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I am reminded of two curious statements which are not normally linked: (0) There could be a Siegel zero [0], like an "indentation" in the Riemann zeta function

I am reminded of two curious statements which are not normally linked: (0) There could be a Siegel zero [0], like an "indentation" in the Riemann zeta function. The prime numbers need to all collaborate to "shift" the zeta function a little in order to "buckle" the zeta function at the Siegel zero, so there's only really enough room in the primes to accomodate one Siegel zero at most. (1) Ultrafinitism [1] comes in many flavors, including a flavor where there is a single largest prime number [2]. In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. Since this would be a true statement about natural numbers, it would also be a sort of conspiracy amongst the primes, with only one instance. Situations like this article remind me of both simultaneously: If there's any sort of conspiracy amongst the primes to forge a Siegel zero, it would have to be the same conspiracy which produces an ultrafinite prime candidate; the primes don't have bandwidth for two conspiracies. [0] https://en.wikipedia.org/wiki/Siegel_zero https://en.wikipedia.org/wiki/Siegel_zero [1] https://en.wikipedia.org/wiki/Ultrafinitism https://en.wikipedia.org/wiki/Ultrafinitism [2] https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/real.pdf https://sites.math.rutgers.edu/~zeilberg/mamarim/mamarimPDF/...

msla 6y ago
> In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. This is why I don't like ultrafinitism: It insists on mixing levels, like an Escher print. The world of pure mathematics isn't the physical world, for all the same reasons Santa Claus isn't real, and trying to bind pure math to some (theoretical!) properties of the physical world is just as wrong-headed as insisting that nobody can ever make songs about Santa because we know nobody can deliver toys that quickly.
- mensetmanusman 6y ago
  But it’s still a useful exercise, because at the interface where math does seem to touch the real world, we can use it to make amazing predictions about the behavior of future events.

msla 6y ago

> In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. This is why I don't like ultrafinitism: It insists on mixing levels, like an Escher print. The world of pure mathematics isn't the physical world, for all the same reasons Santa Claus isn't real, and trying to bind pure math to some (theoretical!) properties of the physical world is just as wrong-headed as insisting that nobody can ever make songs about Santa because we know nobody can deliver toys that quickly.

mensetmanusman 6y ago
But it’s still a useful exercise, because at the interface where math does seem to touch the real world, we can use it to make amazing predictions about the behavior of future events.