## Semantics for Physicists

I once complained that my student Brendan Fong said ‘semantics’ too much. You see, I’m in a math department, but he was actually in the computer science department at Oxford: I was his informal supervisor. Theoretical computer scientists love talking about syntax versus semantics—that is, written expressions versus what those expressions actually mean, or programs versus what those programs actually do. So Brendan was very comfortable with that distinction. But my other grad students, coming from a math department didn’t understand it… and he was mentioning it in practically ever other sentence.

In 1963, in his PhD thesis, Bill Lawvere figured out a way to talk about syntax versus semantics that even mathematicians—well, even category theorists—could understand. It’s called ‘functorial semantics’. The idea is that things you write are morphisms in some category $X,$ while their meanings are morphisms in some other category $Y.$ There’s a functor $F \colon X \to Y$ which sends things you write to their meanings. This functor sends syntax to semantics!

But physicists may not enjoy this idea unless they see it at work in physics. In physics, too, the distinction is important! But it takes a while to understand. I hope Prakash Panangaden’s talk at the start of the Simons Institute workshop on compositionality is helpful. Check it out:

### 6 Responses to Semantics for Physicists

1. Jon Awbrey says:

Been a while, but I seem to recall that compositionality is something you get with context-free languages, affording a mirroring of the object domain in the syntactic domain, but that you lose it as you pass to more general languages in the Chomsky–Schützenberger hierarchy.

2. arch1 says:

Thanks John! I hadn’t thought about it before but it seems the meaning of a ‘while’ statement can be described in terms of its parts (bexp and com) as the function w which maps each program state s to that program state f=w(s) in the sequence (com^n)(s) (n=0,1,…) for which bexp(f)=FALSE and n is minimal. (i.e. basically partitioning the bexp=TRUE states according to which FALSE state is first encountered on repeated iterations of com). I was left curious how this compares to the descriptions of ‘while’ in denotational semantics and in the apparently-somewhat-lamer operational semantics.

The talk also left me wondering about ways in which the challenge of applying denotational semantics / category theory to understanding physical systems from their pieces, might be different from doing this with programs (Prakash seemed unconcerned with the continuous/discrete distinction, which naively seems important).

3. joseph park says:

Thanks for this post! This is Joe (the one who emailed you about your Rosetta paper changing my life). The last person asked him about Tonti diagrams. Am I wrong in my perception that Tonti diagrams are surprisingly unpopular? If not, do have any why they are not more popular? Also, are you aware of any papers (or anything) that speak of Tonti diagrams in terms of category theory (or even speak of Tonti diagrams and category theory in the same paper)?

• John Baez says:

Hi! The last questioner was Morad Behandish, and he subsequently sent me a copy of Enzo Tonti’s book The Mathematical Structure of Classical and Relativistic Physics, which I’m currently perusing. You can learn more here:

• Wikipedia, Tonti diagram.

I’m not ready to say anything about this work.

4. Jack Morava says:

I’m indebted to Emily Riehl and Dominic Verity for informing me about Lawvere’s Hegelian Taco:

https://ncatlab.org/nlab/show/Hegelian+taco

See also E Morehouse, Burritos for the hungry mathematician,
http://sigbovik.org/2015/proceedings.pdf (p 57)

My own work (On the Grothendieck-Teichmüller Chimichanga) is soon to appear in the Cahiers de la Societe Philharmonique de Zanzibar.

• John Baez says:

Hi! You’re in a mood today. I’ve been trying to understand that ‘Hegelian taco’ on and off for years. I still don’t get the point of it. I think I roughly do understand the concept of ‘display categories’ that shows up earlier in that paper, but I lose the thread before I get to the taco.