How real are real numbers? · HackerPapers discussions

How real are real numbers? — mentioned by cwillu

Tue, 28 Jul 2026 05:32:06 GMT

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

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How real are real numbers? — HN submission

Mon, 27 Jul 2026 15:40:17 GMT

How real are real numbers? (2004)

96 points; 103 discussion replies on HN. Paper and source history. A submission is not an endorsement.

How real are real numbers? — HN submission

Wed, 17 Jun 2026 14:32:12 GMT

How real are real numbers? (2004)

3 points; 0 discussion replies on HN. Paper and source history. A submission is not an endorsement.

How real are real numbers? — mentioned by akssri

Sat, 30 Aug 2025 11:49:32 GMT

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 were reflected a half-century later by Turing. https://arxiv.org/abs/math/0411418 Except of-course, while "hyper-Turing" machines that can do magic "post-Turing" "post-Halting" computation are seen as absurd fictions, real-numbers are seen as "normal" and "obvious" and "common-sensical"! It was amusing sometime back to…

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How real are real numbers? — HN submission

Fri, 27 Sep 2024 13:41:23 GMT

How real are real numbers? (2004)

2 points; 0 discussion replies on HN. Paper and source history. A submission is not an endorsement.

How real are real numbers? — HN submission

Fri, 08 Mar 2024 06:59:20 GMT

How real are real numbers? (2004) [pdf]

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How real are real numbers? — mentioned by dia80

Mon, 23 Jan 2023 15:31:37 GMT

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

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How real are real numbers? — mentioned by yamrzou

Wed, 07 Dec 2022 02:14:51 GMT

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 Real? - https://www.quantamagazine.org/the-infinity-puzzle-20160616 - Solution: https://www.quantamagazine.org/the-infinity-puzzle-solution-... Do infinities exist in nature? - https://plus.maths.org/content/do-infinities-exist-nature-0 Does infinity exist? - http://plus.maths.org/content/does-infinity-exist…

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How real are real numbers? — mentioned by matt123456789

Wed, 22 Dec 2021 02:42:39 GMT

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

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How real are real numbers? — mentioned by rstuart4133

Sat, 08 Aug 2020 00:46:39 GMT

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, and for the curious, the reason Godel's theorem is so hard to understand is he needed a computer programming language for his proof but there were none lying around back then, so he invented his own. I strongly suspect he wasn't thinking of it as a series of executable steps, let alone a programming language whose most…

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How real are real numbers? — HN submission

Sun, 02 Aug 2020 16:53:23 GMT

How real are real numbers? (2004)

81 points; 112 discussion replies on HN. Paper and source history. A submission is not an endorsement.

How real are real numbers? — mentioned by carapace

Sat, 11 Jan 2020 21:35:58 GMT

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.…

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How real are real numbers? — mentioned by currymj

Sun, 26 May 2019 03:48:25 GMT

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

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How real are real numbers? — HN submission

Sun, 10 Feb 2019 04:16:59 GMT

How real are real numbers?

3 points; 0 discussion replies on HN. Paper and source history. A submission is not an endorsement.

How real are real numbers? — mentioned by jonathanstrange

Fri, 19 Oct 2018 13:11:58 GMT

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]…

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How real are real numbers? — mentioned by FeepingCreature

Sun, 12 Aug 2018 18:47:38 GMT

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

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How real are real numbers? — mentioned by ttctciyf

Tue, 12 Jun 2018 15:40:51 GMT

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…

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How real are real numbers? — mentioned by jfaucett

Mon, 10 Apr 2017 18:02:32 GMT

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…

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How real are real numbers? — HN submission

Mon, 10 Apr 2017 16:40:09 GMT

How real are real numbers? (2004)

218 points; 265 discussion replies on HN. Paper and source history. A submission is not an endorsement.

How real are real numbers? — mentioned by GregBuchholz

Tue, 15 Nov 2016 19:06:06 GMT

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…

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