Amateur Math Research

Nearly two months ago, I stumbled across this question on Math.StackExchange:

Is it possible to do mathematics WITHOUT background knowledge?… Is it humanly possible to do mathematics on own without research or is the information content too much to discover identities, or methods of proofs on one’s own?… But, how to get started then? How to discover mathematics on my own?

I wouldn’t say that I’ve been “discovering mathematics”, but I definitely have been playing around with many of my original math research ideas, some of which I am happy with. So I introspected a little about my amateur math explorations and how my experience can help people who ask this question. Here is my slightly edited response.

Playful construction is fundamental to research in all fields.

Playful construction is fundamental to research in all fields.

Math research without background knowledge

I argue that it is possible; I have only learned high-school math but I have published a paper that used only vector geometry. I  am also currently working on a research paper that uses, for the most part, basic functions and set theory.

In my (not so well-formed) opinion, mathematical research ability consists of:

  1. (Direction) Initiative to ask new questions, and good questions, charting one’s own direction
  2. (Technique) Foundational skills in commonly-used techniques (e.g. Mathematical Rigor, Arithmetic, Algebra, Geometry, Calculus, any other routine skill – even Programming)
  3. (Problem-solving Skill) The ability to recognize which tools can be applied to attack a problem, view a problem from different angles, as well as persevere through difficult problems.
  4. Past experience in math research

Of the three criteria above, only (2) directly depends on background knowledge. I argue that some research can be done using only skills (1) and (3). However, they are not completely independent of background knowledge, which places a limit to how much can be done. (4) is an issue of honing all (1)-(3) to muscle memory, so I won’t discuss that.

(1): Asking good questions depends to some extent on a wide background – if you know a lot of what already has been done, you can form an “instinct” as to what approach would most likely work. Ramanujan – a first-rate mathematical genius – had a crazy amount of instinct. Besides instinct, of course it would help to avoid dead ends that others have pointed out. The inter-connectivity of Mathematics also frequently allows insights from one field to apply fruitfully in another – such as the application of Elliptic Curves to Fermat’s Last Theorem.

(3): Seeing how others attack problems allows one to learn crucial proof techniques, such as Diagonalization in Set Theory or Problem Reduction in Computational Complexity. In addition, there is a limit to how much problem-solving experience one can have if one does not have many tools (2) to begin with, and one can’t view a problem from many angles if one doesn’t have many angles (2) to begin with.

So, my conclusion for “can research be done without background knowledge?” is:

It is possible to do some research in a certain field provided that one is somewhat familiar with the techniques of that field, and one is also armed with some skill in (1) and (3). However, the scope of the research will be severely limited without a wide background knowledge, especially with regards to the literature review in that field. It will be especially difficult to do anything outside that field.

How do I start?

I suggest that you simply play with the things you learn, which trains (1). When trying to act on the suspicions or goals you have set up, you will naturally have to work on (3).

I may draw out an example from my own limited experience, I began playing with ideas shortly after I was exposed to matrices; I loved it so much that I simply tried to represent everything as matrices and find analogues for matrix operations (now i know this is “finding homomorphisms”).

When I learned complex numbers, I represented them as scaled rotation matrices, and that representation proved rather fruitful (a previous blog post has full details). Eventually I moved on to a homomorphism from complex matrices to real matrices, at which point I realized I had reinvented a wheel.

Play around with ideas, derive interesting things and train (1) and (3), but never stop learning new skills and background knowledge! That way, the moment your (2) shapes up, your (1) and (3) are fully ready to gun down problems with the new ammunition. Meanwhile, you can supplement (2) with Stack Exchange.

However, doing original research in an established field is virtually impossible without rising to the edge of what is already known in the field (that’s what Ph.D.’s are for). Basically,

What is easy and interesting has usually been done before.

However, if you manage to find a niche where nobody is paying much attention (e.g. Classical Geometry, correct me if I’m wrong), it’s highly possible to do good original research with a strong (1) and (3).

Happy research, and I hope this helps!

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