Many famous results were discovered through non-rigorous proofs, with
correct proofs being found only later and with greater difficulty. One that is well
known is Euler's 1737 proof that
$1+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}+\cdots =\frac{\pi^2}{6}$
in which he pretends that the power series for $\frac{\sin\sqrt{x}}{\sqrt{x}}$
is an infinite polynomial and factorizes it from knowledge of its roots.
Another example, of a different type, is the Jordan curve theorem. In this case,
the theorem seems obvious, and Jordan gets credit for realizing that it requires
proof. However, the proof was harder than he thought, and the first rigorous proof
was found some decades later than Jordan's attempt. Many of the basic theorems of topology are like this.
Then of course there is Ramanujan, who is in a class of his own when it
comes to discovering theorems without proving them.
I'd be interested to see other examples, and in your thoughts on what the examples reveal about the connection between discovery and proof.
Clarification. When I posed the question I was hoping for some explanations
for the gap between discovery and proof to emerge, without any hinting from me. Since this hasn't happened much yet, let me suggest some possible
explanations that I had in mind:
Physical intuition. This lies behind results such as the Jordan curve theorem,
Riemann mapping theorem, Fourier analysis.
Lack of foundations. This accounts for the late arrival of rigor in calculus, topology,
and (?) algebraic geometry.
Complexity. Hard results cannot proved correctly the first time, only via a
series of partially correct, or incomplete, proofs. Example: Fermat's last theorem.
I hope this gives a better idea of what I was looking for. Feel free to edit your
answers if you have anything to add.
Best Answer
In 1905, Lebesgue gave a "proof" of the following theorem:
If $f:\mathbb{R}^2\to\mathbb{R}$ is a Baire function such that for every $x$, there is a unique $y$ such that $f(x,y)=0$, then the thus implicitly defined function is Baire.
He made use of the "trivial fact" that the projection of a Borel set is a Borel set. This turns out to be wrong, but the result is still true. Souslin spotted the mistake, and named continuous images of Borel sets analytic sets. So a mistake of Lebesgue led to the rich theory of analytic sets. Lebesgue seemingly enjoyed this fact and mentioned it in the foreword to a book, "Leçons sur les Ensembles Analytiques et leurs Applications", of Souslin's teacher Lusin (as referenced in an AMS review of the book).