Don’t Build New Theories: Build Actual Knowledge Instead

[1] We all deep down want to be like Grothendieck, Iwasawa, Galois, Langlands. Superhuman mathematicians who erected entire new subfields of this vast, amazing body of knowledge we call math.

[2] But for every subfield builder like Iwasawa, there are a thousand Shinichi Mochizukies, John Gabriels, Norman Wildbergers, etc. who pollute mathematics with utter nonsense that barely qualifies as a set of tools, let alone a whole new sovereign subfield of mathematics.

[3] It just can’t happen. If you truly were to create a new subfield of mathematics, you’d need to find: a new variety of objects to study; methods to study them; a basic set of “cohesion” and “nontriviality” theorems; and actual support from other mathematicians who would approve your work as a new distinct subfield. That’s how mathematics works!

[4] You could certainly get away with proving a new theorem on your own, such as improving a bound or showing X property holds for Y family of sheafs or Z property holds for W family of elliptic curves or whatever. That’s something that would be reasonably accepted with not a lot of scrutiny aside from what’s necessitated at the bare minimum.

[5] Let’s briefly address a critique the reader might bring up in my own work as a detour before we return to our current discussion.

But Harland, you yourself have tried to create a new subfield of mathematics. The “Theory of “Concretization!”

[5] (cont) Concretization “theory” was never intended to be a new subfield of mathematics. It IS a set of methods and theorems that is not a new subfield per se, like surgery “theory” a la Milnor. The “theory” in concretization “theory” and surgery “theory” just stands for “set of tools and theorems.” Not “theory” like Galois theory or Iwasawa theory.

[6] Tangent over. Aside from that, trying to create new subfields of mathematics when there are already so many is the domain of crackpots, not actual mathematicians. See how I brought up Johnny G and Norm Wildberger?

[7] Yeah. The chance is slim-to-none, even if you have a PhD in mathematics. If you want to make a new subfield it will consume all of your time as a mathematician, and in the end you’ll likely never make it anyways. The pool of objects that haven’t yet been given their own subfield such that you could be the one who founds it is so small, it might as well be considered 0.

[8] There’s also a psychological aspect to it. An overly excited crank begins working on a “new subfield” of mathematics, thinking he’s going to get published in Annals or Advances, and be seen as a “lone genius” who created their own field like Mochizuki or Galois.

[9] They imagine the awards, the praise, the esteemed Fields Medal….yet, none of it will actually ever come. Because the idea of a “new subfield” was dead on arrival.