Some personal thoughts and opinions on what ``good quality mathematics'' is, and whether one should try to define this term rigorously. As a case study, the story of Szemer'edi's theorem is presented.
Some thoughts after reading
And indeed it was. The opening has lovely long (though possibly incomplete) list of aspects of good mathematics toward which mathematicians should strive. The second section contains an interesting example which looks at the history of a theorem and it’s effect on several different areas. To me most of the value is in thinking about the first several pages. I highly recommend this to all young budding mathematicians.
In particular, as a society, we need to be careful of early students in elementary and high school as well as college as the pedagogy of mathematics at these lower levels tends to weed out potential mathematicians of many of these stripes. Students often get discouraged from pursuing mathematics because it’s “too hard” often because they don’t have the right resources or support. These students, may in fact be those who add to the well-roundedness of the subject which help to push it forward.
I believe that this diverse and multifaceted nature of “good mathematics” is very healthy for mathematics as a whole, as it it allows us to pursue many different approaches to the subject, and exploit many different types of mathematical talent, towards our common goal of greater mathematical progress and understanding. While each one of the above attributes is generally accepted to be a desirable trait to have in mathematics, it can become detrimental to a field to pursue only one or two of them at the expense of all the others.
As I look at his list of scenarios, it also reminds me of how areas within the humanities can become quickly stymied. The trouble in some of those areas of study is that they’re not as rigorously underpinned, as systematic, or as brutally clear as mathematics can be, so the fact that they’ve become stuck may not be noticed until a dreadfully much later date. These facts also make it much easier and clearer in some of these fields to notice the true stars.
As a reminder for later, I’ll include these scenarios about research fields:
- A field which becomes increasingly ornate and baroque, in which individual
results are generalised and refined for their own sake, but the subject as a
whole drifts aimlessly without any definite direction or sense of progress;
- A field which becomes filled with many astounding conjectures, but with no
hope of rigorous progress on any of them;
- A field which now consists primarily of using ad hoc methods to solve a collection
of unrelated problems, which have no unifying theme, connections, or purpose;
- A field which has become overly dry and theoretical, continually recasting and
unifying previous results in increasingly technical formal frameworks, but not
generating any exciting new breakthroughs as a consequence; or
- A field which reveres classical results, and continually presents shorter, simpler,
and more elegant proofs of these results, but which does not generate any truly
original and new results beyond the classical literature.