I have recently had a chance to talk to some mathematicians and other math students about their research; I would hear them describe to me, in intricate details, the subjects they have worked on for numerous hours. Surprisingly, I realized that some of them succeeded in conveying such a clear picture of their topic that it caught my interest and that I, to some extent, was able to immerse myself in their problems. The experience to me has been like acquiring a new language, except at an accelerated pace. Also, what I didn’t fail to notice was the apparent difference in discourse when I would shift to a non-mathematical conversation with my interlocutors; it was as if I had been using a different language. I had the impression that even the way I was thinking in both conversations had changed, which led me to wonder if there is such a term as “mathematical thinking”? If so, how could it be different from other modes of thinking? Also, if this term is well-defined, could there be variants of it, differences that would be dictated by time and places? Ultimately, would this thinking be substitutable to other ways of thinking?
Some of the mathematical conversations involved establishing new formal languages and axiomatic systems and exploring what can be done with those new systems such as checking what statements can be proved from the axioms; also, different mathematical structures were discussed, where one can ask what conditions are needed to go from one structure to another one or whether such conditions could be reasonably formulated. There were also mentions of different maps between structures and weakening of statements in hope to find some results in case the target problem is too hard to tackle at once. While discussing some of the problems, a sense of uncertainty about the outcome, and maybe some doubt, seems to pervade the atmosphere, which, I think, is the drive to work on those problems. However, conversions about where to have lunch or whether the rain would prevent a planned evening outing seemed to be more relaxed in that one does not seem to consider, say, all the hypotheses needed to consider eating in a Japanese restaurant instead of an Italian or all the conditions needed to transform a night spent indoor because of the rain into something more enjoyable. There was no need to set axioms before we could make claims about the weather, the campus, or the city, nor was there any fear that some joke was provable.
This difference suggested to me, and still does, that modern practice of mathematics (I don’t know if this practice is what mathematical thinking is.) only uses deductive reasoning, where the main concern is whether a statement follows from some assumptions. However, I wonder if mathematics has always been done this way. As far as I know, Euclid used this reasoning for geometry, but were there other contemporary mathematicians, either from Greece or other parts of the worlds, who used different modes of reasoning? If yes, what explains that one way, presumably the one influenced by the deductive method, prevails and tends to be how mathematics is taught? Even in our time, I wonder if mathematics is being done only in the deductive tradition; I would find it interesting if it were not the case.