Foundations of Physics and Mathematics

There's obviously some overlap in these fields, because physics depends upon mathematics and so you would think that anyone in one of these fields would be at least cognizant of the other, but are they? There are plenty of people who claim to have answers and they often involve computation. But why? Roger Penrose wrote his book The Emperor's New Mind: Concerning Computers, Minds and The Laws of Physics for these people, I think! In it he touches on the foundations of Mathematics too, but in a somewhat cursory way: his discussion of intuitionistic logic in that book is almost dismissive. I think it is the failure of mainstream physicists and logicians to relate these inter-dependent foundations that makes room for ideas of computation as somehow foundational in itself, rather than as a framework for effective reasoning.

Here's a rather good survey of the idea of inertial frames in physics by Robert DiSalle, who is a Philosopher at UWO: Stanford Encyclopedia of Philosophy - Space and Time: Inertial Frames. I heard about this idea of Ludwig Lange's in a recent interview with Julian Barbour. See his 1982 paper Relational Concepts of Space and Time.

 

In this interview he doesn't get much chance to say what he means by Variety. It's better to read the above paper first, but more or less it's this: Variety is macroscopic phenomena, differentiated individuated etc. The sort of thing you need if you are setting up boundary conditions in matter or energy. Uniformity is the matter or energy itself.

43:42 Here is Barbour's scale-invariant dimensionless quantity he calls Complexity, described entirely in terms of the relative distances between points. The first one neglects masses. The second is close to Newton's gravitational potential multiplied by the size of the system. See Nerdearla - Entrevista con Ken Thompson for a similar-sounding model of space travel that he and Dennis Ritchie programmed in the sixties.

1:15:50 On thermodynamics and the Second Law, see Jade on Ising Models.

See Julian Barbour's home page: http://www.platonia.com.

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The Greeks had this idea of computation as geometry:

We have this idea of Computation as proof. See Emily Riehl on Univalent Foundations:

See https://karagila.org/

I've just discovered that two months before he died Mike Gordon wrote this blog post: Corecursion and coinduction: what they are and how they relate to recursion and induction. In it he mentions this excellent 1997 tutorial An introduction to (co)algebra and (co)induction, by Bart Jacobs and Jan Rutten, published originally in EATCS Bulletin Issue 62. But the paper is quite hard to find now, since it was published as part of a book by CUP. There is still a low-resolution copy available from Bart Jacob's web site: http://www.cs.ru.nl/~bart/PAPERS/JR.pdf. He also mentions Milner and Tofte's Co-induction and Relational Semantics, published in Theoretical Computer Science 87 (1991) 209-220 and Sangiorgi's On the Origins of Bisimulation and Coinduction.

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I wrote this soon after I arrived in Bolivia. I forget the exact date, but I seem to recall it was some time after April 2010. In it I said "knowledge is in fact nothing more than the coherence of the different representations of it. "

Here's Carlo Rovelli interviewed by Curt Jaimungal on 17 Dec 2021

Here is another interview he did with Rovelli, from about a month ago, talking about Loop Quantum Gravity.

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Norman Wildberger is one of those people who do take seriously the connection between foundations of mathematics and physics.

 

See his 2009 tutorial Universal Hyperbolic Geometry. This approach is close to the one Julian Barbour talks about (around 10:40) when he refers to Lange's definition of inertial frames. See also https://www.openlearning.com/wildegg/ where Norman says he will be running a course on Universal Hyperbolic Geometry soon.