Examples of results which were surprising but later shown to be natural. - MathOverflow [closed]

Examples of results which were surprising but later shown to be natural.

2011-02-11T11:22:16Z

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?

Answer by Hans Stricker for Examples of results which were surprising but later shown to be natural.

2011-02-11T12:07:27Z

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".

Answer by Gil Kalai for Examples of results which were surprising but later shown to be natural.

2011-02-11T12:32:38Z

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.

Answer by none for Examples of results which were surprising but later shown to be natural.

2011-02-11T12:35:00Z

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 ;-)