Audio note: this article contains 134 uses of latex notation, so the narration may be difficult to follow. There's a link to the original text in the episode description.

In a recent paper in Annals of Mathematics and Philosophy, Fields medalist Timothy Gowers asks why mathematicians sometimes believe that unproved statements are likely to be true. For example, it is unknown whether <span>_pi_</span> is a normal number (which, roughly speaking, means that every digit appears in <span>_pi_</span> with equal frequency), yet this is widely believed. Gowers proposes that there is no sign of any reason for <span>_pi_</span> to be non-normal -- especially not one that would fail to reveal itself in the first million digits -- and in the absence of any such reason, any deviation from normality would be an outrageous coincidence. Thus, the likely normality of <span>_pi_</span> is inferred from the following general principle:

No-coincidence [...]

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Outline:

(02:32) Our no-coincidence conjecture

(05:37) How we came up with the statement

(08:31) Thoughts for theoretical computer scientists

(10:27) Why we care

The original text contained 12 footnotes which were omitted from this narration.

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First published:
February 14th, 2025

Source:
https://www.lesswrong.com/posts/Xt9r4SNNuYxW83tmo/a-computational-no-coincidence-principle

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Narrated by TYPE III AUDIO.