How real are real numbers?

Greg Buchholz in the site's comments linked to Gregory Chaitin's "How real are real numbers?"[1] and asks, > Isn't it a little bit crazy to talk about computations on real numbers, when most of them are uncomputable/unnameable? [1] https://arxiv.org/pdf/math/0411418.pdf > Experimental physicists know how difficult accurate measurements are. No physical quantity has ever been measured with more than 15 or so digits of accuracy. Mathematicians, however, freely fantasize with infinite-precision real numbers. Nevertheless within pure math the notion of a real number is extremely problematic. We’ll compare and contrast two parallel historical episodes: > 1. the diagonal and probabilistic proofs that reals are uncountable, and > 2. the diagonal and probabilistic proofs that there are uncomputable reals. > Both case histories open chasms beneath the feet of mathematicians.

Lenore Blum has worked on some things in this direction. http://www.cs.cmu.edu/~lblum/PAPERS/TuringMeetsNewton.pdf I think most computer scientists have a natural and justified distrust of the real numbers, though, because when you get down to it they are too bizarre and impossible to actually be real. (i.e. the set of computable reals has measure 0, see Chaitin). https://arxiv.org/pdf/math/0411418.pdf

Ah, okay, so it boils down to the question whether real numbers are physically realizable. There is a debate about that in the philosophy of physics and mathematics. AFAIK, according to the current state of physics real numbers cannot be realized within a finite amount of space, at least Gisin [1] recently argued that way and I've seen other arguments for this view before [2,3,4,5:167-170]. I also remember some argument based on thermodynamics but cannot find the reference. [1] https://arxiv.org/pdf/1803.06824.pdf [2] https://arxiv.org/abs/math/0411418 [3] https://news.ycombinator.com/item?id=14080024 [4] https://ptolemy.berkeley.edu/~eal/pnerd/blog/are-real-number... [5] https://www.sciencedirect.com/science/article/pii/S030439750...

I think there's a legitimate question as to whether every real number is actually physically real. [1] It seems credible that our universe can only actually present us with a finite number of possible sensory experiences, in which case an event with probability 0 actually couldn't ever happen. That said, even if the universe can present you with an infinity of observations you can certainly only consider a finite number of classes of them, leading to the same outcome. [1] https://arxiv.org/abs/math/0411418

The author alludes to the fact that the overwhelming majority of reals have an infinite number of decimal places that constitute an algorithmically random sequence[1] Sure, plenty of real numbers have finite (0.1, 0.01, ...) or non-random (0.111..., 0.0111..., ...) sequences of decimal places, but there are only a countable infinity of these, whereas there is an uncountable infinity of the other, "random," kind. Usually we almost exclusively deal in the only-countably-infinite subset of the reals which are not random and it's counterintuitive that these comprise a tiny tiny archipelago of order in a huge engulfing ocean of randomness, but such is the case. Gregory Chaitin's "How real are real numbers?"[2] has some relatively straightforward further reading on this if you're interested. 1: https://en.wikipedia.org/wiki/Algorithmically_random_sequenc... 2: https://arxiv.org/abs/math/0411418

This was a light and interesting read. Here is the direct link: https://arxiv.org/pdf/math/0411418.pdf "Indeed, the most important thing in understanding a complex system is to understand how it processes information. This viewpoint regards physical systems as information processors, as performing computations. This approach also sheds new light on microscopic quantum systems, as is demonstrated in the highly developed field of quantum information and quantum computation. An extreme version of this doctrine would attempt to build the world entirely out of discrete digital information, out of 0 and 1 bits." It would indeed be quite a blast to discover we are to a high degree of probability in a simulation.

For anyone who might be misled, here are some links that people can use to judge for themselves how "crank" someone is: Professor Norman Wildberger's homepage: http://web.maths.unsw.edu.au/~norman/ ...a paper by Wildberger tackling some of the issues in this thread: Set Theory: Should You Believe? http://web.maths.unsw.edu.au/~norman/papers/SetTheory.pdf Some interesting works by Gregory Chaitin Meta Math! The Quest for Omega https://arxiv.org/abs/math/0404335 Exploring Randomness https://www.cs.auckland.ac.nz/~chaitin/ait/ How Real are Real Numbers? https://arxiv.org/abs/math/0411418 People who are interested in ultrafinitism would also want to check out Doron Zeilberger http://www.math.rutgers.edu/~zeilberg/ and Edward Nelson https://web.math.princeton.edu/~nelson/