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The seventh most popular easily understood unsolved problem on MathOverflow

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Re: The seventh most popular easily understood unsolved problem on MathOverflow

#101

Post by kevinventullo »

Buttons840 wrote: post #98
permo-w wrote: post #23

Earlier quoted context omitted.

>Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist why on earth would you ever think that

I think you're being purposely dense. You pretend that it's unusual that a human would view a pattern that has occured 10 billion times and then expect that it would continue forever. But, I'll answer anyway. It seems that for such a simple problem, involving basic arithmetic and small numbers, 1, 2, and 3, there should be a number N where if you try all examples less than N, you have sufficient "resolution" to revea…

I think what you’re describing is very roughly equivalent to Busy Beaver numbers: https://en.m.wikipedia.org/wiki/Busy_beaver

This kind of assumes that “A simple problem, involving basic arithmetic and small numbers, 1, 2, and 3” can be encoded into some bounded-state Turing machine.

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#102

Post by permo-w »

Buttons840 wrote: post #98
permo-w wrote: post #23

Earlier quoted context omitted.

>Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist why on earth would you ever think that

I think you're being purposely dense. You pretend that it's unusual that a human would view a pattern that has occured 10 billion times and then expect that it would continue forever. But, I'll answer anyway. It seems that for such a simple problem, involving basic arithmetic and small numbers, 1, 2, and 3, there should be a number N where if you try all examples less than N, you have sufficient "resolution" to revea…

if you have absolutely no grounding in maths or science, then sure, it would be unusual to think that way, but if you do, which you clearly did, then it should no longer be unusual whatsoever.

it would be trivial to artificially construct a series that equals 1 until n=10^10 at which point it equals pi/785498. if these series do exist, then - without additional information - why would you ever think that one you're studying is not one of them?

even from a metaepistemological standpoint, had you never come across the notion that, for example, Fermat's Last Theorem could not have been proven simply by showing that 10^10/infinity of the possible outcomes had been shown to follow it?

>It seems that for such a simple problem, involving basic arithmetic and small numbers, 1, 2, and 3, there should be a number N where if you try all examples less than N, you have sufficient "resolution" to reveal all patterns between the numbers

does it? why?

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#103

Post by permo-w »

xanderlewis wrote: post #89
permo-w wrote: post #86

Earlier quoted context omitted.

this analogy illustrates the opposite of what I think you intend it to. if we could artificially iterate the sun rising infinite times, somewhere very high up, maybe after 10^10, it eventually would no longer rise

True, and that did occur to me as I was writing it. The point was more that any length of time greater than a few thousand years is as good as infinite for a human, so (for, you know, everyday purposes) we might as well assume that the sun will keep on rising forever.

well, true, but it seems strange to me to apply such everyday processes to something as comfortably infinite as maths

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#104

Post by xanderlewis »

permo-w wrote: post #103
xanderlewis wrote: post #89

Earlier quoted context omitted.

True, and that did occur to me as I was writing it. The point was more that any length of time greater than a few thousand years is as good as infinite for a human, so (for, you know, everyday purposes) we might as well assume that the sun will keep on rising forever.

well, true, but it seems strange to me to apply such everyday processes to something as comfortably infinite as maths

Then you have better intuition than some!

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#105

Post by eru »

xeonmc wrote: post #88
eru wrote: post #65

Earlier quoted context omitted.

In number theory, 2 is often a special case in a lot of theorems. There's something odd about the even prime.

i.e. the first information-encoding base.

I'm not sure. The Fibonacci numbering system has a smaller effective base.

(And there's always unary.)

See https://en.wikipedia.org/wiki/Fibonacci_coding

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#106

Post by daxfohl »

hansvm wrote: post #100
daxfohl wrote: post #51

> One annoying thing about both the Collatz and Goldbach conjectures is that if they are unprovable, it's also impossible to prove they're unprovable (unless math is inconsistent). This has been proved! So is there a countable order of metaprovability? i.e. are there problems that if they're impossible to prove, it's impossible to prove whether it's possible to prove whether they're unprovable, but _that_ can be prov…

A couple of things: > level 2: if not provable (can't prove level 0) then can't prove undecideable. (Goldbach, Collatz) If you're operating in the ordinary naturals, neither Goldbach nor Collatz are undecidable. The statements are either true or false. The question at hand is whether we can generate a finite proof of that fact in a given axiomatic system. The quote at hand simply said that if no such proof exists (ob…

Fwiw I'm going back to my original hypothesis and saying there's no limit at infinity, but there is a class of questions that are in none of the levels, essentially meaning you can't prove anything about their solvability. It's impossible to prove that any particular question is in that class, by definition, but it may be possible to prove that there exist questions that are. Which opens the philosophical question of why so many questions aren't.

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#107

Post by daynthelife »

1propionyl wrote: post #96
eru wrote: post #65

Earlier quoted context omitted.

In number theory, 2 is often a special case in a lot of theorems. There's something odd about the even prime.

In analysis and its applications, the L^2 norm is also rather special.

My hot take is that it should be called the L^(1/2) norm. Many theorems and formulas become a lot easier to state if you redefine L^p as L^(1/p).

For instance, under this notation, the dual of L^p is just L^(1-p). And Littlewood’s interpolation inequality is a lot easier to remember, since the exponents used come directly from the coefficients in the convex combination:

If r = ap + bq, where a and b are nonnegative and sum to 1, then |f|_r <= (|f|_p)^a (|f|_q)^b

Re: The seventh most popular easily understood unsolved problem on MathOverflow

#108

Post by eru »

daynthelife wrote: post #107
1propionyl wrote: post #96

Earlier quoted context omitted.

In analysis and its applications, the L^2 norm is also rather special.

My hot take is that it should be called the L^(1/2) norm. Many theorems and formulas become a lot easier to state if you redefine L^p as L^(1/p). For instance, under this notation, the dual of L^p is just L^(1-p). And Littlewood’s interpolation inequality is a lot easier to remember, since the exponents used come directly from the coefficients in the convex combination: If r = ap + bq, where a and b are nonnegative a…

You should join the folks at the tau-not-pi club.

And going off on a tangent: in thermodynamics, we should measure coldness (coldness ~ 1 / temperature). It makes all the math come out nicer.

See https://en.wikipedia.org/wiki/Coldness

Coldness handles 'negative temperatures' much better. As Wikipedia puts it:

> Though completely equivalent in conceptual content to temperature, β [= coldness] is generally considered a more fundamental quantity than temperature owing to the phenomenon of negative temperature, in which β is continuous as it crosses zero whereas T has a singularity.[7]

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