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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

#51

Post by daxfohl »

> 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 proven? Or does that mean the same thing? What does it mean if that goes infinitely deep? That we'll never know anything about a problem in that set unless we prove it? I guess it's not possible to prove that a problem is in that set, right? Is it possible to prove that problems do exist in that set at all?

edit: (non-vacuously. Provable problems are obviously vacuously in all the sets).

edit2: oh I see now (I think). Above I'd need an "AND if you can't prove it undecideable ... then you can't prove its undecidability's decideability either, and that can be proved"

So basically each level needs to be conditioned on all 0..n-2 levels not holding, then you can prove level n-1 is undecideable.

level 0: provable

level 1: provably undecideable

level 2: if not provable (can't prove level 0) then can't prove undecideable. (Goldbach, Collatz)

level 3: if it's not in level 0 or 1, then can't prove level 2.

level 4: if it's not in levels 0, 1, or 2, then can't prove level 3.

...

Which means there's no level infinity, since there's no level infinity minus 1 to talk about so it's meaningless. Correct?

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

#52

Post by xanderlewis »

permo-w wrote: post #23
Buttons840 wrote: post #8

One of the comments was "the Collatz conjecture feels like we're missing a branch of mathematics", or something to that effect. I followed that rabbit hole a bit and found the Plya conjecture[0], which was disproven when a counter example was found at approximately 10^361. Here I was naively thinking that if no counter examples were found in the first, say, 10^10 numbers, no counter examples should exist. If only it…

>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

Why do you assume the sun will rise tomorrow?

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

#53

Post by datameta »

bo1024 wrote: post #48
datameta wrote: post #18

This may be a question that misses the point - respectfully, what are some practical applications in physics or engineering for such proofs and/or the search for a conjecture counterexample?

Nothing wrong with asking the question. There might not be any practical applications in those fields, and it could still be a worthwhile, interesting, fun question. But in this case, I'd bet that developing the math needed to solve it would end up with practical consequences. I think of your question sort of like "what are the practical applications of teaching an AI to master Starcraft?" Even if playing Starcraft i…

Ah I see. As we evolve our understanding and toolset of certain fields in math, we may serendipitously happen upon approaches to solve existing problems or solve future ones, perhaps.

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

#54

Post by datameta »

Sharlin wrote: post #37
datameta wrote: post #18

This may be a question that misses the point - respectfully, what are some practical applications in physics or engineering for such proofs and/or the search for a conjecture counterexample?

None that we know of. We still do it because it's worth doing.

I do believe it is worth doing in and of itself. So my question is not trying to get a gotcha if not supplied with a practical application. I think I could certainly improve my intuition in theoretical math fields.

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

#55

Post by xanderlewis »

datameta wrote: post #18

This may be a question that misses the point - respectfully, what are some practical applications in physics or engineering for such proofs and/or the search for a conjecture counterexample?

One thing that makes it an appealing problem to solve is its apparent simplicity. It’s almost embarrassing that our modern mathematical technology isn’t adequate to resolve such an elementary question.

…or maybe the fact that it proves so hard to answer shows that we’re just wrong for thinking that it’s ‘elementary’. Therefore, any solution would necessarily require new breakthroughs that would lead to greater general understanding. So:

1) mathematicians would like to save face

2) you can’t solve difficult problems without accidentally solving other difficult ones that turn out to have immediate ‘practical’ importance

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

#56

Post by kstrauser »

datameta wrote: post #18

This may be a question that misses the point - respectfully, what are some practical applications in physics or engineering for such proofs and/or the search for a conjecture counterexample?

We don’t know.

And a telephone company wanted to connect calls more easily, so they invented the transistor, which changed the world in unimaginably many ways. Sometimes you just don’t know where you’ll end up until you try it.

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

#57

Post by dtgriscom »

gnatman wrote: post #46

>> 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! That’s the most fun thing I’ve read all day!

Prove it.

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

#58

Post by ajkjk »

tzs wrote: post #38

Here's a fairly easily understood problem, which if you could solve it would make you famous in the mathematical world and win you a million dollar prize: For a positive integer n: Let H(n) = 1 + 1/2 + 1/3 + ... + 1/n Let D(n) = the sum of the positive integers that divide n. E.g., D(12) = 1 + 2 + 3 + 4 + 6 + 12 = 28. Prove or disprove that for any positive integer n > 1: D(n) That easy to understand problem turns ou…

I've always felt that the Riemann hypothesis' specific formulation, in terms of the zeroes of zeta on the critical strip, is a very annoying way of expressing it. Like it is clearly related to the prime numbers in a really intrinsic way and therefore the most "intrinsic" expression of the problem shouldn't have anything to do with zeta or complex numbers or roots at all... whatever that is, I wish it's what we treated as the big unsolved problem instead of what we got.

I wonder what the most pleasing equivalent formulation of the RH would be. Gotta assume that some of them are much more directly about prime numbers. Yours seems pretty good. There are some others on here: https://mathoverflow.net/questions/39944/collection-of-equiv... which seem good also.

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

#59

Post by hatthew »

screenoridesaga wrote: post #14

I love how these raw mathematicians consider something proved when they can understand, meanwhile the computer can prove it easily just by counting a finite number of bits. What exactly would be considered proof in this case? Any explanation only mathematicians can understand?

If you ask a computer to compute (1/x) + 10^(-googolplex), it'll look an awful lot like the limit is 0.

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

#60

Post by keithalewis »

klysm wrote: post #17
messe wrote: post #9

Earlier quoted context omitted.

That's just the first counterexample that was proven to exist. The smallest counter example is less than 10^10 (in fact smaller than 10^9). > The Pólya conjecture was disproved by C. Brian Haselgrove in 1958. He showed that the conjecture has a counterexample, which he estimated to be around 1.845 × 10^361.[3] > An explicit counterexample, of n = 906,180,359 was given by R. Sherman Lehman in 1960;[4] the smallest cou…

I believe inability to do arithmetic correctly is more common in folks with maths and other STEM degrees

There are three kinds of mathematicians...
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