The Nature of Mathematical Research

It's been some time since I last made a blog post, mostly because I've been fairly busy. I have a handful blog posts in mind for the not-so-distant future involving Fourier analysis, Greens functions and other topics. This post however is going to be a less mathematical and more expository post in order to shed light upon the nature of mathematical research. I've found that people often think mathematics is stale or is not advancing like other fields of study or that mathematical research is akin to number crunching. Both of these notions are flawed and incorrect and perpetuate because of a public education system which does not highlight the beauty and intricacies of mathematics; instead one is taught to apply formulas blindly without developing a truly fundamental understanding of the mathematics at play and it is easy to develop the notion that mathematics is stale and boring. I was very good at mathematics as a kid but I lived within a box, never venturing outside of it, because no one had really pushed me to discover things on my own. It wasn't until the end of my senior year in high school and freshman year of undergraduate that I began to truly appreciate mathematics. My goal isn't to persuade the reader to pursue mathematics (though that would be a nice unintended effect) but instead to explain what mathematical research truly is, thereby explaining what mathematics truly is.

At the most fundamental level, mathematics derives from axioms - things we assume to be true that cannot be proved. We postulate that certain things are true and build up mathematics upon that. Without making initial assumptions about how number systems work or, more generally, how sets work, there is no meaningful structure to work with. It would be akin to placing van Gogh in a pitch black maze with an easel in one corner of the maze, placing oil paints in another corner of the maze and asking him to not only find the easel and oils but to proceed to paint Sunflowers in the pitch black of the maze. Without the axioms, mathematics cannot be done and so we make assumptions about how certain objects work and can then build up theorems, lemmas, corollaries and propositions. Axioms are like the frame of a building and mathematics is the building itself. Without the frame, the building cannot exist.

Most research being done in mathematics does not operate at the level of axioms (like in axiomatic set theory) but there are still many mathematicians working at this level. Most mathematicians do research in mathematics at a much higher level but still rely on the axioms underlying mathematics. Despite mathematics being based upon logic, there is an immense amount of freedom to explore. To develop an understanding of how mathematics is done now, it is instructive to look at modern mathematics in its early years.

As I stated in my post "The Importance of Mathematics", calculus was not on solid foundation for over one hundred years, all the while many mathematicians were devising ways to make it rigorous. Until the 1900s, a lot of mathematicians and physicists shot from the hip and took many things for granted mathematically. While it turned out that a lot of these men and women had the right idea, they took liberties with rigor in order to obtain results. Newton and Leibniz both used the calculus tools they developed to solve mathematical and physical problems before we even had a rigorous definition of a limit; Fourier used infinite series expansions in terms of exponentials before such expansions were made rigorous. Both of these leaps of faith ended up giving physically relevant results and provided the basis for a lot of rich theory to explore. Mathematicians like Cauchy and Weierstrass provided the foundations for real analysis and helped start a more formal treatment of advanced mathematics. For over one hundred years, some of the world's most brilliant men toiled with something as simple as the definition of a limit.

With the definition of a limit, analysis quickly took off and became a fully-fledged field of mathematics in its own right. As rigor became the standard for advanced mathematics, mathematics bloomed because paradoxically the somewhat rigid framework of rigor actually provided freedom. Mathematicians had a clear idea of exactly what the constraints they were working under were and having basic definitions and axioms established paved a way to constructing the building from the frame.

In mathematics (like other fields of study), there are different approaches. Some choose to tackle open problems like the Millennium Problems which have a lot of implications for mathematics, some choose to find a much smaller niche and look into very specific problems which are less profound but still very important, others still observe a general framework and break it apart to find its defining features and build an even more general framework from that. Personally, I feel like I fall into the third category.