TITLE: Non-rigorous reasoning in rigorous mathematics
QUESTION [34 upvotes]: I was wondering what role non-rigorous, heuristic type arguments play in rigorous math. Are there examples of rigorous, formal proofs in which a non-rigorous reasoning still plays a central part?
Here is an example of what I am thinking of. You want to prove that some formula $f(n)$ holds, and you want to prove this by induction. Based on heuristic arguments, you conjecture what the correct formula is. Then you prove it by induction. But, if you had just given the induction proof on its own, then you would have to pluck this mysterious formula out of thin air.
I am interested in situations in which there is a heuristic argument which is valid and can be formalized. I am more interested in cases in which there is a heuristic argument and a separate (or complementary) rigorous argument, but the heuristic argument is more enlightening and more explanatory.

REPLY [7 votes]: To me, the quintessential example of the employment of non-rigorous reasoning to arrive at rigorous proofs is Archimedes' On method, where he employs a "mechanical" method, usually called the method of indivisibles, as a heuristic, to arrive at results which he then proves with the method of exhaustion. Even better, Archimedes acknowledges and explains this way of proceeding in the work itself, where he draws the distinction between discovering new results on the one hand (θεωρεῖν), and the subsequent rigorous proving of the results (ἀποδεικνύναι). 
Somehow, this feels remarkably similar to Grothendieck's description of his theory of motives as an instrument of discovery.