Bourbaki’s Hypothetico-Deductive Structuralism
In the 1930s, Bourbaki finally defended the view that mathematics does not deal with the traditional mathematical objects at all, but that objectivity is solely based on the stipulation of structures.
Bourbaki inaugurated an axiomatic-structural point of view that could seemingly work without the need of metamathematics in Hilbert’s sense. Indeed, metamathematics being “finite” and contentual, it would be an exception to the slogan that mathematics is only about formal structures.The hypothetico-deductive Bourbakist foundations were explicitly designed as neutral with respect to philosophical foundations and opposed to this effect to the practical turn in philosophy of mathematics. In fact, the practical turn is positioned in a field of tension between pragmatism and the working mathematician: Wittgenstein and Bourbaki. What unites both is that they refuse the hypostasis of mathematical objects either from a philosophical perspective or from the point of view of mathematical practice. The ‘working mathematician’[156] Henri Cartan, one of the founders of Bourbaki, wrote in 1943:
The mathematician does not need a metaphysical definition; he must only know the precise rules to which are subject the use he has in mind [...] But who decides upon the rules? (Cartan 1943, 1b [transl. G.H.]).
This sounds Wittgensteinian but is not really so: according to Cartan, the first mathematical reasoning on a certain area intuitively obeys certain rules and if difficulties arise, the use is adapted, etc. Thus, a mathematical reality is created through practice. What is the criterion of practice and rules that result in that reality? In his historical notice on set theory, Bourbaki
“recognized that the ‘nature’ of mathematical objects is ultimately of secondary importance, and that it matters little, for example, whether a result is presented as a theorem of a ‘pure’ geometry or as a theorem of algebra via analytical geometry. In other words, the essence of mathematics [...] appeared as the study of relations between objects which do not of themselves intrude on our consciousness, but are known to us by means of some of their properties, namely those which serve as the axioms at the basis of their theory” (Bourbaki 1968, 316-317).
Bourbaki considers the fact of addressing ‘the problem of the nature of beings’ or of ‘mathematical objects’ as a ‘naive point of view”, “half-philosophical, half-mathematical” (Bourbaki 1948, 40) and abandoned the philosophical problem of object-individuation in favor of structures.
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