Martin-Löf Conference on the Foundations of Mathematics: Day 3
One of the most engaging talks was Colin McLarty’s ‘What are the things of mathematics?’, a motivational blurb for structuralism-cum-category theory with some neat examples. McLarty wanted (at least in part) to make the case that mathematicians in practice talk about existence and identity in a manner closer to categorical set-theory à la Lawere rather than ZF speak. In principal, ‘elements’ of mathematical theories ought to be individuated not by ZF but by participation in structures. The foundations, then, is not membership as irreducible, but invariance up to isomorphism. McLarty’s preferred framework is CCAF, a classical first-order categorical set theory.
The major part of the talk was spent trying to deal with a prima facie problem for his foundational view. Mathematicians standardly talk about natural numbers as if they are identical to corresponding objects in the set of integers (similarly for integers with respect to the set of rationals, etc). However, from a structural point of view this equation is unfortunate. As far as these elements live inside different structures, equating them is a misunderstanding; the appropriate strategy would be to relate them by looking categorically at relations between the structures; and CCAF offers precisely such a perspective.
Since I spent a number of hours discussing with Stewart Shapiro on our way to Uppsala, his talk didn’t contain many surprises. It did, however, engender one of the more lively discussions during the conference. Stewart’s training in classical model-theory doesn’t prevent him from enjoying the occasional excursion into intuitionistic terrain. More precisely, Stewart subscribes to something along the lines of the Hilbertian slogan: “Consistency entail existence”. If it’s an interesting mathematical structure (‘interesting’ here largely a pragmatic issue), it matters little what the underlying logic is. His favourite example: smooth infinitesimal analysis which are classically inconsistent, and, furthermore, different from Bishop style constructive analysis (which is simply a sub-theory of classical real analysis) and intuitionistic analysis.
(And when Stewart had first opened the door to non-classicality, he didn’t shy away from opening for possibly interesting theories which require paraconsistent logics. Of course, as he pointed out, in that case ‘consistency entails existence’ ought to be ‘non-triviality entails existence’!)
In Stewart’s view, all of these intuitionistic structures are interesting in the sense that they not only underlie consistent mathematical theories, but in some cases they provide ‘better’ results than their classical counterparts when we consider pretheoretic intuitions about what’s being modelled. What does Stewart take to be the moral of this observation? Logical relativism: the issue of a One True Logic for which validity is a matter of truth-preservation in all structures is moot. Rather, Stewart is after a roughly naturalistic understanding of logic which allows for variations in what is correct reasoning depending on what counts as mathematically interesting structures. Clearly, this Hilbertian relativism has certain affinities with what is sometimes called pluralism, but Stewart prefers relativism so as not to confuse the position with the one recently advocated by JC Beall and Greg Restall.
The major part of the talk was spent trying to deal with a prima facie problem for his foundational view. Mathematicians standardly talk about natural numbers as if they are identical to corresponding objects in the set of integers (similarly for integers with respect to the set of rationals, etc). However, from a structural point of view this equation is unfortunate. As far as these elements live inside different structures, equating them is a misunderstanding; the appropriate strategy would be to relate them by looking categorically at relations between the structures; and CCAF offers precisely such a perspective.
Since I spent a number of hours discussing with Stewart Shapiro on our way to Uppsala, his talk didn’t contain many surprises. It did, however, engender one of the more lively discussions during the conference. Stewart’s training in classical model-theory doesn’t prevent him from enjoying the occasional excursion into intuitionistic terrain. More precisely, Stewart subscribes to something along the lines of the Hilbertian slogan: “Consistency entail existence”. If it’s an interesting mathematical structure (‘interesting’ here largely a pragmatic issue), it matters little what the underlying logic is. His favourite example: smooth infinitesimal analysis which are classically inconsistent, and, furthermore, different from Bishop style constructive analysis (which is simply a sub-theory of classical real analysis) and intuitionistic analysis.
(And when Stewart had first opened the door to non-classicality, he didn’t shy away from opening for possibly interesting theories which require paraconsistent logics. Of course, as he pointed out, in that case ‘consistency entails existence’ ought to be ‘non-triviality entails existence’!)
In Stewart’s view, all of these intuitionistic structures are interesting in the sense that they not only underlie consistent mathematical theories, but in some cases they provide ‘better’ results than their classical counterparts when we consider pretheoretic intuitions about what’s being modelled. What does Stewart take to be the moral of this observation? Logical relativism: the issue of a One True Logic for which validity is a matter of truth-preservation in all structures is moot. Rather, Stewart is after a roughly naturalistic understanding of logic which allows for variations in what is correct reasoning depending on what counts as mathematically interesting structures. Clearly, this Hilbertian relativism has certain affinities with what is sometimes called pluralism, but Stewart prefers relativism so as not to confuse the position with the one recently advocated by JC Beall and Greg Restall.
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