Summary. This article develops two principal points. First, the so-called rivals of logical foundations, associated with Zermelo, Hilbert, and Brouwer, are here regarded as variants; specifically: to simplify, refine, resp. extend Frege’s scheme. Each of the variations is seen as a special case of a familiar strategy in the pursuit of knowledge. In particular, the extension provided by Brouwer’s intuitionistic logic concerns the class of propositions considered: about incompletely defined objects such as choice sequences. In contrast, Frege or, for that (...) matter, Aristotle thought that only propositions about precisely defined terms lent themselves to anything like logical theory.. This view of intuitionistic logic is fairly consistent with the mature Brouwer’s own research practice, but not at all with his rhetoric. Secondly, the article explains the silent majority’s scepticism about all variants of logical foundations, and, in note 7, sketches a genuine alternative which is almost explicit in the manifesto [#3] of Bourbaki with the aim of exploring the architecture of mathematics and our intuitive resonances to it. The alternative sees the weakness of logical foundations not in any logical defects, such as lack of precision, but in its sterility: stress on the logical elements of mathematics and mathematical reasoning draws attention away from discoveries of more significant features; more significant even for validity and reliability, though these matters are less prominent here than in logical foundations. The weakness in question is relevant beyond the present topic of foundations because this kind of weakness affects the whole analytic tradition in philosophy. Finally, for reasons explained there, an Appendix goes into some relations between the present paper and [17]. (shrink)

In a modal system of arithmetic, a theory S has the modal disjunction property if whenever $S \vdash \square\varphi \vee \square\psi$ , either $S \vdash \square\varphi$ or $S \vdash \square\psi. S$ has the modal numerical existence property if whenever $S \vdash \exists x\square\varphi(x)$ , there is some natural number n such that $S \vdash \square\varphi(\mathbf{n})$ . Under certain broadly applicable assumptions, these two properties are equivalent.