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arXiv:0903.0340quant-ph

Physics, Topology, Logic and Computation: A Rosetta Stone

John C. Baez, Mike Stay

Abstract

In physics, Feynman diagrams are used to reason about quantum processes. In the 1980s, it became clear that underlying these diagrams is a powerful analogy between quantum physics and topology: namely, a linear operator behaves very much like a "cobordism". Similar diagrams can be used to reason about logic, where they represent proofs, and computation, where they represent programs. With the rise of interest in quantum cryptography and quantum computation, it became clear that there is extensive network of analogies between physics, topology, logic and computation. In this expository paper, we make some of these analogies precise using the concept of "closed symmetric monoidal category". We assume no prior knowledge of category theory, proof theory or computer science.

arXiv comments
73 pages, 8 encapsulated postscript figures
Journal reference
Physics, topology, logic and compIn New Structures for Physics, ed. Bob Coecke, Lecture Notes in Physics vol. 813, Springer, Berlin, 2011, pp. 95-174
DOI
doi:10.1007/978-3-642-12821-9_2
Subjects
quant-phQuantum Physics math.CTCategory Theory

On Hacker News

Linked by 12 identifiable commenters in 14 comments across 14 threads, from Jun 2015 to Oct 2026. Found in 9 HN submissions from Oct 2009 to Apr 2026. Open submission discussions.

Comments linking this paper, per year.

Discussed in comments14 threads

  1. How to Formalize All of Mathematics 1 comment

    • Thanks for further recommendations. Programming Language Foundations in Agda. The Diamond Age by Neal Stephenson. I made a note to look into these. Oh I see one of the authors of PLFA is Philip Wadler, whose name I recognize from Haskell and various papers on programming…

      lioetersView on HNParent context

  2. Is Math Big or Small? 1 comment

    • Even before I started the video, I had a feeling it was going to lead to a kind of "introspective" mathematics that can reason about its own reasoning. I was not disappointed, thank you. Physics, Topology, Logic and Computation: A Rosetta Stone - https://arxiv.org/abs/0903.0340

      lioetersView on HNParent context

  3. Vine: A programming language based on Interaction Nets 1 comment

    • I like Inverse types! While it's not novel idea, this seems like a very concise implementation of it. Inverse types are also known as negative types (https://legacy.cs.indiana.edu/~sabry/papers/rational.pdf) or holes. This concept is what needed to make a concrete category for…

      sw1shView on HNParent context

  4. Why mathematicians study knots 1 comment
  5. Category theory is a universal modeling language 1 comment

    • See e.g. Baez & Stay 2009, "Physics, Topology, Logic, and Computation: A Rosetta Stone". [1] [1] https://arxiv.org/abs/0903.0340

      benrbrayView on HNParent context

  6. Programming with Categories 1 comment

    • Wow, what a combo! I'd be interested in anything from each of the lecturers, but all 3 at the same course, is amazing. Category Theory could be the rosetta stone of many things(this is relevant https://arxiv.org/pdf/0903.0340.pdf).

      random3View on HNParent context

  7. Mirth – a type-safe concatenative purely functional programming language 1 comment

    • See https://arxiv.org/abs/0903.0340 for a basic introduction to the issues, and https://arxiv.org/abs/0908.3347 for a slightly more extensive treatment, also mentioning proof nets. This does not explicitly say that these diagrams can be understood as the equivalent to sequences…

      0815testView on HNParent context

  8. The Mathematics of Quantum Mechanics [pdf] 1 comment

    • This line of thought is very interesting indeed. Understanding the similarity/contrast to the join operator is really at the heart of what quantum is about. It comes down to, what do you mean by "and" ? Check out the breezy writing in this paper: https://arxiv.org/abs/1602.07618…

      mathgeniusView on HNParent context

  9. Logic is Metaphysics (2011) [pdf] 1 comment

    • What I'm saying is that linear logic is exactly a technical exploration directly pertinent to the questions Quine and Dummett considered. And who cares if the paper is specifically about Quine and Dummett? What's the point of resuscitating such a dialogue without informing it of…

      danharajView on HNParent context

  10. Applied Category Theory 1 comment
  11. What is backpropagation and what is it doing? [video] 1 comment

    • Isn't that what category theory is about, on the meta level, and in the result in case of specific isomorphisms, too? edit: at that I still have John C. Baez, Mike Stay - "Physics, Topology, Logic and Computation: A Rosetta Stone" on my reading list…

      posterboyView on HNParent context

  12. Basic Category Theory 1 comment
  13. The Holy Trinity: Logic, Languages, Categories (2011) 1 comment

    • An interesting published review of this relationship is "Physics, Topology, Logic and Computation: A Rosetta Stone", by John C. Baez, Mike Stay https://arxiv.org/abs/0903.0340. The first author alos has lots of interesting stuff related to (quantum) computation on his website.

      orbifoldView on HNParent context

  14. Category Theory for Scientists and Engineers (2013) [pdf] 1 comment

    • > So it would be useful to have category textbooks for CS folks which spent less time on topology, and more time on familiar categories. Agreed, and I should have been clearer in my post above that I was just airing some thoughts on category theory rather than meaning to imply…

      JadeNBView on HNParent context

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