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Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two

Others have addressed why it matters (or doesn't matter) when viewed from outside mathematics. But within mathematics, unsolved problems usually matter for two reasons: (1) When lots of really smart people spend lots of time trying to solve something, and fail to do so, it becomes even more interesting for other smart people. A well-known example is the Collatz Conjecture[0] which, by most accounts is a meaningless problem, but endlessly fascinates many for its renowned difficulty. (2) In mathematics, you often stumble upon problems that are equivalent to other problems. If X is difficult to solve so you give up and go work on Y, only to find out that Y as a problem is basically the same to X, or they have some other intimate relationship, and it makes X that much more tantalizing. A common example for computer scientists is the whole class of NP complete problems[1] for which we don't know if there exists a polytime algorithm to decide. Every time we discover a new problem that lives in the realm of NP-complete, the P vs NP problem gets a little more tantalizing. [0] https://en.wikipedia.org/wiki/Collatz_conjecture https://en.wikipedia.org/wiki/Collatz_conjecture [1] https://en.wikipedia.org/wiki/NP-completeness https://en.wikipedia.org/wiki/NP-completeness

sillysaurusx 5y ago
This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of histograms and empirical data supporting the conjecture. My question is, why is this empirical data not "good enough" for mathematicians? As something closer to a physicist, I do wonder what the fascination is with trying to prove conclusively that 3n+1 xor n/2 will eventually encounter 1. It seems a bit like trying to prove conclusively that the gravitational constant is so-and-so, when it seems the best you can do is to measure it as precisely as possible: > Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics." As someone who loved mathematics but was never much good at it, what's the fascination here? (I'm only trying to understand the motivations of people smarter than I am.)
- ivanbakel 5y ago
  >My question is, why is this empirical data not "good enough" for mathematicians? There are two reasons: firstly, because mathematics is not an empirical discipline (well, unless you're a number theorist...), so it is possible to be certain of mathematical truth, unlike the inherent uncertainty of physical truth; secondly, because every finite bound on the natural numbers may as well be 0 when compared to the numbers that remain. There is simply no way to take any empirical measurements of the natural numbers (or the reals) that would let you estimate anything "for all numbers", since the proportion of numbers you failed to sample is infinite. You may be interested in reading the answers to this question: https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples/ https://math.stackexchange.com/questions/514/conjectures-tha... which describes some problems which seemed true "up to some large number", but later turned out to be false.
  - sillysaurusx 5y ago
    https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples/ https://math.stackexchange.com/questions/514/conjectures-tha... was excellent. Thank you. > 𝑛^17+9 and (𝑛+1)^17+9 are relatively prime > The first counterexample is 𝑛=8424432925592889329288197322308900672459420460792433 I think this helped me appreciate the difficulty of being satisfied with the Collatz conjecture.
- svat 5y ago
  No amount of empirical data is ever "good enough" for the mathematical standard of proof. (It may be enough for mathematicians to "believe" something in some informal sense, but not enough to consider it "proved".) You can see some examples at the answers to these questions; maybe at least one of them will be interesting to you: - https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples https://math.stackexchange.com/questions/514/conjectures-tha... - https://math.stackexchange.com/questions/111440/examples-of-patterns-that-eventually-fail https://math.stackexchange.com/questions/111440/examples-of-... (and maybe https://mathoverflow.net/questions/11517/computer-algebra-errors https://mathoverflow.net/questions/11517/computer-algebra-er... ) - https://mathoverflow.net/questions/15444/examples-of-eventual-counterexamples https://mathoverflow.net/questions/15444/examples-of-eventua... (There are over a hundred examples at those questions; I guess this counts as a lot of empirical data that empirical data is not enough! However, sometimes empirical data can be enough for a mathematical proof; for instance if you know somehow that a polynomial f is of degree less than n and you have shown that f(x)=g(x) at n distinct points, you can indeed conclude that f=g and so on — see the "Proofs by Example?" section of the first "Proof Machines" chapter of the book "A=B" available online https://www2.math.upenn.edu/~wilf/AeqB.html https://www2.math.upenn.edu/~wilf/AeqB.html .)
thehappypm 5y ago
I mean, at its simplest, the Collatz conjecture is a type of random walk, with the sequence either exponentially growing or exponentially shrinking, basically at random. It's statistically impossible that that any random walk will continue one direction forever, and once it randomly decays to 1 you're done.
- quickthrower2 5y ago
  Loops are the other option you’d need to prove don’t exist. I think I can see why trying to solve this problem is popular. It almost feels like it should be easy to solve but it just quite isn’t!

sillysaurusx 5y ago

This is an excellent answer. I have a mostly unrelated question. In the case of the Collatz conjecture, it seems all but proved: > If the conjecture is false, it can only be because there is some starting number which gives rise to a sequence that does not contain 1. Such a sequence would either enter a repeating cycle that excludes 1, or increase without bound. No such sequence has been found. There are lots of histograms and empirical data supporting the conjecture. My question is, why is this empirical data not "good enough" for mathematicians? As something closer to a physicist, I do wonder what the fascination is with trying to prove conclusively that 3n+1 xor n/2 will eventually encounter 1. It seems a bit like trying to prove conclusively that the gravitational constant is so-and-so, when it seems the best you can do is to measure it as precisely as possible: > Jeffrey Lagarias stated in 2010 that the Collatz conjecture "is an extraordinarily difficult problem, completely out of reach of present day mathematics." As someone who loved mathematics but was never much good at it, what's the fascination here? (I'm only trying to understand the motivations of people smarter than I am.)

ivanbakel 5y ago
>My question is, why is this empirical data not "good enough" for mathematicians? There are two reasons: firstly, because mathematics is not an empirical discipline (well, unless you're a number theorist...), so it is possible to be certain of mathematical truth, unlike the inherent uncertainty of physical truth; secondly, because every finite bound on the natural numbers may as well be 0 when compared to the numbers that remain. There is simply no way to take any empirical measurements of the natural numbers (or the reals) that would let you estimate anything "for all numbers", since the proportion of numbers you failed to sample is infinite. You may be interested in reading the answers to this question: https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples/ https://math.stackexchange.com/questions/514/conjectures-tha... which describes some problems which seemed true "up to some large number", but later turned out to be false.
- sillysaurusx 5y ago
  https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples/ https://math.stackexchange.com/questions/514/conjectures-tha... was excellent. Thank you. > 𝑛^17+9 and (𝑛+1)^17+9 are relatively prime > The first counterexample is 𝑛=8424432925592889329288197322308900672459420460792433 I think this helped me appreciate the difficulty of being satisfied with the Collatz conjecture.
svat 5y ago
No amount of empirical data is ever "good enough" for the mathematical standard of proof. (It may be enough for mathematicians to "believe" something in some informal sense, but not enough to consider it "proved".) You can see some examples at the answers to these questions; maybe at least one of them will be interesting to you: - https://math.stackexchange.com/questions/514/conjectures-that-have-been-disproved-with-extremely-large-counterexamples https://math.stackexchange.com/questions/514/conjectures-tha... - https://math.stackexchange.com/questions/111440/examples-of-patterns-that-eventually-fail https://math.stackexchange.com/questions/111440/examples-of-... (and maybe https://mathoverflow.net/questions/11517/computer-algebra-errors https://mathoverflow.net/questions/11517/computer-algebra-er... ) - https://mathoverflow.net/questions/15444/examples-of-eventual-counterexamples https://mathoverflow.net/questions/15444/examples-of-eventua... (There are over a hundred examples at those questions; I guess this counts as a lot of empirical data that empirical data is not enough! However, sometimes empirical data can be enough for a mathematical proof; for instance if you know somehow that a polynomial f is of degree less than n and you have shown that f(x)=g(x) at n distinct points, you can indeed conclude that f=g and so on — see the "Proofs by Example?" section of the first "Proof Machines" chapter of the book "A=B" available online https://www2.math.upenn.edu/~wilf/AeqB.html https://www2.math.upenn.edu/~wilf/AeqB.html .)

thehappypm 5y ago

I mean, at its simplest, the Collatz conjecture is a type of random walk, with the sequence either exponentially growing or exponentially shrinking, basically at random. It's statistically impossible that that any random walk will continue one direction forever, and once it randomly decays to 1 you're done.

quickthrower2 5y ago
Loops are the other option you’d need to prove don’t exist. I think I can see why trying to solve this problem is popular. It almost feels like it should be easy to solve but it just quite isn’t!