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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 in

by 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 .)

sillysaurusx 5y ago
That was another question in the back of my mind -- famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility). At the time, it was controversial as a "proof" since it didn't really take the usual form of a proof. I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally infeasible. But that raises the question: Suppose the four-color theorem was only able to prove 95% of cases rather than 100%. Isn't it at least sort of valuable to do so? Or is that last 5% all the difference? I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions. But that's an odd kind of debate, and not too productive. I just can't help but wonder about stuff like this.
- schoen 5y ago
  The four-color theorem had the preliminary challenge of creating a method to identify all the relevant cases. That also required a mathematical theory. (I don't actually understand how it was done!) If you didn't have that, you would have an infinite search for possible maps that violate the conjecture, because you wouldn't be able to divide them into a finite number of equivalence classes, or enumerate a finite number of possible counterexamples, or whatever. For Collatz, I don't think we have any lemma that gives a path to saying "a counterexample must be one of these 2¹⁰⁰ integers" or the like. So without such a thing, checking every possibility would mean checking every integer. (Well, there are definitely lemmas that make it unnecessary to check large numbers of integers, so I should say instead that checking every possibility would still mean checking an infinite number of cases.)
  - kevinventullo 5y ago
    As an interesting example where we did have a finite bound that was still out of reach to computationally exhaust, see the Weak Goldbach Conjecture: https://en.m.wikipedia.org/wiki/Goldbach%27s_weak_conjecture https://en.m.wikipedia.org/wiki/Goldbach%27s_weak_conjecture It was known to have at most finitely many counter-examples by the 1930’s, and by the 1950’s it was known that the largest counter-example had to be < 3^3^15. In 2002 that was lowered to about 10^1346. Still well out of reach of any computer! Only in 2012 was an unconditional proof given.
- helen___keller 5y ago
  It can be valuable, but it's not as valuable as covering all cases. If I could prove collatz for 99% of numbers it would be great. If i could additionally prove which 99% converge, let's say all numbers not divisible by 100, it would be enormous because even showing this much is likely to be a stepping stone to a full proof (what property is it about the number 100 that excludes all non-divisors from diverging?) > I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions. Right, and that's the whole point and beauty of it.
- 5y ago

by svat 5y ago

sillysaurusx 5y ago

That was another question in the back of my mind -- famously, the four color theorem was proved by computers through exhaustive analysis (checking every possibility). At the time, it was controversial as a "proof" since it didn't really take the usual form of a proof. I've often wondered "Why can't we do something like that, but for all instances of things like the Collatz conjecture?" Of course, it's computationally infeasible. But that raises the question: Suppose the four-color theorem was only able to prove 95% of cases rather than 100%. Isn't it at least sort of valuable to do so? Or is that last 5% all the difference? I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions. But that's an odd kind of debate, and not too productive. I just can't help but wonder about stuff like this.

schoen 5y ago
The four-color theorem had the preliminary challenge of creating a method to identify all the relevant cases. That also required a mathematical theory. (I don't actually understand how it was done!) If you didn't have that, you would have an infinite search for possible maps that violate the conjecture, because you wouldn't be able to divide them into a finite number of equivalence classes, or enumerate a finite number of possible counterexamples, or whatever. For Collatz, I don't think we have any lemma that gives a path to saying "a counterexample must be one of these 2¹⁰⁰ integers" or the like. So without such a thing, checking every possibility would mean checking every integer. (Well, there are definitely lemmas that make it unnecessary to check large numbers of integers, so I should say instead that checking every possibility would still mean checking an infinite number of cases.)
- kevinventullo 5y ago
  As an interesting example where we did have a finite bound that was still out of reach to computationally exhaust, see the Weak Goldbach Conjecture: https://en.m.wikipedia.org/wiki/Goldbach%27s_weak_conjecture https://en.m.wikipedia.org/wiki/Goldbach%27s_weak_conjecture It was known to have at most finitely many counter-examples by the 1930’s, and by the 1950’s it was known that the largest counter-example had to be < 3^3^15. In 2002 that was lowered to about 10^1346. Still well out of reach of any computer! Only in 2012 was an unconditional proof given.
helen___keller 5y ago
It can be valuable, but it's not as valuable as covering all cases. If I could prove collatz for 99% of numbers it would be great. If i could additionally prove which 99% converge, let's say all numbers not divisible by 100, it would be enormous because even showing this much is likely to be a stepping stone to a full proof (what property is it about the number 100 that excludes all non-divisors from diverging?) > I suppose what bugs me about math is, physicists are proved wrong all the time. Mathematicians are rarely proved mistaken, because they construct assumptions that you can't disagree with. There's no chance for empirical data to falsify one's assertions. Right, and that's the whole point and beauty of it.
5y ago