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Poincare": Mathematics &; Logic &; Intuitiont

C OL IN M C L A R T Y *

But how have we attained rigor? It is by restraining the part of intuition inscience, and increasing the part of formal logic... we have attained perfect

rigo r. (Poincare' 1899, 129)

What does the word exist mean in mathem atics? It means, I say, to b e freefrom con tradiction. (Poincare' 1905-06, 297/474)

Recent work on Poincare''s philosophy of mathematics illuminates his con-troversy with R ussell and such affinity as he has with Brouwer. B ut the re isanother side. Poincar6's view of logic is very like Russell's, and often takenfrom Russell with acknowledgement. A broader reading revises the view

that 'his papers on foundations are disconnected from his positive work inmathematics' (Goldfarb 1988, 62) or 'His philosophic comments [on math-ematics] are almost exclusively concerned with basic number theory, settheory, and logic' (Folina 1992, xi). And we can relate Detlefsen's 1992 and1993 account of Poincare on intuition to Poincar^'s intuitive mathematicalpractice.

Contrast three kinds of intuitionism: banal, expansive, and restrictive.The banal merely says research and teaching require something beyond for-mal rigor—perhaps 'motivation'. Expansive intuitionism claims the actual

content of mathematics goes beyond any formalization. The restrictive re-jects some stand ard ma thematics as inaccessible to intuition. Poincar6 wasan expansive intuitionist. His problem, which we inherit here, was to avoidlapsing into the banal. He was not restrictive. He objected to none ofclassical mathem atics. He considered standard formal logic the gua ran torof rigor in mathematics. He found Peano, Russell, and Cou tura t's newlogistic cumbersome and prey to fundamental confusion—agreeing closelywith Russell on this except for the prospect of future reform. He had little

t Research supp orted by a gra nt from the National Endowment for the Hu ma nities.I thank Michael Detlefsen, Janet Folina, Michael Friedman, Warren Goldfarb, GregoryMoore, and Michael Resnik for comments improving the paper.

* Department of Philosophy, Case Western Reserve University, Cleveland, Ohio 44106,U. S. A. cxm70pop.cwru.edu

PHILOSOPHIA MATHEMATICA (3) Vol. 5 (1997), pp. 97-115.



98 McLARTY

objection to Cantor's set theory, and was among the first to use it . His ob-

jection to Zennelo's axioms (apart from a late ambiguous one to the axiom

of choice) was only that he was unwilhng to trust formal axioms without

either a consistency proof or a principled account of why just these axiomswere chosen, especially when a key technical term remained unclear.

1 . Poincar^ Champ, ' >ns Logic

Brouwer got Poincare r ight and Russell wrong when he wrote:

How far Poincare' is from taking the intuitive construction as the only basisof his criticism, appears from his words: 'Mathematics is independent of theexistence of material objects; in mathematics the word exist can have onlyone meaning, it m eans free from con tradiction'. It might almost have been

w ritt en by his opponent Russell. (Brouwer 1907, 96, quo ting Poincare' 1905-06, 819/454)

Poincare understood the logicists better , and chided Couturat for not tak-

ing consistency as the proof of existence (1905-06, 297/474).

After defining the ari thmetized continuum Poincare dismisses the de-

mand for a constructive definition, saying 'no one will doubt the possibility

of the operation, unless from forgett ing that possible, in the language of

geometers, simply means free from contradiction' (1893, 27/44). He also

defends non-Euclidean geometries this way: 'A mathematical enti ty exists ,

provided its definition implies no contradiction' (1921, 61). He says suchthin gs repe ated ly. For his one possible reversal of pos ition see section 7

below.

He valued logic as a counterpoise to intuition:

We see we have progressed towards rigor; I would add that we have attainedit and our reasonings [as opposed to many older proofs—CM] will not appearridiculous to our descendents; I refer, of course, to those of our reasonings thatsatisfy us.

Bu t how have we attained rigor? It is by restraining the part of intuition

in science, and increasing the part of formal logic. Before, one began with alarge number of concepts regarded as primitive irreducible and intuitive; suchwere the concepts of whole number, fraction, continuous magnitude, space,point, line, surface, etc. Today only one remains, that of whole number; allthe others are only combinations, and at this price we have attained perfectrigor. (1899, 129)

Po inca re well knew th e accuracy and pro du ctivity of W eierstrassian an al-

ysis. He is often quoted saying: 'Heretofore when a new function was in-

vented it was for some practical end; today they are invented expressly to

pu t a t fault th e reaso ning of our fathers; a nd one will never get mo re from

the m th a n th a t ' (1899, 130, repe ated in 1904, 264 /435 ). Bu t cri t ique is

som ethin g too , an d Poincar6 less famously affirmed th e need for it: 'O ur

fathers th ou gh t th ey knew w hat a fraction was, or continuity, or the area

of a curved surface. We have found they did not know it ' (1904, 265/437).



P0INCAR6 99

To see how serious he was, see how he clarifies the fundamentals of curvedsurfaces in all his mathematics cited below.

One more of many passages on this implies a comment on Poincar6's

own work:Now in the analysis of today, when one cares to take th e troub le to be rigorous,ther e can be no thing bu t syllogisms or appeals to this intuition of pure number,the only intuition which cannot deceive us. It may be said today th at absoluterigor is atta ine d. (1900, 122/216)

2. Poincar<5's Style

Poincar6 did not take the trouble to be rigorous. His famous series ofarticles 'On curves denned by a differential equation' (Oeuvres, Vol. 1, 1-

222) offers a brilliant array of new, detailed analytic theorems. Proofs arehasty or absent, all untouched by W eierstrassian method s. Many statedtheorem s are plainly false. They are supposed to be 'generically tru e' insome unspecified sense of 'the generic case'. Most have been rigorouslyexplicated since by a considerable effort. Some are simply wrong. Some areimpossible to interpret from the sketchy statements and proofs. And thiswork founded modern topological dynamics.

The equally famous series 'On analysis situs' (Oeuvres, Vol. 6, 189-498)reasons very subtly about the continuum, again without arithmetic defini-

tions, and gets crucial points wrong. Wild nonsequiturs and plain errorsare richly documented in Dieudonn6 1989. And th is founded algebraictopology.

Poincare knew he published many incorrect proofs, leaving the 'care' toothers. His talk of 'reasonings that satisfy us ' was an ironic contrast tohis own work. On his principles faulty proofs are worthless, but not tooliterally worthless:

In mathematics rigor is not everything, but without it there is nothing. Ademonstration which is not rigorous is nothingness. I think no one will contest

this tru th . But if it were taken too literally, we should be led to conclude thatbefore 1820, for example, there was no mathematics; this would be manifestlyexcessive; the geometers of that time understood voluntarily what we explainby prolix discourse. This does not mean they did not see it at all; but that theypassed over it too rapidly, and to see it well would have necessitated takingthe pains to say it. (1908, 171/374)

Darboux, Poincare^s doctoral thesis advisor, tells us: 'It must be said,if one wants to give an accurate idea of how Poincare worked, that manypoints of [his thesis] needed correction or explication. Poincare was an

intuitif. Once at the summit he never went back over his steps' (Darboux1913, xxi). Poincare corrected what D arboux asked him to, bu t said he wasthinking of other things as he did it.

Poincare's streng th lay in passionate intuition . His nephew describesthis:



100 McLARTY

Research as he understood it had to be a duel. It is han d-to-han d w ith fugitiveand rebellious reality, which one aims to strike to the heart . In such a duelthe re is no place for w itnesses. Intuition , the means of discovery, is a directcommunion, with no possible intermediaries, of mind and tru th . (Boutroux1921, 148)

3 . Intuition: Three Mathematical Examples

Goldfarb f inds that : 'For Poincar6, to assert that a mathematical t ruth is

given to us by intui t ion amounts to nothing more than that we recognize

its truth and do not need, or do not feel a need, to argue for it ' (63). But

Po incar6 's exam ples show otherwise. His neph ew 's image of 'a du el ' was

closer to the point.

On e example ha s a part icularly r ich history. (See M onn a 1975, an d theappendix to Bottazzini 1986):

I shall take as m y second exam ple [of the role of intuition] D irichlet's principleon which rest so many theorems of mathematical physics; today we establishit by reasonings very rigorous but very long; heretofore on the contrary, wewere content with a very summary proof. A certain integral depending onan arbitrary function can never vanish. Hence it is concluded it must have aminimum Th e flaw in this reasoning strikes us imm ediately, since we use th eabstract term function and are familiar with all the singularities functions can

present when the word is used in the most general sense (1900, 119/213).This 'Dirichlet principle' was named by Riemann, who learned it from

Dirichlet, tho ug h it ha d been used before. T he arg um en t Poincare" cites

was given by Dirichlet and Riemann and others and is obviously fallacious,

but not for the reason Poincar6 gives.

The fallacy has nothing to do with smoothness of functions but is only

this: A lower bound to the integrals does not imply that one of them has

a minim al value. C om par e polygo ns enclosing a circle. Th eir area s are

bou nde d below by the area of th e circle bu t none is m inim al. Ev eryon e

involved knew th e poin t. Ga uss used just such fallacious reason ing wh enhe ha d to an d criticized it when he could. Dirichlet faulted Steine r for using

it t o prove a circle ha s th e gre ate st are a for a given circum ferenc e. (See

Monna 1975, 11-25 and 37-40.) And Felix Klein says:

Weierstrass once had occasion to tell me that Riemann put no decisive valueon proving his existence theorem throu gh the 'Dirichlet p rinciple'. An d soWeierstrass's criticism of the 'Dirichlet principle' did not particularly impresshim. (quoted in Monna 1975, 34)

Riemann felt the argument offered some insight or he would not have

given it . At least i t played th e expository role th a t pa rtia l, or slightly

abusive, proofs often play in math texts today. And he certainly wanted a

rigorous one. In the meantime he put 'decisive value' on the whole theory—

a huge array of partial proofs and independently verifiable consequences



POINCAR& IOI

th at rang well with his exper t expecta t ions . For example see the Riem ann-

Roch theorem as in McLarty (forthcoming).

There were other critics of the principle and other arguments for it,

including physical analogies. It is not clear which of these were meant asr igorous, or in what sense. Various efforts, including those by Weierstrass

and by Poincare, led to notable progress in analysis (Monna 1975, 43-44) .

We come back to Poincare 's mention of very general functions. R ath er

t h a n the simple fallacy by Riem ann et a/., Poincare addresses 'object ions

concerning the continuity of functions defined by the calculus of var ia t ions '

arising precisely in various attempted proofs (1890, 33).

A s to the outcome: In today ' s te rms the 'Dirichlet principle' says there is

a minimal value for the integral . The related 'Dirichlet problem' is to show

certain boundary conditions define harmonic functions. It is not clear whichPoincar6 meant . Poinca re was believed to have solved the Dirichlet problem

for fairly general boundary conditions. (See Poincare 1890. H a d a m a r d

in Oeuvres, Vol. 9, 284-88, calls it 'one of the most beautiful t r iumphs

of Poincare's genius' . ) But there were technical errors only corrected by

Lebesgue in 1937.

The first good proof of the Dirichlet principle was given by Hilbert to the

Deutsche Mathematiker-Vereinigung in September 1899. Poincare was no

avid journal reader and probably did not see the published notice (Hilbert

1900) before his August 1900 talk quoted above but he had surely heard ofth e proof.

For another example Poincare' says

You know what Poncelet understood by the principle of continuity. What is

true of a real quantity, said Poncelet, should be true of an imaginary quantity;what is true of the hyperbola whose asymptotes are real, should then be trueof the ellipse whose asymptotes are imaginary. Poncelet was one of the mostintuitive minds of this century; he was passionately, almost ostentatiously so;

he regarded the principle of continuity as one of his boldest conceptions, and

yet this principle did not rest on the evidence of the senses. l b assimilate the

hyperbola to the ellipse was rather to contradict this evidence. It was onlya sort of precocious and instinctive generalization which, moreover, I have no

desire to defend. (1900, 122/215, the summary of the principle is not in the

original. Cf. Kline 1972, 840-845.)

This principle is plainly false as stated . Real quan ti t ies are not imagi-

nary. Hyperbolas are not ellipses. But correctly taken it powerfully unified

the geometry of lines and conies. Poncelet engaged in a prominent debate :

The other members of the Paris Academy of Sciences criticized the princi-

ple of continuity and regarded it as having only heuristic value. Cauchy, inparticular, criticized the principle but unfortunately his criticism was directedat applications made by Poncelet wherein the principle did work. . . the notesPoncelet made in prison show th at he did use analysis to test the soundness of

the principle. These notes, incidentally, were written up by Poncelet and pub-



102 McLAKTY

lished by him in two volumes entitled. Applications d'analyse et de g6omStrie(1862-64). (Kline 1972, 843-44)

Those notes were not only to persuade othe rs. Poncelet aimed to pro-

mote synthetic over analytic geometry. To weaken his case by using anal-ysis shows he was honestly testing his method—building arguments by itssuccesses—not writing propaganda. Poincare, at the center of the Parisianmathematical world a few decades later, surely knew the passionately in-tuitive Poncelet had backed his intuition with these arguments.

This principle was eventually transformed into complex projective ge-ometry. There, among other things, the ellipse and hyperbola merge inthe single non-degenerate conic. Its further development was tied w ith ourthird example. Poincare's work on the connection was barely the beginning.

(See especially Oeuvres, Vol. 6, 373-434.)Poincare had a special fondness for his own analysis situs, now topology.

He says 'geometric facts are only algebraic or analytic facts expressed inanother language'; but to think geometry offers nothing new 'would be tofail to recognize the importance of well constructed language':

The problems of analysis situs would perhaps not have suggested themselvesif the analytic language alone had been spoken; or, rather, I am mistaken,they would have occurred surely, since their solution is essential to a crowd ofquestions in analysis, but they would have come singly, one after another, and

without our being able to perceive their common bond. (1908, 180-81/380-81)Without using the term 'intuition' this passage deals with 'a direct sense

of what constitutes the unity of a piece of reasoning, of what makes, soto speak, its soul and inmost life', which is Poincare's account of intuitionin (1900, 128/220). And 'analysis situs... is the true domain of geometricintuition' (1912a 502/42).

Yet he says each single fact of analysis situs would eventually have beenproved in analysis, without geometric intuition. Nor does having intuitionrelieve us of a need to argue for its results. He gives detailed though often

sloppy proofs, and violates his general rule against revising work by re-turning to improve them again and again in the 'Complements' (Oeuvres,Vol. 6, 290-498).

A nineteenth-century mathematician who felt no need to argue for Di-richlet's principle, or Poncelet's, or for Poincare's analysis situs was guar-anteed to learn little from them. None was terribly compelling on its face.None was quite correct. Only expert judgm ent could tell their uses; andthis was gained only by exploring and checking them as far as possible.Intuition, for Poincare, has nothing to do with not needing arguments.



POINCAR£ 103

4. Intuition: Philosophical Account

Poincar6 wrote of several kinds of intuit ion. The intuit ion of number gives

the principle of mathematical induction. Poincar^ understood that Russell

also took this principle as a synthetic judgement. He says:

On this point all seem agreed, but what Russell claims, and what seems tome doubtful, is that after these appeals to intuition [i.e., the few principles oflogistic, including induction], that will be the end of it; we need make no othersand can build all mathematics without the intervention of any new element.(1905-06 830/462, emphasis in original)

The continuum is ari thmetizable in logist ic, using induction, but i t orig-

inated in sensible intuit ion. This is the base of Poincare 's theory of visual ,

tactile, and motor space, which is outside the scope of this essay, but see

(1893) and (1912a) along with much else. Within pure mathematics, thisintuition was productive even when it fooled us, as when it told us every

curve has a well defined tangent at each point (1900, 118/213). And it was

more productive yet when i t was corrected.

Analysis made huge progress when 'The vague idea of continuity, which

we owe to intuition, resolved itself into a complicated system of inequalities

referring to whole numbers ' (1900, 120/214). Poincare says:

Th e analysts axe none the less right in defining the co ntinuum as they d o . . . Butthis is enough to apprise us that the veritable mathematical continuum is avery different thing from that of the physicists and metaphysicians. (1893,27/43-44)

The mathemat ical cont inuum would never have occurred to anyone, and

its complicated inequali t ies are incomprehensible even now, without the

sensible intuit ion.

Further, the sensible intuit ion of the continuum has aspects, such as

various orde rs of infinitesimals, which are not yet ari thm etized . Poin care

saw no use for these, except to show sensible intuition goes beyond the

current ar i thmet izat ion (1893, 3 3 ^ / 5 0 - 1 ) .

Po incare also calls analogy within ma them atics a kind of intuit ion (1900,

128/220). He draws all these into his theory of:

a faculty which makes us see the end from afar, and intuition is this faculty...This view of the aggregate is necessary for the inventor; it is equally necessaryfor whoever wishes really to comprehend th e inventor. (1900, 125-6/218-9)

He says:

In these com plicated edifices raised by the m asters of ma them atical Science, itis not enough to affirm the solidity of each part and admire the mason's work,

one must understand the architect's plan.Now, to understand a plan, one must see all its parts at once, and only

intuition can give us the m eans to take in all at a glance. (1900, 125/-.Similar passage in 1904, 264/436.)

Folina sees in Poincare a Kantian theory of a priori intuit ion which



104 McLAKTY

lets us 'gloss over' irrelevant details and 'supplies organization' to largeareas of knowledge (Folina 1992, 88). But G olfarb (1988, 63) notes th eKantian appa ratu s of sensibility and th e categories is absent from Poincare^

Indeed, organizing and viewing the whole are functions of reason and notintuition for Kant. However that may be, Folina (1994) discusses Poincar6'sintuitions of induction and the continuum in a way that shows they aim atthe large scale organization of arithmetic and geometry.

I refer to Detlefsen (1993) for further philosophic exposition of Poinca-rean intuition as insight into 'architectu re'. Detlefsen sees it as givinga generally Kantian epistemology for mathematics, not specifically corre-sponding to Kant ian intuition. I think Detlefsen exaggerates the opposi-tion between intuition and formal logic; but he makes the crucial point

that Poincar6's 'challenge to logicism was not of the form that some par-ticular proof or kind of proof is not fonnalizable' (Detlefsen 1993, 27). Infact, formalization reveals the mistakes of intuition and is the only meansof assuring its insights. Yet Detlefsen shows well that for PoincarS formalproof never itself yields knowledge—neither for the inventor nor for the stu-dent. Mathematical knowledge comes from mathematical reasoning, whichis contentual and not formal.

Darboux had this sense of intuition as overview in mind when he calledPoincar6 an intuitif. The details Poincar6 neglected were not irrelevant, to

Darboux or to Poincare, only they were not Po incar6's special gift. Theycould be left to others. And Poincar6's most distinctive contribution , bistopology, was precisely an effort to unify broad reaches of mathematics con-tentually. Hubert, for example, would seek unity in a means of expression,the axiom atic method. Poincare sought to simplify and organize mechanics,differential equations, complex function theory, and group theo ry (see 1901,322-23) by bringing out the content of our deepest topological intuition.

A last point on this topic: Goldfarb claims that 'intuition' in Poincare(1905-06) 'is a psychological term '. He finds 'Poincar6 's concern is explicitly

with the psychology of mathematical thinking', and 'Poincar£ first raisesthe charge that logicism does not accurately portray the psychology ofmathem atical thinking [and] then claims to foreswear the ch arge ' (Goldfarb1988, 63-4) . Th is despite the fact that the syllable 'psych' occurs just oncein Poincar6's lengthy essay: Russell and Couturat offer no proof of theconsistency of arith metic, so 'a psychological question arises: how is it twosuch knowledgeable logicians have not noticed this gap?'. Poincare tracesCouturat's negligence to faith in Russell, and Russell's to a simplified viewof the principle of induction which weakened it so its consistency seemed

trivial. (See 1905-06, 834-35/-.)Goldfarb finds psychologism most explicit in the critique of Couturat's

definition of 'two' by a propositional function with variables x and y.PoincarS says 'I continue to think tha t M. C ou tura t defines the clear by the



POINCAR6 105

obscure, and that one cannot speak of x and y without thinking two' (1905-06, 294/-). Goldfarb claims the distinction 'between "clear" and "obscure"is a psychological one' and the reference to thought 'highlights the psycho-

logistic nature of this argument' (Goldfarb 1988, 66-7). But I see Poincare'as saying, in the aphoristic style that suits this showy sarcastic passage,the same thing Godel would say about Principia Mathema tica (quoted insection 9 below). That is, Couturat's definitions presuppose arithmetic inwh at we now call their metatheory.

Goldfarb rightly says 'The logical system Prege or Russell proposes ismeant to be the universal language, inside of which all reasoning takesplace. There is no metatheoretical stance available or needed' (Goldfarb1988, 69). Bu t without denying the intent, Poincare" and Godel deny the

achievement. They deny the systems can function apa rt from their in tu-itive explanations in ordinary language— though Poincare argued this wouldapply to any formal language, while Godel did no t. In fact the formal inad-equacy of the systems Poincare" discussed was noncontroversial among thelogicists themselves a few years later—and Prege famously recognized hisown system 's failure. It is no question of psychology.

Poincare' studied the psychology of mathematics, even of debates on thefoundations of mathematics. Bu t his theory of intuition does not reduce topsychology.

5. Logic and the Actual Infinite

Poincar^ asks 'May the ordinary rules of logic be applied without changewhen we consider collections comprising an infinite number of objects?' andanswers that they not only may, they m ust. In particular we must followthe ordinary rule that 'the classification which is adopted [in any piece ofreasoning] be immutable':

The antinomies which have been revealed all arise from forgetting this very

simple condition: a classification was relied on which was not immutable andwhich could not be so; the precaution was taken to proclaim it as immutable;but this precaution was insufficient. It was necessary to render it immutablein fact, and there are cases in which this is not possible. (1909, 461/45)

He happily gives an example which he and Russell agree involves onlyfinite integers. Russell thought Poincare" blamed the antinomies on infinitesets, and sought to confound him by this example (1909, 462/46). It is theparadox of 'the smallest integer which cannot be denned by fewer than onehundred English words'.

Poincare" sees it this way: No classification of integers into those thatARE so definable and those NOT can be immutable because it explicitlydepends on what means of definition are available. Any classification weadopt becomes a new means of definition, and the smallest integer in theNOT category then IS denned in a few English words using this very clas-



106 McLAKTY

sification.Russell had already found not all descriptions define sets or classes, and

baptized the problem: descriptions 'which do no t define classes I propose to

call non-predicative; those which do define classes I shall call predicative'.He claims all the non-predicative involve 'what we may call self-reproductiveprocesses'. He then offers three possible solutions: the zig-zag theory, thetheory of limitation of size, and the no-classes theory (Russell 1905, 141 ff.).

Poincare' (1905-06, 305/478 ff.) adopts the terminology. His at titu de wasalways th at Russell 'seeks the origin [of the paradoxes] and finds it correctlyin a sort of vicious circle' (1909, 467/50). But he rejects the particular cir-cles proposed by Russell in favor of self-reference. Th is is close to Russell'slater m otive for ramified type theory. (See the succinct description in Godel

1944, 133-35.) Both Poincar6 and Russell saw predicativity as implicit intraditional logic, and wrongly ignored in the first logicist efforts.

Poincar6 did believe it was easier to fall into this fallacy when dealingwith infinite sets. ' There is no actual infinite; the Cantorians forgot this,and they fell into contradictions' (1906, 316). But he does not deny thatinfinite sets actually exist. He never doubted there is a set of na turalnumbers, its power set, a set of arithmetized real numbers, etc. His work intopological dynamics explicitly involves infinite sets of curves, each curvebeing an infinite subset of a continuum. He had no tendency a t all to avoid

infinite sets.Rather, in rejecting the 'actual infinite' Poincar6 rejects the belief that

all the members of an infinite set can be seen as 'given' independently ofspecific definitions for them:

The word all has a clear meaning when applied to a finite number of objects; forit still to have one when the objects are infinite in number would require hatthere be an actual infinity. Otherwise all these objects cannot be conceived asgiven prior to their definition and if the definition of a notion N depends onall the objects A, it might be spoiled by a vicious circle, if some of the objects

A cannot be defined without using the notion N itself. (1905-06, 316/-)

Poincare' does not specify the ontology an d/or epistemology of 'givenness'but we saw it function in the paradox of integers definable hi under onehundred words. We can not regard the integers as 'all given prior to' such aclassification, since any such purported classification itself 'gives' us a newinteger escaping it.

Poincare' applies the same analysis to the diagonal arguments in set the-ory. For example, any purported enumeration of the poults of the contin-uum itself 'gives' us a continuum point no t in the enumeration. He sayswe legitimately conclude that 'the power of the continuum is not the powerof the integers' because 'it is impossible to establish between these twosets a law of correspondence which will be free from this sort of disrup-tion' (1912b, 4/6 8). But Poincar6 also concludes that any determination of



POINCARfi 107

infinite cardinality is relative to particular means of expression.

So wh en he denies th e existence of Aleph one, or higher infinite card inals,

he does not deny that the sets supposed to have these cardinal i t ies exist .

We ju st saw him affirm t h a t th e continuum exists as a set , an d ha s notth e power of th e integ ers. R ath er he denies th at th e theo ry of infinite

cardin alit ies is well defined. Th is parallels the Skolem para do x, tho ug h I

do not know how seriously to take the resemblance.

On the point of definitions (though here not specifically tied to the infi-

nite), Fblina cites Poincare^s view that a definit ion must have two parts:

The first part of the definition, common to all the elements of the set, willteach us to distinguish them from elements which are alien to this set; thiswill be the definition of the set; the second part will teach us to distinguish

th e different elements of the set from one another. (Folina 1992, 114, qu ote ofPoincare' 1909, 478/61)

She interprets the second part in terms of Neo-Kantian construct ivism or

ant i -Pla tonism and P oincar6 's theory of intui t ion. However th at m ay be, I

would note a pract ical mathematical motive as wel l .

In his analysis situs Poincare' speaks of curves, homotopy loops, and

homology cycles . Ident ical curves occupy exact ly the same poin ts . Tw o

curves which can be continuously deformed into one another are identical

homotopy loops. They are identical homology cycles if together they form

th e bou nd ary of som e surface. He was very fond of thi s com pariso n. (See

e.g., Oeuvres, Vol. 6, 239-246 and 449.)

Today we define homotopy classes and homology cycles as equivalence

classes of curves. And Poincar6 sometimes spoke that way. But he generally

spoke as if curves, loops, and cycles are the same things with different

ident i ty condi t ions.

6. Zermelo's Axioms

Probably Poincare ' ' s most famous quote on mathematics is that ' la ter gen-erations will regard set theory as a disease from which one has recovered' .

But Gray (1991) shows there is no reliable source for this quote—and he

shows a trail of unreliable rumors growing into the story that Poincar£ said

it .

In fact Poinc ar6 objected to nothing in C an tor ' s own set theory. See

the cruelly witty passage on Burali-Forti 's logistic versus Cantor in (1905—

06, 824/458-459). He was an ear ly advocate of set theory. For example,

his 'On curves defined by a differential equation' talks about the points in

which a trajectory in space might repeatedly cross a given reference plane.These 'form a set of points which I call P, using the nota t ion adopted by

M . C an tor. I will use P' for the derived set of P, that is to say the set

of points whose every neighborhood contains an infinity of points of P'

(Oeuvres, Vol. 1, 142). Citing several papers by Cantor, he uses derived



108 McLAKTY

sets to help describ e th e large scale beh avior of a trajecto ry. He has not

the slightest tendency to avoid infinite sets.

He did object to Zermelo's axioms. Folina claims:

Poincare''s arguments against set theory are arguments against the truth find/or unrestricted application of these 'nonconstructive' axioms [infinity, powerset, choice]. (Folina 1992, 112)

She identif ies precision in mathematics with Cantor 's set theory, and both

with formalization, in a way Poincare ' would hardly have done:

It was not that he was 'against' precision or the fonnalisation of mathematics(as some of his rem arks suggest). Indeed he was one of the first to employCantor's theory of sets. Rather, he objected to the realist, or nonconstructiveinterpretation of the existential axioms. (Folina 1992, 112)

She cites no passag e wh ere Poincare" objects to such inter pre tat ion , and I

believe there are none, except an ambivalent one on the axiom of choice.

In ste ad , Po inca r6 found Zermelo's axiom s shod dy. In (1909) he gives

them in full . He notes that axioms:

can be regarded as nothing but arbitrary decrees which are nothing but thedisguised definitions of fundamental notions. It is thus that Mr. Hilbert [pro-ceeds] at the beginning of geometry...

For this to be legitimate, it is necessary to prove that the axioms thus in-

troduced are not contradictory, and Mr. Hilbert has succeeded perfectly as faras geometry is concerned, because he assumed [arithmetized] analysis to bealready e stabli shed ... Mr. Zermelo did not prove his axioms were exempt fromcontrad ictions, and he could not do so, for, in order to do so, he would have t ouse as a basis othe r tr ut hs already established. But as for tru th s already es-tablished and a science already completed—he supposes there are none as yet;he sweeps everything away and wants his axioms to be entirely sen" sufficient.(1909, 472-73/55-56)

He says ' the postulates cannot therefore owe their value only to a sort of

arbitrary decree; i t is necessary that they be self-evident ' (1909, 473/56;

the negation is missing in 1963).

Apart from the axiom of infinity, he finds the axioms indeed self-evident

for finite sets. B ut for finite sets so is an eig hth axio m ' 8 . Any objec ts

form a set ' (1909, 475/58), which we know will not do for infinite sets. So

how does Zerm elo find which tru th s will ex ten d to th e infinite? Poincare'

charges that Zermelo has no grounds:

Now, by what m echanism were [Zermelo's axioms] cons tructed ? Those axiomswere taken which are true for finite collections; they could not all be extendedto infinite collections, this extension was made only for a certain number ofthem , chosen m ore or less arbitrarily. (1909, 481-82 /61-62; the w ord 'all' ismisplaced in 1963.)

Further, Zermelo sought to avoid the paradoxes with his separation ax-

iom. This axiom says that given a set A and a 'definite ' property P there



POINCAR£ 109

is a set of all those elements of A that have P. But Zermelo's expla-nation of the term 'definite' struck Poincar6 and many others as terriblyvague. Poincare" supposed it m eant som ething like 'predicative' but noted

'the use made of it by Mr. Zermelo shows the synonymy is not perfect'(1909,476 /59). Poincare' could not find just wh at Zermelo mean t.

Today we usually settle the matter in Skolem's way: a 'definite' pro-perty is one expressible hi the first-order language of pure set theory. Butthis is so far from Zermelo's own intent that he objected to it in just theway Poincare would have if he, Poincar6, had lived to see it: this solutiondepends on a prior theory of the syntax of first-order logic and so it, likeFraenkel's proposal, 'depends on the concept of finite number whose expli-cation should rather be one of the main goals of set theory' (Zermelo 1929,

340; cf. 342). Poincare' did not object to theories tha t depend on arithm etic,as for example Hilbert's geometry did, but he Like Zermelo would say sucha theory can not be the foundation of mathematics.

In fact, Poincare' and Russell suspected Zermelo's axioms were inconsis-ten t; and each preferred his own solutions to the paradoxes. (See Moore1982, 160.) But the objection Poincare leaned on was that Zermelo couldnot justify his axioms nor even state them clearly.

Poincare" dealt with the axiom of choice twice at length. First he favoredit, quoting Russell on its unse ttled st atu s and its usefulness. His only

disagreement is that 'Russell still hopes one could prove deductively fromother postula tes, that Zermelo's axiom is false, or th at it is tru e. Thereis no need to say how this hope appears to me illusory'. Poincar6 finds itis 'a synthetic judgement a priori, without which the "theory of cardinals"would be impossible, as well for finite numbers as infinite' (1905-06, 312-13/-) . An undated letter from Poincar6 to Zermelo cautiously approves theaxiom but says he will need more tune 'before I adopt a definitive solution'(Moore 1982, footnote p. 146). His second look at the axiom came in oneof his last papers.

7. The Last 'Logique de PInfini'

A few months before his death Poincare' gave the talk which became his1912b. He declines to repeat his argum ents on the infinite. The argumentis stalled since neither side listens to the oth er. Ra ther he will study thepsychological origin of the schools of thought he calls 'pragmatists' and'Ca nto rian s'. He lays out their views in fictional debates.

He identifies only Zermelo as a Cantorian, only Hermite as an opponentof Cantorism . He says nothing of himself. We naturally link him to his'pragmatists' as they, like Poincare", admit only objects defined hi a finitenumber of words, and they share his doubts on the Alephs (1912b, 5/68-69). But we should be warned when he concludes:

There is therefore no hope of seeing harmony established between the pragma-



110 McLAKTY

tists and Cantorians. Men do not agree because they do not speak the samelanguage, and there are languages which cannot be learned. (1912b, 11/74)

Poincar6 throughout the talk speaks both languages in turn.

Poincar6 says in his authorial voice, not attributing it to either pa rty inthe debate, what he said throughout his career:

A definition by postulate has value only when the existence of the objectdefined has been proved. In mathematical language, this means that the pos-tulate does not imply a contradiction; we do not have the right to neglect thiscondition. (1912b, 6/69)

But this time he continues:

Some pragmatists [certains pragmatists] will be more exacting; in order for

them to consider a definition as legitimate, it is not sufficient that it does notlead to a contradiction in terms; they will require that it have a meaning ac-cording to their particular point of view which I tried to define above. (1912b,6/69-70)

Is Poincar6 among these certain pragm atists? He does not say and heimmediately drops the issue.

I will mention that in 1911 Poincar6 answered a letter from Brouwer ontopology. (See Alexandrov 1954.) But it is unlikely th at he had Brouwerin mind as a pragm atist. Brouwer's philosophy of mathematics was then

little developed and was published only in Dutch.As to the axiom of choice, this paper 'did not repudiate the Axiom

directly' (Moore 1982, 177). It only says pragmatists will make noth ingof the well-ordering theorem as long as no one can, for example, explicitlywell-order the real num bers. Th e proof will seem to them 'bouillie pourles chats' (empty talk, literally pap fit for catfood, 1912b, 3/66). In short,'Poincare' had probably come to reject the Axiom' (Moore 1982, 177).

This short article, Poincar^'s closest brush with constructivism, is pur-posely inexplicit. It is far from typical of his earlier writing. And it chal-

lenges no specific mathematics except the well-ordering principle.

8. Couturat

Poincare"'s deba te style is always ironic. Sarton p ut this in a nice light, bu ttrailed off in his own ellipsis, saying: 'His irony was always trem endouslydiscreet and benevolent. There was almost no one bu t the logisticians towhom he was a little r u d e .. . ' (Sarton 1913, 45). By 1905 Poincar6 was sim-ply hostile towards Louis Couturat— and could be very funny. (See 1905-06,

294-5; shortened in 1921, 472-3.) Cou turat by then disliked Poincar6 andwas contemptuous of his grasp of logistic. (See Schmid 1983.) Even Russellin his autobiography has to say: 'Couturat was for a time a very ardentadvocate of my ideas on mathematical logic, but he was not always veryprudent, and in my long duel with Poincar61 found it sometimes something



P0INCAR6 in

of a burden to have to defend Couturat as well as myself' (Russell 1975,

136).Goldfarb says Poincar6 1905-06 seems 'particularly outraged at the lo-

gicists' claim to have conclusively refuted Kant's philosophy of science'(Goldfarb 1988, 61). But as I read it, Poincar6 scorns all Couturat 'sclaims—especially that logistic gives invention 'stilts and wings'. Poincar6looks at the output and reasonably says 'How is that? You have had wingsfor ten years, and have never yet flown!' (1905-06 295/472, my transla-tion). He notes th at logistic claims to give safer proofs, but it never beforeclaimed to be more inventive than ordinary practice. Dieudonne (1983)stresses how little Couturat appreciated the great advances of nineteenth-century mathematics.

But Poincare"'s attitude to Couturat is not his attitude to all logistic.He genuinely admired Hilbert's work in the foundations of geometry (see

Poincare' 1902, and recall Hilbert was early considered a logicist). I believehe means it when he writes 'Russell and Hilbert have each made a vigorouseffort; they have each w ritten a work full of original views, profound and wellw arran ted ... Among their results, some, many even, are solid and destinedto live' (1905-06, 34/470). Yet logicism will never capture mathematics.

9. Poincare' with the Logicians

Poincar6 often sides with Hilbert and Russell. He quotes Hilbert:

in the usual expositions of the laws of logic certain fundamental concepts of

arithmetic are already employed; for example, the concept of aggregate, in part

also the concept of number. We fall thus into a vicious circle and therefore to

avoid paradoxes a partly simultaneous development of the laws of logic and

arithmetic is requisite. (1905-06, 17/464. Cf. Hilbert 1904, 131.)

Prege was defenseless against the paradoxes of set theory, paradoxes exempli-

fied by consideration of the set of all sets, and which seem to me to establish

that the notions and methods of the usual logic do not yet have the precision

and rigor required by set theory. (1905-06, 19/-. Cf. Hilbert 1904, 130.)

Poincar6 adds 'we have seen that what Hilbert says of logic in the usualexposition applies likewise to the logic of Russell'; and he goes a step fartherapplying it as well to Hilbert's logic (1905-60, 17/464 ff.). Hilbert wouldlater agree. It was decades before Hilbert's school had a metamathematicsthat could limit the amount of arithmetic assumed.

Poincare quotes Russell on each of Russell's possible definitions of pred-icativity. Specifics of the zig-zag theory 'have to be exceedingly compli-

cated and cannot be recommended by any intrinsic plausibility'. They haveno guiding principle but the ad hoc avoidance of contradictions (1905-06,306/479; quoting Russell 1905, 147). On limitation of size 'A great dif-

ficulty of this theory is that it does not tell us how far up the series of

ordinals it is legitimate to go' (1905-06 306/-; quoting Russell 1905, 153).



POINCARjfe 113

References

English translations of PoincarS's philosophy are edited from French collections,themselves edited from originals now crumbling on high acid paper. Omitted are

largely the mathematical examples, technical logic, and the sharpest polemics.I generally cite the original and an English translation, quoting the translation.Cita tions by dat e alone are to Poincare\ So (1908, 171/374) refers to page 171of the original (Poincare' 1908) entered below, and page 374 of the translationlisted in that entry. A dash as in (1900, 125/-) means the passage is not in thetranslation. Poincare' 1916-1956 is cited as Oeuvres.

ALESANDROV, P. S. 1954: 'Poincare' and topology '. Speech to Interna tionalCongress of M athem aticians, A msterdam, reprin ted in Browder 1983, 245-55.BOTTAZZINI, U. 1986: The higher calculus: A h istory of real and complex analysis

from Eider to Weierstrass, W. Van Egmond (trans.). Berlin: Springer-Verlag.BOUTROUX, P. 1921: 'Lettre de M. Pierre B outroux a M. Mittag-Leffler' in ActaMathematica 38 , 197-201. Citation is to the reprint in Poincar^'s Oeuvres,Vol. 11, pp. 146-151.

BRO UWER, L. E. J. 1907: Thesis 'On the foundations of mathematics', in A. Hey-ting (ed.), Collected Works, Vol. 1. Amsterdam: North-Holland, 1975.BRO WDER, F. (ed.) 1983: The mathematical heritage of Henri Poincare', 2 vol-umes. Providence, R. I.: American Mathematical Society.DARBOUX, G. 1913: 'Eloge historique d'Henri PoincarS' reprinted in Poincar6

Oeuvres, Vol. 2, vii-lxxi.DETLEFSEN, M. 1992: 'Poincare against the logicians', Synthese 90, 349-378.1993: 'Poincare' vs. Russell on the role of logic in m athe m atics', Philoso-

phia Mathematica (3) 1, 24-49.DIEUBONNE, J. 1983: 'Louis Couturat et les mathematiques de son 6poque', inColloque International consacr^ a l'oeuvre de Louis Couturat, L'Oeuvre de LouisCouturat, Paris: Presses de l'ecole normale supdrieure, pp. 97-112.

1989: A history of algebraic and d ifferential topology , 1900-1960, Basel:Birkhauser.FOLINA, J. 1992: Poincare" and the philosophy of mathematics. New York: St.

Martin's.1994: 'Poincar^'s conception of the objectivity of m athe m atics ', Philoso-

phia Mathematica (3) 2, 202-27.G O D E L , K. 1944: 'Russell's mathematical logic', in P. Schilpp (ed.), The Philo-sophy ofBertrand Russell. La Salle, 111.: Open Court, pp. 123-154.GOLDFARB, W. 1988: 'Poincare

1against the logicists', in W. Asprey and P. Kitcher

(eds.), History and philosophy of modern mathematics. Minneapolis: Universityof Minnesota Press, pp. 61-81.

GRAY, J. 1991: 'Did Poincare say "set theory is a disease"?', Mathematica/ Intel-ligencer 13, 19-22.HILBERT, D. 1900: 'Uber das Dirichlet'sche Princip' in Jahresberichte der Deut-schen Mathematiker-Vereinigung 8 Erstes H eft. Th is was published 6 April 1900according to its cover. I know nothing about Europ ean distribution, b ut HarvardCollege catalogued its copy on 2 June 1900.

1904: 'Uber die Grundlagen der Logik und der Arithmetik', Verhand-



114 McLARTY

lungen des Dritten Internationalen M athematiker-Kongress in Heidelberg vom 8.bis 13 August. Leipzig: Teubner, pp. 174-185. Translated in van Heijenoort, J.(ed.), From Frege to Godel, Cambridge, Mass.: Harvard University Press, 1971,pp . 129-138.M CLARTY, C. forthcoming: 'Vbir-dire in the case of m athe m atica l progress ' inH. Breger and E. Grosholz (eds.), Proceedings of the conference: The growth ofmathematical knowledge.MONNA, A. F. 1975: Dirichlet's principle: A mathematical comedy of errors.Utrecht: Oosthoek, Scheltema & Holkema.M O O R E , G. H. 1982: Zermelo's axiom of choice, New York: Springer-Verlag.PoiNCARi:, H. 1890: 'Sur les equa tions aux de ri v es partielles de la physiquemathe'matique', American Journal of Mathematics 12, 211-294. Citation is tothe reprint in Oeuvres, Vol. 9, 28-113 .

1893: 'Le continu math^matique', Revue de M6taphysique et de MoraleI, 26-34. Translated with alterations as pp. 43-52 of 'M athem atical m agni-tude and experience' in Poincare' 1921. Some confusion between Kronecker andDedekind is not corrected but merely reversed in 1921.

1899: 'La logique et l'intuition dans la science mathe'matique et dansl'enseignement', L'enseignement mathe'matique I, 157-163. Citation is to thereprint in Oeuvres, Vol. 11, pp . 129-133, incorrectly d ated 1889.

1900: 'Du role de l'intuition et de la logique en mathe'matiques', inComptes Rendus II Congres International des Math6m aticiens, Paris 1900. Paris:Gauthier-Villars, pp . 115-130. Translated with slight alterations as 'Intuition and

logic in mathematics' in Poincare' 1921, 210-222.1901: 'Analyse de ses travaux scientifiques', a description of his own

work w ritten for M ittag-Lefler, Ada Mathematics 38 (1921), 36-135. Citation isto the reprint in Browder 1983, Vol. 2, pp. 257-357. Excerpts scattered throughthe Oeuvres do not include the section on analysis situs.

1902: 'Les fondements de la geome'trie', Bulletin des Sciences Math6-matiques (2) 26 , 249-272. Rep rinted in Oeuvres, Vol. 11, pp . 92-1 13.

1904: 'Les definitions g^n^rales en mathe'matiques', L'Enseignementmath&natique 6, 257-283. Translated with some examples omitted as 'Mathe-matical definitions and teaching' in Poincare' 1921, pp. 430-447.

1905-06: 'Les mathe'matiques et la logique' in Revue de MStaphysiqueet de Morale XIII, 815-835, and XIV, 17-34, 294-317. Translated with extensivetechnical and polemical omissions as 'Mathematics and logic', "The new logics'and "The latest efforts of the logisticians' in Poincare' 1921, pp. 448-485.

1908: 'L'avenir des mathematiques', Atti del TV Congresso Interna-zionale dei Ma tematici, Rom a 6-11 Aprile. Rome: Accademia dei Lincei, pp. 16 7-182. Published in several journals the same year, see Browder 1983 bibliography.Translated om itting some subjects, w ith slightly expanded description of analysissitus, as "The future of math em atics' in Poincare 1921, pp . 369-382.

1909: 'La logique de Pinfini', Revue de M6taphysique et de Morale

X V I I, 461-482. Translated in Poincare 1963 as 'The logic of infinity', p p. 45-64.1912a: 'Pourquoi l'espace a trois dimensions', Revue de Me"tapiiysique

et de Morale XX, 481-504. Translated as 'Why space has three dimensions' inPoincare 1963, pp . 25-42.



P O I N C A R £ 1 1 5

1912b: 'La logique de l'infini', Scientia 12, 1-11. Translated as 'Math-ematics and logic' in Poincare' 1963, pp. 65-74.

1916-1956; Oeuvres de Henri Poincare'. (G. Darboux et a/, eds.), 11volumes. Paris: Gauthier-Villars.

1921: Tie foundations of science, G. B. Halstead (trans.). New York:The Science Press.

1963: Mathematics and science: Last essays, J. Bolduc (trans.). NewYork: Dover.RUSSELL, B. 1903: Principles of Mathematics. Cambridge, Cambridge U niversityPress.

1905: 'On some difficulties in the theory of transfinite numbers andorder type s'. Reprinted in D. Lackey (ed .), Essays in Analysis. New York: GeorgeBraziller, 1973.

1975: The autobiography of Bertrand Russell. London: Unwin.SARTON, G. 1913: 'Henri Poincare (1854-1912)' in del et T erre, Bulletin de laSocie'te' b eige d'Astronomie, pp. 1-11 and pp. 37-48.SCHMID, A-F. 1983: 'La correspondence inedite Couturat-RusseU' in ColloqueInterna tional consacr£ a l'oeuvre d e Louis Cou turat, L'Oeuvre de Louis Couturat.Paris: Presses de l'ecole nonnale superieure, pp. 81-96.ZERMELO, E. 1929: 'Uber den Begriff der Definitheit in der Axiomatik', Funda-menta Mathematicae 14, 339-344.

ABSTRACT. Poincar6 often insisted existence in mathematics means logical con-sistency, and formal logic ia the sole guaranto r of rigor. Th e pape r joins th is

to his view of intuition a nd his own m athem atics. It looks at predicativity andthe infinity Poincar^'s early endorsement of the axiom of choice, and Cantor'sset theo ry versus Zermelo's axioms. PoincarS discussed construc tivism sym pa-thetically only once, a few months before his death, and conspicuously avoidedcommitting himself. We end with Poincare' on Couturat, Russell, and Hilbert.

poincaré_mathematics_ logic_ intuition

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