> There are lots of histograms and empirical data supporting the conjecture.
There's a very interesting implicit question there: why should counterexamples be small? [1][2] Certainly, counterexamples to many conjectures about infinite sets can be found with a brute-force search, even if the problem is merely semidecidable. But isn't that simply an example of selection bias?
There is a certain subjective beauty in having counterexamples like 9 or 341 or even 23338590792. But is that just anthropocentrism? After all, no matter how many cases we check, we have made absolutely no progress in exhausting the whole set of natural numbers! We can never reach even reasonably easily constructable numbers like 3↑↑↑3 (using Knuth arrow notation [3]), and still almost all[4] natural numbers are bigger than that.
In physics, there's an (often implicitly made) assumption that more evidence in support of a hypothesis makes it more likely that the hypothesis is supported by any future evidence as well. But why should we be able to make that assumption? We do, because it seems to work, but why should it still work tomorrow? This is, of course, the famous philosophical problem of induction [5]. But math is basically what happens when you explicitly reject inductive reasoning and then start to explore the space of things that can still be reached, using purely deductive reasoning!
There's a very interesting implicit question there: why should counterexamples be small? [1][2] Certainly, counterexamples to many conjectures about infinite sets can be found with a brute-force search, even if the problem is merely semidecidable. But isn't that simply an example of selection bias?
There is a certain subjective beauty in having counterexamples like 9 or 341 or even 23338590792. But is that just anthropocentrism? After all, no matter how many cases we check, we have made absolutely no progress in exhausting the whole set of natural numbers! We can never reach even reasonably easily constructable numbers like 3↑↑↑3 (using Knuth arrow notation [3]), and still almost all[4] natural numbers are bigger than that.
In physics, there's an (often implicitly made) assumption that more evidence in support of a hypothesis makes it more likely that the hypothesis is supported by any future evidence as well. But why should we be able to make that assumption? We do, because it seems to work, but why should it still work tomorrow? This is, of course, the famous philosophical problem of induction [5]. But math is basically what happens when you explicitly reject inductive reasoning and then start to explore the space of things that can still be reached, using purely deductive reasoning!
[1] https://math.stackexchange.com/questions/449886/the-largest-...
[2] https://math.stackexchange.com/questions/111440/examples-of-...
[3] https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation
[4] https://en.wikipedia.org/wiki/Almost_all
[5] https://en.wikipedia.org/wiki/Problem_of_induction