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

Re:Physics, Topology, Logic and Computation: A Rosetta Stone... by baq on Friday August 19, 2016 @06:45AM (#12319120)

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.

[ Parent | on HN ]

Re:Physics, Topology, Logic and Computation: A Rosetta Stone... by torustic on Friday August 19, 2016 @09:38PM (#12324643)

> 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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Re:Physics, Topology, Logic and Computation: A Rosetta Stone... by torustic on Sunday August 21, 2016 @01:25PM (#12331746)

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...