Timeline for Why are some heuristics successful?
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17 events
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| Nov 20 at 4:31 | comment | added | Timothy Chow | @semisimpleton The Bateman-Horn conjecture is perhaps one such precise conjecture. It doesn't encompass everything about the Cramér model, though. | |
| Nov 20 at 1:18 | comment | added | semisimpleton | @TimothyChow Cramér's expectations of which conjectures in number theory are likely to be true were not "fed into" the construction of the model, but the model nevertheless predicts the truth of various conjectures. I feel like this means it should be possible to translate the hypotheses underlying the Cramér model into a very general conjecture (or several), and this general conjecture would be a precise version of "nothing weird happens" (in the context of the distribution of primes). Has this been attempted? I have edited the main post to include this question. | |
| Nov 20 at 1:10 | history | edited | semisimpleton | CC BY-SA 4.0 |
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| Nov 20 at 0:58 | comment | added | semisimpleton | @TimothyChow If the assumption that "nothing weird happens" can be used to predict several statements that are later shown to be true, then surely one would like to be able to translate "nothing weird happens" into a precise mathematical statement. In context, this could mean proving a very powerful theorem that closely reflects the hypotheses underlying the Cramér model, so that the Riemann Hypothesis, Twin Prime Conjecture, etc. can be deduced from this theorem. In effect, this would amount to "upgrading" Cramér model itself to a theorem, or set of theorems. | |
| Nov 19 at 23:41 | comment | added | semisimpleton | @PeterLeFanuLumsdaine I definitely didn't mean it that way. I did compare heuristics to reading tea leaves, but not in a negative way. I don't believe for a moment that the shape of my tea leaves can have something to do with the outcome of a football match, but I do believe that probabilistic heuristics are meaningfully connected to the truth of a mathematical statement. But with both tea leaves and heuristics, there is no clear connection between the "method of divination" and actual fact, so how can I justify this feeling that using probabilistic heuristics is not complete nonsense? | |
| Nov 19 at 19:17 | comment | added | Peter LeFanu Lumsdaine | Your second paragraph insinuates a dichotomy, comparing anything short of proven theorems to “reading tealeaves”. This is rank chauvinism: outside mathematics, almost everything falls short of mathematically rigorous proof, and all good science/methodology is about developing reliable practices and intuitions for such things nonetheless. | |
| Nov 19 at 18:00 | history | became hot network question | |||
| Nov 19 at 16:08 | comment | added | Alessandro Della Corte | I think that some historically important examples of what you search are the heuristics described by Archimedes in the Method and developed into theorems in On the Sphere and Cylinder. | |
| Nov 19 at 14:18 | comment | added | Timothy Chow | I think a lot of heuristics are of the form, "Nothing weird happens." The Cramér model says that the sequence of primes doesn't contain any weird structure. Similarly, the only thing stopping many heuristic arguments from being rigorous is that some bizarre counterexample isn't ruled out. So when the rigorous proof arrives, and the ghosts are banished, I'm not sure that it "explains" the heuristic beyond confirming that what we thought would be too bizarre to be true is indeed too bizarre to be true. That is, the explanatory value is more "negative" than "positive." | |
| Nov 19 at 14:10 | comment | added | Timothy Chow | Related: Rigorous version of heuristic argument for genus-degree formula? | |
| Nov 19 at 13:38 | answer | added | Carlo Beenakker | timeline score: 7 | |
| Nov 19 at 13:00 | comment | added | Iosif Pinelis | I think almost every nontrivial proof is preceded by successful heuristics. Such heuristics may sometimes be also called ideas. | |
| Nov 19 at 12:53 | history | made wiki | Post Made Community Wiki by Stefan Kohl♦ | ||
| Nov 19 at 10:44 | history | edited | YCor |
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| Nov 19 at 10:02 | comment | added | Carl-Fredrik Nyberg Brodda | Here’s a heuristic for why heuristics work: because they’ve worked in the past. | |
| Nov 19 at 10:00 | comment | added | Alessandro Della Corte | I'd say that's what happens most of the times when heuristics lead to correct predictions and then to theorems: the proof grows often up from the heuristics themselves. | |
| Nov 19 at 9:54 | history | asked | semisimpleton | CC BY-SA 4.0 |