Mathematics as a human endeavour∗

The second half of the twentieth century presents a panorama of the philos-ophy of mathematics quite different from the first half. Philosophy is domi-nated by the reaction to neopositivism, mathematics lives in the aftermath of the great Bourbaki’s synthesis. Let us consider the state of the philosophy of mathematics at the out-set of the post-Bourbaki era, that is at the beginning of the second half of the twentieth century. The traditional foundational schools of logicism, formalism and intuitionism still filled the stage, but they were barren and stagnating, according to the picture made popular by Reuben Hersh1. Their grand programs had failed; Bourbaki’s organization of mathematics was less ambitious, but it was itself a source of uneasy feelings; we can see now that it was a kind of patchwork, an opportunistic mixture of platonism and formalism: set theory was used to fix the concept of “structure”, but it was really only a grammar. As a grammar it was not so comfortable as advertised, since new concepts such as those of category theory struggled in vain to find a place in the picture; from a logical point of view it was preposterous to ignore the ramification given by the independence proofs, while for the working mathematician too much real mathematics was cut out from its abstract treatment