[FOM] How real are real numbers?

[Joe Shipman]

Here’s a short summary of attempts to make non-absolute mathematics physically relevant since Banach-Tarski convinced people that it wasn’t:

1) Pitowsky used AC and CH to create an anti-Fubini set whose iterated integrals did not commute in order to give a local realistic model of Quantum Mechanics that resolves the EPR and Bell puzzles at the cost of assuming infinite precision could be physically meaningful
2) Freiling argued in the other direction against CH by formulating axioms of symmetry related to infinitely precise selection of a point in a surface perfectly coordinatized by [0,1]^2
3) I showed in my thesis that it was consistent, and also followed from RVM, that Pitowsky-style anti-Fubini sets did not exist, and clarified the related independence phenomena to some extent, and later argued that RVM or Fubini axioms were attractive alternatives to some other proposed extensions of ZFC
4) Ben-David et al propose that certain machine learning tasks are connected to CH, but are criticized for using models that unrealistically suppose that a machine can be “given” an infinite amount of information in finite time. (This reminds me of an old paper by Adleman and Blum which formulated inductive inference tasks with Turing degrees intermediate between 0 and 0’ — a different kind of learning-related unsolvability which also depended on finitely-unverifiable definitions).

I think we need a metatheorem that certain types of mathematical questions cannot be relevant to physical experimental results if the ultimate physical theories are anything like today’s, and are therefore unknowable simply by resorting to experiment, or else an extension of the Church-Turing thesis that covers phenomena not precisely formalized by the current versions.


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