Timeline for True by accident (and therefore not amenable to proof)
Current License: CC BY-SA 3.0
13 events
| when toggle format | what | by | license | comment | |
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| Nov 5, 2023 at 4:55 | comment | added | Daniel Asimov | Junkie wrote: "... if [π] is normal, the reason is just that it's random." But what does the word "random" mean here? | |
| Nov 4, 2023 at 18:41 | comment | added | Daniel Asimov | It seems entirely possible that the Twin Prime Conjecture might be something that just happens to be true. | |
| Aug 26, 2011 at 14:28 | comment | added | user9072 | Junkie, thank you for the examples. In particular, the quote on $\pi$, which I in principle knew but forgot. Let us look a bit more than a century back on the transcendence of certain numbers (not pi itself, but in Hilbert 7th problem). Perhaps somebody could have said...since there would need to be a reason for them to be algebraic, and there is not, they will be transcendental, but no proof in sight (if I remeber well Hilbert was quite pessimistic in his prediction when 7th prob. would be solved.) Yet, 'suddenly' came Gelfond--Schneider and a 'reason' for transcendence of these numbers. | |
| Aug 26, 2011 at 1:26 | comment | added | Junkie | "it is true because it has no reason to be false." Zagier had a similar comment, say about why $\pi$ is normal; if it isn't, there must be some deep reason, but if it is normal, the reason is just that it's random. For the other part, you can model many things by random sense (like $\mu(n)$ as random $0,\pm 1$ coefficients of a Dirichlet series, for RH, or Cramer's model of primes, for gaps), and the question should be whether the general randomness persists in the problem at hand. Another similar is Goldbach (partitio numerorum for general) as said. | |
| Aug 26, 2011 at 0:09 | comment | added | user9072 | Peter, thinking about it, I agree with your assertion; since I am so speculative in an optimistic way in others parts, I should have been consistent and phrased this less absolutely. | |
| Aug 25, 2011 at 21:39 | comment | added | Peter LeFanu Lumsdaine | @quid: Agreed greatly with most of your answer; but I wouldn’t share the pessimism of your last paragraph. Analogously, before the development of formal logic, someone might have said “‘Proven’ is a time-dependent notion; it seems thus difficult for me to imagine that its spirit can be captured in a formal theory.” But now provability is formalised, and we can prove independence results which tell us things about the time-dependent notion of ‘proven’. Surely some imaginative logician might yet come up with a formalisation which says something about this intuitive idea of ‘true by accident’? | |
| Aug 25, 2011 at 18:58 | comment | added | user9072 | The above comment was written before Alex comment so let me highlight a point is view of it. Alex, it seems you misunderstand my point with 3-AP. There is a reason, the density, which suffices to explain the phenomenon. (It's not most sets and the primes happen to be one, it is all sets.) | |
| Aug 25, 2011 at 18:53 | comment | added | user9072 | Yes, my main point is what Timothy Chow high-lighted. Perhaps, the illustrating ex. are not choosen optimal. Alex, regarding 3-AP my intent was to point out that the fact is completely explained by density. So, I would not say it is a 'no reason to be false' plus luck situation as all sets having this density (conjecturally) have the property. So one knows a reason why it is true. For Goldbach, first density cannot suffice but this is irrelevant, but second what I meant was that this super-advanced state of AC I dreamed-up would give a reason for what you refer to as 'luck'. | |
| Aug 25, 2011 at 18:46 | comment | added | Alex B. | @Timothy Yes, I wasn't trying to address that part at all, since the wonderful answers already given address the provability issue very nicely. I was just joining quid in trying to understand what exactly the OP has in mind when he talks about accidents. What I am trying to say is that even though a theorem is provable, it can feel like it's true for no special reason other than the absence of a reason to be false. And the example quid gave seems to me to exemplify exactly this situation. | |
| Aug 25, 2011 at 18:22 | comment | added | Timothy Chow | @Alex: However, the OP also says that accidental truths cannot be proved. This puts an additional spin on the topic. I think quid is right that one's feeling of what sorts of things "cannot be proved" are informed mainly by one's sense of what current mathematical technology is or is not capable of achieving. | |
| Aug 25, 2011 at 18:16 | comment | added | Alex B. | Similarly with the Goldbach conjecture: there are so many primes that if Goldbach doesn't accidentally fail for some small number, then a probabilistic argument says that it has a very low chance of failing at all. This might not be a special property of the primes, but simply a property of a generic sufficiently dense set, and the fact that it doesn't accidentally fail for the primes is "luck". | |
| Aug 25, 2011 at 18:14 | comment | added | Alex B. | Maybe I am misunderstanding the OP's take on these things, but the way I read it, the fact that any set of positive integers with the density of the primes should have the same property is precisely the sort of "accident" the OP was talking about: given how ubiquitous the primes are, a suitable random model would tell you that the statement has a very low chance of failing. In other words, it is not true for a special reason, it is true because it has no reason to be false. | |
| Aug 25, 2011 at 18:07 | history | answered | user9072 | CC BY-SA 3.0 |