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The view that mathematics is about abstract structure is quite deeply rooted in mathematical practice, with further philosophical views about the nature of structures emerging more recently. This essay argues, first, that Poincaré’s views about structure are properly philosophical, since they go beyond basic claims about the general subject matter of mathematics. Second, it proposes that these further views align Poincaré with a strong version of structuralism—one typically associated with realism. This raises a question since he is a constructivist about mathematics, supporting a broadly Kantian conception of mathematical knowledge and existence. There is thus an apparent tension in Poincaré’s philosophy of mathematics, and a third goal is to resolve this tension.

The mathematical developments of the 19th century seemed to undermine Kant’s philosophy. Non-Euclidean geometries challenged Kant’s view that there is a spatial intuition rich enough to yield the truth of Euclidean geometry. Similarly, advancements in algebra challenged the view that temporal intuition provides a foundation for both it and arithmetic. Mathematics seemed increasingly detached from experience as well as its form; moreover, with advances in symbolic logic, mathematical inference also seemed independent of intuition. This paper considers various philosophical responses to these changes, focusing on the idea of modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics. It is argued that far from clinging to an outdated paradigm, programs based on new conceptions of intuition should be seen as motivated by important philosophical desiderata, such as the truth, apriority, distinctiveness and autonomy of mathematics.

Although it is clear that Sir William Rowan Hamilton supported a Kantian account of algebra, I argue that there is an important sense in which Hamilton's philosophy of mathematics can be situated in the Newtonian tradition. Drawing from both Niccolo Guicciardini's (2009) and Stephen Gaukroger's (2010) readings of the Newton–Leibniz controversy over the calculus, I aim to show that the very epistemic ideals that underpin Newton's argument for the superiority of geometry over algebra also motivate Hamilton's philosophy of algebra. Namely, Hamilton's defense of algebra, like Newton's defense of geometry, is driven by the claim that a mathematical science must have a proper object and thus a basis in truth. In particular, Hamilton aims to show that algebra is not a mere language, or tool, or a mere “art”; instead, he argues, algebra is a bona fide mathematical science, like geometry, because its methods also provide true and accurate insight into a genuine subject matter, namely, the pure form of temporal intuition.

Though logical positivism is part of Kant's complex legacy, positivists rejected both Kant's theory of intuition and his classification of mathematical knowledge as synthetic a priori. This paper considers some lingering defenses of intuition in mathematics during the early part of the twentieth century, as logical positivism was born. In particular, it focuses on the difficult and changing views of Hermann Weyl about the proper role of intuition in mathematics. I argue that it was not intuition in general, but his commitment to twodifferent types of intuition, which explains his rather unusual and tormented philosophy of the mathematical continuum.

Jules Henri Poincare is famous for his “conventionalist” philosophy of science. But what exactly does this mean? Poincare invented the category of convention because he thought that there are some central principles in science that are neither based on intuition , empirical data, nor that are arbitrary stipulations. His views here resemble those of Wittgenstein , in particular, as presented in On Certainty. The invention of convention is lauded (for example, by Robert DiSalle ) as a genuine philosophical discovery. But it is also critiqued (for example by Michael Friedman ) as yielding a vision of science that is too rigid – one that is refuted by general relativity . This paper aims to defend Poincare’s views about conventions by focusing on his central idea that conventional choices, though “free”, are “guided” by experience . I will argue that conventionalism is not a commitment to fixed a priori stipulations, as DiSalle and Friedman propose. Rather, it mandates empirically motivated shifts in (even geometric) conventions – a view surprisingly in accord with Friedman’s “relativized a priori”, and thus more consistent with general relativity than is generally thought.

There are several distinct components of the realist anti-realist debate. Since each side in the debate has its disadvantages, it is tempting to try to combine realist theses with anti-realist theses in order to obtain a better, more moderate position. Putnam attempts to hold a realist concept of truth, yet he rejects realist metaphysics and realist semantics. He calls this view “internal realism”. Truth is realist on this picture for it is objective, rather than merely intersubjective, and eternal. Putnam introduces a concept of epistemic idealization — epistemically ideal conditions, or idealized justification — to try to ground the strong objectivity of truth without sliding into metaphysical realism. I argue that the concept of ideal conditions to which Putnam appeals does not cohere with his (anti-realist) commitment to an assertability conditions account of meaning.

Introductions to the philosophy of mathematics rarely focus on the period 1790–1870, if they mention it at all. Aside from Mill’s empiricist program, what was happening in the philosophy of mathematics between Kant and Frege? Certainly there were crucial mathematical developments. For example, non-Euclidean geometry was discovered in the 1830s, and this paved the way for later conventionalist and formalist philosophies of mathematics such as those offered by Poincare and Hilbert. In addition, the rigorization of analysis during our period was an important precedent to logicism, a philosophical program espoused by Frege, Dedekind, and Russell just after 1870. But where is the influential philosophy? Of course, there were philosophers, so there was also philosophy of mathematics. One approach to this period would be to survey what philosophers such as Hegel, Herbart, Fries, Mill, and Comte said about mathematics. However, few of these authors are known for their contributions to philosophy of mathematics, with the notable exception of Mill. Moreover, most of this philosophical work seems far removed from the revolutionary developments in mathematics. On the other hand, entwined in the mathematical developments are crucial and fascinating philosophical arguments. In fact, it is precisely in its close connection to mathematical developments that the relevant philosophy is so exciting, for it shows that the border between mathematics and its philosophy is both permeable and dynamic. Philosophical issues arise in mathematics, and mathematicians deal with them, explicitly and implicitly, in their creative work and in their teaching. Furthermore, how the philosophical issues are treated affects mathematics.

In his first philosophy book, Science and Hypothesis, Poincaré gives us a picture which relates the different sciences to different kinds of hypotheses. In fact, as Michael Friedman has pointed out (Friedman 1995), Poincaré arranges this book—chapter by chapter—in terms of a hierarchy of sciences. Arithmetic is the most general of all the sciences because it is presupposed by all the others. Next comes mathematical magnitude, or the analysis of the continuum, which presupposes arithmetic; then geometry which presupposes magnitude; the principles of mechanics which presuppose geometry; and finally experimental physics which presupposes mechanics. Poincaré's basic view was that experiment in science depends on fixing other concepts first. In particular he believed at the time that our concept of space had to be fixed before we could discover truths about the objects in space.

Poincare is often called a ‘pre-intuitionist’ or a’ semi-intuitionist’ because he influenced the foundations of intuitionism in the philosophy of mathematics. Intuitionism is a form of constructivism; and constructivism is the general view that mathematical objects (numbers, domains and so on) are mental constructions. That is, it is the view that mathematical objects have no existence independent of the minds of mathematicians. Poincare was certainly a constructivist, but whether or not he can be grouped with the intuitionists needs to be made clear. And this leads to the further question: if he cannot be grouped with the intuitionists, is his ‘constructivism’ coherent and defensible?

In Chapters 4 and 7 I argued that Poincare’s theory of predicativity is a consequence of his constructivist account of what is required in order to define a set. More generally, it is a consequence of his theory of meaning. The central component of Poincare’s theory of meaning is the notion of ‘verifiability in principle’. This notion is, however, far from clear; and we must now enquire as to whether it is coherent. The notion of ‘verifiability in principle’ is one of the cornerstones of Poincare’s philosophy of mathematics (and of science, in general). If either it or the more basic notion of ‘in principle possible’ (for instance, in ‘constructible in principle’, ‘provable in principle’, ‘decidable in principle’) cannot be regarded as possessing a clear, definite sense (as the strict finitist argues), then Poincare’s philosophy as a whole is in danger of losing all significance. The argument of this chapter is that Poincare’s weak ‘verificationist’ theory of meaning is coherent, and when interpreted in the light of his background theory of the synthetic a priori, provides a stable position from which to fend off strict finitist attacks. Moreover, it provides a viable foundation for analysis.

Following Kant, Poincare believed mathematics to be synthetic a priori. He thus endorses Kant’s view that not all true mathematical propositions are obtainable as theorems of logic alone, even when the logic is supplemented by appropriate definitions of the non-logical vocabulary involved. Poincare’s sense of a synthetic proposition is essentially that of Frege. Whereas Frege might have held that the entire corpus of mathematics (with the possible exception of geometry) is analytic, Poincare held that it is synthetic in precisely Frege’s sense. For Poincare argues against Frege that in order to provide a proof of certain key mathematical claims (even number-theoretic ones) it is necessary to ‘[make] use of truths which are not of a general logical nature, but belong to a sphere of some special science.’1 Like Kant, Poincare maintained that mathematical knowledge is knowledge ‘a priori’. Where he differs from Kant is in exactly what he takes to be ‘synthetic a priori’. For Kant, the synthetic a priori includes the intuitions of Euclidean, three-dimensional space and time (the forms of experience); and these provide the content of (the intuitions for) geometric and number-theoretic truths. (In addition, certain general statements about substance, cause and effect, and so on, are also synthetic a priori.) In contrast, Poincare’s theory of the synthetic a priori is much more minimal. It is that which is imposed upon us with such a force that we could not conceive of the contrary proposition, nor could we build upon it a theoretical edifice.2

Poincare’s theory of predicativity is a central and exciting component of his general philosophical position. As is well known, his philosophy of mathematics was foundational for intuitionism. It is also well known that he was concerned about the set-theoretic paradoxes, and that he was one of the first to write about the ‘Vicious Circle Principle’ (VCP). Just what constitutes Poincare’s version of the VCP, the theory of predicativity which underlies it, and his contribution to the solution of the contradictions of classical mathematics, is much more obscure. To be sure, his work in this area ought to be regarded as ancestrally related to modem programmes in predicative analysis and predicative set theory. However, just as it is wrong to consider a modern formalised intuitionism as a natural extension of his general philosophical views, so is it a mistake to consider a predicative version of axiomatised set theory as a programme he would have unequivocally endorsed. In fact, in view of the formality of both of these programmes, he probably would have opposed them.

The key to understanding Poincare’s philosophy of mathematics is to realise that he defends Kant’s epistemological view that mathematics is synthetic a priori, but that the details of Poincare’s theory of the ‘synthetic a priori’ are quite distinct from Kant’s. Indeed, Poincare’s theory might be sufficiently distinct so as not to be subject to the pitfalls to which Kant’s theory is so vulnerable.1 Since Poincare adopts the Kantian terminology, though he adapts the theory, my first task is to outline Kant’s theory, to enable the necessary comparison to be made. Poincare explicitly rejects Kant’s thesis that Euclidean geometry is synthetic a priori. He even disagrees with Kant’s more minimal thesis that the three-dimensionality of space is synthetic a priori. He holds that these are, rather, ‘conventional’ matters. However, he follows Kant in asserting that the theorems and the acceptable axioms of pure number-theoretic mathematics have the synthetic a priori status.

In the same way that Poincare’s theory of arithmetic intuition is exhibited in his negative arguments against the logicist, his theory of geometric intuition (of continuity) is exhibited in his negative arguments against the set theorist. Thus we must examine his attack on set theory in order to uncover the details of his theory of geometric intuition. Set theory is a foundationalist programme (begun by Cantor and developed by Russell, Zermelo and others); and its aim is to provide an explicit and precise method for characterising acceptable mathematical objects (especially those which are infinite) and acceptable mathematical inference, in terms of axioms about sets or collections alone. Interestingly, it was found that in order to reproduce the main body of mathematical results (for example, results pertaining to the continuum), certain unintuitive axioms had to be accepted. That is, strongly existential, nonconstructive axioms such as the axiom of infinity (there exists an infinite set), the unrestricted power set axiom (for any set there also exists the set of all its subsets), and the axiom of choice (for any set, x, of sets there exists another set which consists of exactly one element from each of the sets in x) had to be accepted in order for set theory to be powerful enough to produce all the mathematical results perceived as important.

The claim that mathematics is synthetic a priori is most commonly and most famously attacked by the logicists. Logicists usually argue against the syntheticity of mathematics in order to argue against Kant’s thesis that mathematics possesses an extra-logical subject matter. Frege, Russell, and others endeavoured to show that Kant was wrong about the content of mathematics by showing how all true mathematical statements can be derived, once certain primitives (e.g., for Frege, for arithmetic, ‘ number’, ‘immediate predecessor’, ‘0’, and the ‘ancestral’ relation) have been defined in logical terminology alone.1 This is generally taken as an attempt to show that mathematics is analytic a priori. (This label is not accurate for Russell’s logicism, as he believed even logic to be synthetic. However, the following general point still stands.) If successful, logicism would have shown mathematics to be independent of ‘intuitions’ or extra-definitional content, for its foundation would essentially be that of logic. Logicism and the related foundational programme of set theory are addressed below, in the following chapters.

Poincare argued that logicism fails because the logicist cannot show that the epistemological source of the concepts required for the logicist derivation of arithmetic is really logic. Indeed, Poincare believed that the concept of indefinite iterability — of the ability to iterate or repeat (for example, the application of a rule, like ‘+’) without stopping — is foundational, not only for arithmetic, but for all systematic thinking; and its epistemological source is synthetic a priori intuition. I call this intuition ‘arithmetic’ intuition; and the claim made by Poincare is that it underlies all systematic thinking because it underlies our ability to generalise, as in our understanding of ‘and so on’, of potential infinity, ‘etc.’, etc. On this view, knowledge that the principle of induction is satisfied by the numbers is not strictly knowledge of logical or analytic truths, for induction is not merely part of the definition of ‘number’. Rather, any definition of ‘number’ requires the employment of induction (or a principle at least as strong as induction) outside of it, so to speak; and this means that to define ‘number’ we require prior epistemological access to (something strong enough to yield) induction. Poincare therefore concludes that induction is a synthetic (a priori) principle, and it holds in any nonempirical domain which can be regarded as a model of an indefinitely iterable rule.

Contributors: Michael Detlefsen, Michael D. Resnik, Stewart Shapiro, Mark Steiner, Pirmin Stekeler-Weithofer, Shelley Stillwell, William J. Tait, Steven J. Wagner