Physics, Topology, Logic and Computation: A Rosetta Stone

by John C. Baez, Mike Stay

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’: a manifold representing spacetime, going between two manifolds representing space. This led to a burst of work on topological quantum field theory and ‘quantum topology’. But this was just the beginning: 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.

Genres: Mathematics, Physics, Computer Science, Science

73 pages, ebook

First published March 2, 2009

About the author

http://en.wikipedia.org/wiki/John_Baez

JOHN BAEZ is a mathematical physicist working on quantum gravity using the techniques of "higher-dimensional algebra". A professor of mathematics at the University of California, Riverside, he enjoys answering physics questions on the usenet newsgroup sci.physics.research, and also writes a regular column entitled "This Week's Finds in Mathematical Physics". (The Third Culture: http://www.edge.org/3rd_culture/bios/... )

Reviews

[Mark Moon · Rating 5 out of 5]

This is probably one of the most important texts I've ever read. It was my introduction to category theory, and the case it makes for cat theory (as I now call it) being important to the future of computer science, pure math, and especially physics, is incredibly compelling.

Recommended to anyone who's into consilience and isn't afraid of a little abstraction. No significant background in any of the relevant fields is assumed or required. If you've been looking for a way to start thinking about these kinds of fields, this is a great place to start; if you've already done a lot of thinking about any or all of these fields, this is a great place to see where to go next.

[Wigners Friend · Rating 5 out of 5]

In this beautiful piece John Baez is like King Midas, physics, logic, computer science, topology all he touches becomes category theory, beautiful like gold, but some will say as useless. It's a great work of science commutation as communication between sciences as it gives like it title says "A Rosetta Stone" to speak a common language: Category theory.
For me it was a nice introduction to some topics I wanted to learn about for a long time.
It has the appeal of a popular science book to make a big picture of different "cool" sounding subjects, without having its problem of being informal and associative.
A great first introduction to category theory and insight into many disciplines and a motivation to further study.

P.S. Is it a shame that CS Peirce didn't witness the rise of category theory since this mixture of studying abstraction by relations and drawing cool diagrams would be his thing.

[Adriano · Rating 5 out of 5]

This is actually a free paper: https://arxiv.org/abs/0903.0340 which I would wish would turn into a book with more details, and more contributors, in the style of the open IAS project by Vladimir Voevodsky et al. for Homotopy Type Theory: Univalent Foundations of Mathematics https://homotopytypetheory.org/book/ -- yielding a formal road map for deductive reasoning. (The future challenge is the formal generalization of inductive reasoning, say by, probabilistic structure over graphs.)