Stein-Logos Logic Logistike

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    Howard Stein

    Logos, Logic, and Logistike:Some Philosophical Remarks onNineteenth-Century Transformation

    of Mathematics

    iM athematics u nderw ent, in the nineteenth century, a transforma tion

    so profound that it is not too much to call it a second b irth of the subjectits first birth having occurred among the ancient Greeks, say from thesixth through the fourth century B.C. In speaking so of the first bi r th ,I am taking the word mathematics to refer, not merely to a body ofknowledge, or lore, such as existed for example among th e Babyloniansmany centuries earlier than the time I have mentioned, but rather to asystematic discipline with clearly defined concepts and with theoremsrigorously dem onstrated . It follows that the birth of m athematics can alsobe regarded as the discovery of a capacity of the human mind, or of humanthought hence i ts t remendous importance for philosophy: it is surelysignificant that, in the semilegendary intellectual tradition of the Greeks,Thales is named both as the earliest of the philosophers and the first prov-er of geometric theorems.

    As to the "second birth," I have to emphasize that it is of the verysame subject. O ne might m aintain with some plausibility that in the t imeof Aristotle th ere w as no such science as that we call physics; that Platoand Aristotle w ere acquainted with mathem atics in our own sense of theterm is beyon d serious controversy: a mathem atician today, reading theworks of Archimedes, or Eudoxos's theory of ratios in Book V of Euclid,will feel that he is reading a co ntemp orary. Then in w hat consists the "sec-ond birth"? There was,of course, an enormous expansion of the sub-ject, an d th at is relevant; but that is not qu ite it: The expansion w as itselfeffected by the very same capacity of thoug ht that the Greeks d iscovered;but in the process, something new was learned about the nature of thatcapaci tywhat it is, and what it is not. I believe that what has been

    238

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    LOGOS, L O G I C , A ND LOGISTIKg 239learned, when properly understood, constitutes one of the greatest advan-ces of philosophy although i t , too, like th e advance in mathematics itself,has a close relation to ancient ideas (I intend by the word logos in thetitle an allusion to the philosophy of Plato). I also believe that, whenproperly understood, this philosophical advance should conduce to a cer-tain modesty: one of the things we should have learned in the course ofit is how much we do not yet understand about the nature of mathematics.

    IIA few decades ago, the reigning cliche in the philosophy of mathematics

    was "the three schools": logicism (Frege-Russell-Carnap), formalism(Hilbert), intuitionism (Brouwer). Many people would now hold that allthree schools have failed: Hilbert 's program has not been able to over-come th e obstacle discovered by Godel; logicism was lamed by the para-doxes of set theory, impaired more critically by Godel's demonstrationof the dilemma: either paradoxes i.e., inconsistency or incompleteness,and (some would add) destroyed by Quine's criticisms of Carnap; intui-t ionism, finally, has limped along, and simply failed to deliver the lucidand purified new mathem atics i t had prom ised. In these negative ju dg -ments, there is a large measure of truth (which bears upon th e lesson ofmodesty I have spoken of); although I shall argue later that none of thethree defeats has been total, or necessarily final. But my chief interest herein the "three schools" will be to relate their positions to the actual develop-ments in mathematics in the preceding century. I hope to show that insome degree the usua l view of them has suffered from an excessive preoc-cupation with quasi-technical "phi losophica l" or , perhaps better, ide-ological issues and oppositions, in which perspective was lost of the math-em atical interests these arose f rom. To this end , I shall be mo re concernedwith the progeni tors or foreshadowers of these schools than with theirlater typical exponents; more particularly, with Kronecker rather thanBrouwer , and w ith Dedek ind m ore than Frege. Hilbert is another m atter:his program was very much his own creation. Yet in a sense, Hilberthimself, in his mathematical work before the turn of the century, standsas th e precursor of his own later foundational program; and in the samesense, as we shall see, Dedekin d is a very important precursor of Hilbertas well as of logicism. Since Kro neck er and Frege, too, both contributedessential ingredients to Hilbert 's program Kronecker on the philosoph-ical, Frege on the technical s idethe web is quite intr icate.

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    240 Howard SteinIII

    It is far beyond both the scope of this paper and the competence ofits author to do justice, even in outline, to the complex of interrelatedinvestigations and mathematical discoveries (o r "inventions") by whichmathematics itself was deeply transformed in the nineteenth century; Iam going to consider only a few strands in the vast fabric. A good partof the interest will center in the theory of numbersand in more or lessrelated matters of algebra and analysisbut with a little attention togeometry too.

    Let us begin with a very brief glance at the situation early in the cen-tury, when the transformation had just commenced. A first faint prefigura-tion of it can be seen in work of Lagrange in the 1770s on the problemsof the solution of algebraic equations by radicals and of the arithm eticaltheory of binary quadratic forms. On both of these problems, Lagrangebrought to bear new m ethods, involving attention to transfo rm ations ormappings and their invariants, and to classifications induced by equiva-lence-relations. (That Lagrange was aware of the deep importance of hismethods is apparent from his remark that the behavior of functions ofthe roots of an equation under permutations of those roots constitutes"the true principles, and, so to speak, th e metaphysics of the resolutionof equations of the third and f ou rth degree." Bourbaki1 suggests that onecan see here a first vague intuition of the modern concept of structure.}This work of Lagrange w as continued, on the num ber-theoretic side, byGauss; on the algebraic side, by Gauss, Abel, and Galois; and, despitethe failure of Galois's investigations to attract attention and gain recogni-tion until nearly fifteen years after his death, the results attained by theearly 1830s were enough to ensure tha tborrowing Lagrange's wordthe new "metaphysics" would go on to play a dominant role in algebraand number-theory. (I should not omit to remark here that a procedureclosely related to tha t of classification was the introduc tion of new "ob-jects" for consideration in general, and for calculation in particular. Inthe Disquisitiones Arithmeticae of Gauss, explicit introduction of new ideal"objects" is avoided in favor of the device of introducing new "quasi-equalities" [congruences]; Galois, on the other hand, went so far as tointroduce "ideal roots" of polynomial congruences modulo a primenumber, thus initiating the algebraic theory of finite fields.2)

    By the same period, the early 1830s, decisive developments had alsotaken place in analysis and in geometry: in the former, the creation

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    LOGOS, L O G I C , AND LOGISTIKE 241privately by Gauss, publicly by Cauchyof the theory of complex-analyticfunctions; in the latter, besides th e emergence of projective geo m etry,above all the discovery of the existence of at least one geom etry alternativeto that of Euclid. For although Bolyai-Lobachevskyan geom etry m adeno great immediate impression on the community of mathematicians,Gausswho, again privately, had discovered it for himselfwell under-stood its importance; and Gauss's influence upon several of the mainagents in this story was profound and direct.

    IVAs pivotal figures for the history now to be discussed, I would name

    Dirichlet, Riemann, and Dedekind all closely linked personally to oneanother, and to Gauss. Among these, Dirichlet is perhaps the poet'spoet better appreciated by mathematicians, more especially number-theorists, with a taste for original sources, than by any wide public. Itis not too m uch to characterize Dirichlet's influence, not only upon thosewho had direct contact with h i m a m ong those in our story: Riemannand Dedekind, Kum m er and Kroneckerbut upo n a later generation ofm athematicians, as a spiritual o ne (the G erman geistig w ould do better).Let me cite, in this connection, H ilbert and M inko w ski. In his G ottingenaddress of 1905, on the occasion of Dirichlet's centenary, M inkow ski, nam -ing a list of m athematicians who had received from Dirichlet "the strongestimpulse of their scientific aspiration," refers to Riemann: "What math-ematician could fail to und erstand that the luminous path of Riemann,this gigantic meteor in the m athematical heaven, h ad its starting-point inthe constellation of Dirichlet"; and he then remarks that although theassumption to wh ich Riem ann gave the name "Dirichlet's Principles"we m ay recall that this had just a few years previously been put on a soundbasis by Hilbertwas in fact introduced not by Dirichlet but by the youngWilliam Thom son, still "the m odern period in the history of mathematics"dates from what he calls "the other Dirichlet Principle: to conquer th eproblems with a m inim um of blind calculation, a m axim um of clear-seeingthoughts."3 Just four years later, in his deeply moving eulogy of Min-kowski, Hilbert says of his friend: "H e strove first of all for simplicityand clarity of thought in this Dirichlet and H erm ite were his models."4And for perhaps the strongest formulation of Minkowski's "other Dirich-le t Principle," consider this passage quoted by Otto Blumenthal, in hisbiographical sketch of Hilbert, from a letter of Hilbert to Minkowski:

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    242 Howard Stein"In our science it is always and only the reflecting mind der uberlegendeGeist], not the applied force of the formula, that is the condition of asuccessful result."5

    VI am going to make a sudden jump here: Why did Dedekind write his

    little monograph on continuity and irrational numbers? To be sure, thatwork is in no need of an excuse; but I have long been struck by these cir-cumstances: (1)Dedekind himself tells us in his foreword that when, somedozen years earlier, he first found himself obliged to teach the elementsof the differential calculus, he "felt more keenly than ever before the lackof a really scientific foundation of arithmetic." He goes on to expresshis dissatisfaction with "recourse to the geometrically evident" for theprinciples of the theory of limits.6 (Note that Dedekind speaks of a "foun-dation of arithmetic" rather than of analysis.) (2) In the foreword to thefirst edition of his monograph on the natural number, Dedekind invokes,as something "self-evident," the principle that every theorem of algebraand of the higher analysis can be expressed as a theorem about the naturalnumbers: "an assertion," he says, "that I have also heard repeatedly fromthe mouth of Dirichlet."7 (3) From antiquity (e.g., Aristotle, Euclid)through the late eighteenth and early nineteenth centuries (e.g., Kant,Gauss), a prevalent view was that there are two distinct sorts of "quanti-ty": the discrete and the continuous, represented mathematically by thetheories of number and of continuous magnitude. Gauss, for instance,excludes from consideration in the Disquisitiones Arithmeticae "fractionsfo r the most part, surds always."8 But Dirichlet, in 1837, succeeded inproving that there are infinitely many prime numbers in any arithmeticprogression containing two relatively prime terms, by an argument thatmakes essential use of continuous variables and the theory of limits. Thisfamous investigation was the beginning of analytic number theory; andDirichlet himself signalizes the importance of the newmethods he has in-troduced into arithmetic: "The method I employ seems to me above allto merit attention by the connection it establishes between the infinitesimalAnalysis and the higher Arithmetic [I'Arithmetique transcendante];9

    I have been led to investigate a large number of questions concerningnumbers [among them, those related to the number of classes of binaryquadratic forms and to the distribution of primes] from an entirelynew point of view, which attaches itself to the principles of infinitesimal

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    LOGOS, LOGIC, AND LOGISTIKE 243analysis and to the rem arka ble properties of a class of in fin ite seriesand infini te products.10From all this it seems reasonable I would even say, inevitable to con-clude that a part at least of Dedekind's motive was that of clearing

    "arithmetic" in the strict sense, i.e. the theory of the rational integers,from any taint of reliance upon principles drawn from a questionablesource. In any case, this motive is explicitly stressed by Dedekind's greatrival, Kronecker, in the first lecture of his course of lectures on numbertheory (published by Hensel in 1901). He there quotes from Gauss's prefaceto the Disquisitiones Arithmeticae: "The investigations contained in thiswork belong to that part of m athematics wh ich is concerned with the wholenumbersfract ions excluded for the most part, and surds always." But ,says Kronecker, "the Gaussian d i c t u m . . . is only then justified, if thequantities he wishes to exclude are borrowed from geometry or mechan-ics...." Thenafter referring to Gauss's ow n treatment, in the Disquisi-tiones, of cyclometry (thus, irrational numbers) and of forms (thus,"algebra" or Buchstabenrechnung) he cites elem entary exam ples (Leib-niz's series for 7r/4; partial fraction expansion of z tan(z)) to suppor t th eclaim that analysis has its roots in the theory o f the who le nu m be rs (with ,he says, th e sole exception of the concept of the l imitas to how this isto be dealt with, he unfortunately leaves us in the dark); and goes on toconclude:

    Thus arithmetic cannot be demarcated from that analysis which hasfreed itself from its original source of geometry, and has been devel-oped independently on its own ground; all the less so, as Dirichlet hassucceeded in attaining precisely the most beauti ful and deep-lyingarithm etical results thro ug h th e combination of methods of both dis-ciplines.11Finally, the epistemological connection is mad e explicit by K ron eck er inanother placehis essay "Uber den Zahlbegri ff" again with a referenceto Gauss:

    The difference in principles between geometry and mechanics on theone hand and the remaining mathematical disciplines, here comprisedunder the designation "arithmetic," consists according to Gauss in this,that the object of the latter, Nu m ber, is solely the product of our m ind,whereas Space as well as Time have also a reality, outside our mind,whose laws we are unable to prescribe completely a priori.12

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    L O G O S , LOGIC, AND LOGISTIKE 245Principle": "a minimum of blind calculation, a maximum of clear-seeingthoughts." Here is how Dedekind puts it, in the version of his theory thathe published in French in 1877:A theory based upon calculation would, as it seems to me, not offer

    the highest degree of perfection; it is preferable, as in the modern theoryof functions, to seek to draw the demonstrations, no longer fromcalculations, but directly from the characteristic fundamental concepts,and to construct the theory in such a way that it will, on the contrary,be in a position to predict the results of the calculation (for example,the composition of decomposable forms of all degrees). Such is the aimthat I shall pursue in the following Sections of this Memoir.15The reference to "the modern theory of functions" is, unmistakably,

    to Riemann; but the methods by which Dedekind pursues his stated aimare distinctly his own. They may be summed up, in a word, as structural'.Emmy Noether, in her notes to the Mathematische Werke, recognizesplainly and with evident enthusiasm the strong kinship of Dedekind's pointof view with her own (and I hope it may be presumed that the influenceof Noether's point of view upon the mathematics of our century is itselfwell known).

    VIIPerhaps, however, a distinction should be made: One theme associated

    with the name of Emmy Noether is that of the "abstract axiomatic ap-proach" to algebra; a second is that of attention to entire algebraic struc-tures and their mappings, rather than to, say, just numerical attributesof those structures (as a notable example, in topology, attention to homo-logy groups and their induced mappings, not just to Betti numbers andtors ion coefficients). O ne might consider that, of these themes, the secondl inks Noether to Dedekind, the first rather to Hilbert. I want to considerboth themes; but I begin with the second, which certainly is present infull measure in Dedekind. And I shall here take up this theme in connec-tion with the monograph Was sind und was sollen die Zahlenl

    In the foreword to the first edition of that work, Dedekind speaks of"the simplest science, namely that part of logic which treats of numbers."16Arithmetic, then, is a "part of logic"; but what is logic?

    The same question can be asked of other philosophers w ho claim thatmathematics is, or belongs to, logic; the inquiry tends to be frustra t ing.For example, Frege says17 that his claim that arithmetical theorems areanalytic can be conclusively established "only by a gap-free chain of in-

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    246 Howard Steinferences, so that no step occurs that does not conform to one of a smallnumber of modes of inferen ce recognized as logical"; but he quite failsto tell us how su ch recognition occursor what its content is. The trou-ble lies in the reigning presumption that to recognize a proposition as a"logical truth" is to identify its epistemological basis', and this is a trou -ble becauseif I may be forgiven fo r pontificating on the pointno cogenttheory of the "epistemological basis" of any kind of knowledge has everbeen formulated.On the other hand, some things can be said about Dedekind's claimand even, I think, some epistemological things. In a general way, we shouldremember, "logic" in the period in question was taken to be concernedwith the "laws of thought." I have said in my opening remarks that th ediscovery of mathematics was also the discovery of a capacity of the humanm ind. Dedek ind tells us what , in his opinio n, this capacity is. His earliestpublished statement dates from 1879 (by coincidence, the year of Frege'sB egriffsschrift) and occurs in 161 of the E leventh Supplement toDirichlet's Zahlentheorie, third edition:18 in the text he introduces the no-tion of a mapping; in a footnote he remarks , and repeats th e statementin the foreword to the first edition of Was sind und was sollen die Zahlen? 19that the whole science of num bers rests upo n this capacity of this mindthe capacity to envisage mappings without which no thinking at all ispossible. So the claim that arithmetic belongs to logic is the claim thatthe principles of arithmetic are essentially involved in all thoughtwith-out anything said about an epistemological basis. Moreover, the principlesinvolved are indeed those em ployed explicitly in Dedek ind's algebraic-num-ber-theoretic investigations: the formation of what he calls "systems"fields, rings (or "orders"), modules, ideals and mappings.

    But there is another aspect of Dedekind's view that should not beoverlooked. What is his answer to the question posed by the title of hismonograph? But first, what is the question? The version given by theEnglish translator of the work, The Nature and Meaning of Numbers,is (setting aside its abandonment of the interrogative fo rm ) quite mislead-ing: th e German idiom "Was soil [etwas]?" is much broader than th eEnglish "What does [something] mean?" What it connotes is always, insome sense, "intention"; but with th e full ambiguity of the latter term.We can, however, see the precise sense of Dedekind's question from thequite explicit answer he gives:

    M y general answer [or "principal answer": Hauptantwort ] to the ques-

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    LOGOS, LOGIC, AND LOGISTIKE 247tion posed in the title of this tract is: numbers are free creations of thehuman mind; they serve as a means for more easily and more sharplyconceiving the diversity of things.2 0

    The title, therefore, asks: What are numbers, and what are they for? (Whatis their use, their function?)

    The answer is not very satisfying. I think Dedekind makes a mistakeby assuming that numbers are "for" some one thing. But it is a venialmistake that lies really only on the surface of his formulation. It is, in-deed, part of the great discovery of the nineteenth century that mathe-matical constructs m ay have manifold "uses," and uses that lie far fromthose envisaged at their "creation." On the other hand, this formulationof Dedekind's is so general and vague that perhaps it could be stretchedto cover absolutely any application of the concept of number .

    What I think is more interesting in all this is that it is not what numbers"are" intrinsically that concerns Dedekind. He is not concerned, like Frege,to identify numbers as particular "objects" or "entities"; he is quite freeof the preoccupation with "ontology" that so dominated Frege, and hasso fascinated later philosophers. Dedekind's general answer to his firstquest ion, "Numbers are free creations of the human mind," later takesthe following specific fo rm : He definesor, equivalently, axiomatizes thenotion of a "simply infinite system"; and then says (in effect: this is myown free rendering) it does not matter what numbers are; what mattersis that they constitute a simply infinite system. He addsand here Itranslate l i teral ly "In respect of this freeing of the elements from anyfurther content (abstraction), one can justly call the numbers a free crea-tion of the human mind."21 O f course it follows from this characteriza-t ion that numbers are "for" any use to which a simply infinite systemcan be put; i t is because this answer does follow, and because it is theright one, that I described Dedekind's mistake about this as venial.

    It should be noted that a very similar point applies to Dedekind'sanalysis of real numbers. Here again th e contrast with Frege is instruc-tive. In the second volume of his Grundgesetze der Arithmetik, Fregemoves slowly toward a definition of the real numbers.2 2 H e does not quitereach i t tha t was reserved for the third v olum e, which never appeared;but what i t would have been is pretty clear. Frege's idea was that realnumbers are "for" representing ratios of measurable magnitudes (as, forhim, whole numbers Anzahlen a r e "for" representing sizes of setsin his terminology, sizes of "extensions of concepts"); and he wants the

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    248 Howard Steinreal numbers to be "objects" specifically adapted to that function. Thatis wh y he rejects Dede kind 's, Cantor ' s , and Weierstrass's constructions.But Dedekind says, again, that it does not matter what real numbers "are."In particular, he does not define them as cuts in the rational line. He says,rather, "Whenever a cut (Ai,A 2) is present that is induced by no rationalnumber, we create a new, an irrational num ber a, which we regard as com-pletely defined by the cut (Ai,A2)."23 In a letter to his friend and col-laborator Heinrich Weber,24 he defen ds this point, together with the relatedone concerning th e integers, in a rather remarkable passage. Weber hasevidently suggested that the natural numbers be regarded primarily as car-dinal rather than ordinal nu m bers, and that they be defined as Russelllater did define them. Dedekind replies:

    If one wishes to pursue your wayand I wou ld strongly recomm endthat this be carried out in detail I should still advise that by num-ber ... there be understood not the class (the system of all mutuallysimilar finite systems), but rather something new (corresponding to thisclass), which the mind creates. We are of divine species [wir sind go'tt-lichen Geschlechtes] and without doubt possess creative power not mere-ly in material things (railroads, telegraphs), but quite specially in in-tellectual things. This is the same question of which you speak at theend of you r letter concerning my theory of irrationals, w here you saythat the irrational number is nothing else than the cut itself, whereasI prefer to create something new (different from the cut), which cor-responds to the c u t . . . . We have the right to claim such a creativepower, and besides it is m uch m ore suitable, for the sake of the homo-geneity of all numbers , to proceed in this manner.

    Dedekind continues w ith essentially the points m ade in a we ll-known paperof Paul Benacerraf: there are many attributes of cuts that would soundvery odd if one applied them to the corresponding numbers; one will saymany things about the class of similar systems that one would be mostloath to hangas a burdenupon th e number itself. And he concludeswith a reference to algebraic num ber theory: "On the same ground s I havealways held Kum m er's creation of the ideal numbers to be entirely justified,if only it is carried out r igorously" a condition that Kummer, inDedekind's opinion, had not fully satisfied.In contrast with this last remark of Dedekind's, no less cultivated am athematician than Felix Klein, as late as the 1910s (when his invaluablelectures on the development of mathematics in the nineteenth century weredelivered), felt it necessary to demystify Kum m er's "creation" by insisting

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    L OGOS, LOGIC, AND LOGISTIK& 249on the fact that one canalthough not in a uniquely distinguished (asone would now say, "canonical") fashion identi fy Kummer's "idealdivisors" with actual complex algebraic numbers in the ordinary sense.25There can be no doubt in the mind of anyone acquainted with the laterdevelopment of the subject t ha t a t least in point of actual practice (butI would argue, also in principle) it w as Dedekind in 1888, not Klein thirtyyears later, who had this right.

    VIIIDedekind's term "free creation" also deserves some attention. (The

    theme has, again, some Dirichletian resonance, since Dirichlet in hisworkon trigonometric series played a significant role in legitimating the notionof an "absolutely arbitrary function," unrestricted by any necessaryreference to a formula or "rule.") It is very characteristic of Dedekindto wish to open up the possibilities for developing concepts, and to wishalso that alternative, and new, paths be explored. W e have just seen himurging Weber to develop his own views on the natural numbers; herepeatedly urged Kronecker to make known his way of developing alge-braic number theory; in the foreword to his fourth (and last) edition ofDirichlet's Vorlesgunen uber Zahlentheorie, containing his final revisionof the Eleventh Supplement, he expresses the hope that one of Kronecker'sstudents may prepare a complete and systematic presentation of Kroneck-er's t h eo r y an d also recommends the attempt to simplify the founda-t ions of his own theory to younger mathematicians, w ho enter the fieldwithout preconceived notions, and to whom therefore such simplificationmay be easier than to himself .2 6 (This was in September 1893. Kroneckerhad recently died; Hilbert had just begun to work on algebraic numbertheory.)

    If this has come to sound too much like a panegyric on Dedekind, Ican only say that that is because he does seem to be a great and true pro-phet of the sub jec t a genuine philosopher, of and in mathematics.

    IXIn his brief account of Dirichlet, Felix Klein mentions27 as "a particular

    characteristic" of Dirichlet's number-theoretic investigations the type ofproo f , which he w as the first to employ, that establishes the existence ofsomething without furnishing any method for finding or constructing it.(One recalls that when, some forty-five years later, Hilbert publishedproofs of a similar type in the algebraic theory of invariants, they were

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    250 Howard Steinregarded as unprecedented; and that Paul Gordan is said to have pro-nounced, "This is not mathematics, it is theology!"28) The nonconstruc-tive character of Dirichlet's theorem on the arithmetic progression is notedby Kronecker in his lectures, where he is able to tell his students, withlegitim ate pride , that he had himself succeeded in the year 1885 in repair-ing this defect and that the more complete result is to be presented forthe first t ime in the course of those lecture.29

    What is perhaps most notable about this is the absence, in these pub -lished lectures of Kronecker's, of any polemical tone in his comments onconstructive vs. nonconstructive methods, andin th e passage I quotedearlier his positive emphasis upon the role of irrational numbers andlimiting processes in number theory. It is of course possible that therestrained tone of these com m ents, wh ich contrast so markedly with whatis generally reported about Kronecker (and also with th e m ore explosivereaction of Gordan to Hilbert), is partly conditioned by his reverence forDirichlet (it is a rather pleasing fact that the two great num ber-theo reticrivals and philosophical opponents, K roneck er and Ded ekind , each editedpublications of work of Dirichlet), and partly by the exigencies of the sub-ject itself (it is clear enough that Kronecker hoped to reduce everythingto a constructive and finite basis, but clear also that he w as far from hav-ing any definite idea of how to do this for the theory of limits).30 It ispossible also, since the work as published was assembled from a varietyof manuscript sources,31 that Hensel in editing this material exercised somemoderating influence. But the foreword by Hensel does provide us witha more substantive clue to Kronecker's philosophical stand on the non-constructive in mathematics:32

    He believed that one can and must in this domain formulate each defini-tion in such a way that its applicability to a given quantity can be as-sessed by means of a finite num ber of tests. Likewise that an existenceproof for a quantity is to be regarded as entirely rigorous only if it con-tains a method by which that quantity can really be found. Kroneckerwas fa r from the position of rejecting entirely a definition or proof thatdid not meet those highest demands, but he believed that in that casesomething remained lacking, and held a completion in this directionto be an important task, through which our knowledge would be ad-vanced in an essential point.No mathematician could quarrel with the statement that a construc-

    tive definition or proof adds something to our knowledge beyond what

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    LOGOS, L O G I C , A ND LOGISTIK& 251is contained in a nonco nstructive one. Ho wev er, K rone cker's public standon the work of others w as certainly a m ore repressive one than Hensel'scharacterization suggests. I think th e issue concerns definitions rathermore crucially than proofs; but let me say, borrowing a usage from Pla-to, that it concerns the mathematical logos, in the sense both of "dis-course" generally, and of definition i.e., th e formation of concepts inparticular.Kronecker 's vehe m ent polemic against th e ideas and methods of Can-tor is more or less notorious. So far as I am aware, that polemic is notrepresented in the published writings. The first place I know of in whichKronecker published strictures against nonconstructive definition is a paperof 1886, "Uber einige Anwendungen der Modulsysteme auf elementarealgebraische Fragen"; and here it is in the first instance th e apparatus ofDedekind's algebraic num ber-theory "jene Dedekind'sche Begriffsbil-dungen w ie 'Modul', 'Ideal', u .s .w" that comes under attack.33 Unfor-tunately, this attack of K roneck er's contains no hint that nonconstructivedefinitions can be accepted at least provisionally (or as a Platonic "sec-ond best") even where , as in the case of analysis, Kroneck er has in factnothing to offer that meets his "higher" demands. It is possible, ther efo re,that Hensel's reading is off on this point: it is possible, and seems to ac-cord with the fact , that w hereas K ronecker w as willing to acknowledgeth e provisional value of a nonconstructive argument like Dirichlet's, hemeant to exclude more rigidly any introduction of concepts by non-constructive logoi. (I t must be confessed, again, that this seems to leaveanalysis, for Kronecker, in l imbo.)34

    XOn the other han d, Kronecker appears uniform ly to exempt "geometry

    and mechanics from his stringent requirements.35 Are these still to be con-sidered parts of m athematics? K roneck er does not seem to want to ex-clude them; but perhaps he regards them (e.g., on the basis of Gauss'sremarks) as in some m easure em pirical sciences.From this a Dedekindian question arises: "Was ist und was soil dieMathematik?" As in the case of number, I immediately repudiate the ideaof seeking a definite formula to state "what mathematics is fo r " i t is"for" whatever it proves to be useful for. Still, what m ay that be?

    Riemann's wonderful habilitation-lecture begins with a characteriza-tion of an "-tuply extended m agnitude" in term s that i t would not be

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    252 Howard Steinunreasonable to describe as belonging to "logic":36 th e points of such amanifold are "modes of determination [or "specification"] of a generalconcept"; examples of concepts w hose modes of specification constitutesuch a manifold are found both in "ordinary life" places in sensible ob-jects; colors) and within mathematics (e.g., in the theory of analytic func-tions). In the concluding section of that paper, Riem ann considers the ques-tion of the bearings of his great generalization of geometry upon ourunderstanding of ordinary physical space; this, he says, is an empiricalquestion, and must remain open, subject to what developments may oc-cur in physics itself: "Investigations wh ich, like that conducted here, pro-ceed from general concepts, can serve only to en sure that this wo rk shallnot be hindered by a narrow ness of conceptions, an d that progress in theknowledge of the connections of things shall not be hampered by tradi-tional prejudices."37If I may paraphrase: Geometry is not an empirical science, or a partof physics; it is a part of mathematics. The role of a mathematical theoryis to explore conceptual possibilities to open up the scientific logos ingeneral, in the interest of science in general. One might say, in the languageof C. S. Peirce, that mathematics is to serve, according to Riemann, amongother interests (e.g., that of facilitating calculation), th e interest of "ab-duct ion"of providing th e means o f formulating hypotheses or theoriesfo r the empirical sciences.

    The requirement of constructiveness is the requirement that allmathem atical notions be effectively computable; that m athematics be fun-damentally reduced to what the Greeks called logistike: to processes ofcalculation. Kronecker's concession to geometry and mechanics of freedomfrom this requirement is tacit recognition that there is no reason to assumea priori that structural relationships in nature are necessarily all of an ef-fectively computable kind. But then, when one sees, with Riemann, theusefulness of elaborating in advance, that is, independently of empiricalevidence (and, in this sense, a priori), a theo ry of structure s that need notbut may prove to have empirical application, Kronecker's limitation to"geometry and m echanics" of the license he offers to th ose sciences losesmuch of its plausibility.Another aspect of this point: Kant made a distinction between logic,which, concerned exclusively with rules of discursive thought in abstrac-tion from all "content," is a "canon" but not an "organon"not aninstrument for gaining knowledge; and mathematics, which is an organon,

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    254 Howard Steinwith all basic terms replaced by "meaningless" symbols, is the elimina-tion of the Kantian "intuitive" t h e "proceeding to concepts"; hence th eaxioms taken together constitute a definition. Of what? Of a species ofstructure. Now, by this, we need not mean "a set." It is a point of in-terest, and a very useful one, th at the notion of set proves so serviceableboth as a tool for concept-formulation within a mathematical discipline(ideals, etc.) and as affording a general framework for many or allmathematical disciplines; but set theory is a tool, not a foundation. ForHilbert, it is the axiomatized discourse itself that constitutes the mathe-matical logos; and the only restriction upon it that he recognizes is thatof formal consistency.

    As to "ideas": Kant associates ideas with completeness or totality. ForHilbert, I believe, this connoted the survey of the general characteristicsof the structures of a species and of related speciesmore explicitly, inthe geometrical case, the study of the problems of consistency andcategoricity of the axioms, their independence, and the sorts of alternativesone gets by changing certain axioms; in sh ort, something very m uch likemodel-theory. (O f course, when one now gets dow n to brass tacks, setsare pretty m uch indispensable; and because of the essential incom pletabilityof set theory, the "absolute completeness or totality" fails: Kantian"dialectical illusion" has not after all been avoided.)

    In the correspondence of Frege and Hilbert,4 2 it is amusing or ex-asperating, depending upon one's mood, to see Frege, wondering whatHilbert can mean by calling his axioms "definitions," come in his pon-derous but thorough wa y to the conclusion that //they are definitions,they must define what he calls a "concept of the second level" and thenmore or less drop this notion as implausible or u ninteresting to pursue;and to see Hilbert, not very interested in Frege's terminology or his pointof view, fail to und erstand w hat Frege's undervalued insight really was:for a Fregean "second-level concept" simply is the concept of a speciesof structure. So: a tragically or comically missed chance for a meetingof minds.But now, what of Hilbert 's "program"? I think it is unfortunate thatHilbert, in his later foun datio nal period, insisted on the form ulatio n thatordinary mathematics is "meaningless" and that only finitary mathematicshas "meaning." Hilbert certainly never abandoned the view that math-ematics is an organon for the sciences: he states this view very stronglyin the last paper reprinted in his Gesammelte Abhandlungen, called

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    LOGOS, LOGIC, A ND LOGISTIK& 255"Naturerkennen und Logik" (1930);43 and he surely did not think thatphysics is meaningless, or its discourse a play with "blind" symbols. H ispoint is, I think, this rather: that th e mathematical logos has no respon-sibility to any imposed standard of meaning: not to Kantian or Brouwerian"intuition," not to finite or effective decidability, not to anyone's meta-physical standards for "ontology"; its sole "formal" or "legal" respon-sibility is to be consistent (o f course, it has also what one might call a"moral" or "aesthetic" responsibility: to be useful, or interesting, orbeautiful; but to this it cannot be constrained poetry is not producedthrough censorship).

    In proceeding to his "program," however, Hilbert set as his goal th emathematical investigation of th e mathematical logos itself, with th e prin-cipal aim of establishing its consistency (o r "their" consis tencyfor hedid not envisage a single canonical axiom-system for all of m athem atics).And here he made essential use both of Frege and of Kronecker . ForFrege's extremely careful and minute regimentation of logical languageor "concept-writ ing" did not, as Frege himself thought it wo uld, serveas a guarantee of co nsistency; but the techniques he used for that regimen-tation did render the formal languages of mathematical theories, and theirformal rules of derivation, subject to m athematical study in their own righ t,regarded as purely "blind" symbolic systems. M oreov er, the regimenta-tion w as itself of such a kind that the play with symbols was a speciesof calculationof logistike. Without Frege, proof-theory in Hilbert's sensewould have been impossible.

    Of course, the final irony of this story, and the collapse of Hilbert 'sdream of establishing the consistency of the logic of the logos by meansrestricted to logistike, lies in the discovery by Godel, Post, Church, andTuring that there is a general theory of logistike, and that this theoryis nonconstruct ive; in particular, that neither the notion of consistencynor that of provability is (in general) effective; and further that all suf -ficiently rich consistent systems fall short of the Kantian " idea l" areincomplete.

    This leaves us with a mys te rya subject of "wonder," in which, ac-cording to Aristotle, philosophy begins. H e says it ends in the contrarystate; I am inclined to believe, as I think Aristotle's master Plato did, thatphilosophy does not "end," but that mysteries become better under-stood and deepe r. The m ysteries we now have about m athematics arecertainly better understood and deeper than those that con fronted Kant

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    LOGOS, LOGIC, AND LOGISTIKE 257

    25. Felix K lein, Vorlesungen iiber die Entwicklung der Mathematik im 19ten Jahrhundert,vol. 1, ed. R. Courant and O. Neugebauer (reprinted New York: Chelsea 1956), p. 322.26. Dedekind, Werke, vol. 3, p. 427. Ano ther passage is worth quoting as an illustrationboth of Dedekin d's generous atti tude toward alternative constructions and of the Dirichle-tian ideal that governs his own preferences; it comes, again, from th e foreword to Was sindund was sollen die Zahlen?, but refers back to h is procedure in constructing the real numb ers.Dedekind recognizes that th e theories of Weierstrass and of Cantor are both entirely ade-quate possess com plete rigor; but of his own theory, he says that "it seems to me somewhatsimpler, I might say quieter," than th e other two. (See Werke, vol. 3, p. 339. Italics addedDedekind 's word is ruhiger; the pub lished English translation, "easier," loses the point ofthe expression.)27. Klein, Entwicklung der Mathematik, vol. 1, p. 98.28. See Hilbert, Gesammelte Abhandlungen, vol. 3, pp. 394-95.29. Kronecker , Vorlesungen uber Zahlentheorie, vol. 1, p. 11.30. Kroneck er had, of course, a generally w orkable technique for dealing w ith algebraicirrationals; but when he speaks in Lecture 1 of the definitions that occur in analysis, heremarks that "from th e entire domain of this branch of mathematics, only th e concept oflimit or bound has thus far remained al ien to number theory" (ibid., vol. 1, pp. 4-5).31. Ibid., p. viii.32 . Ibid . , p. vi.33. Dedekind, Werke, vol. 3, p. 156 n. W hat K rone cker criticizes in the second instancehere is "the various con cept-fo rma tions wi th the help of whic h, in recent times , it has beenattempted from several sides (first of all by Heine) to conceive and to ground the 'irrational'in general." That Kronecker mentions Heinewho in fact adopted, with explicit acknow ledg-ment, the method introduced by Cantor as "first" is str iking; was his antipa thy to Cantorso great that he refused to m ention the latter's name at all or, indeed, suppressed his recollec-tion of it? The suspicion that some degree of pathological aversion is involved here is in-creased by an example cited by Fraenkel in his biographical sketch of Cantor appended toZermelo's edition of Ca ntor 's w orks (Georg Cantor, Abhandlungen mathematischen undphilosophischen Inhalts, ed . Ernst Zermelo [Berlin, 1932; reprinted Hildesheim: Georg OlmsV erlags buc hhan dlun g, 1966], p. 455, n. 3). Cantor published in 1870 his fam ou s theoremon the uniqueness of trigonometric series (see his Abhandlungen, pp. 80ff.) . In the follow-in g year, he published the first stage in his extension of that result (allowing a finite n u m b erof possibly exceptional points in the interval of periodicity); this paper also contains asimplification of his earlier proof, which he acknowledges as due to "a gracious oral com-municat ion of Herr Professor Kronecker" (ibid., p. 84). What Fraenkel repor t swith anexclamation pointis that in his posthumously published lectures on the theory of integrals,Kronecker cites th e problem of the uniqueness of trigonometric expansions as still open!It is w orth remem bering that Cantor achieved his broadest generalization of his uniq uene sstheorem in 1872, in the same paper (ibid.) pp. 92ff.) in which he first presented his con-struction of the real numbers and initiated th e study of the topology of point sets. As toDedekind, he responds, in his own quiet style, to Kro neck er 's footnote attack in a footn oteof his own to 1 of Was sind und was sollen die Zahlen? (see Dedekind, Werke, vol. 3, p. 345).34 . This point deserves a little further emphasis. Kroneck er had aprogram for the elimina-tion of the nonconstructive from mathem atics. Such a program was of unquestionably greatinterest. It continues to be so, in the doub le sense that a radical e l imina t ion showing howto render constructive a sufficient body of m athematics, and at the same time so greatlysimplifying that body, as to make it plausible that this constructive part is really all thatdeserves to be studied and that the constructive point of view is the clearly most fruitfulway of studying i twould be a stupendous achievement; whereas in the absence of suchradical elimination, i t remains always im portant to investigate how far constructive me th-ods may be applied. B ut K ronec ker 's own published ideas for constructivization appear tohave extended only to the algebraic, as indeed the rem ark cited in n. 28 above explicitlyacknowledges (for wh en K ronecke r says that "the concept o f the l i m i t . . . has thus far re-mained alien to number theory," that last phrase has to be taken to mean irreducible to

    E

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    258 Howard Steinthefinitary theory of the natural numberswhich reducibility is w hat K ronecker's constructiveprogram aimed at). K roneck er, therefore, in his documented strong statemen ts against "thevarious concept-format ions with th e help o f whi ch . . . i t has been at tempted . . . to conceiveand ground the 'irrational' in general," is in the position of arguing that, in the absenceof a co nstruct ive defin i t ion, nonco nstruct ive ones are nevertheless to be avoided ; that i t i sbet ter to have no definition at all. This is the posit ion that seems to me to follow fromKro neck er 's s tatements , al though i t confl icts wi th the report of Hensel .35. See K ronecker , Werke, vol . 3, pp. 252-53; Vorlesungen fiber Zahlentheorie, vol . 1,pp. 3, 5.36. Bernhard Riemann 's Gesammelte Mathematische Werke, ed. H einrich Weber w iththe assistance of Richard Dedekind, 2d ed. (1892; reprinted New York: Dover, 1953), pp.273-74.37. Ib id . , p. 286.38. See Logik nach den Vorlesungen des Herrn Prof. Kant im Sommerhalbenjahre 1792,in Diephilosophischen Hauptvorlesungen Immanuel Kants, ed. Arnold Kow alewski (M unich1924; reprinted Hildesheim: Georg Olms Verlagsbuchhandlung, 1965), pp. 393-95.39. Hilbert , Gesammelte Abhandlungen, vol . 3, pp. 402-3.40 . Dedekind , Werke, vol. 3, p. 479.4 1. K ant , Critique of Pure Reason, A702/B730.4 2. Gottlob Frege, On the Foundations of Geometry and Formal Theories of Arithmetic,trans. Eike-Henner W. Kluge (New Haven: Yale University Press, 1971), pp. 6-21. See esp.p. 19 (where Frege says quite distinctly, "It seems to me that you really intend to definesecond-level concepts"); but cf. also Frege's subsequent remark in the first of his two papers,"On the Foundat ions of Geometry" (ibid. , p. 36): "If any concept is defined by meansof [Hilbert 's axioms], i t can only be a second-level concept. It must of course be doubtedwhether any concept is defined at all, since not only the word 'point' but also the words'straight line' and 'plane' occur." In the latter passage, Freg e's notion seems to be that Hilbertmust have set out to define the (second-level) concept of a concept of apoint; and similarlyfo r line, etc. The earlier passage, in his letter to Hilbert , w as preceded by this: "The char-acterist ics which yo u state in your axioms . . . do no t provide an answer to the ques t ion ,'What proper ty mus t an object have to be a point , a straight line, a plane, etc.?' Insteadthey contain, for example, second-level relations, such as that of the concept of 'point' tothe concept 'straight line'." It is not far from this to the conclusion that what Hilbert 'saxioms do define is a single second-level relation, n am ely, that relation a m ong a w hole systemof first-level concepts corresponding to the u nde fined term s of the axiom atization, 'point','line', 'plane', 'between', and 'congruent ' which qualifies the whole as a Euclidean geometry.But however near that conclusion l ies , Frege evident ly failed to draw it.4 3. Hilbert , Gesammelte Abhandlungen, vol. 3, pp. 378-87.44. As I have suggested earlier in this paper, the role that may yet be played in fur therclari f icat ion of the nature of mathematics by the views of the "three schools" cannot beregarded as certain. My own opinion is that all three views (if they can really be called "three":within each "school," there have been q ui te s ignif icant di fferences) have m ade useful con-tributions to our understanding, but also that each will remain as merely partial and notfully sat isfactory. Y et , as I have also rem arked (above, n . 34), one cannot exclude th e pos-sibility that a genuine construct ivizat ion of mathematics will yet be achieved (al though Iconsider the arg um ents of sect ion X ab ove as cou nt ing rather s t rongly against this). In thesame w ay, i t remains possible that a recognizably construct ive proof of the consis tency ofanalysis, or even of (some version of) set theory, will be found; and such an event wouldafford great imp etus to the revival of (a mo dified form of) the Hi lbert program . As to"logicism," as it seems to m e the vaguest of the three doctr ines, I find it hardest to envisageprospects for i t , and am mo st s t rongly inclined to see i ts posi tive contr ibut ion as exhaustedby the insight that m athema tics is in some sense "about" conceptual possibil i t ies or con-ceptual s t ructure an insight , as I have remarked, already clearly present in Riemann, butm u c h more fully w o r k e d out af ter th e s t imulus provided by Ded ekind , Frege, R ussel l , and

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    LOGOS, L O G I C , ANDLOGISTIKE 259Whitehead. Nevertheless it is (barely, I think) conceivable that progress in understandingthe structure of "knowledge" will succeed in isolating a special kind of knowledge thatreasonably deserves to be considered "logical," thus c onferring new interesting content u ponth e question whether mathematics does or does not belong to logic.