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I added a new section to Bayesian reasoning, Exchangeability, which outlines the de Finetti Representation theorem. As indicated, there’s a multivariate version. This was used to talk about Bose-Einstein statistics.
I wonder if anything interesting would happen with a HoTT rendition of statistical meachanics.
I don’t know whether there’s anything interesting for math or physics, but I think that there are interesting things to be said regarding the philosophical literature surrounding particle individuality in classical and quantum contexts. I’ve been starting to think about this recently, mostly inspired by learning about species in HoTT, and the discussion at your n-café post on the covariance of coloured balls. I think that it would be helpful in getting philosophers (of physics, anyway) on board with higher identities to link them up with Simon Saunders’ work on Leibniz’s identity principles. I think that philosophers of physics might be convinced when they see how useful higher identities are in treating what they call “weak discernibility”, following Saunders, and how this is related to the treatment of counting via groupoid cardinalities.
Thinking of species and your mention of Bose-Einstein statistics reminds me of a puzzling quote from Connes that John Baez has mentioned a number of times (e.g., here). I don’t really have anything to contribute to solving that puzzle, I’m just resurrecting it because I’ve been running into it a lot lately.
I wonder how far reflecting on indiscernibility of particles takes us. Maybe HoTT thinking can counteract the tendency to rely on classical, set theoretic modes of thought. Then again, there’s the danger in overlooking the achievement of even extracting a particle account from a QFT, as we’re being warned at particle.
Perhaps better to bring species into the game by working out the connections between: Fock space, groupoidification, geometric function theory, dependent linear type theory, tangent infinity-topos, Goodwillie calculus, …, and maybe stuff types meets categorified Heisenberg algebra.
Perhaps better to bring species into the game by working out the connections between: Fock space, groupoidification, geometric function theory, dependent linear type theory, tangent infinity-topos, Goodwillie calculus, …, and maybe stuff types meets categorified Heisenberg algebra.
The royal road to general abstract accounts of particle statistics should be via the exponential modality, which interconnects at least five of the keywords that you mention.
I was going to point out that a chunk of the relations are written out in section 5.5. of dcct, including
5.5.3 Exponential modality, Linear spaces of states and Fock space.
I came across this old comment (well only 16 months old):
So we have a refined kind of logic – linear homotopy-type theory – which describes quantum field theoretic processes in their probabilistic nature and is at the same time such that making propositions about these QFT systems means “wave function collapse”. All this from the logical substrate, nothing put in “by hand”. Seems to be rather beautiful to me. (But I guess this point deserves to be further elaborated on eventually…)
There would be a good project for philosophers to get involved in.
There would be a good project for philosophers to get involved in.
If you see an opportunity to make a philosopher colleague of yours interested in this, that might be worthwhile. My impression is that there is much to be gained here.
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