Mathematics

When mathematicians make conjectures which they believe to be true but are not yet able to prove, what exactly supports their belief?

Comments

There are a few kinds of support. One is that one can prove certain special cases of the conjecture that seem inherently unrelated, so one thinks that these special cases must really be true because a certain generalization of them is true---and that's what one conjectures. But conjectures are often based upon a dim and hard to express appreciation for "what is going on", so that it just sort of seems as if the thing ought to be true. One can sometimes give reasons to think things ought to work out that way, but they wouldn't be the kinds of reasons that would count as a proof.

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