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January 26, 2009

Abstract Stone Duality

Posted by David Corfield

Paul Taylor, perhaps best known to readers for his Practical Foundations of Mathematics and for his macros, has a survey paper out – Foundations for Computable Topology. This provides an introduction to a programme he’s been working on for several years, which offer us

… a purely recursive theory of elementary real analysis in which, unlike in previous approaches, the real closed interval [0,1][0, 1] is compact.

Judging by the following description of the programme, there ought to be many points of common interest with Café visitors:

The new theory axiomatises continuity directly. It is based on ideas concerning the duality of algebra and topology that were introduced by Marshall Stone. The resulting theory rests on computable rather than set-theoretical foundations.

Instead of set theory, it relies on an abstract formulation of Stone duality (hence the name ASD) using a monadic adjunction, along with an algebraic equation that characterises the way in which the Sierpinski space uniquely classifies open subspaces.

Philosophy tends to raise its head when people are rethinking basic concepts.

The philosophical thesis of this paper is that we can make Logic the servant of (a particular discipline in) Mathematics, and employ logical methods as tools. Complex numbers, and the differential calculus that lies behind Méchanique Céleste, came from mathematics and not logic, but they are now fully incorporated into our notation. We argue that modern mathematical ideas such as those from algebra and topology can similarly be built in to new language, but that logical methods are needed to carry this out.

These ideas are explored in an interesting methodological section – 3. Method and critique, which includes the complaint against funding agencies:

Nowadays, one is asked to give advance notice of all of the theorems that one intends to prove. Such planning may be possible when building a house, but it can be done if and only if there are no original ideas. A mathematician with a plan for a theorem wants to carry it out straight away, and the only pieces of equipment that are needed are a clear head and a clear blackboard. We don’t put our lives at stake as Columbus did when we embark on scientific experiments or try to prove mathematical theorems, but if there is no intellectual risk of failure in a proposed piece of research, then it is redundant, and probably not worthy of funding.

Sounds like Taylor might make common cause with Ronnie Brown.

Posted at January 26, 2009 9:55 AM UTC

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Re: Abstract Stone Duality

No one shall expel us from the Paradise that Cantor has created.

It is impressive how well he articulates the problem of searching for the ‘right’ axioms. Thanks for the link: I am enjoying it immensely, although he meanders about a bit too much.

Posted by: Kea on January 27, 2009 12:40 AM | Permalink | Reply to this

Re: Abstract Stone Duality

Yes!, I love Taylor's work, although (for the reason Kea notes) it can be hard to read until you get used to the ideas. (But once you're used to them, I find it very easy to read; I can't see how else one would want it written.)

I'm starting to wonder how much of mathematics in general one can do using ASD as the only foundation. This should be a handicap, as ASD is explicitly intended as a foundation for topology and not for mathematics as such. And yet, it can be unexpectedly (to me) expressive.

(A related question that I've also been wondering about: How much can be done internal to a pretopos whose every polynomial endofunctor has an initial algebra and has a final coalgebra? Again, a surprising amount. Yet neither of these can fully express logical negation!)

Posted by: Toby Bartels on February 6, 2009 8:35 PM | Permalink | Reply to this

Re: Abstract Stone Duality

My paper “A lambda calculus for real analysis”, which is the introduction to ASD for the general mathematicianm, has now appeared in the Journal of Logic and Analysis (vol 2, no 5).
It is still available from my webpage too using this link under my name below.

David’s posting is actually about the paper “Foundations for computable topology”, which I hope will appear in a book edited by Giovanni Sommaruga soon.

Posted by: Paul Taylor on August 21, 2010 11:42 AM | Permalink | Reply to this

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