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After Ramanujan formulated his conjectures on the Tau-function, and after the importance of the function was realized, it took the development of the theory of Modular forms for the complete resolution and understanding of the conjectures and the function itself.(For example, it was only later that people could explain the appearance of the mysterious index 24 in its definition.)

Another example is the problem of constructibility of regular polygons. The Ancient Greeks must have pondered the reason for their inability to construct certain polygons. But after Gauss, it now seems natural why one cannot construct a 11-gon using only a compass and straightedge.

In the above two cases, there is a common feature. There is a discovery which at first seems surprising or baffling. Only later, after sufficient developments in theory, was the mystery lifted. Are there any other such examples?

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    $\begingroup$ I disagree with your account of history in the first paragraph. The appearance of the 24 was explained by Jacobi well before Ramanujan formulated his conjectures. Deligne's resolution of Ramanujan's conjecture on the growth of tau did not "take the development of the theory of modular forms" except in a very broad sense. It used the Weil conjectures in an essential way. $\endgroup$
    – S. Carnahan ♦
    Commented Feb 11, 2011 at 11:44
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    $\begingroup$ It seems to me that this could be an impossibly big list. Here are some more famous examples: the seeming impossibility of proving the parallell postulate, explained by the introduction of hyperbolic geometry. The Weil conjectures hinting at the existence of a kind of cohomology and Lefschetz trace formula for varieties over arbitrary fields, explained by Grothendieck's étale cohomology. The fact that the Fourier coefficients of the j-function are given by small integer linear combinations of dimensions of irreps of the Monster group, explained by the construction of the Monster module... $\endgroup$
    – Dan Petersen
    Commented Feb 11, 2011 at 11:45
  • $\begingroup$ ..., the theory of vertex operator algebras, and ideas from string theory. $\endgroup$
    – Dan Petersen
    Commented Feb 11, 2011 at 11:45
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    $\begingroup$ I agree with Dan Petersen: a substantial proportion of all mathematical results could reasonably answer this question, which to me indicates it is too broad. I have voted to close. $\endgroup$
    – Pete L. Clark
    Commented Feb 11, 2011 at 12:58
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    $\begingroup$ Young man, in mathematics you don't understand things, you just get used to them. -- John von Neumann $\endgroup$
    – Allen Knutson
    Commented Feb 12, 2011 at 21:48

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In his Indiscrete Thoughts Gian-Carlo Rota writes:

Every mathematical theorem is eventually proved trivial. The mathematician's ideal of truth is triviality, and the community of mathematicians will not cease its beaver-like work on a newly discovered result until it has shown to everyone's satisfaction that all difficulties in the early proofs were spurious, and only an analytic triviality is to be found at the end of the road.

According to Rota what you ask for is the normal case. He - and others - do not even rule out the possibility that some day Fermat's Last Theorem turns out to be "trivial".

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A favorite example of this kind for me is the result that the set of triangulations of an n gon (or, equivalenty, the set of interpretations of the nonassociative product $x_1x_2\dots x_{n-1}$) has the structure of a convex (n-1)-dimensional polytope. (It is called the associahedron or the Stasheff Polytope.) This looked (to me) as a curiosity at first but it turned out to be very basic and natural construction which is related to a lot of exciting mathematics. See Ziegler's lecture on the associahedron.

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Gödel's incompleteess theorems -- knocked over the Hilbert program to prove mathematics was consistent and complete.

Smale's sphere eversion theorem - originally thought to be a counterexample showing an error in the proof, but the eversion is really there.

Butterfly effect (Edward Lorenz) -- looked like a numerical artifact in an ODE integrator, the idea of nonperiodic solutions hadn't been forseen.

Banach Tarski paradox

Monty Hall problem (ducks head)

Barrington's theorem for branching programs

PCP theorem

Rationality of Legendre's constant ;-)

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  • $\begingroup$ I have no idea how the Banach Tarski paradox was received at the time; I can't imagine it must have been very surprising, but I may very well be wrong. $\endgroup$
    – Thierry Zell
    Commented Feb 15, 2011 at 3:30

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