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.
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.
> 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
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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.
In retrospect, my other comment was stupidly obtuse. Both too technical (in the sense of specificity) and too unstructured (in the sense of presentation order). A more appropriate path from CS might be analogous (well, inverse if anything) to the path Robert Goldblatt has taken. It dips into nonstandard analysis, but not totally without reason. Some subset of the following, with nLab and Wikipedia supplementing as necessary:
0. Milewski's "Category Theory for Programmers"[0]
1. Goldblatt's "Topoi"[1]
2. McLarty's "The Uses and Abuses of the History of Topos Theory"[2] (this does not require [1], it just undoes some historical assumptions made in [1] and, like everything else by McLarty, is extraordinarily well-written)
3. Goldblatt's "Lectures on the Hyperreals"[3]
4. Nelson's "Radically Elementary Probability Theory"[4]
5. Tao's "Ultraproducts as a Bridge Between Discrete and Continuous Analysis"[5]
6. Some canonical machine learning text, like Murphy[6] or Bishop[7]
From there you should see a variety of paths for mapping (things:Uncertainty) <-> (things:Structure). The Giry monad is just one of them, and would probably be understandable after reading Barr/Wells' "Toposes, Triples and Theories"[10].
The above list also assumes some comfort with integration. Particularly good books in line with this pedagogical path might be:
9. Any and all canonical intros to real analysis
10. Malliavin's "Integration and Probability"[11]
11. Segal/Kunze's "Integrals and Operators"[12]
Similarly, some normative focus on probability would be useful:
> It was then realized that the loose analogy between flow charts and Feynman diagrams could be made more precise and powerful with the aid of category theory
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.
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.
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/).
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...
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.
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.
> 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...
In retrospect, my other comment was stupidly obtuse. Both too technical (in the sense of specificity) and too unstructured (in the sense of presentation order). A more appropriate path from CS might be analogous (well, inverse if anything) to the path Robert Goldblatt has taken. It dips into nonstandard analysis, but not totally without reason. Some subset of the following, with nLab and Wikipedia supplementing as necessary:
0. Milewski's "Category Theory for Programmers"[0]
1. Goldblatt's "Topoi"[1]
2. McLarty's "The Uses and Abuses of the History of Topos Theory"[2] (this does not require [1], it just undoes some historical assumptions made in [1] and, like everything else by McLarty, is extraordinarily well-written)
3. Goldblatt's "Lectures on the Hyperreals"[3]
4. Nelson's "Radically Elementary Probability Theory"[4]
5. Tao's "Ultraproducts as a Bridge Between Discrete and Continuous Analysis"[5]
6. Some canonical machine learning text, like Murphy[6] or Bishop[7]
7. Koller/Friedman's "Probabilistic Graphical Models"[8]
8. Lawvere's "Taking Categories Seriously"[9]
From there you should see a variety of paths for mapping (things:Uncertainty) <-> (things:Structure). The Giry monad is just one of them, and would probably be understandable after reading Barr/Wells' "Toposes, Triples and Theories"[10].
The above list also assumes some comfort with integration. Particularly good books in line with this pedagogical path might be:
9. Any and all canonical intros to real analysis
10. Malliavin's "Integration and Probability"[11]
11. Segal/Kunze's "Integrals and Operators"[12]
Similarly, some normative focus on probability would be useful:
12. Jaynes' "Probability Theory"[13]
13. Pearl's "Causality"[14]
---
[0] https://bartoszmilewski.com/2014/10/28/category-theory-for-p...
[1] https://www.amazon.com/Topoi-Categorial-Analysis-Logic-Mathe...
[2] http://www.cwru.edu/artsci/phil/UsesandAbuses%20HistoryTopos...
[3] https://www.amazon.com/Lectures-Hyperreals-Introduction-Nons...
[4] https://web.math.princeton.edu/%7Enelson/books/rept.pdf
[5] https://www.youtube.com/watch?v=IS9fsr3yGLE
[6] https://www.amazon.com/Machine-Learning-Probabilistic-Perspe...
[7] https://www.amazon.com/Pattern-Recognition-Learning-Informat...
[8] https://www.amazon.com/Probabilistic-Graphical-Models-Princi...
[9] http://www.emis.de/journals/TAC/reprints/articles/8/tr8.pdf
[10] http://www.tac.mta.ca/tac/reprints/articles/12/tr12.pdf
[11] https://www.springer.com/us/book/9780387944098
[12] https://www.amazon.com/Integrals-Operators-Grundlehren-mathe...
[13] http://www.med.mcgill.ca/epidemiology/hanley/bios601/Gaussia...
[14] https://www.amazon.com/Causality-Reasoning-Inference-Judea-P...
If you like this Propositions As Types by Wadler is also an excellent read. In fact, the author cites PTLC: Rosetta.
http://m.cacm.acm.org/magazines/2015/12/194626-propositions-...
> It was then realized that the loose analogy between flow charts and Feynman diagrams could be made more precise and powerful with the aid of category theory
This is fascinating!
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.