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Mathematicians sometimes use heuristics to form expectations about what might be true or false. For examples, see Matthew Emerton's answer to Why should I believe the Mordell Conjecture?, this blog post by Terence Tao, and this expository article by Barry Mazur.

Such heuristics occupy a funny position in the epistemology of mathematics. We tend to think that using such heuristics somehow makes "more sense" than, say, reading tea leaves to decide which football team to bet on. We know they can be misleading, yet we allow ourselves to be influenced by them. Indeed, this MO question provides an example of heuristics leading to two contradictory conclusions.

Question: Has it ever happened that a conjecture supported by heuristics was proved, and then an explanation was found for the success of the heuristics? Concretely, this might mean that the heuristic argument was developed into a rigorous proof.

Example: When the umbral calculus was first discovered, it was mysterious that the seemingly nonsensical operation of "turning indices into exponents" could yield correct identities.

Example: The Cramér probabilistic model of the prime numbers predicts that the Riemann hypothesis is true. After the Riemann hypothesis is proved, as no doubt it eventually will, it would remain to be understood why the Cramér model was successful (at least in the case of RH).

Timothy Chow suggests in the comments that heuristics like the Cramér model are, in some sense, precise expressions of the imprecise hypothesis that "nothing weird happens".

Question: Have any attempts been made to formulate a conjecture that reflects the hypotheses underlying the Cramér model? Such a conjecture would be very powerful indeed, since it would imply the Riemann Hypothesis, Legendre's Conjecture, the Twin Prime Conjecture, and more.

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    $\begingroup$ 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. $\endgroup$
    – Alessandro Della Corte
    Commented Nov 19 at 10:00
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    $\begingroup$ Here’s a heuristic for why heuristics work: because they’ve worked in the past. $\endgroup$
    – Carl-Fredrik Nyberg Brodda
    Commented Nov 19 at 10:02
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    $\begingroup$ I think almost every nontrivial proof is preceded by successful heuristics. Such heuristics may sometimes be also called ideas. $\endgroup$
    – Iosif Pinelis
    Commented Nov 19 at 13:00
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    $\begingroup$ Related: Rigorous version of heuristic argument for genus-degree formula? $\endgroup$
    – Timothy Chow
    Commented Nov 19 at 14:10
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    $\begingroup$ 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." $\endgroup$
    – Timothy Chow
    Commented Nov 19 at 14:18

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The Weil conjectures could qualify as a set of heuristics developed into a rigorous proof by Deligne and others:

What was really eye-catching, from the point of view of other mathematical areas, was the proposed connection with algebraic topology. Given that finite fields are discrete in nature, and topology speaks only about the continuous, the detailed formulation of Weil (a heuristic based on working out some examples) was striking and novel. It suggested that geometry over finite fields should fit into well-known patterns relating to Betti numbers, the Lefschetz fixed-point theorem and so on. The analogy with topology suggested that a new homological theory be set up applying within algebraic geometry. This took two decades (it was a central aim of the work and school of Alexander Grothendieck) building up on initial suggestions from Serre.

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    $\begingroup$ More generally, the analogy between function fields and number fields provides a heuristic: "what is true in one setting should be true in the other setting as well." However, whether we now possess an "explanation for the success" of this heuristic, as the OP requested, is maybe up for debate. $\endgroup$
    – Timothy Chow
    Commented Nov 19 at 14:22
  • $\begingroup$ @TimothyChow I think a theory can be considered a "full explanation" of the function field analogy only if it can be used to prove the number field Riemann Hypothesis in a manner analogous to the proof of the Weil Conjectures $\endgroup$
    – semisimpleton
    Commented Nov 20 at 1:22
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    $\begingroup$ Unless I'm mistaken, @CarloBeenakker's answer is not about the analogy between number fields and function fields, but rather about the analogy between some algebro-geometric concepts and some topological concepts, which I think has been fully realised as a precise theory $\endgroup$
    – semisimpleton
    Commented Nov 20 at 1:31

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