## Mathematical Structure and Intuition, from Kant to Poincaré

Abstract:

As a fully-fledged philosophy of mathematics, structuralism is quite young. This contribution will explore one issue connected to its pre-history: the evolution of conceptions of mathematical intuition after Kant.
In the nineteenth century mathematics became more abstract; yet mathematicians still often appealed to intuition for various explanatory purposes. In order for this to make sense, conceptions of mathematical intuition had to shift—away from a strict Kantian conception—along with the new ways of thinking about mathematical content. I will argue that one trend at this time was to connect intuition with insight into mathematical structure rather than with the construction of mathematical objects.
I will focus on the philosophy of Henri Poincaré. The interpretation of Poincaré as a structural realist about natural science has been well known since Worrall (1989). Poincaré was also a conventionalist regarding geometry and some of the fundamental principles of physics. Finally he espoused constructivist views, and defended intuition, about (pure) mathematics.
I will argue that in addition to all of these labels, one can also interpret Poincaré as a structuralist (or pre-structuralist) about mathematics. As he put it: “Mathematicians do not study objects, but the relations between objects; to them it is a matter of indifference if these objects are replaced by others, provided that the relations do not change. Matter does not engage their attention; they are interested in form alone.” (Science and Hypothesis, p.
20). This certainly sounds very structuralist. The aim of this contribution is to both better understand this remark and to provide some additional context for it. In particular, I will consider both how structuralism fits with Poincaré’s other philosophical views and how it relates to similar views of other mathematicians.
Regarding Poincaré’s broader philosophy, he regarded intuition as necessary for both the content and knowledge of mathematics. Induction is “mathematical reasoning par excellence,” (SH, p. 9) required for mathematics to be a science (p. 16). And our knowledge of induction depends on the intuition of “the power of the mind which knows it can conceive of the indefinite repetition of the same act, when once this act is possible” (p. 13). If mathematics is about relations between objects rather than objects themselves; and if intuition is what governs mathematical content and our knowledge of it; then intuition, for Poincaré, must be something like insight into abstract structure.
Regarding the broader historical context, I will compare Poincaré’s remarks about mathematical structure with similar remarks from other mathematicians from at least the early 19th century. Thus, I will argue that Poincaré’s more “cognitive”, structuralist, conception of mathematical intuition was “in the air” at this time.