhaft philosophisch. Morespecifically, w.r.t. (logical) foundations attention is given to the idea(l)s ofknowledge which were embraced by the pioneers (and certainly continueto strike many a minds naked eye).

1. One such idea(l), on understanding-in-principle, was related in 2(a) ofthe last section to a pun on principles given in advance. Another suchideal, aiming at mere truth, was highlighted at the outset by contrast withTENOS on knowledge having other properties (besides truth). In the lightof those ideals the possibility of a constant intervention by (Kants) intu-ition in reasoning appears as a spectre; in the first case, at least as long asno principles determining such intervention are available. But, to me, muchmore strikingly there is plenty of TENOS on capacities actually available(to many) for the exercise of such intuition: they differ (strikingly) amongdifferent subjects so that mere truth soon becomes relatively marginal (forunderstanding). Digression. This aspect of (Kants) subjectivity is a bitat odds with his idea(l) of universal laws. But so is his sound bite onmankind being made of crooked timber, which brings to (my) mind suchunsuitable laws as all timber being inflammable if dry. Like many of Kants(splendidly) flashy ideals those above are tempered by tacit understandingsprovided by practical reason. End of digression.

Russell himself described Principia memorably as a parenthesis in therefutation of Kant. Perhaps it is or can be turned into one with a lit-


tle formal care if that staple of Kants business, preoccupation withpossibilities-in-principle, is taken at all literally. Otherwise 3(b) of the lastsection is good enough to be prepared for reservations about the refuta-tion in turn (by TENOS on a Pyrrhic victory), for example, by remindersaround the logical (categorical) definition (in Peanos axioms) of the suc-cession of the natural numbers. Now, the axioms state properties (of naturalnumbers as previously understood) used in number theory. One is thecontrapositive of the principle of induction known as Kstners principleat the time of Kant. But also, first, the validity of those axioms was notrecognized only by logical deduction, and, secondly, there is the moredemanding matter of TENOS: in more than a century of familiarity withthose axioms and of a great deal of progress in number theory there havebeen few occasions for referring to the axioms as a definition, sometimesaka explanation; in contrast to TENOS in the case of, say, groups (in placeof natural numbers); cf. 2(b) below and the disclaimer and reminder in 3(b)of the last section.

2. Around 1950 Bourbaki published 2 articles already mentioned in thePreamble about logical foundations (with and without this word in thetitle), in(to) which I now read much more than I did (even) at the timeof Gdels piece. By TENOS, for example, in reviews (at the time) bylogicians, also some others had, let us say, similar blindspots to mine. Theywere (for me) removed as follows.

(a) In Larchitecture des mathmatiques (Bourbaki 1950), aka Bour-bakis manifesto, some strong language is used, but adequate if PeircesLaw in 3(c) of the last section is remembered. In particular, the founda-tional ideal of unity is embraced (w.r.t. mathematical knowledge), onlythe logical variety proclaimed by Russell, cf. (4) in the last section is declared to be the wrong kind. Logic, tacitly, its foundational side,7 isdubbed useful, but also the least interesting side of mathematics. Bour-bakis as it were competing offer was an abstract side of mathematics, witha different kind of unity; (to be) achieved by suitable combinations of so-called structures mres, aka basic structures, including groups, topologicalspaces and the like. On a tacit understanding the offer was to be effectiveacross the board of contemporary elementary, aka basic, pure mathematics,and correspondingly marginal in more demanding situations.

At the time I did not read that understanding into the manifesto nordid an open-ended list of basic structures fit the idea(l) of unity formed bymy minds (pretty) naked eye. Furthermore the structures themselves weredefined in logical terms, in particular, of sets; with symbols for operationson sets being ubiquitous. This seemed interesting (not only to the eye in

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question). NB By the early 60s, there was no doubt about the successof abstract mathematics, especially, in comparison with the enterprise ofinfinitistic mathematics, popular in the 20s (around Fund. Mathematicae,genitive singular). The question was how to say what was known; cf. also(4) below.

To say it in terms of the Preamble, its reminder about the diversityof the world is good enough to be prepared for tempering the idea(l)of unity, by the alternative of relatively few basic structures suitable forrelatively many situations (actually encountered, not merely, possible-in-principle, when a suitable understanding of relatively would be required).As to orders of interest of logical knowledge, by prominence, again oneis prepared for (relatively few) distinctions. But within available TENOSvery few occasions have been encountered in abstract mathematics whereall the knowledge of sets available (in familiar axioms) is rewarding. NBThis is not interpreted here as reducing dubious doubts over ontologicalcommitments, but to specify the kind of interest that Bourbaki found lack-ing: rewards in abstract mathematics for intensive analysis of principlesfor sets (at an opposite extreme to Peirces Law). Again, it is a matter ofTENOS where the evolution of suitable combinations, aka compatibilityof basic structures proceeds smoothly. It does, for example, in combi-nation of (the logical structure) order and group (in algebra) inasmuch asin (suitably) ordered groups the operation is merely required to preservethe order, and this has been good enough in many (not all) situations inalgebra. Suitable orders of interest are a bit more demanding in broaderareas of knowledge, even within (very parochial) proof theory (cf. Zbl.815.03036 cited early on in the Preamble). More memorably, there is theformal conflict between the logical ideal of a universal well-ordering andTENOS (from algebra) of suitable orderings, many of which, includingR,are not well-founded.

(b) Foundations for the Working Mathematician (1949) also by Bour-baki, is here viewed as a footnote to the manifesto in (a), incidentally,with second thoughts in endnote 7 (with a so to speak less sterile idea(l)for using logical knowledge). The emphasis is not on principles (for sets),but as in 2(a) of the Preamble on a selection of particular properties of,and notation for, sets (decidedly at odds with Quines quip in 3(c) of thelast section about the ideal of, tacitly, analytical philosophy). The selectionis (cl)aimed to be suitable across the board, evidently, a matter of TENOS.

As in the case of basic structures in (a), the list of (logical) items insuch foundations suitable for the working mathematician remains open-ended. Candidates from a broader logical education (than mere set theory)have crossed not only my mind. But my knowledge of broad math-


ematics is not good enough for TENOS, and thus (by endnote 5) not for(my being) interest(ed in my impressions). Digression on some familiarliterature about not only this kind of uncertainty. This had a dramaticside for Keats (cf. [N1(d)] on negative capability) or for Russell (cf. p. 11of History of Western Philosophy, Allen and Unwin, 1946). In contrast, for some of us, who remember the diversity of the world, other things areto be found in it for which ones knowledge at hand or within onesreach is good enough; as always, depending on ones resources andappetites. [N3] (re)views these last two paragraphs as an exercise of thenon-academic option for using ones logical education.

3. Wittgenstein, like Bourbaki, was given to strong language about logicalfoundations; at least by the time I met him in the 40s. (By that time hecertainly paid attention to the diversity of the world, but was not one of those of us mentioned for contrast at the end of (2) above; almost everyside he looked at seemed to have a tragic or at least dramatic elementfor him.) By TENOS, of which more in (d) below, I know too little forscholarly (cl)aims regarding details of his ideas. But what I know is goodenough for the following:

(a) In Tractatus logico-philosophicus, aka as an ode to propositionallogic, Wittgenstein had presented a logical ideal(ization) of the world andof knowledge; tacitly, with finitely many simples (in the 20s, the decade fa-mous for its craving for crisis). In our first conversation, but also elsewhere,he emphasized that his later ideas were in sharp contrast to that logi-cal idea(lization); without specifying any particular aspects that presentparticularly striking differences.

Now, in Cantors terms (used for explaining his idea of sets), a logicalview is a subject, a set as it were, that can be grasped as a unity, but not itscomplement, where Wittgensteins contrasts are to be found. This is satire,and easy; in Juvenals words difficile est saturam non scribere. But it isgood enough as a reminder of an alternative: to discover relatively few keywords adequate for labelling relatively many contrasts as they come up; asalways, at a price, for example, in quality of literary form (of this articlecompared to Wittgensteins writings).

(b) Wittgensteins oft-quoted key words our language(s?), never cameup in our conversations. If they had they would have reminded me ofMephisto (in Goethes Faust), who had enchanted me in my mid-teens;especially, his response to Fausts grand-standing about his troubles with words: Willst Du was in Worten sagen, musst Du Worten nachjagen.

One to me, particularly unsatisfactory element in Wittgensteinswritings is his very frequent reliance on imagined situations. They are OK,

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and occasionally singularly attractive, if used by such singularly giftedpeople as Einstein, especially w.r.t. celestial and other inaccessible matters(cf. [N2]). Popular and correspondingly primitive speculations on influ-ences (and similar causes, cf. endnote 3) aside, in retrospect I view mypresent emphasis on TENOS not only in this article as a contrast to thatelement; again, at a (here, social) price, described already in the Preambleat the end of (2), for the sake of TENOS recorded in some of my reviews.

(c) If Wittgensteins writings are interpreted as a philosophers non-philosophical miscellanies (cf. endnote 4 and its surroundings), a greatdeal fits (a) above; sometimes more memorably if background is added(provided one has it, deconstructionist doctrine being more suitable forthose lacking suitable background). Thus somewhere Wittgenstein writeseloquently about phenomena of reading. Here suitable background is inthe introduction to: Was sind und was sollen die Zahlen (Dedekind 1888),which states a counterpart to the logical idea(l) of rigour in the case ofreading: spelling out words (letter by letter); not haphazardly, but as aparallel to the rigour provided by logical deduction from his now calledPeanos axioms for arithmetic. NB This quotation is, by TENOS, oftentaken as ridicule. For the non-academic option of the Preamble, such mis-cellanies are comparable to (miscellanies of) contrasts between dynamicidea(lization)s of liquids (in motion) and other aspects of liquids; dynamicidea(l)s corresponding to logical idea(l)s and liquids to the world or toknowledge at large. Relatively little need be known about properties ofso-called ideal liquids to see that they are not suitable for understandingsuch striking phenomena as drag or turbulence in hydrodynamics, wetness,chemical composition, etc. Such a miscellany is not a theory (to which thenon-academic option adds the reminder that not all effective thought, letalone, all understanding of a diverse world, is theoretical). But, at leastfor me, who does not happen to be one of Littlewoods irreconcilablepersons, a miscellany by an acute observer can be a sight more rewardingthan Tripos Exercises in mechanics or scholastic exercises in formal dress,aka philosophical logic.

Given Wittgensteins limited knowledge of both logic and mathematicsthere is (fittingly) nothing (I know of) in his writings that corresponds to2(b) above, on foundations for the working mathematician. Admittedly,at the time in the mid 40s, I knew some snippets in proof theory which Ithought of as candidates for what I now, not then, call such foundations;in contrast to what, by above, I have realized since then. Be that as it may Istressed particularly the shift of telos involved in (new) uses of consistencyproofs; Wittgenstein took all this in (his stride), and seemed to find it allcongenial; but cf. (d) below.


(d) By a fluke I dragged some comments by Wittgenstein into footnote4 on p. 281 of Fund. Math. 37 (1950) concerning formal incompleteness,including an anticipation of what later became known as Henkinsproblem. The paper was a write-up of something I had done in the early40s, but published only after the war. The background to this footnotewas a relatively short conversation with Wittgenstein, in which he wantedme to tell him Gdels proof. (He told me he had never read it, havingbeen put off by the introduction.) NB As I see it now at length in (3) ofthe Preamble, certainly not then a formal refutation of logical founda-tions, which he had already recognized as crassly defective, would fittinglyarouse his passions. At the time, by reflex I described the proof in termsof diagonalizing sequences (of partial functions) without (ab)use of self-reference; in short, as a jolly piece of mathematical logic. (I have describedhis enthusiasm on various occasions.)

Until recently, I lacked TENOS striking enough (for me) to consider theextent to which the ideas that interested him in our conversations wereactually taken up by him or at least fitted those he pursued when on hisown. Recently, in the socalled Wiener Edition some pages with marginalcomments by Wittgenstein to Hardys Pure Mathematics are reproduced.By a fluke his copy was published in 1941 (though this date was not men-tioned loc. cit.), and my extended conversations with Wittgenstein aboutthis very book took place in the first 5 months of 1942. So those commentswere certainly not written long before the conversations, which I have oftenhad occasion to relate to matters of (to me) continuing interest. In contrastId not merely have been bored by the published marginalia, but havefound them barmy (as I still do).

Added in proof. Only in autumn 1997 did I come across a typescript (ofa MS in Wittgensteins Nachla) with the title: Mathematik und Logik. Itstarts on 6.I.1943, almost exactly a year after the conversations just men-tioned started, and continues beyond 3.III.1944. In para. 197 near the endhe refers to transformations of proofs, which had been prominent in ourconversations (and seemed to me, at the time, to satisfy him). But loc. cit.he recognised in such transformations a strong temptation to (mis)under-stand them as contributing to his idea(l) of philosophy, adumbratedimmediately before in para. 196; cf. a fuller statement of this ideal (inanother typescript) on 30.XII.1942. Quite simply, philosophy (of mathe-matics) should not add new (mathematical) knowledge, but linger, albeitcritically, over what are called in this article views taken by many aminds naked eye.

Patently, my disclaimers in the opening paragraph of the present sub-section (3) were sound; the new knowledge of Wittgensteins feelings about

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the matter is needed as a threshold for informed views on it. Correspond-ing disclaimers apply to conversations with Gdel, for example, in [Nl(e)]or (4) below under the heading milksop foundations.

4. In terms of the Preamble the (cl)aims of logical foundations are viewstaken by many a minds naked eye, and so Wittgensteins observationsin (3) are viewed as reservations about those (cl)aims; in effect, noton purpose, they paraphrase Bourbakis strong language (about the leastinteresting side). As other such language it becomes rewarding when tem-pered by a suitable reminder; it leaves open where which kind and degreeof elaboration of those foundations is suitable or at least good enough; cf.[N2] on situations where a geocentric view is suitable (with due regard forthe temptation of Tripos Exercises on epicycles). Anecdote. As describedon top of p. 613 of [N1(d)], 40 years ago I was blind to any interest ofany foundational idea(l)s, and delighted by my interpretation of pointsmade by Gdel in conversations on the potential of some of them. The timeseems ripe for the following comments around that interpretation.

(a) A milksop view of the foundational debates about different ismsis a reminder of the truism that most knowledge has aspects involvingobjects, ideas, subjects and even formal notation (particularly prominent insymbolic data processing). Einstein may have compared those debates tooo (cf. Aesops Fable CLXVIII), but his own debateswith Bohr over the foundations of quantum mechanics had a distinctlysimilar general flavour. (They differed from most foundational debates bythe intellectual vitality of the specific ideas for areas of interest to Bohrand Einstein, and missing in the scholastic logical tradition).

(b) A milksop view of debates on logical laws regards them as an op-portunity for a tour de force on the theme of diversity; even within themodest domain of minding ones ps and qs (in classical and intuitionisticlogic). Later, in the 50s, in line with the medieval tradition of lumping logicand rhetoric together (among the trivia), one had (Lorenzens) dialogues;where scoring Debating Points replaced the search for Truth.

(c) A milksop counterpart to Gdels specific (cl)aim for Wesensschau(to discover axioms of infinity suitable for settling CH) was broader; itwent back to various (elementary, relatively neglected) epistemologicalnotions regarding properties of proofs and to somewhat related notionsin the area of choice sequences. As expected this was good enough forseeing similar flaws of course, not mere lack of precision in popularepistemological notions (cf. 1(c) of the next section), and saying what isseen.


Remark for readers interested in details how this is largely superseded byevents outside logic. J.R. Moschovakis (1994) returned recently to socalledlawless sequences providing an occasion for a review that goes into suchdetails; cf. Zbl. 795.03083. It should be added that a little later, on p.829, the author corrected an oversight (about free variables in schemata),which A.S. Troelstra has pointed out repeatedly and conscientiously, butapparently not memorably enough.

Sundry items from Vol. III, preceded by 3 loose ends from Vol. II, re-viewed (too) briefly in [N1]. Wherever available, trade jargon and refer-ences will be used; allowing readers to test if they have a suitable back-ground and interest: they have the former when the jargon is familiar,and the latter, when they follow up the references. (Here the market forillusions of understanding is neglected since, once again, I do not knowenough to be interested in any ideas I might have about it.)

1. Hardcore foundations follow the idea in Quines quip about sustainedanalysis; they are at an opposite extreme to milksop foundations in (4) ofthe last section.

(a) On p. 124 of Vol. II Gdel reports (his) amazement at . . . logicalintuitions concerning . . . being . . . being self-contradictory. At the time,in the mid 40s, he related this to the familiar paradoxes, later (on p. 376 ofVol. III) relegated to epistemology; cf. also (b) and (c) below. A milksopview replaces speechlessness for example, Weyls (cf. p. 632 of [N1(d)])at (t)his amazement; it is enough to remember suspicions (rather than intu-itions) in the 19th century, even of such modest objects as Cantors abstractsets, where here and below, object is used rather than being, let alone,beable.

W.r.t. objects, specifically, in abstract mathematics, Hilbert establisheda suitable milksop view in his bestseller on the Foundations of Geometry:their nature does not matter. It turned out by TENOS that contrary to, forexample, Freges (usual) convictions by and large enough is known: it isenough that there be infinitely many objects (in the situation) considered.But also and this elementary reminder, too, is part of a milksop view in any model of an abstract geometry the objects (such as points andlines) are specified as part of the data. This loss of drama or, according totemperament, feeling of enlightenment is tempered as follows (at least,for those with a suitable education).

The milksop view above is not good enough in some parts of the algebraof free uncountable abelian groups; by TENOS, it is not always easy torecognize such parts on sight. NB. A milksop view is good enough to be

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prepared for this possibility; of needing a closer look at the objects (thegroups and the free bases). Samples Such a look is needed in the caseof Whiteheads problem; cf. P. Eklof (1976). It is not needed in the caseof another uncountable abelian group, which has (all) bounded sequencesof integers as elements and the operation of pointwise addition (shownin the 50s by Specker, to be free, when the objects are elements from acollection satisfying CH); cf. G. Nbling (1968), and (with a twist) G.Bergman (1972). (The twist consists in a shift to matters of discoveredinterest away from the particular group; including transfinite induction onsuitable pseudo-well-orderings.)

(b) In footnote h on p. 275 of Vol. II Gdel touches another familiarfoundational idea(l), although he does not give it a familiar name; as inthe case of ramified sets, which he called constructible (for his particularvariant). The material below is complementary to p. 615 at the end of 5in [Nl(d)] on the idea(l) of reductive proof.

In familiar terms, purity of method has struck many a minds naked eyeas an ideal, at least since those Greeks who objected to the use Archimedesmade of (his knowledge of) 3 dimensional cones for knowledge of ellipsesetc., which are conic sections. The peroration of Hilberts Foundations ofGeometry propounds the ideal as a principle, which Hilbert rarely followedin practice. (In the Foundations he has an entertaining exercise regardingDesargues theorem in 2 and 3 dimensional projective geometry.) In thiscentury it was popular in number theory under the heading of elementary,aka direct, proofs of the prime number theorem, and so forth; cf. (Kreisel1951, 248). It may also be viewed as a local version of [A476/B504], citedon p. 294 of Vol. III, on self-contained bodies of knowledge, once againexpounded so wahrhaft philosophisch (in Kants manner).

Here a milksop view recalls, first, realizations of the ideal, and, sec-ondly, TENOS that has accumulated around them; for example, about theircontributions (on second thoughts) compared to (original) expectations; cf.the case of real algebra in 3(b) at the Preamble. But also one will recallthe diversity of the world, perhaps by now effectively (at least, here), interms of the following elementary, decidedly milksop reminder.

Every relation between an object and any (other) thing is a property ofthat object, where, as in (a), a milksop view of objects is taken.

NB. This is certainly elementary, and seems, by TENOS, to many a mindsnaked eye innocent; deceptively, inasmuch as it is quite good enough forreservations about the broad ideal of completeness; cf. (3) of the Pream-ble. It thus prepares one for limitations of formal (in) completeness results,but does not replace what has been said above in detail. In a related way,


the milksop view is good enough for correcting totally unrealistic expec-tations of the ideal of purity of method, and especially of Tripos Exercisesaround it. Reminder. As usual, once the limitations of the ideal itself havebeen grasped, material (with intellectual vitality) evolved in the course ofpursuing the ideal, may be discovered to serve a more suitable discoveredtelos (which, by TENOS, will generally not be as vivid to many a mindsnaked eye as the ideal itself).

Samples. First, in the case of the prime number theorem mentioned a mo-ment ago, direct proofs were given in the 40s (by Selberg and Erdos); real-istically speaking, helping to put the ideal in its place. Less well known isa later interpretation, aka restructuring, of those proofs on pp. 102103 ofW.J. Ellisons Les nombres premiers, Paris, 1975. This exhibits vividly atleast one element lacking in direct proofs, which for some of us is mademore memorable by contrast with the familiar idea(l) of Beziehungsreich-tum. Secondly, at another extreme, Gentzens cut-free rules for elementary,aka first-order, logic satisfy the ideal with knobs on; not merely in thecrude form that logical theorems ideally have (ordinary) logical proofs.Only (logical) parts (in the technical sense of subformulae) of the formulaproved appear in a cut-free proof, which is thus reductive in a colloquialsense, too.8 Here, a (heavy) price is well known: the use of lemmas andof proving specific cases from general theorems is excluded by the ideal.Shifts of telos have been known (but not always so seen, let alone, suitablypursued); for reading off numerical bounds from general theorems at oneextreme, and general theorems from computations at another (for some 60and 10 years respectively; cf. also [A2 a(iii)]).

Remarks. Hilberts programme requiring finitist proofs for finitist(icallystated) theorems fits the ideal, and thus his peroration loc. cit. Now the(epistemological) notion of finitist proof, which like other such notionsstrikes many a minds naked eye, has long been known; enough (about it) torecognize that the programme was carried out up to the hilt for real algebra.For a milksop view this (TENOS from algebra) has, at least, some epis-temological value; some compared to the immense value proclaimedon p. 85 (or p. 112) of Vol. III: Falls das ursprngliche Hilbertsche Pro-gramm durchfhrbar gewesen wre, so wre das zweifellos von un-geheurem erkenntnistheoretischen Wert gewesen. The milksop view recog-nizes but does not enter into the separate question: Where, outside the(subsidized) trade of epistemology, are the orders, aka values, which haveevolved in it, suitable for, aka compatible with, other demands on knowl-edge, here, around proofs? Readers with suitable background may look

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at [A2(d)] for a little more detail. I do not know enough to be interestedin my ideas around the fact that Einstein himself, to whose generaltheory of relativity the citation by the Nobel Prize Committee attributed(great) epistemological value, did less well in foundational debates aroundthe quantum theory, after having done very well in establishing quantumeffects. (The behauptende Kraft of his ideas may have been different forhim and for those irreconcilable persons who assume that power to beindependent of subjects, aka objective; for TENOS, cf. Zbl. 815.03036.)

(c) On p. 150 of Vol. II epistemology comes up (in a lecture in the mid40s at Princeton University): . . . the great importance of general recursive-ness (or Turing computability) . . . is largely due to the fact that with thisconcept one has succeeded in giving an absolute definition of an interestingepistemological notion.

This hard core view of recursiveness certainly fits Gdels ideas inour conversations beginning some 10 years later. It also fits a quip, re-ported on p. 128 of Quines Quiddities (as a bit odd, if not cheeky), bya mathematician at (the Institute of Advanced Study at) Princeton; if notin the 40s, certainly by the early 60s. Roughly, Gdels interests wouldbe more suitable for the School of Historical Studies than for the Schoolof Mathematics (to which Gdel belonged); at least, on the then prettydominant parochial view of (pure) mathematics, noted in Zbl. 795.03083(and cited in 4(a) of the last section). Less flippantly, Gdels would-bebold formulation of his view is below the threshold established by thenon-academic option in the preceding pages.

Reminders. The assumption (in that view) of some universal order of im-portance or some privileged place for orders evolved in epistemology neglects the matter of compatibility with orders evolved in other options.This neglect has already been adumbrated at one extreme, in foundationsfor the working mathematician, and in the view of philosophy as a repos-itory of ideas that strike many a minds naked eye, at another. There isa patent parallel between logical foundations of mathematics and socalledrecursion-theoretic foundations, a.k.a. the logical idea(lization), of compu-tation, and thus the parallel option of foundations for the working (humanor electronic) computer. Further, it must be remembered that experiencemeant by the household word computation does not consist only of eval-uating expressions, aka execution of programmes; no more than experi-ence meant by proof (in mathematics) consists only of formal deduction:the discovery of suitable properties of (suitable) objects for mathematicalstudy and of suitable descriptions by programmes is, by TENOS, generallymore demanding (and correspondingly rewarding). A familiar reminder


stresses the balance between contributions and distraction (from more re-warding aspects) by attention to those logical aspects of computation. (By[N2] that TENOS is fitting inasmuch as the computation meant here, incontrast to the epistemological variety, is a terrestrial business.) Readersfamiliar with the subject of finitely generated groups will remember that(in the jargon of 2(b) of the last section) some simple facts about recursive-ness have belonged to the foundations for the working group theorist (whoworks on those groups) for more than 30 years; cf. G. Higman, Subgroupsof Finitely Presented Groups, (1961). In other parts of group theory otheridea(lization)s of computability have found a place, for example, by finiteautomata, whence the name automatic groups; cf. Word processing andgroups by Cannon, Epstein, Holt and Paterson.

For any of those working foundations there is little difference betweenGdels own and Turings descriptions of recursiveness: GdelHerbrandequations and Turings universal machine, resp. Unflinchingly, ever sincethe mid 30s, Gdel himself stressed the advance made by Turing over hisown description (but also over Churchs in terms of the -calculus). Milk-sop foundations dotted some is and crossed some ts w.r.t. those equations,and established additional relations between various descriptions; for ex-ample, by matching steps in the corresponding evaluation procedures, akaChurchs superthesis. But milksop foundations have nothing memorableabout aspects of the kind involved in the advance, which Gdel saw(without describing it). One candidate for such a description focuses onideas that strike many a minds naked eye. Thus Turings idealization iscertainly more vivid to such eyes (than Gdels equations); cf. also Gdelsexpression finite mind used in his Gibbs lecture quoted in (2) below. Asusual and this is very much a matter of TENOS , the naked eye ishere, too, not dramatically in error, but it is often a bit short-sighted; inthe present case, without a clue of the often (more) laborious path from avivid description to properties of discovered interest; here from Turingsto such properties in the case of recursiveness, for example, in Higman(1961) above.

In short, orders of importance (of such properties), evolved in epis-temology and in the other trades considered, generally conflict; with thepotential for the kind of distractions (from other contributions) stressedthroughout. Those with suitable intellectual reflexes will take this in stride.But, by TENOS, not all; in ways related to the following asymmetry.

The epistemological tradition is so undemanding that Gdel (tacitly,following it) could properly use titles containing the phrases some basictheorems or, as already mentioned, the modern development of the foun-dations of mathematics when talking about incompleteness and recursive

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functions or recursively enumerable sets of axioms. This is in sharp con-trast to the reservations above about the notion of completeness itself in(b), as a corollary to diversity , and about recursiveness as a logical, buthardly realistic, idea(lization) of computation. This asymmetry is generallyreflected in the markets for which these different traditions are suitable.

Reminder of the famous exception mentioned in (b) above: Einsteins the-ories of relativity (tacitly, particularly his introductions) and their (great)epistemological value; so to speak in contrast to (marginal) relativistic ef-fects known at the time. Now, an exception does not prove a rule, neitherin the sense of establishing nor of testing it; often it draws attention to therule. But the present exception is good enough not to pontificate generallyagainst attention to the epistemological tradition; in particular, attentionby those with Einsteins particular gifts, who admittedly are not likely tobe discouraged by any such pontification anyway.

2. Brains and faster computers. On pp. 304323 of Vol. III a draft or thetext of the 1951 Gibbs Lecture of the American Math. Soc. is printed forthe first time. The broad outline has long been known (and more recentlybeen, let us say, popularized by critics of socalled strong AI). In the late40s, Turing (often regarded as a patron saint of that kind of AI) had pub-lished his exercise in the cult of black boxes, enshrined in what has cometo be called Turings test.

Gdels and Turings views are formally at opposite extremes, and so at least according to what is said about extremes would be expected totouch. They do; by neglecting points particularly prominent in this article,including the warnings (in [N2]) about views taken by many a mindsnaked eye; here, views of the mind concerned or of other minds, and corre-sponding practical consequences. Thus expanded TENOS but not only from scientific studies, say, of brain damaged subjects would be expectedto provide idea(lization)s around phenomena meant by such householdwords as intelligence or, more generally, mind, which are more suitable for such expanded TENOS than (and therefore different from) thoseimagined; at least, imagined by those of us without Einsteins particulargifts. This expectation is here viewed as parallel to the emphasis atthe end of the Preamble w.r.t. Life Itself, but also in (1) above w.r.t.computation and proof. Readers with a little background in number theorywill know examples of the evolution of more suitable ideas in the lightof TENOS also in the humble area of diophantine problems; cf. Mazursquestion in [A2 c (ii)].


It would be unrewarding (for me) to review the vast literature aroundGdels and Turings views in the last 45 years; let alone, belabour thesound bites on diverse aspects of those views in [N1]; on mechanics being(non-)mechanical (in [N1(a)]), on ideal(izations of) mathematicians (in[N1(b)]), on Pyrrhic victories of strong Al (in [N1(c)]), or on Turingsphilosophical error in (in [N1(d)]). Instead, some items in Gdels GibbsLecture will be used to underline some of those neglected points, includingof course the potential of those items for additions to the repository (inendnote 3). One of Gdels conclusions, prominent on p. 290 of Vol. III,gives fair warning; not to look in the Gibbs Lecture for contributions toaspects (emphasized in this article as) outside that repository:either . . . the human mind . . . infinitely surpasses the powers of any finite mind, or elsethere exist absolutely unsolvable diophantine problems.

(a) As to a difference between Turings and Gdels descriptions of re-cursiveness adumbrated in 1(c) above, the epistemological idea(l) of afinite mind is more vivid at least, to my mind when one thinksof Turings machine than of Gdels equations. As to the business of infinitely, or at least demonstrably surpassing the powers of this orthat, a memorable example is provided by familiar relations betweenTurings machine and any finite automaton; cf. 1(c), too. Readers mayremember here (and in (b) below) Juvenal quoted in 3(a) of the lastsection.

(b) Peirces Law, applied to the words absolutely unsolvable, is goodenough to be certain of Gdels conclusion; at least, for those of usfor whom these words are sufficiently vague. But once again theyprovide fair warning against assuming as some of those popularizersmentioned earlier do that the first alternative in that conclusion, aboutminds and machines, is established outright in the piece (by the incom-pleteness theorem). NB It is a matter of formal routine to refute thatassumption formally (for those with suitable formal resources); cf. [A2d (i)] for some items in the Gibbs Lecture that have been adapted inthe literature for logical exercises on diverse epistemological notions. The next point is less parochial.

(c) In footnote 13 on p. 309 of Vol. III Gdel envisages future advances inbrain physiology which would establish with empirical certainty thatthe brain is a machine in the sense of Turing. (Fortunately, I heard this let us say, innocent vision from him some 40 years ago, leaving meplenty of time for second thoughts.) Now, given Gdels preoccupa-tion with non-scientific meditation exercises it would be unrealistic toexpect much of his idea(l)s on empirical certainty. So one option here

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is to apply again Peirces Law; in particular, (in my case) given (my)ignorance of any specific aspects of the brain that are even candidatesfor such advances.

But too much is known (also, by me) about the infinitistic characterof recursiveness to be satisfied with this option. In terms of (3) in thePreamble, Gdels footnote is comparable to envisaging future advancesin, say, scientific carpentry to establish with empirical certainty that thelength of a diagonal of a unit square is rational. NB. In contrast to Gdelsfootnote, here specific (geometric) aspects are stated.Reminder: Every finite sequence of (pedantically, hereditarity finite) datais recursive, just as every finite (initial segment of a) decimal or binaryexpansion is a rational real number. In this connection Gdel liked toquote (in conversation) the hoary canard about all theory involving ideal-izations, but was impatient with elementary second thoughts about someidealizations being less suitable than others.

For milksop foundations the emphasis shifts to TENOS on where, if any-where, such infinitistic distinctions do have empirical interpretations, forexample, as follows:

(i) Interpretations aside, the mathematical apparatus used for some sci-entific theory may be better understood by attention to infinitistic(including infinitesimal) distinctions. A recent spectacular exampleuses algebraic (among analytical) solutions of the differential equa-tions of Yang and Mills; as usual, by a suitable shift to (previouslyneglected) discovered aspects of those equations. It is a separate mat-ter in which way(s), if any, such mathematical understanding findsan empirical counterpart; for example, in an interpretation of a new(discovered) parameter such as the degree of those algebraic solutions. The next point is more general, and hence demands less background.

(ii) In my teens there was a wide-spread superstition that infinitistic dis-tinctions, including discontinuities (in continuous parameters), werein-principle empirically meaningless. (A distinction by Hadamardwas so (mis)interpreted.) Various atomic and more generally particletheories are counter examples. Here the phenomena considered are assumed to be described in integral terms, and so approximateknowledge of empirical effects such as a potential is good enough tobe interpreted by an integral result (on the number of particles). Lesselementary uses of infinitesimal properties, by M. V. Berry, are quotedon p. 640 of [N1(d)]; quite apart from discontinuities in continuummechanics, when, for example, discrete drops drip from a tap.


In short, the weakness of Gdels footnote 13 loc. cit. is here seen notas a mere matter of principle, but in not being rooted in (enough) knowl-edge around brain physiology. Reminder. For many of us there is an air ofunreality in attempts to use Turings idealization of a computer for strongAI, which is engaged in barely doing what people and other animals dowell. The alternative is to use a few properties of that idea in foundationsfor the working software engineer, using sensible, aka robust, AI for robotsthat do their jobs better than people etc. As in other engineering, this oftenrequires very different (material) structures.

(iii) Some of the literature around (i) and (ii) above is rewarding; tacitly(as always), for those with suitable resources. At one extreme there iswork by Pour El and Richards (cf. [N1(a) and (d)]), on equations oc-curring in (well established) physics. Some of their solutions turn outto be formally non-recursive; only formally inasmuch as the aspects,aka parameters, considered are found, on closer inspection, not to besuitable for the empirical interpretation envisaged (cf. pp. 900902 ofJSL 47, 1982, on an abuse of socalled initial values in the case ofhyperbolic equations). At another extreme there are aberrations by oth-erwise realistically minded applied mathematicians (like V. I. Arnold)who ask teratological questions around recursiveness; roughly, aboutsubsets of Q , where in fact subsets of R are suitable, and appropriatedefinitions (of recursiveness) are available; cf. also da Costa and Doria(1993).

(iv) Brain physiology aside, the broad contrast between minds and ma-chines is also familiar from the foundational debates in the 20s be-tween Brouwer and Hilbert: formal operations (on formal objects)are mechanical while not only those of higher mathematics, but even logical operations are interpreted intuitionistically to have mentalconstructions (including proofs) as arguments and values. Gdel putthis into formal dress in the 30s in terms of (formal and intuitionistic)provability, cf. [Al b (iii)]; with second thoughts on using proofs aspart of the mental data (but leaving open specifics; cf. [A2 (b)]).

NB. Though Kleene presented recursive realizations as formal exercisesthese can be (and were) interpreted in the terms above; for example, thefamiliar closure condition (of many systems, with the usual notation): If .8x 2 !/ .9y 2 !/ A.x; y/, then, for some recursive V ! ! ! andeach n!,

AT Nn; .n/U; where Nn is the numeral of n:(This is in sharp contrast to classical rules for suitable501-formulae A.)

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In plain English, some thought is needed to check that the properties(enshrined in intuitionistic logic) of those mental operations are notblatantly non-recursive (despite striking differences in general laws forformal and other provability loc. cit.). In retrospect, this closure propertyof familiar, not of all intuitionistically valid, formal systems was related toanother (closure) property:If .9y 2 !/B.y/, and .9y 2 !/B.y/ is closed then, for some n!, B. Nn/;cf. JSL 37 (1972), p. 327, where NIE should be replaced by NID in l.-20and -19. (For validity, B. Nn/ need not be proved in the same, necessarilyincomplete, formal system.)

Now, academic traditions have evolved in the last 25 years so as to makethe (academic) option of continuing such exercises viable (for making anacademic living). But for the non-academic option adopted in this article,second thoughts have priority. At one extreme, (even) the old exercisespresent an alternative to a cult of black boxes. Thus enough is knownaround those mental operations (put in black boxes) for, let us say, modesttheory; cf. [A 2 d(i)] for more, comparable to exercises on neural nets. Atanother striking extreme, there is the topic of suitable areas for dif-ferences between (not necessarily conscious) mental operations and thoseof (Turing) machines. Unless the primary interest is in those exercises onintuitionistic logic, one would not start by assuming that differences in theareas of mathematics, let alone, of logic are particularly rewarding. It maywell be that differences between humans and other animals are striking inthese areas. But some elements common to humans and (suitable) animalsseem to differ enough from Turing machines at least without fancy sen-sors for establishing empirical differences (if one wants to do this); withmore scope for experimentation.

Reminders of the non-academic option for using ones logical education;cf. 3(c) of the Preamble. Above, it was used for co

kreisel, 1998 - second thoughts around some of göde's writings

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