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In this video njwildberger argues that the idea that we can uniquely factor any integer into primes is problematic. His central argument appears to involve skepticism that we can say anything meaningful about very large numbers even if we can define them in some sense.

It seems crazy to doubt such a core part of mathematics. But is there some self-consistent world where his argument about very large numbers makes sense? Or is there another flaw in his reasoning?

matt_black
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  • On the normal number line you can prove that there is no smallest positive integer with multiple prime factorization, hence all positive integers greater than 1 factor uniquely into primes. https://en.wikipedia.org/wiki/Fundamental_theorem_of_arithmetic. i'm not sure what this would be for a-adic numbers though for a non prime which are arguably larger then any integer. – shai horowitz Oct 28 '16 at 21:45
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    Is this some strategy to raise up the views of that video? –  Oct 28 '16 at 21:45
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    It's not crazy, and formally it's called Ultrafinitism: https://en.wikipedia.org/wiki/Ultrafinitism – Alex R. Oct 28 '16 at 21:46
  • To gain credence, Njwildberger should submit his paper to a mathematical journal. Any journal editor would be proud to publish a result even more momentous than Gödel's famous result. – John Bentin Oct 28 '16 at 21:47
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    His argument sounds equivalent to the rabbits in watership down that they can't count larger than four, so any amount larger than four is simply referred to as "hrair." I am perfectly content to talk about numbers larger than four even if rabbits aren't able to conceive them. I am also perfectly content to talk about numbers larger than ten despite not having enough fingers to count them with (without using more intricate techniques). Indeed, I am content to talk about numbers larger than whatever limit he thinks is in place. "Dark numbers" being impractical doesn't imply nonexistence. – JMoravitz Oct 28 '16 at 22:00
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    @mathbeing Nope. I came across it randomly and wondered if any of his logic could actually hold up. I don't have enough mathematics to judge myself so I thought I'd ask people who did. – matt_black Oct 28 '16 at 22:01
  • It might be of interest that there are extensions of the integers in which uniqueness of prime factorization fails. For this previous Question Bill Dubuque gives a brief account and some references. – hardmath Oct 28 '16 at 22:02
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    @JohnBentin um, he's not a quack, and he's published quite a lot in legitimate journals and is an associate professor at the university of new south wales. He is an ultrafinitist. He doesn't say the fundamental theorem is wrong. He says it is problematic relying a pure constructivism. Ultrafinitism is a very minor view point but it isn't tin foil hat territory. (It should be maybe... but it isn't.) – fleablood Oct 28 '16 at 22:17
  • @fleablood: OK. He's a real mathematician. It's almost trivial that, if you can't say anything general about numbers bigger than 4, or a googoloplex, or whatever (I don't know exactly where ultrafinitists draw the line), then most theorems of mathematics, unqualified by this arbitrary bound, are unprovable. It's not so much crazy as just plain boring. – John Bentin Oct 29 '16 at 13:31
  • @JohnBentin heh, heh. I won't argue with that. What's the fun of mathematics if its not speculative. And I have to admit I can't really comprehend ultrafinitism. – fleablood Oct 29 '16 at 15:14

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His argument seems inexact. Rather than "The number $10\Delta 10 + 23$ does not have a prime factorization", his argument seems to be "The prime factorization of $10\Delta 10 + 23$ is not computable." He makes a fairly good case for the latter, and the former does hold when interpreted in certain logical realms such as Ultrafinitism. The first statement does not hold when interpreted classical logic and the ordinary understanding of prime factorization.

Larry B.
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