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Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

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Re: Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

#2

Post by mrcactu5 »

Homotopy Type Theory by Vladimir Voevodsky is another possibility. This is an attempt to link Topology and Computer Science

https://homotopytypetheory.org/book/

Back in the day there was Feynman's Lectures on Computation. Hint: pdf can be found by searching

https://www.amazon.com/Feynman-Lectures-Computation-Richard-...

See also nLab

https://ncatlab.org/nlab/show/higher+category+theory

one should never forget Jacob Lurie's "Higher Topos Theory" which is 1000 pages just like that

http://www.math.harvard.edu/~lurie/papers/croppedtopoi.pdf

Actually I recommend against readin it as it only covers 2 of the 4 topics you discuss (Topology and Logic). However it certainly has applications to the other two.

Re: Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

#4

Post by fmap »

mrcactu5 wrote: post #2

Homotopy Type Theory by Vladimir Voevodsky is another possibility. This is an attempt to link Topology and Computer Science https://homotopytypetheory.org/book/ Back in the day there was Feynman's Lectures on Computation. Hint: pdf can be found by searching https://www.amazon.com/Feynman-Lectures-Computation-Richard-... See also nLab https://ncatlab.org/nlab/show/higher+category+theory one should never forget Jacob L…

The infinity groupoid models of type theory have already revolutionized our understanding of equality in type theory. So far, the 21st century has been an incredible time for logicians.

There are also older, and very different topological models for typed lambda calculi (see e.g. http://www.cs.bham.ac.uk/~mhe/papers/entcs87.pdf). These motivate things like Escardo's "seemingly impossible functional programs" (http://math.andrej.com/2007/09/28/seemingly-impossible-funct...) and, along different lines, Abstract Stone Duality (http://www.paultaylor.eu/ASD/).

Re: Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

#5

Post by baq »

excerpt:

> At present, the deductive systems in mathematical logic look like hieroglyphs to most physicists. Similarly, quantum field theory is Greek to most computer scientists, and so on.

   Category Theory    Physics    Topology    Logic          Computation
   --------------------------------------------------------------------
   object             system     manifold    proposition    data type
   morphism           process    cobordism   proof          program
yup, as a computer scientist by education, sounds about right - Greek and hieroglyphs (and those are just names!)

while we're at it, we need an update with statistics, data science and machine learning.

Re: Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

#7

Post by erik998 »

Also checkout Steve Awodey's Category Theory book.

https://global.oup.com/academic/product/category-theory-9780...

On a personal note, I remember Awodey from my Senior Thesis seminar. He was very amicable and affable. During my presentation, I remember having to demonstrate Cantor's diagonalisation argument. Luckily for me it was one of the things I spent a good amount of time studying. Had a great time there.

Re: Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

#8

Post by Koshkin »

erik998 wrote: post #7

Also checkout Steve Awodey's Category Theory book. https://global.oup.com/academic/product/category-theory-9780... On a personal note, I remember Awodey from my Senior Thesis seminar. He was very amicable and affable. During my presentation, I remember having to demonstrate Cantor's diagonalisation argument. Luckily for me it was one of the things I spent a good amount of time studying. Had a great time there.

Also, a book by B.Pierce [ISBN 9780262660716] is a very good short introduction to Category Theory for programmers.

Re: Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

#9

Post by torustic »

mrcactu5 wrote: post #2

Homotopy Type Theory by Vladimir Voevodsky is another possibility. This is an attempt to link Topology and Computer Science https://homotopytypetheory.org/book/ Back in the day there was Feynman's Lectures on Computation. Hint: pdf can be found by searching https://www.amazon.com/Feynman-Lectures-Computation-Richard-... See also nLab https://ncatlab.org/nlab/show/higher+category+theory one should never forget Jacob L…

Lurie's "On Infinity Topoi"[0] preceded the book and is, I quote[1],

> 50 pages of pure cake, beautifully and informally written. The book is 1000 pages long. There is some new cake there, but not 20 times as much.

[0] https://arxiv.org/abs/math/0306109

[1] https://mathematicswithoutapologies.wordpress.com/2015/05/13...

Re: Physics, Topology, Logic and Computation: A Rosetta Stone (2009) [pdf]

#10

Post by torustic »

baq wrote: post #5

excerpt: > At present, the deductive systems in mathematical logic look like hieroglyphs to most physicists. Similarly, quantum field theory is Greek to most computer scientists, and so on. Category Theory Physics Topology Logic Computation -------------------------------------------------------------------- object system manifold proposition data type morphism process cobordism proof program yup, as a computer scien…

> while we're at it, we need an update with statistics, data science and machine learning.

See this post and its comments[0] to get some foundations upon which to think about these correspondences.

[0] https://golem.ph.utexas.edu/category/2014/10/where_do_probab...

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