Poincaré Replies to Hilbert: On the Future of Mathematics ca. 1908

Years Ago David E. Rowe, Editor

Poincare Repliesto Hilbert:On the Futureof Mathematicsca. 1908JEREMY GRAY

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IIn April 1908, Henri Poincare boarded a train for Rome,where he planned to deliver a plenary lecture at theInternational Congress of Mathematicians (ICM) on the

present state of mathematics and its prospects for the future.This was his first opportunity to reply to David Hilbert’saddress, delivered at the ICM in Paris in 1900 (Hilbert 1900),in which Hilbert had presented a by now famous list ofproblems for the new century. Poincare could not havereplied at the intervening ICM held in Heidelberg in 1904,because he had chosen to go to the International Congress ofArts and Sciences in St. Louis instead, where he spoke on thepast, present, and future of physics. However, by 1908 heknew that Hilbert’s problems were beginning to drawattention, and like many French mathematicians, he waswell aware that the Germans did things differently – andrather well. Hilbert was by now attracting many talentedstudents toGottingen, including several foreigners. Poincare,as the universally recognized leader of the French mathe-matical community, clearly felt prompted to respond toHilbert’s challenge.

Personal illness intervened, however, preventing Poin-care from presenting his lecture at the Rome Congress.Instead, the paper was presented on his behalf by GastonDarboux, a long-standing supporter. Poincare was able tokeep his promise to the readers of Le Temps by writing areport on the Congress after talking to other guests in hishotel,1 but the illness was, in fact, serious and was still trou-bling Poincare a year later when he went to Gottingen todeliver the first series of Wolfskehl lectures he too, was not inthe best of health at this time. Since he was recovering from about of depression, he chose not to attend the ICM in Rome.However, his friend Hermann Minkowski was present andpassed along news to him about what had transpired.Remarking on the text that Darboux read, Minkowski calledit an ‘‘entirely weak remake of your Paris lecture’’ (‘‘ein ganzschwahlicher Abklatsch Deines Pariser Vortrages’’) (Min-kowski to Hilbert, 4 May 1908).

Poincare’s paper at the Rome ICM, ‘‘L’avenir des Math-ematiques,’’ was promptly published in the Revue generaledes sciences pures et appliquees and then in the Atti del IVcongresso internazionale dei matematici, 1909. As was Poin-care’s practice, it was afterward published in a modified formin a book of essays, Science et Methode, from which twoEnglish translations were made, one in the American journalthe Monist, volume 20, 1910, pages 76–92, and another in theEnglish edition of the book: Science and Method. What is notapparent from these latter publications is that they differsubstantially from the original in that almost all the detailedmathematical comments have been removed; whole sec-tions have, in fact, been deleted. Because these books are

1Le Temps, 21 April 1908, nr. 17102.

� 2012 Springer Science+Business Media, LLC, Volume 34, Number 3, 2012 15

DOI 10.1007/s00283-012-9299-7

usually the only sources cited today, this has contributed to amisleading picture of Poincare’s famous lecture. The pur-poseof this essay is to restore the fuller pictureofwhat hehadto say about the future of mathematics in 1908 by firstsketching the main themes that surface in Poincare’s lecture,followed by a presentation of the entire text in a new Englishtranslation. At the same time, I will suggest the reason thatthis speech can be seen as Poincare’s reply to Hilbert’s evenmore famous lecture, ‘‘Mathematische Probleme,’’ deliveredeight years earlier in Paris.

The first people to realise the existence of the two verydifferent texts in recent times seem to have been Philip J.Davis and David Mumford in their article of 2008 entitled‘‘Henri’s crystal ball’’, which used the translation of the fulltext available at http://portail.mathdoc.fr/BIBLIOS/PDF/Poincare.pdf. Their article takes what might be called atelescopic view, looking at developments after 1908 with aview toward seeing what Poincare got right and what hemissed. This article aims instead at gaining a closerunderstanding of how he saw the future of mathematics inthe year 1908. It thus takes what might be called a micro-scopic view of the full text and its context.

The quality of the Portail translation, which seems to beanonymous, leaves much to be desired. Far from capturingPoincare’s elegant prose, it is written in garbled, substan-dard English, as is clear from its very beginning:

The true method of forecasting the future of mathe-matics lies in the study of its history and its present state.And have we not here for us mathematicians, a task insome sort professional? We are accustomed to extrapo-lation, that process which serves to deduce the futurefrom the past and the present and I ’so well know itslimitations that we run no risk of being deluded with itsforecasts.

The well-known but much abbreviated translation is at leastfree of a strong hint of Hercule Poirot, but in view of thedefects of both translations, it seems worthwhile to publish afresh and complete translation. What follows makes only a

small number of changes to the familiar English version andrestores the missing passages. A commentary on the textfollows the translation.

The Future of Mathematics by Henri Poincare[I thank Claude Baesens for help with some passages in thenew translations; any mistakes in them are, however, mine.]

If we wish to foresee the future of mathematics, ourproper course is to study the history and present conditionof the science.

For us mathematicians, is not this procedure to someextent professional? We are accustomed to extrapolation,which is a method of deducing the future from the past andthe present; and since we are well aware of its limitations,we run no risk of deluding ourselves as to the scope of theresults it gives us.

In the past there have been prophets of ill. They tookpleasure in repeating that all problems susceptible of beingsolved had already been solved, and that after them therewould be nothing left but gleanings. Happily we are reas-sured by the example of the past. Many times already menhave thought that they had solved all the problems, or at leastthat they had made an inventory of all that admit of solution.And then the meaning of the word solution has beenextended; the insoluble problems became the most inter-esting of all, and other problems were proposed that one hadnot dreamt of.2 For the Greeks, a good solution was one thatemployed only ruler and compass; later it became oneobtained by the extraction of radicals, then one in which onlyalgebraic functions and logarithms appeared. Thus thepessimists found themselves continually passed over, con-tinually forced to retreat, so that at present I really believethere are none left.

My intention, therefore, is not to refute them, since theyare dead. We know very well that mathematics will con-tinue to develop, but we have to find out in what direction.I shall be told ‘‘in all directions’’, and that is partly true; butif it were altogether true, it would become somewhatalarming. Our riches would soon become embarrassing,and their accumulation would soon produce a mass just asimpenetrable as the unknown truth was to the ignorant.

The historian, the physicist himself, must make a selec-tion of facts. The scientist’s brain, which is only a corner ofthe universe, will never be able to contain the whole uni-verse; whence it follows that, of the innumerable factsoffered by nature, we shall leave some aside and retainothers. The same is true, a fortiori, in mathematics. Themathematician similarly cannot retain pell-mell all the factsthat are presented to him, the more so in that it is he himself– I was going to say his own caprice – that creates them. Itis he who constructs a new combination of the pieces byassembling the elements and from top to bottom; it isgenerally not brought to him ready-made by nature.

No doubt it is sometimes the case that a mathematicianattacks aproblem to satisfy some requirement of physics, thatthephysicist or the engineer asks him tomakea calculation inview of some particular application. Will it be said that we

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

AU

TH

OR JEREMY GRAY is a Professor of the History

of Mathematics at the Open University, and

is an Honorary Professor at the University of

Warwick, where he lectures on the history

of mathematics. His most recent book is

Plato’s Ghost: The Modernist Transfor-mation of Mathematics, Princeton Univer-

sity Press (2008), and he is presently finishing

a scientific biography of Henri Poincare.

Faculty of Mathematics, Computing

and Technology

The Open UniversityWalton Hall, Milton Keynes MK7 6AA

UK

e-mail: [email protected]

2Text amended or inserted in this translation is shown in italics.

16 THE MATHEMATICAL INTELLIGENCER

http://portail.mathdoc.fr/BIBLIOS/PDF/Poincare.pdf

http://portail.mathdoc.fr/BIBLIOS/PDF/Poincare.pdf

geometricians are to confine ourselves to waiting for orders,and, instead of cultivating our science for our own pleasure,to have no other care but that of accommodating ourselves toour clients’ tastes? If the only object ofmathematics is to cometo the help of those who make a study of nature, it is to themwe must look for the word of command. Is this the correctview of the matter? Certainly not; for if we had not cultivatedthe exact sciences for themselves, we should never havecreated the mathematical instrument, and when the word ofcommand came from the physicist we should have beenfound without arms.

Similarly, physicists do not wait to study a phenomenonuntil some pressing need of material life makes it an absolutenecessity, and they are quite right. If the scientists of theeighteenth century had disregarded electricity, because itappeared to them merely a curiosity having no practicalinterest, we should not have, in the twentieth century, eithertelegraphyor electro-chemistry or electro-traction. Physicistsforced to select are not guided in their selection solely byutility. What method, then, do they pursue in making aselection between the different natural facts? [I haveexplained this in the preceding chapter.]3 We can reply eas-ily: the facts that interest them are those that may lead to thediscovery of a law, those that have an analogy with manyother facts and do not appear to us as isolated, but as closelygroupedwithothers. The isolated fact attracts the attentionofall, of the layman as well as the scientist. But what the truescientist alone can see is the link that unites several factswhich have a deep but hidden analogy. The anecdote ofNewton’s apple is probably not true, but it is symbolical, sowe will treat it as if it were true. Well, we must suppose thatbefore Newton’s day many men had seen apples fall, butnone had been able to draw any conclusion. Facts would bebarren if there were not minds capable of selecting betweenthemanddistinguishing thosewhichhave somethinghiddenbehind them and recognizing what is hidden – minds which,behind the bare fact, can detect the soul of the fact.

In mathematics we do exactly the same thing. Of thevarious elements at our disposal we can form millions ofdifferent combinations, but any one of these combinations,so long as it is isolated, is absolutely without value; oftenwe have taken great trouble to construct it, but it is ofabsolutely no use, unless it be, perhaps, to supply a subjectfor an exercise in secondary schools. It will be quite dif-ferent as soon as this combination takes its place in a classof analogous combinations whose analogy we have rec-ognized; we shall then be no longer in the presence of afact, but of a law. And then the true discoverer will not bethe workman who has patiently built up some of thesecombinations, but the man who has brought out theirrelation. The former has only seen the bare fact, the latteralone has detected the soul of the fact. The invention of anew word will often be sufficient to bring out the relation,and the word will be creative. The history of science fur-nishes us with a host of examples that are familiar to all.

The celebrated Viennese philosopher Mach has said thatthe role of science is to effect economy of thought, just as a

machine effects economy of effort, and this is very true.The savage calculates on his fingers, or by putting togetherpebbles. By teaching children the multiplication table wesave them later on countless operations with pebbles.Someone once recognized, whether by pebbles or other-wise, that 6 times 7 are 42, and had the idea of recordingthe result, and that is the reason why we do not need torepeat the operation. His time was not wasted even if hewas only calculating for his own amusement. His operationonly took him two minutes, but it would have taken twomillion, if a million people had had to repeat it after him.

Thus the importance of a fact is measured by the returnit gives – that is, by the amount of thought it enables us toeconomise.

In physics, the facts that give a large return are those thattake their place in a very general law, because they enableus to foresee a very large number of others, and it is exactlythe same in mathematics. Suppose I apply myself to acomplicated calculation and with much difficulty arrive at aresult, I shall have gained nothing by my trouble if it hasnot enabled me to foresee the results of other analogouscalculations, and to direct them with certainty, avoiding theblind groping with which I had to be contented the firsttime. On the contrary, my time will not have been lost ifthis very groping has succeeded in revealing to me theprofound analogy between the problem just dealt with anda much more extensive class of other problems; if it hasshown me at once their resemblances and their differences;if, in a word, it has enabled me to perceive the possibility ofa generalization, then it will not be merely a new result thatI have acquired, but a new force.

An algebraic formula that gives us the solution of a typeof numerical problem, if we finally replace the letters bynumbers, is the simple example that occurs to one’s mind atonce. Thanks to the formula, a single algebraic calculationsaves us the trouble of a constant repetition of numericalcalculations. But this is only a rough example: everyonefeels that there are analogies that cannot be expressed by aformula, and that they are the most valuable.

If a new result is to have any value, it must unite ele-ments long since known, but till then scattered andseemingly foreign to each other, and suddenly introduceorder where the appearance of disorder reigned. Then itenables us to see at a glance each of these elements in theplace it occupies in the whole. Not only is the new factvaluable on its own account, but it alone gives a value tothe old facts it unites. Our mind is as frail as our senses are;it would lose itself in the complexity of the world if thatcomplexity were not harmonious; like the short-sighted, itwould only see the details, and would be obliged to forgeteach of these details before examining the next, because itwould be incapable of taking in the whole. The only factsworthy of our attention are those that introduce order intothis complexity and so make it accessible to us.

Mathematicians attach a great importance to the ele-gance of their methods and of their results, and this is notmere dilettantism. What is it that gives us the feeling of

3Sentences such as this one which were added in the book versions are set apart in brackets.


elegance in a solution or a demonstration? It is the harmonyof the different parts, their symmetry, and their happyadjustment; it is, in a word, all that introduces order, all thatgives them unity, that enables us to obtain a clear com-prehension of the whole as well as of the parts. But that isalso precisely what causes it to give a large return; and infact the more we see this whole clearly and at a singleglance, the better we shall perceive the analogies withother neighbouring objects, and consequently the betterchance we shall have of guessing the possible generaliza-tions. Elegance may result from the feeling of surprisecaused by the unlooked-for occurrence together of objectsnot habitually associated. In this, again, it is fruitful, since itthus discloses relations until then unrecognized. It is alsofruitful even when it only results from the contrast betweenthe simplicity of the means and the complexity of theproblem presented, for it then causes us to reflect on thereason for this contrast, and generally shows us that thisreason is not chance, but is to be found in some unsus-pected law. Briefly stated, the sentiment of mathematicalelegance is nothing but the satisfaction due to some con-formity between the solution we wish to discover and thenecessities of our mind, and it is on account of this veryconformity that the solution can be an instrument for us.This aesthetic satisfaction is consequently connected withthe economy of thought. It is in this way that the caryatidsof the Erechtheum, for example, seem elegant to us becausethey bear a heavy load with suppleness and, one might saysay, nimbly, and so they give us the feeling of economy ofeffort. [Again the comparison with the Erechtheum occursto me, but I do not wish to serve it up too often.]

It is for the same reason that, when a somewhat lengthycalculation has conducted us to some simple and strikingresult, we are not satisfied until we have shown that wemight have foreseen, if not the whole result, at least its mostcharacteristic features. Why is this? What is it that preventsour being contented with a calculation that has taught usapparently all that we wished to know? The reason is that,in analogous cases, the lengthy calculation might not beable to be used again, whereas this is not true of the rea-soning, often semi-intuitive, which might have enabled usto foresee the result. This reasoning being short, we can seeall the parts at a single glance, so that we perceive imme-diately what must be changed to adapt it to all the problemsof a similar nature that may be presented. And since itenables us to foresee whether the solution of these prob-lems will be simple, it shows us at least whether thecalculation is worth undertaking.

What I have just said is sufficient to show how vain itwould be to attempt to replace the mathematician’s freeinitiative by a mechanical process of any kind. In order toobtain a result having any real value, it is not enough togrind out calculations, or to have a machine for puttingthings in order: it is not order only, but unexpected order,that has a value. A machine can take hold of the bare fact,but the soul of the fact will always escape it.

Since the middle of the last century, mathematicians havebecome more and more anxious to attain to absolute exact-ness. They are quite right, and this tendency will becomemore and more marked. In mathematics, exactness is not

everything, but without it there is nothing: a demonstrationthat lacks exactness is nothing at all. This is a truth that I thinkno one will dispute, but if it is taken too literally it leads us tothe conclusion that before 1820, for instance, there was nosuch thing as mathematics, and this is clearly an exaggera-tion. The geometricians of that day were willing to assumewhat we explain by prolix dissertations. This does not meanthat they did not see it at all, but they passed it over toohastily, and, in order to see it clearly, they would have had totake the trouble to state it.

Only, is it always necessary to state it so many times? Thosewho were the first to pay special attention to exactness havegiven us reasonings that we may attempt to imitate; but if thedemonstrations of the future are to be constructed on thismodel, mathematical works will become exceedingly long,and if I dread length, it is not only because I am afraid of thecongestionofour libraries, but because I fear that as theygrowin length, our demonstrations will lose that appearance ofharmony that plays such a useful part, as I have just explained.

It is economy of thought that we should aim at, andtherefore it is not sufficient to give models to be copied. Wemust enable those that come after us to do without themodels, and not to repeat a previous reasoning, but sum-marize it in a few lines. And this has already been donesuccessfully in certain cases. For instance, there was a wholeclass of reasonings that resembled each other, and werefound everywhere; they were perfectly exact, but they werelong. One day someone thought of the term ‘‘uniformity ofconvergence’’, and this term alone made them useless; it wasno longer necessary to repeat them, since they could now beassumed. Thus the hair-splitters can render us a double ser-vice, first by teaching us to do as they do if necessary, butmore especially, by enabling us as often as possible not to doas they do, and yet make no sacrifice of exactness.

Oneexample has just shownus the importance of terms inmathematics; but I could quote many others. It is hardlypossible to believe what economy of thought, as Mach usedto say, can be effected by a well-chosen term. I think I havealready said somewhere that mathematics is the art of givingthe same name to different things. It is enough that thesethings, though differing in matter, should be similar in form,to permit of their being, so to speak, run in the same mould.When language has been well chosen, one is astonished tofind that all demonstrations made for a known object applyimmediately to many new objects: nothing is required to bechanged, not even the terms, since the names have becomethe same.

One example presents itself to the mind before all others,and that is quaternions, upon which I need not insist. Awell-chosen term is very often sufficient to remove theexceptions permitted by the rules as stated in the oldphraseology. This accounts for the invention of negativequantities, imaginary quantities, decimals to infinity, and Iknow not what else. And we must never forget thatexceptions are pernicious, because they conceal laws.

This is one of the characteristics by which we recognizefacts that give a great return: they are the facts that permitthese happy innovations of language. The bare fact, then,has sometimes no great interest: it may have been notedmany times without rendering any great service to science;


it only acquires a value when some more careful thinkerperceives the connection it brings out, and symbolizes it bya term.

The physicists also proceed in exactly the same way.They have invented the term ‘‘energy’’, and the term hasbeen enormously fruitful, because it also creates a law byeliminating exceptions; because it gives the same name tothings that differ in matter, but are similar in form.

Among the terms that have exercised the most happyinfluence I would note ‘‘group’’ and ‘‘invariant’’. They haveenabled us to perceive the essence of many mathematicalreasonings, and have shown us in how many cases the oldmathematicians were dealing with groups without knowingit, and how, believing themselves far removed from eachother, they suddenly found themselves close together with-out understanding why.

Today we should say that they had been examining iso-morphic groups. We now know that, in a group, the matter isof little interest, that the form only is of importance, and thatwhen we are well acquainted with one group, we know bythat very fact all the isomorphic groups. Thanks to the terms‘‘group’’ and ‘‘isomorphism’’, which sumup this subtle rule ina few syllables, and make it readily familiar to all minds, thepassage is immediate, and can be made without expendingany effort of thinking. The idea of group is, moreover, con-nected with that of transformation. Why do we attach somuch value to the discovery of a new transformation? It isbecause, from a single theorem, it enables us to draw ten ortwenty others. It has the same value as a zero added to theright of a whole number.

This is what has determined the direction of the move-ment of mathematical science up to the present, and it isalso most certainly what will determine it in the future. Butthe nature of the problems that present themselves con-tribute to it in an equal degree. We cannot forget what ouraim should be, and in my opinion this aim is a double one.Our science borders on both philosophy and physics, andit is for these two neighbours that we must work. And sowe have always seen, and we shall still see, mathematiciansadvancing in two opposite directions.

On the one side, mathematical science must reflect uponitself, and this is useful because reflecting upon itself isreflecting upon the human mind that has created it; themore so because, of all its creations, mathematics is the onefor which it has borrowed least from outside. This is thereason for the utility of certain mathematical speculations,such as those that have in view the study of postulates, ofunusual geometries, of functions with strange behaviour.The more these speculations depart from the most ordinaryconceptions, and, consequently, from nature and applica-tions to natural problems, the better will they show us whatthe human mind can do when it is more and more with-drawn from the tyranny of the exterior world; the better,consequently, will they enable us to know this mind itself.

But it is to the opposite side, to the side of nature, thatwe must direct our main forces.

There we meet the physicist or the engineer, who says,‘‘Will you integrate this differential equation for me?; I shallneed it within a week for a piece of construction work thathas to be completed by a certain date’’. ‘‘This equation’’, we

answer, ‘‘is not included in one of the types that can beintegrated, of which you know there are not very many.’’‘‘Yes, I know; but, then, what good are you?’’ More oftenthan not, a mutual understanding is sufficient. The engineerdoes not really require the integral in finite terms, he onlyrequires knowledge of the general behaviour of the integralfunction, or he merely wants a certain figure that would beeasily deduced from this integral if we knew it. Ordinarilywe do not know it, but we could calculate the figurewithout it if we knew just what figure and what degree ofexactness the engineer required.

Formerly an equation was not considered to have beensolved until the solution had been expressed by means of afinite number of known functions. But this is impossible inabout ninety-nine cases out of a hundred. What we canalways do, or rather what we should always try to do, is tosolve the problem qualitatively so to speak – that is, to tryto know approximately the general form of the curve thatrepresents the unknown function.

It then remains to find the quantitative solution of theproblem. But if the unknown cannot be determined by afinite calculation, we can always represent it by an infiniteconverging series that enables us to calculate it. Can this beregarded as a true solution? The story goes that Newtononce communicated to Leibnitz an anagram somewhat likethe following: aaaaabbbeeeeii, etc. Naturally, Leibnitz didnot understand it at all, but we who have the key know thatthe anagram, translated into modern phraseology, means,‘‘I know how to integrate all differential equations’’ and weare tempted to make the comment that Newton was eitherexceedingly fortunate or that he had very singular illusions.What he meant to say was simply that he could form (bymeans of indeterminate coefficients) a series of powersformally satisfying the equation presented.

Today a similar solution would no longer satisfy us fortwo reasons – because the convergence is too slow, andbecause the terms succeed one another without obeyingany law. On the other hand, the H series appears to us toleave nothing to be desired, first, because it converges veryrapidly (this is for the practical man who wants his number

Jules Henri Poincare (1854 - 1912). AIP Emilio Segre Visual

Archives.


as quickly as possible), and second, because we perceive ata glance the law of the terms, which satisfies the aestheticrequirements of the theorist.

There are, therefore, no longer some problems solvedand others unsolved, there are only problems more or lesssolved, according as this is accomplished by a series ofmore or less rapid convergence or regulated by a more orless harmonious law. Nevertheless an imperfect solutionmay happen to lead us toward a better one.

Sometimes the series is of such slow convergence that thecalculation is impracticable, and we have only succeeded indemonstrating the possibility of the problem. The engineerconsiders this absurd, and he is right, since it will not helphim to complete his construction within the time allowed. Hedoesn’t trouble himselfwith thequestionofwhether itwill beof use to the engineers of the twenty-second century. Wethink differently, and we are sometimes more pleased athaving economized a day’s work for our grandchildren thanan hour for our contemporaries.

Sometimes by groping, so to speak, empirically, we arriveat a formula that is sufficiently convergent. ‘‘What morewould you have?’’ says the engineer; and yet, in spite ofeverything, we are not satisfied, for we should have liked tobe able to predict the convergence. And why? Because if wehadknownhow topredict it in the one case,we should knowhow to predict it in another. We have been successful, it istrue, but that is little in our eyes if we have no real hope ofrepeating our success.

In proportion as the science develops, it becomes moredifficult to take it in its entirety. Then an attempt is made tocut it in pieces and to be satisfied with one of these pieces – ina word, to specialize. Too great a movement in this directionwould constitute a serious obstacle to the progress of sci-ence. As I have said, it is by unexpected concurrencesbetween its different parts that it can make progress. Toomuch specializing would prohibit these concurrences. Let ushope that congresses, such as those of Heidelberg and Rome,by putting us in touch with each other, will open up a view ofour neighbours’ territory and force us to compare it with ourown, and so escape in a measure from our own little village.In this way they will be the best remedy against the danger Ihave just noted.

But I have delayed too long over generalities; it is timeto enter into details.

Let us review the different particular sciences that go tomake up mathematics; let us see what each of them has done,in what direction it is tending, and what we may expect of it. Ifthe preceding views are correct, we should see that the greatprogress of the past has been made when two of these sci-ences have been brought into conjunction, when men havebecome aware of the similarity of their form in spite of thedissimilarity of their matter, when they have modelled them-selves upon each other in such a way that each could profit bythe triumphs of the other. At the same time we should look toconcurrences of a similar nature for progress in the future.

I. Arithmetic

The progress of arithmetic has been much slower than thatof algebra and analysis, and it is easy to understand thereason. The feeling of continuity is a precious guide that

fails the arithmetician. Every whole number is separatedfrom the rest, and has, so to speak, its own individuality;each of them is a sort of exception, and that is the reasonwhy general theorems will always be less common in thetheory of numbers, and also why those that do exist will bemore hidden and will escape detection longer.

If arithmetic is backward as compared with algebra andanalysis, the best thing for it to do is to try to model itself onthese sciences, in order to profit by their advance. Thearithmetician then should be guided by the analogies withalgebra. These analogies are numerous, and if in manycases they have not yet been studied sufficiently closely tobecome serviceable, they have at least been long fore-shadowed, and the very language of the two sciencesshows that they have been perceived. Thus we speak oftranscendental numbers, and so become aware of the factthat the future classification of these numbers has already amodel in the classification of transcendental functions.However, it is not yet very clear how we are to pass fromone classification to the other; but if it were clear, it wouldbe already done and would no longer be the work of thefuture.

The first example that comes to mind is the theory ofcongruences, in which we find a perfect parallelism withthat of algebraic equations. We shall certainly succeed incompleting this parallelism, which must exist, for instance,between the theory of algebraic curves and that of con-gruences with two variables. When the problems relating tocongruences with several variables have been solved, weshall have made the first step toward the solution of manyquestions of indeterminate analysis.

Another example where, however, the analogy was onlyseen later, is provided for us by the theory of fields and ideals.To have a counterpart to this, consider curves traced on asurface: to existing numbers correspond complete intersec-tions, to ideals incomplete intersections, to prime idealsindecomposable curves; the various classes of ideals also havetheir analogues.

There is no doubt that this analogy can explain the theoryof ideals, or that of surfaces, or perhaps both at once.

The theory of forms, in particular that of quadraticforms, is intimately linked to that of ideals. If, amongarithmetical theories, it has been one of the first to takeshape, it was when one was led to introduce unity into it byconsidering groups of linear transformations.

These transformations have allowed a classificationand, consequently, the introduction of order. Perhaps onehas harvested all the fruit one can hope to; but, if thesetransformations are the parents of perspectives in geometry,analytical geometry provides us with many other transfor-mations (as, for example, the birational transformations ofan algebraic curve) of which it will be advantageous to seekthe arithmetical analogues. Without any doubt they willform discontinuous groups, of which one must first seek thefundamental domain, which is the key to everything. In thisstudy, I do not doubt that one will have to make use ofMinkowski’s Geometrie der Zahlen.

One idea from which we have not drawn all that itcontains is Hermite’s introduction of continuous variablesinto the theory of numbers. We know now what it signifies.


Let us take as our point of departure two forms F and F’, thesecond quadratic definite, and apply the same transfor-mation to them. If the transform of F’ is reduced, we saythat the transformation is reduced and also that thetransform of F is reduced. It follows that if the form F can betransformed into itself, it can have several reduced forms;but this inconvenience is essential and cannot be avoidedin any way; and besides it does not prevent these reducedforms from leading to a classification of forms. It is clearthat this idea, which up to now has only been applied tovery particular forms and transformations, can be exten-ded to nonlinear groups of transformations, because it is ofmuch greater import and has not been exhausted.

One domain of arithmetic in which unity seems to beentirely lacking is the theory of prime numbers; we have onlyfound asymptotic laws, and we must hope for others; but theselaws are isolated and one only comes upon them by differentroutes that do not seem to communicate with each other. Ithink I have glimpsed where one might draw the desiredunity, but I have only glimpsed it vaguely; everything willcome down without doubt to a family of transcendentalfunctions that will permit, by the study of their singular pointsand an application of the method of Darboux, calculatingcertain functions of very large numbers asymptotically.

II. Algebra

The theory of algebraic equations will long continue to attractthe attention of geometricians, because the sides bywhich thetheory may be approached are so numerous and so different.The most important is the theory of groups, to which we shallreturn. But there is also the question of the numerical calcu-lation of the roots and that of the discussion of the number ofreal roots. Laguerre has shown that not everything was said onthis point by Sturm. There is a place for studying a system ofinvariants that do not change sign when the number of realroots remains the same. One can also form power series rep-resenting functions that have as their singular points thevarious roots of an algebraic equation (for example, rationalfunctions whose denominator is the first term of that equa-tion); the coefficients of the higher-order terms provide us withan approximation to a greater or lesser degree of accuracy;there is the germ of a procedure there for numerical calcula-tion that one could study systematically.

Some forty years ago it was the study of the invariants ofalgebraic forms that seemed to absorb all of algebra; today itis abandoned, the matter, however, is not exhausted; only it isnecessary to extend it and not restrict oneself, for example, tothe invariants of linear transformations but to tackle thosethat belong to an arbitrary group. The theorems acquiredformerly will thus suggest to us other more general ones thatwill group themselves around them like a crystal feeding itselfin a solution. And as for Gordan’s theorem that the numberof distinct invariants is bounded, and which Hilbert has sohappily simplified the proof of, it seems that this theorem leadsus to ask a much more general question: if one has an infinityof polynomials that depend algebraically on a finite numberof them, can one always derive them from a finite number ofthem by addition and multiplication?

It must not be supposed that algebra is finished becauseit furnishes rules for forming all possible combinations;

finding interesting combinations still remains, that is, thosethat satisfy such-and-such conditions. Thus there will bebuilt up a kind of indeterminate analysis, in which theunknown quantities will no longer be whole numbers butrather polynomials. So this time it is algebra that will modelitself on arithmetic, being guided by the analogy of thewhole number, either with the whole polynomial withindefinite coefficients, or with the whole polynomial withwhole coefficients.

III. Differential Equations

We have already done a lot with linear differential equa-tions, and it only remains to perfect what we have done.But as for what concerns nonlinear differential equations,we are much less advanced. The hope of integrating themby means of known functions was lost a long time ago; it istherefore necessary to study for themselves the functionsdefined by these differential equations and to attempt first asystematic classification of these functions; the study of theway they grow in the neighbourhood of a singular pointwill without doubt furnish the first elements of this classi-fication, but we will only be satisfied when we have found acertain group of transformations (for example, the Cre-mona transformations), which plays with respect todifferential equations, the same role that the group of bi-rational transformations plays for algebraic curves. We willthen be able to place in the same class all the transforms ofthe same equation. We will then have as our guide a theoryalready constructed: that of the birational transformationsand of the genus of an algebraic curve.

One can propose to reduce the study of these functions tothat of single-valued functions, and that in two ways: weknow that if y = f(x), one can, whatever the functionf(x) is, write y and x in terms of single-valued functions ofan auxiliary variable t; but, if f(x) is the solution of a dif-ferential equation, in what case do the auxiliary functionsthemselves satisfy a differential equation? We do not know;we do not even know in what case the general integral canbe put in the form F(x, y) = an arbitrary constant, F(x, y)being single-valued.

I will insist on the qualitative discussionofcurves defined bydifferential equations. In the simplest case, when the equationis of the first order and the first degree, this discussion leads tothe determination of the number of limit cycles. It is very del-icate, and what can enable us to do it is the analogy with thestudy of the number of real roots of an algebraic equation;when a fact makes the nature of this analogy evident, we canbe certain in advance that it will be fertile.

IV. Partial Differential Equations

We have recently made considerable progress in our knowl-edge of partial differential equations, following the discoveriesof M. Fredholm. Now, if one examines the essence of thesediscoveries closely, one sees that they consist in modelling adifficult theory on another much simpler one: that of deter-minants and systems of the first degree. In most of the problemsin mathematical physics, the equations to be integrated arelinear; they allow us to determine unknown functions ofseveral variables, and those functions are continuous. Why?


Because we have written those equations while regardingmatter as continuous. But matter is not continuous; it is madeof atoms, and, if we were to have wanted to write the equationsas they would be written by an observer sufficiently perceptiveto see the atoms, we would not have a small number of dif-ferential equations serving to determine certain unknownfunctions, we would have a great number of algebraicequations serving to determine a great number of unknownconstants. And these algebraic equations would be linear, sothat one could, with infinite patience, directly apply themethod of determinants.

But, as the shortness of our lives does not permit us theluxury of infinite patience, it is necessary to proceed differ-ently: it is necessary to pass to the limit by supposing matter tobe continuous. There are two ways of generalising the theoryof first-degree equations by passing to the limit. One canconsider an infinite number of discrete equations withanother infinity, equally discrete, of unknowns. This, forexample, is what Hill did in his theory of the Moon. One thenhas infinite determinants, which are to ordinary determi-nants as series are to finite sums.

One can take a partial differential equation, represent-ing, so to speak, a continuous infinity of equations, and useit to determine an unknown function representing a con-tinuous infinity of unknowns. One then has other infinitedeterminants, which are to ordinary determinants whatintegrals are to finite sums. That is what Fredholm did; hissuccess also leads to the following fact: if in a determinantthe elements on the principal diagonal are equal to 1, andthe other elements are considered as homogeneous of thefirst order, one can arrange the development of the deter-minant by collecting in a single group the terms that arehomogeneous of the same degree. Fredholm’s infinitedeterminant can be expanded in this fashion, and ithappens that one obtains in this way a convergent series.

Has this analogy, which certainly guided Fredholm, givenall that it can give? Certainly not; if its success derives from thelinear form of the equations, one must be able to apply ideasof the same kind to all problems that relate to equations of alinear form, and even to ordinary differential equations,because their solution can always be reduced to that of a first-order linear partial differential equation.

For some time, people have tackled the Dirichlet problemand other connected problems by another means, by goingback to Dirichlet’s original idea and seeking the minimumof a definite integral, but this time by rigorous procedures. Ido not doubt that one can without great difficulty bring thetwo methods together, taking account of their mutualrelationships, and I do not doubt that both have much togain thereby. Thanks to M. Hilbert, who has been doublythe initiator, we will proceed on that path.

V. Abelian Functions

One knows the principal question that remains to beresolved in the theory of Abelian functions. The Abelianfunctions generated by a curve are not the most general;they are only a particular case that we can call the specialAbelian functions. How are they related to the generalfunctions, and how can we classify the latter? Not long ago,the solution seemed remote. Today I consider the problem as

virtually solved since MM. Castelnuovo and Enriques havepublished their memoir on the integrals of total differentialson varieties of more than two dimensions. We now knowthat there are Abelian functions attached to a curve andothers to a surface and that it will never be necessary to passto varieties of more than two dimensions. By combiningthis result with those that follow from Wirtinger’s work, wewill get without doubt to the end of our difficulties.

VI. Theory of Functions

It is above all of the theory of functions of two or more variablesthat I wish to speak. The analogy with the theory of functions ofa single variable is a precious guide but is insufficient. There isan essential difference between the two sorts of functions, andevery time one attempts a generalization passing from one tothe other, one comes on an unexpected difficulty that onesometimes overcomes by special tricks, but which has oftenremained unsurpassable until now. We must therefore studythe facts, which are of the kind that will illuminate for us theessence of the difference between functions of one variableand those that contain several. It is first necessary to look clo-sely at the tricks that have been used in particular cases andthen to see what they have in common. Why is conformalrepresentation usually impossible in a domain of fourdimensions, and what is necessary to put in its place? Is not thetrue generalization of functions of one variable the harmonicfunctions of four variables, of which the real parts of functionsof two variables are only particular cases? Can one include inthe theory of the transcendental functions of several variableswhat one knows about algebraic and rational functions? Or,in other terms, in what sense can one say that transcendentalfunctions of two variables are to transcendental functions ofone variable, as rational functions of two variables are torational functions of one variable?

Is it true that if z = f(x, y) then one can write x, y, z asuniform functions of two auxiliary variables or, to use anexpression that has begun to be consecrated by use, can oneuniformise functions of two variables as one uniformisesfunctions of one? I restrict myself to raising the question, towhich the immediate future may perhaps provide thesolution for us.

VII. Theory of Groups

The theory of groups is an extensive subject on which there ismuch to say. There are several sorts of groups, and whateverthe classification adopted, one always finds new groups thathave not been included. I want to restrict myself here andspeak only of Lie’s continuous groups and Galois’s discon-tinuous groups, which one is accustomed to qualify in eachcase as groups of finite order, although ‘‘finite order’’ does notat all have the same sense in the one case and in the other.

In the theory of Lie groups, one is guided by a particularanalogy: a finite transformation is the result of infinitelymany infinitesimal transformations. The simplest case isthe one in which the infinitesimal transformations reduceto multiplication by 1þ e; e being very small. The repetitionof this function generates the exponential transformation;this is how Napier came to it. We know that the exponentialfunction can be represented by a very simple and rapidly


convergent series, and this analogy can show us the way tofollow. This analogy can be expressed by a very specialsymbolism, and the reader will forgive me if I do not insiston it here. We have made great progress here, thanks to Lie,Killing, and Cartan. It only remains to simplify the proofsand to coordinate and classify the results.

The study of Galois groups is much less advanced, and thatexplains itself; it is for the same reason that arithmetic is lessadvanced than analysis, because continuity confers greatadvantages from which one has profited. But happily there isa manifest parallelism between the two theories, which onemust insist on and must bring more and more into evidence.The analogy is quite similar to what we have noted betweenarithmetic and algebra and makes the same inclusion.

VIII. Geometry

It would seem that geometry can contain nothing that is notalready contained in algebra or analysis, and that geometricfacts are nothing but the facts of algebra or analysisexpressed in another language. It might be supposed, then,that after the review that has just been made, there wouldbe nothing left to say having any special bearing ongeometry. But this would imply a failure to recognize thegreat importance of a well-formed language, or to under-stand what is added to things themselves by the method ofexpressing, and consequently of grouping, those things.

To begin, geometric considerations lead us to set ourselvesnew problems. These are certainly, if you will, analyticalproblems, but they are problems we should never have setourselves on the score of analysis. Analysis, however, profitsby them, as it profits by those it is obliged to solve in order tosatisfy the requirements of physics.

One great advantage of geometry lies precisely in the factthat the senses can come to the assistance of the intellect andcan help to determine the road to be followed, and manyminds prefer to reduce the problems of analysis to geometricform. Unfortunately our senses cannot carry us very far, andthey leave us in the lurch as soon as we wish to pass outsidethe three classical dimensions. Does this mean that, when wehave left this restricted domain in which they would seem towish to imprison us, we must no longer count on anythingbut pure analysis, and that all geometry of more than threedimensions is vain and without object? In the generation thatpreceded ours, the greatest masters would have answered‘‘Yes’’. Today we are so familiar with this notion that we canspeak of it, even in a university course, without exciting toomuch astonishment.

But of what use can it be? This is easy to see. In the firstplace it gives us a very convenient language, which expres-ses in very concise terms what the ordinary language ofanalysis would state in long-winded phrases. More than that,this language causes us to give the same name to things thatresemble one another, and states analogies that it does notallow us to forget. It thus enables us still to find our way inthat space that is too great for us, by calling to our mindcontinually the visible space, which is only an imperfectimage of it, no doubt, but is still an image. Here again, as in allthe preceding examples, it is the analogy with what is simplethat enables us to understand what is complex.

This geometry of more than three dimensions is not asimple analytical geometry, it is not purely quantitative, butalso qualitative, and it is principally on this ground that itbecomes interesting. There is a science called Geometry ofPosition, which has for its object the study of the relations ofpositionof thedifferent elements of a figure, after eliminatingtheir magnitudes. This geometry is purely qualitative; itstheorems would remain true if the figures, instead of beingexact, were rudely imitated by a child. We can also constructa Geometry of Position of more than three dimensions. TheimportanceofGeometry of Position is immense, and I cannotinsist upon it too much; what Riemann, one of its principalcreators, has gained from it would be sufficient to demon-strate this. We must succeed in constructing it completely inthe higher spaces, and we shall then have an instrument thatwill enable us really to see into hyperspace and to supple-ment our senses.

The problems of Geometry of Position would perhapsnot have presented themselves if only the language ofanalysis had been used. Or rather I am wrong, for theywould certainly have presented themselves, since theirsolution is necessary for a host of questions of analysis, butthey would have presented themselves isolated, one afterthe other, and without our being able to perceive theircommon link.

What has contributed above all to recent progress ingeometry is the introduction of the notion of transformationsand groups. It is thanks to this that geometry is no longer acollection of more or less curious theorems that follow eachother without any resemblance, but geometry has achieved aunity. And on the other side, the history of science must notforget that it is in geometry that we began to study continuoustransformations systematically, so that pure geometry hasplayed its part in the development of the idea of a group that isso useful in other branches of mathematics.

The study of groups of points on an algebraic curve in themanner of Brill and Noether still gives us useful results, eitherdirectly or by providing a model for analogous theories. It is inthis way that we have developed a whole chapter in the theoryof geometry where curves on a surface play a role similar tothat of points on a curve. In this way we can hope today toclear up the final mysteries that belong to the theory of sur-faces and that appear so persistent.

Geometers therefore have a vast field to explore, and I mustbe careful not to forget enumerative geometry and infinites-imal geometry, cultivated with such skill by M. Darboux andto which M. Bianchi has made such useful contributions. If Ido not say anything on this subject, it is because I havenothing to tell you after M. Darboux’s brilliant lecture.

IX. Cantorism

I have spoken previously of the need for returning continuallyto the first principles of our science, and of the advantage ofthis process to the study of the human mind. It is this need thathas inspired two attempts that have each held a very greatplace in the most recent history of mathematics. The first isCantorism, and one knows the services it has rendered toscience. [Here Poincare added some text for the later editions.]One of the characteristic features of Cantorism is that, insteadof rising to the general by erecting constructions that are more


and more complicated, and defining by construction, Canto-rism starts with the genus supremum and only defines, as thescholastics would have said, per genus proximum et differ-entiam specificam. Hence exists the horror that it has some-times inspired in certain minds, such as Hermite’s, whosefavourite idea was to compare the mathematical with thenatural sciences. For most of us, these prejudices have beendissipated, but it has come about that we have run againstcertain paradoxes and apparent contradictions, which wouldhave rejoiced the heart of Zeno of Elea and the school ofMegara. Then everyone sought a remedy, each man his ownway. Formypart I think, and I amnot alone in so thinking, thatthe important thing is never to introduce any entities but suchas can be completely defined in a finite number of words.Whatever be the remedy adopted, we can promise ourselvesthe joy of the doctor who is called in to follow a fine patho-logical case.

X. The Search for Postulates

Attempts have been made, from another point of view, toenumerate the axioms and postulates more or less con-cealed, which form the foundation of different mathematicaltheories, and in this direction Mr Hilbert has obtained themost brilliant results. It seems at first that this domain must bestrictly limited, and that there will be nothing more to dowhen the inventory has been completed, which cannot belong. But when everything has been enumerated, there willbe many ways of classifying it all. A good librarian alwaysfinds work to do, and each new classification will beinstructive for the philosopher.

I here close this review, which I cannot dream of makingcomplete for a number of reasons, above all because I havealready abused your attention too much. I think that theseexamples will have been sufficient to show the mechanism bywhich the mathematical sciences have progressed in the past,and the direction in which they must advance in the future.

Remarks on ‘‘L’avenir des Mathematiques’’The generalities with which Poincare introduced his subjectare well-known, but nonetheless still interesting. His open-ing words ‘‘To predict the future of mathematics the propermethod is to study its history and its present state’’, allowedhis audience to decide whether he agreed or disagreed withHilbert’s approach eight years earlier. But as Poincare wenton to say, he did not fear that mathematics was about tobecome exhausted, but rather that it would grow so muchthat it would produce ‘‘a mass just as impenetrable as theunknown truth was to the ignorant’’. Therefore, he deduced,we are forced to make a selection of the facts, especially ifone is a mathematician who creates those facts. The groundsfor selection should not be narrowly utilitarian – evenphysicists, he pointed out, built their theories in advance ofthe electrical technologies that could not have been discov-ered without them – nor should mathematicians take theirinstructions from natural scientists.

Poincare’s style was always to make points delicately,but these opening remarks were not mere commonplaces.Hilbert had opened his address with a long historical reach,

going back to Fermat and his ‘‘last’’ theorem and JohannBernoulli’s derivation of the curve of quickest descent.Poincare chose instead to put an accent on more recentdevelopments in physics. Thus he pointed to Maxwell andHertz, physicists on whose theories of electricity, magne-tism, and optics Poincare was an acknowledged authority.He noted the unexpected implications of these theories forwireless telegraphy, another topic with which he was inti-mately acquainted. One of his positions was professor ofelectrical theory in the Ecole professionnelle superieure desPostes et des Telegraphes in Paris, where he lectured onthe propagation of electric current (published 1904), tele-phony (1907), and wireless telegraphy (1908, 1911).Whether this difference of emphasis was a merely personalquestion of taste, or a significant difference of opinion withHilbert about the scope and nature of mathematics, Poin-care did not presume to say.

Poincare was no great advocate of facts, whether inmathematics or physics, and instead advocated Mach’sprinciple of economy of thought. ‘‘The importance of a fact’’,he said, ‘‘is measured by the return it gives – that is, by theamount of thought it enables us to economise’’. In line withhis lifelong insistence on the importance of discovery inmathematics, he emphasised that what mathematicians callelegance in a good proof is a reflection of an underlyingharmony, which in turn introduces order and unity. This‘‘enables us to obtain a clear comprehension of the whole aswell as its parts. But that is alsopreciselywhat causes it to givea large return.’’ Far from being an aesthetic matter, he sug-gested that our emotional satisfaction derives from aconformity between the sought-for solution and the neces-sities of our mind, whereas a lengthy calculation, even if itleads to a striking result, is not satisfying until it is coupledwith an explanation that, albeit retrospectively, allows us topredict at least the characteristic features of the result.

For Poincare efficiency was the key, though it should alsobe accompanied by rigour. He noted that standards of rigourhad risen steadily in mathematics and that this would surelycontinue, since ‘‘a demonstration that lacks rigour is nothing.’’Hilbert, too, had insisted on rigour and had argued stronglyagainst the idea that rigour was the enemy of simplicity. Hehad insisted that arithmetic and analysis held no monopoly onrigorous arguments, contending that geometry and evenpartsof physics were capable of being placed on similarly firmfoundations. In his Paris address and long afterward, axio-matics emerged as his watchword for upholding newstandards of rigour in all of mathematics and beyond. Newsymbols and new signs would be invented, he said, to handlenew concepts. Poincare ’s emphasis was quite different,though he, too, noted the importance of modern conceptu-alisation. Looking to the recent past, he put his trust in well-chosen terms that encapsulate progress, while preventingrigorous proofs from becoming so long as to be almostincomprehensible: these key concepts included ‘‘uniformconvergence’’, ‘‘group’’, and ‘‘invariance’’ (whichwere amonghis lifelong themes), or the notion of ‘‘energy’’ in physics.

Poincare made at least an indirect nod to Hilbert at theoutset of his speech, in a passage in which he noted thatmathematics is bordered by philosophy on one side andphysics on the other:


On the one side, mathematical science must reflect uponitself, and this is useful because reflecting upon itself isreflecting upon the human mind which has created it; themore so because, of all its creations,mathematics is the onefor which it has borrowed least from outside. This is thereason for the utility of certain mathematical speculations,such as thosewhich have in view the studyofpostulates, ofunusual geometries, of functions with strange behaviour.Themore these speculationsdepart from themost ordinaryconceptions, and, consequently, from nature and appli-cations to natural problems, the better will they show uswhat the human mind can do when it is more and morewithdrawn from the tyranny of the exterior world; thebetter, consequently, will they make us know this minditself.All these matters had deeply preoccupied Hilbert, whose

‘‘unusual geometries’’ were also a source of some discomfortfor Poincare. He had reviewed Hilbert’s Grundlagen derGeometrie in highly positive terms in 1902 and 1903, but hisown view of geometry was intimately tied to a philosophythat explained how knowledge of the external world waspossible at all, and Poincare badly missed any considerationby Hilbert of the psychological origins of the axioms (1903,23): ‘‘The axioms are postulated; we do not know where theycome from . . .: His work is then incomplete; but this is not acriticism which I make against him. Incomplete one mustindeed resign one’s self to be.’’ The mind, the knowingsubject that constructs the world around it, was a concern ofPoincare’s but not one of Hilbert’s.

Turning from philosophy to physics, Poincare next madehis own priorities plain. ‘‘But it is to the opposite side, to theside of nature’’, he went on, ‘‘that we must direct our mainforces.’’ Here, he said, we find that ninety-nine percent of theproblems we face cannot be solved exactly in terms ofknown functions; moreover, the convergence of powerseries solutions is often too slow to be of any use, so progresscan only be made qualitatively. Our standards have changedover the decades: ‘‘There are, therefore, no longer someproblems solved and others unsolved, there are only prob-lems more or less solved . . ..’’ Poincare may not have knownthat Hilbert was actively lecturing in Gottingen on topics inphysics, drawn in partly by the enthusiasm of his friendMinkowski (see Corry 1999, 2004). But Poincare’s publicinvolvement with the subject was patent – formally, at least,he was professor of general astronomy at the Ecole Poly-technique, and had been since 1904 – and his ideas regardingqualitative progress stood in sharp contrast to Hilbert’s quitedifferent message to his fellow mathematicians, namely thatevery well-posed problem has a solution. Hence, for Hilbert,the challenge was clear: we should either be able to solve anysuch problem or to prove it is unsolvable.

Following these generalities, Poincare turned to specif-ics, and at this point, the text in Science et Methode is muchreduced from the original, obscuring the fact that Poincarewas alluding to his own early work.

He started with arithmetic, and here he was of the opinionthat progress had been slow because one could not appeal tocontinuity. He therefore concluded that arithmetic should beguided by the numerous analogies with algebra. He offeredsome analogies between the theory of congruences and that

of algebraic curves, predicting that the solution of problemsabout congruences in several variables would lead to con-siderable progress in indeterminate analysis. He also drewon an analogy between fields and ideals on the one hand,and curves on surfaces on the other, noting that both couldbe explored to their mutual advantage. Likewise the theoryof quadratic forms was intimately connected with the theoryof ideals. This was a subject Poincare had explored in hisarticle (1885), but neither then nor in 1908 did he choose toseparate out his contributions from those of Dedekind,which had by then been swept up into the panorama ofHilbert’s Zahlbericht. Poincare’s way of thinking aboutmathematics relied heavily on analogy as a means for sug-gesting proofs, but the explicit analogy with curves onsurfaces is unexpected and reflects his interest in contem-porary work, both in France and Italy, on this subject.Indeed, another reasonPoincare came toRomewas to awardthe first Guccia medal of the Circolo matematico di Palermoto Severi for his work on algebraic geometry.

It would seem likely that Poincare had in mind the anal-ogy between the ring of algebraic functions on a curve orsurface and the rings of algebraic integers in a number field.This had been the motivating analogy in Dedekind andWeber’s paper on Riemann surfaces (1882). The extension ofthese ideas to algebraic surfaces and higher-dimensionalvarieties is explicit in Kronecker’s famous Grundzuge(1882), and the extension was amplified in the small litera-ture devoted to explaining what Kronecker had not beenable to say in a way people had found easy to follow (e.g., inKonig 1903). By 1908, the best that had been done in thecontext of ideals in polynomial rings in several variables wasLasker’s paper (1905) where he established the theorem onthe factorisation of any ideal into its primary components.This work was heavily algebraic, and the Italian geometersCastelnuovoandEnriques complemented thegeometric sidewith their work on algebraic surfaces, which in turn fed intothe prolonged attempts by Picard and Simart to produce ageneralisation of Riemann’s theory to complex functions oftwo complex variables.

It is likely that Poincare knew of Picard’s work and that ofthe Italians, but it is much less certain that he knew of theGerman work. He had a reputation of not being very wellread, and his citations of Germanworks are sparse. Whatevermay be the modern implications of the analogy that Poincaremade, its vagueness on the day of the lexture may also reflectthat he had not thought about it very deeply. In any case, hedid not go on to exploit it himself.

What unity Poincare could find in arithmetic had comeabout through the use of transformations; indeed, this hadbeen the approach that guided his own study of forms ofdegrees 3 and 4 in his earliest work on number theory(Poincare 1881, 1882). He noted that geometry offeredexamples of many types of transformations, not all of themlinear, and he suggested how these could be applied inarithmetic. The groups that would arise would, withoutdoubt, be discontinuous, and should be analysed by lookingat their fundamental domains, where the most importantcontribution was to be found in Minkowski’s Geometrie derZahlen. It is amusing to see how Poincare was quite far intohis speechbefore heutters thenameof a mathematician (and


that it is not that of Hilbert!). He then added some wordsabout one of his own earliest interests, Hermite’s theory offorms with real coefficients, before concluding with somerather more perfunctory remarks about the study of primenumbers, where he could find no unity in the few asymptoticlaws that were known.

In algebra, he found that the subject had long beendominated by the theory of equations. This topic, he said,was best approached by way of the theory of groups, whichhe would discuss later, but there was also the question of thenumerical solution of equations. Less trite were his remarksabout the theory of invariants, which he observed hadunfolded over some 40 years but had recently been aban-doned, although the subject was far from exhausted. Therewas, for example, the problem of determining the invariantsof an arbitrary group (this is Hilbert’s 14th problem), butthere were also still important problems to be solved inconventional invariant theory, despite Hilbert’s happy sim-plification of Gordan’s algorithmic work. Furthermore,Poincare raised the prospect of a study of the algebra ofpolynomials modelled on arithmetic. This, although he didnot say so, was already the hope and intention of severalGerman mathematicians (Kronecker, Dedekind, and Hil-bert), who looked for analogies between fields of functionsand number fields.

Turning to ordinary differential equations, Poincareremarked that although linear differential equations were bynow well understood, nonlinear equations were not. A starthad been made on the study of their singular points, butPoincare held out hope that a fruitful analogy would befound with Cremona transformations, which are used tosimplify the study of the singular points of algebraic curves.This idea was taken up, perhaps independently, by HenriDulac, who documented some work with it in his paper(1908), and by Poincare’s nephew Pierre Boutroux in lec-tures he gave at the College de France in the same year. It isnot possible here to survey what came of this idea, but animportant recent development came in the 1980s when theBrazilian school took up the study of singular points of ho-lomorphic differential equations (see Cerveau 2006, 2010,and the articles cited therein). True to his custom of men-tioning very few names, Poincare overlooked that Picard haddiscussed this topic in the 1890s in his Trait�e d’analyse, vol. 3,Ch. IV. Instead, Poincare noted that there were importantunsolved problems in the uniformisation of the solutionfunctions. The qualitative study of the solutions, even in thecase of first-order, first-degree equations, had led to delicatequestions about limit cycles and included analogies with thestudy of the real roots of algebraic equations. That, in turn,was connected with the theme of Hilbert’s 16th problem,which, however, Poincare neglected to mention.

Taking up the study of partial differential equations,Poincare expressed his admiration for the pioneering workof Fredholm. Mittag-Leffler had brought this to his attentionsome years earlier, and he offered some thoughts on theunderlying reason for Fredholm’s success. This was tied, heemphasized, to the interesting question as to why so manyof the partial differential equations used in physics werelinear. Poincare claimed this derived from the modellingarguments that were used to construct the equations. These

methods turned the behaviour of infinitely many discretemolecules into a few continuous processes, for example,via the theory of infinite determinants, which appeared inboth Fredholm’s work as well as in that of the Americanlunar theorist George William Hill (1877). The latter hadprovided inspiration to Poincare for his work on celestialmechanics. But much remained to be done in this area, forexample, on the Dirichlet problem and on convexdomains. Here Poincare pointed to recent progress that hadbeen made thanks to Hilbert’s initiatives.

Hilbert had already signalled his interest in such matters inhis ICM address, where the closing problems 19, 20, and 23deal with the calculus of variations and related boundary-valueproblems, including theDirichlet problem.But by 1908he was even more prominently associated with work onfunction spaces and functional analysis, so here it is inter-esting to note that Poincare took every opportunity topromote Fredholm’s merits. This was partly because herecognised that Fredholm had seen more deeply into theproblems of partial differential equations and mathematicalphysics than he had, but also because he appreciated thatFredholm had come up with a simple and productive newformulation of the subject, one that conformed to his ownMachian views about the economy of thought. He was toreiterate his high opinion of Fredholm’s work in the first ofhis six Gottingen lectures in 1909 – it would be interesting toknow how such an unorthodox opinion was accepted – andagain when he gave the Bolyai Prize of the HungarianAcademy of Sciences to Hilbert in 1910. For Poincare, Fred-holm was the discoverer, however great the subsequentachievements of Hilbert and his school might have been.

There follow some brief, and it must be said, obscureremarks about Abelian functions in the context of the Schottkyproblem.An Abelian function in p variables is a functiononC

p

that is periodic with respect to a lattice of real dimension2p. The point at issue is that, as Riemann had shown in hispaper on Abelian functions, the period relations for an alge-braic curve of genus p [ 1 gives rise to a p 9 p symmetricmatrix that determines a lattice in p-dimensional complexspace C

p: The matrix reflects the complex structure of thecurve,which is in turn determined by its 3p - 3moduli, so thespace of all possible such lattices arising from algebraic curvesin this fashion has dimension at most 3p - 3. However, thespace of all p 9 p symmetric matrices is of dimension 1

2 pðpþ1Þ;which isgreater than3p - 3as soonasp C 4.TheSchottkyproblem, first raised in 1888, asks for a characterisation of thelattices that arise from curves. As Poincare noted, the study ofAbelian functions, and especially those that do not arise fromalgebraic curves, had beenblocked for some time, but now henoted that it had been almost completely solved in the recentwork of Castelnuovo and Enriques. This, together with Wirt-inger’s results (no mention of Picard or Painleve!), resolvedmany of the outstanding difficulties.

Poincare himself was to publish a major paper in this areain 1910, so onewonders if he already had such a contributionin mind 2 years earlier. The notion of the genus of an alge-braic curve generalises in two ways to algebraic surfaces: thegeometric genus pg is the dimension of the space of holo-morphic 2-forms, and the arithmetic genus pa (which is toocomplicated to define here) captures some of the


singularities of the surface when it is given as an embeddinginC

3:Even taken together, these two genera do not suffice tocharacterise a surface up to birational equivalence; whenthese numbers are not the same, the surface is said to beirregular. In 1904 Severi had showed that a surface wasirregular if it admitted nonzero holomorphic 1-forms, andEnriques had promptly established the converse. Almost atonce, Severi then showed that the irregularity, the differencebetween the two genera, was r - q, where q and r were thedimensions of the spaces of holomorphic 1-forms and1-forms with simple poles (this was among the papers forwhich he was awarded the first Guccia medal in 1908).Finally, Castelnuovo and then Severi presented differentproofs that showed that q = pg - pa and r = 2(pg - pa),although their arguments hinged on a result of Enriques thatwas not wholly convincing. Picard and Simart gave consid-erable prominence to this result in the secondvolumeof theirTh�eorie des fonctions alg�ebriques de deux variablesind�ependants in 1906, because the result made a connectionbetween the geometrical theory of the Italians and the ana-lytical or transcendental theory in France. As Castelnuovoand Enriques remarked in an appendix they contributed tothat book (Picard, Simart 1906, 495): ‘‘This result is thereforethe fruit of a long series of researches, to which the tran-scendental methods of M. Picard and the geometricalmethods used in Italy contributed equally’’. It is this theoremthat Poincare proved analytically in his article (1910), whenhe showed that the irregularity is the maximum number ofholomorphic 1-forms on the surface. However, all was notwell with the Italian approach, specifically with Enriques’sless than convincing result. This whole story has recentlybeen told in Babbitt and Goodstein (2011) and Mumford(2011), but until Mumford’s (1966) paper, the only acceptedproof was the one Poincare gave in 1910, an argument thatCastelnuovo was to describe almost 30 years later as dis-playing ‘‘the indelible mark of [Poincare’s] universalgenius’’.4

Poincare’s next topic was the general study of functionsof two complex variables, which, as he knew very well, wasdifferent at every turn from functions of a single variable.Among the many disparities, Poincare mentioned the lack ofa uniformisation theorem in two or more variables. Hisremark that ‘‘conformal representation is usually impossiblein a domain of four dimensions’’ probably does not refer toan old result of Liouville’s, but rather alludes to his ownpaper (1907) on holomorphically inequivalent, but topo-logically equivalent domains in C

2: This implies the failureof the Riemann mapping theorem in two or more complexdimensions.

At this stage, he took up the theory of groups, whichinterestingly appears as a topic separate from algebra. Itwas also much too large, so Poincare restricted himself toLie’s continuous groups and Galois’s discrete or finite ones.Even so, the result is curiously perfunctory for someonewhose interest in groups and geometry was so profound.Poincare had long admired Lie’s work, although he onlycontributed to it after Lie’s death in 1899 with his version of

what came to be called the Campbell-Baker-Hausdorfftheorem and the related Poincare-Birkhoff-Witt theorem.Poincare merely observed that much had been done by Lie,Killing, and Cartan that now needed to be simplified andcoordinated. The study of finite groups lagged for the samereason that arithmetic lagged behind algebra: the lack ofcontinuity; but Poincare hoped that the analogy betweenthe two kinds of groups would prove fruitful. This oddremark may reflect Poincare’s situation as a French math-ematician for, despite what should have been the enormousboost to the subject given by Jordan in his Traite des Sub-stitutions et des Equations Algebriques of 1870 and otherworks, it was Serret’s older and less forward-lookingaccount of algebra that had prevailed in French highereducation. The study of finite groups fared much better inBritain with Burnside, in Germany with Frobenius, and inthe United States with Dickson, whereas Galois theoryserved as a metaphor and guiding theme in Hilbert’sZahlbericht (1897).

Poincare then turned to geometry, and the text of Sci-ence et Methode carries all but the final remarks, where,indeed, Poincare attributed much of the recent progress tothe introduction of transformation groups. Indeed, Poin-care’s only criticism of Hilbert’s Grundlagen der Geometriehad been that he had not looked at the role groups play inthis new geometry.

The original text and the later published versions nowagree to the end, including the penultimate topic: Poincare’sfamous critique of Cantorism, the mathematical study ofinfinity. The controversies initially causedbyCantor’s theory,said Poincare, stemmed from the fact that its starting pointwas one of complete generality. This had provoked a truesense of horror in Hermite’s mind, but by now most mathe-maticians had come to terms with it. However, this was onlythe subjective side of the problem, for there were now:

certain paradoxes and apparent contradictions, whichwould have brought joy to the heart of Zeno of Elea andthe school of Megara. Then began the business ofsearching for a remedy, everyone in their own way. Formy part I think, and I am not alone in so thinking, that theimportant thing is only to introduce such entities as can becompletely defined in a finite number of words. Whateverbe the remedy adopted, we can promise ourselves the joyof the doctor called in to follow a fine pathological case.

This passage is the supposed source of a famous remarkthat Poincare did not make.5 The only-too-familiar quoteruns: ‘‘Later generations will regard Mengenlehre as adisease from which one has recovered.’’ It implies thatPoincare was strongly opposed to the study of the theory ofsets. As we can see, Poincare’s position was much moreinteresting, if less colourful.

Poincare concluded his address with some remarks onwhat he called the search for postulates, or what we mightcall the axiomatic method, noting that ‘‘in this direction Mr.Hilbert has obtained the most brilliant results’’. He thenoffered the hope ‘‘that these examples will have beensufficient to show the mechanism by which the

4Castelnuovo, 1938, Bologna ICM, I, 196, quoted in Poincare Oeuvres 6, 178 n.15For an account of how E. T. Bell came to make this particular error, see Gray (1991).


mathematical sciences have progressed in the past, and thedirection in which they must advance in the future.’’

Hilbert and Poincare : Contrasting Visions

Poincare’s complete address, unlike the much truncatedversion in Science et Methode, offers considerable insight intohis approach to mathematics now that he had become asenior statesman of science. Its central theme – that mathe-matical understanding is expressed in the productiveorganisation of facts and the recognition of analogies –indeed reflects how Poincare had guided his own work foralmost 30 years. In fact, his text follows rather closely the arcof his career, from his early work on number theory underthe influence of Hermite to his clashes over foundations withCouturat, Russell, and Zermelo after 1900. He mentionedrelatively few people by name, but even so those namesmake an interesting list: Minkowski, Hilbert of course, Lieand Killing, Castelnuovo and Enriques, Hermite and Dar-boux; but neither Picard nor Painleve appear, and onlyCartan among thenext generationof Frenchmathematicians.

There is a curious way in which Hilbert and Poincare maybe said to have danced around each other, and while doingso, they reached unexpected positions. The French view ofGerman mathematics, expressed forcefully by Hermite, andshared, for example, by Picard, was that it was needlesslyabstract; it did not grow out of interesting problems butimposed a way of thinking from above that might notaccomplish anything worth doing. This was Hermite’s viewof Cantor’s transfinite sets: the mathematics was not wrong; itwas worse than wrong, it was an affront to the sensibilities ofa mathematician who deeply admired work in the traditionof Jacobi and Weierstrass. Cantor’s theory was perhaps anextreme example, but in their different ways many Germanmathematicians were open to the charge of excessiveabstraction: Dedekind’s structural approach to number the-ory would be another example. Yet this was not how hisGerman contemporaries saw Hilbert. They saw him as a manof problems, someone who could bring formidable skill tothe resolution of difficult challenges. To be sure, his ICMaddress in Paris was animated by a vision of the unity ofmathematics, by the interplay he saw between theory andproblems, but it was, first and foremost, a presentation ofproblems. Whereas some were general, others were quitespecific (the continuum hypothesis, the Riemann hypothe-sis), and most drew attention to a specific topic in a way thatwould enable an ambitious mathematician to say that he orshe had eventually solved a Hilbert problem.

That was surely not the case with Poincare’s address in1908. He might suggest an analogy between two fields, hemight even challenge others to find one, but it would bemuch more difficult for anyone to say he or she had solved a‘‘Poincare problem’’. The breakthrough Poincare obviouslyfound most profound was Fredholm’s: here was a way ofposing problems as integral equations that would surely befruitful in many settings. This was Poincare’s most cherishedbelief, often asserted in his writings: the right place to standwas the most important thing a mathematician can possess.Once attained, all manner of facts and problems can be seenin an orderly and productive way. This is what he asked of

himself and, according to his nephew Pierre Boutroux (1914,1921), this is what he asked of others. Clearly, Henri Poincaremoved on a high plane of generality. In his address, heinvited his listeners and later readers to contemplate surfacesand the curves on them, equations algebraic or differentialand their singular points, transformations such as those thathad been so productive in geometry, various analogies thatmight be invoked when continuity is not available. In hisown work across many fields there is little by way of spe-cifics: a theory of Fuchsian groups, for example, and of theirdifferent types, but never a detailed study of a particularFuchsian group; theories of various types of differentialequations, but very seldom any illustrative examples. Even inhis work on physics, Poincare disdained examples and wasonly brought to ground by specific failures to match theorywith experiment.

Comparing these two lectures, we can discern what weresurely temperamental differences, but also striking differ-ences when it came to priorities. Hilbert began his survey ofkey problems with foundational questions, whereas Poin-care ended his with reflection about such issues. Otherdifferences were also at play. Hilbert was comfortable withpeople around him and was stimulated by the competitiveatmosphere in Gottingen. Poincare preferred his own com-pany, or that of a select few. As a result, Hilbert could spin offideas that others would then take up afterward. Poincareclearly leftmuch for others to do as well, but Paris was not theplace for ambitious students who sought the kind of intenseguidance several universities in Germany could offer tothem. This was changing with the presence of Emile Boreland the people around him, but they were working on topicsin real and complex analysis that held little interest forPoincare.

There is still another way in which these two addressesdiffer. No one could mistake the seriousness of Hilbert’sintentions, one measure of which was the scope of his 23 Parisproblems. With fewexceptions they werewell chosen: centralto their field, undoubtedly difficult, and well worth tackling.Poincare’s address could more easily be taken for politeremarks, although it was certainly much more than that. Whatit did not do, unlike Hilbert’s address, was offer a clear view ofwhat mathematics is: certain theories in which certain keyproblems are presently central. Hilbert’s was an open-endedwish list of problems whose solutions would mark break-throughs in the subject. Poincare’s address, by contrast,offered a view of what it meant to do mathematics, how themind can be organised most productively. It tells us about thebest ways to think so that good mathematics can be done. Inthe main, Hilbert selected problems that others found impor-tant and mathematics did advance when they were solved.Poincare offeredadvice,howeverelusive, thatwhenever it canbe taken will always enable mathematics to advance.

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Poincaré Replies to Hilbert: On the Future of Mathematics ca. 1908

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