How real are real numbers?

https://ar5iv.labs.arxiv.org/html/math/0411418 for the in-browser html rendering

cwilluJul 28, 2026View on HNParent context

Theory of Computation wasn't around when all this "exciting" stuff was developed in Mathematics. Given their non-constructive nature "real" numbers are unsurprisingly totally incompatible with computation. Chaitin has a great paper on this and shows how Cantor's constructions…

akssriAug 30, 2025View on HNParent context

You are going to love this: How real are real numbers? https://arxiv.org/abs/math/0411418

dia80Jan 23, 2023View on HNParent context

Thanks for sharing. I'm going to add here some links that I've been curating regarding the "existence" of infinity: The physics of infinity - https://www.nature.com/articles/s41567-018-0238-1.epdf?share... - Discussion: https://news.ycombinator.com/item?id=17599087 Is infinity…

yamrzouDec 7, 2022View on HNParent context

Most reals are not computable anyway: https://arxiv.org/pdf/math/0411418.pdf

matt123456789Dec 22, 2021View on HNParent context

Yes, that Scott Aaronson paper was an eye opener for me. It wasn't because of the correspondence between Godel's theorems and the halting problem, although for a computer programmer Aaronson's formulation turns Godel's inscrutable theorem into something easily accessible. (Oh…

rstuart4133Aug 8, 2020View on HNParent context

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 >…

carapaceJan 11, 2020View on HNParent context

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…

currymjMay 26, 2019View on HNParent context

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…

jonathanstrangeOct 19, 2018View on HNParent context

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…

FeepingCreatureAug 12, 2018View on HNParent context

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…

ttctciyfJun 12, 2018View on HNParent context

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…

jfaucettApr 10, 2017View on HNParent context

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…

GregBuchholzNov 15, 2016View on HNParent context