being abandoned. Let us analyse some of these events and issues in greater detail. The Crisis in the Foundations of Mathematics 1 Early Foundational Questions There is evidence that in 1899 Hilbert endorsed the view- Jos´e Ferreir´os point that came to be known as logicism. Logicism was the thesis that the basic concepts of mathematics are definable by means of logical notions, and that the key principles of The foundational crisis is a celebrated affair among math- mathematics are deducible from logical principles alone. ematicians and it has also reached a large nonmathemati- Over time this thesis has become unclear, based as it cal audience. A well-trained mathematician is supposed to seems to be on a fuzzy and immature conception of the know something about the three viewpoints called “logi- scope of logical theory. But historically speaking logicism cism,” “formalism,” and “intuitionism” (to be explained was a neat intellectual reaction to the rise of modern math- below), and about what Godel¨ s incompleteness results tell ematics, and particularly to the set-theoretic approach and us about the status of mathematical knowledge. Profes- methods. Since the majority opinion was that set theory sional mathematicians tend to be rather opinionated about is just a part of (refined) logic,1 this thesis was thought to such topics, either dismissing the foundational discussion be supported by the fact that the theories of natural and as irrelevant—and thus siding with the winning party— real numbers can be derived from set theory, and also by or defending, in principle or as an intriguing option, some the increasingly important role of set-theoretic methods in form of revisionist approach to mathematics. But the real algebra, and real and complex analysis. outlines of the historical debate are not well-known and Hilbert was following Dedekind in the way he under- the subtler philosophical issues at stake are often ignored. stood mathematics. For us, the essence of Hilbert’s and Here we shall mainly discuss the former, in the hope that Dedekind’s early logicism is their self-conscious endorse- this will help bring the main conceptual issues into sharper ment of certain modern methods, however daring they focus. seemed at the time. These methods had emerged gradually Usually, the foundational crisis is understood as a rela- during the nineteenth century, and were particularly asso- ciated with G¨ottingen mathematics (Gauss, Dirichlet); tively localized event in the 1920s, a heated debate between they experienced a crucial turning point with Riemann’s the partisans of “classical” (meaning late-nineteenth- novel ideas, and were developed further by Dedekind, Can- century) mathematics, led by Hilbert, and their crit- tor, Hilbert, and other lesser figures. Meanwhile, the influ- ics, led by Brouwer, who advocated strong revision ential Berlin school of mathematics had opposed this new of the received doctrines. There is, however, a second, trend, Kronecker head-on and Weierstrass more subtly. and in my opinion very important, sense in which the (The name of Weierstrass is synonymous with the introduc- “crisis” was a long and global process, indistinguishable tion of rigor in real analysis, but in fact, as will be indicated from the rise of modern mathematics and the philosophi- below, he did not favor the more modern methods elabo- cal/methodological issues it created. This is the standpoint rated in his time.) Mathematicians in Paris and elsewhere from which the present account has been written. also harbored doubts about these new and radical ideas. Within this longer process one can still pick out some The most characteristic traits of the modern approach noteworthy intervals. Around 1870 there were many dis- were: cussions about the acceptability of non-Euclidean geome- tries, and also about the proper foundations of complex (i) acceptance of the notion of an “arbitrary” function analysis and even the real numbers. Early in the twenti- proposed by Dirichlet; eth century there were debates about set theory, about the (ii) a wholehearted acceptance of infinite sets and the concept of the continuum, and about the role of logic and higher infinite; the axiomatic method versus the role of intuition. By about (iii) a preference “to put thoughts in the place of calcula- 1925 there was a crisis in the proper sense, during which tions” (Dirichlet), and to concentrate on “structures” the main opinions in these debates were developed and characterized axiomatically; and turned into detailed mathematical research projects. And Riemann Cantor in the 1930s G¨odel proved his incompleteness results, which 1. One should mention that key figures like and disagreed (see Ferreir´os 1999). The “majority” included Dedekind, could not be assimilated without some cherished beliefs Peano, Hilbert, Russell, and others. 1 2 Princeton Companion to Mathematics Proof (iv) a reliance on “purely existential” methods of proof. tions.2 This neat conceptual definition appeared objection- able to Weierstrass, as the class of differentiable functions An early and influential example of these traits was had never been carefully characterized (in terms of series algebraic number Dedekind’s approach (1871) to representations, for example). Exercising his famous crit- theory number fields —his set-theoretic definition of and ical abilities, Weierstrass offered examples of continuous ideals , and the methods by which he proved results such functions that were nowhere differentiable. as the fundamental theorem of unique decomposition. In It is worth mentioning that, in preferring infinite series a remarkable departure from the number-theoretic tradi- as the key means for research in analysis and function tion, Dedekind studied the factorization properties of alge- theory, Weierstrass remained closer to the old eighteenth- braic integers in terms of ideals, which are certain infinite century idea of a function as an analytical expression. On sets of algebraic integers. Using this new abstract concept, the other hand, Riemann and Dedekind were always in together with a suitable definition of the product of two favor of Dirichlet’s abstract idea of a function f as an “arbi- ideals, Dedekind was able to prove in full generality that, trary” way of associating with each x some y = f(x). (Pre- within any ring of algebraic integers, ideals possess a unique viously it had been required that y should be expressed in decomposition into prime ideals. terms of x by means of an explicit formula.) In his letters, The influential algebraist Kronecker complained that Weierstrass criticized this conception of Dirichlet’s as too Dedekind’s proofs do not enable us to calculate, in a par- general and vague to constitute the starting point for any ticular case, the relevant divisors or ideals: that is, the interesting mathematical development. He seems to have proof was purely existential. Kronecker’s view was that this missed the point that it was in fact just the right frame- abstract way of working, made possible by the set-theoretic work in which to define and analyze general concepts such methods and by a concentration on the algebraic proper- as continuity and integration. This framework came to ties of the structures involved, was too remote from an be called the conceptual approach in nineteenth-century algorithmic treatment—that is, from so-called constructive mathematics. methods. But for Dedekind this complaint was misguided: Similar methodological debates emerged in other areas it merely showed that he had succeeded in elaborating the too. In a letter of 1870, Kronecker went as far as say- principle “to put thoughts in the place of calculations,” ing that the Bolzano–Weierstrass theorem was an “obvi- a principle that was also emphasized in Riemann’s theory ous sophism,” promising that he would offer counterexam- of complex functions. Obviously, concrete problems would ples. The Bolzano–Weierstrass theorem, which states that require the development of more delicate computational an infinite bounded set of real numbers has an accumula- techniques, and Dedekind contributed to this in several tion point, was a cornerstone of classical analysis, and was papers. But he also insisted on the importance of a general, emphasized as such by Weierstrass in his famous Berlin conceptual theory. lectures. The problem for Kronecker was that this theo- The ideas and methods of Riemann and Dedekind rem rests entirely on the completeness axiom for the real became better known through publications of the period numbers (which, in one version, states that every sequence 1867–72. These were found particularly shocking because of nonempty nested closed intervals in R has a nonempty of their very explicit defence of the view that mathemat- intersection). The real numbers cannot be constructed in ical theories ought not to be based upon formulas and an elementary way from the rational numbers: one has to calculations—they should always be based on clearly for- make heavy use of infinite sets (such as the set of all pos- mulated general concepts, with analytical expressions or sible “Dedekind cuts,” which are subsets C ⊂ Q such that calculating devices relegated to the further development of p ∈ C whenever p and q are rational numbers such that the theory. p<qand q ∈ C). To put it another way: Kronecker was To explain the contrast, let us consider the particularly drawing attention to the problem that, very often, the accu- clear case of the opposition between Riemann’s and Weier- mulation point in the Bolzano–Weierstrass theorem cannot strass’s approaches to function theory. Weierstrass opted be constructed by elementary operations from the rational systematically for explicit representations of analytic (or holomorphic) functions by means of power series of the 2. Riemann determined particular functions by a series of inde- ∞ − n ana- pendent traits such as the associated Riemann surface and the behav- form n=0 an(z a) , connected with each other by lytic continuation.