The question is not even "How do #mathematicians make progress in #mathematics?" Rather, as a more explicit (and leading) form of the question, I prefer "How do mathematicians advance human understanding of mathematics?"
This question brings to the fore something that is fundamental and pervasive:that what we are doing is finding ways for people to understand and think about mathematics. The rapid advance of computers has helped dramatize this point, because computers and people are very different. For instance, when Appel and Haken completed a proof of the 4-color map theorem using a massive automatic computation, it evoked much controversy. I interpret the controversy as having little to do with doubt people had as to the veracity of the theorem or the correctness of the proof. Rather, it reflected a continuing desire for human understanding of a proof, in addition to knowledge that the theorem is true. On a more everyday level, it is common for people first starting to grapple with computers to make large-scale computations of things they might have done on a smaller scale by hand. They might print out a table of the first10,000 primes, only to find that their printout isn’t something they really wanted after all. They discover by this kind of experience that what they really want is usually not some collection of “answers”—what they want is understanding․
#math http://arxiv.org/pdf/math/9404236v1.pdf
