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charset="Windows-1252" Content-Transfer-Encoding: quoted-printable Can real analysis be done point-free, for locales? That would be daunting t= o attempt in terms of frames and frame homomorphisms, but it has long been = understood that an alternative is to use points and reason geometrically so= that one is not just working with the global points (which may be insuffic= ient). My student Ming Ng developed real exponentiation and logarithms in this sty= le. I have now arXived a paper (arXiv:2312.05228) that shows how to differe= ntiate them. It relies on first proving the Fundamental Theorem of Calculus= , after which the development is more or less conventional. The style is quite natural, and reminiscent of Bishop=92s constructive reas= oning, though with a different mathematical trajectory. However, it is stil= l an open question whether it is truly constructive in the sense of extract= ing algorithms from proofs. I have some notes on this, formatted as slides,= at https://www.cs.bham.ac.uk/~sjv/GeologConsSlides.pdf Steve Vickers. You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message. View group files | Leave group | = Learn more about Microsoft 365 Groups --_000_LNXP265MB10499B45897C1F42ED7EFFB7958FALNXP265MB1049GBRP_ Content-Type: text/html; charset="Windows-1252" Content-Transfer-Encoding: quoted-printable

= Can real analysis be done point-free= , for locales? That would be daunting to attempt in terms of frames and frame homomorphisms, but it has long bee= n understood that an alternative is to use points and reason geometrically = so that one is not just working with the global points (which may be insuff= icient).


My student Ming Ng developed real ex= ponentiation and logarithms in this style. I have now arXived a paper (arXi= v:2312.05228) that shows how to differentiate them. It relies on first proving the Fundamental Theorem of Calculus, afte= r which the development is more or less conventional.


The style is quite natural, and reminiscent of Bishop= =92s constructive reasoning, though with a different mathematical trajectory. However, it is still an open question= whether it is truly constructive in the sense of extracting algorithms fro= m proofs. I have some notes on this, formatted as slides, at


  https://www.cs.bham.ac.uk/~sjv= /GeologConsSlides.pdf


Steve Vickers.

 
 
You're receiving this message because you're a member of the Categories mai= ling list group from Macquarie University. To take part in this conversatio= n, reply all to this message.
 
View group files   |   Leav= e group   |&n= bsp;  Learn more about Microsoft 365 Groups
 
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