Well, the mathematical method is pretty much typified by its propensity to engage in derivation of equivalent statements. This is simply what people do when they construct a "proof", which is of course not a proof, but that is another issue altogether. Alonzo Church says that derivation can entirely be done by (1) renaming (=alpha) (2) application (=beta) and (3) substitution (=eta). I think that Alonso Church is right. That summarizes the only technique that pure mathematicians seem to be capable of, which is indeed a bit poor. So, I agree with Dijkstra: don't go into computing if that is all you can do.
Just like an artist only has the techniques: move pencil horizontally, move pencil vertically, move pencil toward or away from the paper. The art comes from knowing what to draw and how to combine these steps based on a vision in your head. And the real part of pure mathematics is envisioning new structures and new relations that nobody has used before, at which point the proof itself can become a detail. IT's not through lack of trying that formal computer proof systems haven't been able to touch a milli-part of modern mathematics
> We are all taught that “proof” is the central feature
of mathematics, and Euclidean geometry with its careful array of axioms and propositions has provided
the essential framework for modern thought since
the Renaissance. Mathematicians pride themselves on
absolute certainty, in comparison with the tentative
steps of natural scientists, let alone the woolly thinking
of other areas.
> It is true that, since Gödel, absolute certainty has
been undermined, and the more mundane assault of
computer proofs of interminable length has induced
some humility. Despite all this, proof retains its cardinal role in mathematics, and a serious gap in your
argument will lead to your paper being rejected.
> However, it is a mistake to identify research in mathematics with the process of producing proofs. In fact,
one could say that all the really creative aspects of
mathematical research precede the proof stage. To take
the metaphor of the “stage” further, you have to start
with the idea, develop the plot, write the dialogue, and
provide the theatrical instructions. The actual production can be viewed as the “proof”: the implementation
of an idea.
> In mathematics, ideas and concepts come first, then
come questions and problems. At this stage the search
for solutions begins, one looks for a method or strategy. Once you have convinced yourself that the problem has been well-posed, and that you have the right
tools for the job, you then begin to think hard about
the technicalities of the proof"
