Doing maths without background knowledge

Question

My mathematics work is proceeding beyond my wildest hopes, and I am even a bit worried - if it's only in prison that I work so well, will I have to arrange to spend two or three months locked up every year?

And so wrote, André Weil from Rouen prison.

Is it possible to do mathematics WITHOUT background knowledge? I do not intend this to be a philosophical/argumentative discussion.

Suppose, I shut myself up in a room and try to do mathematics on my own. Further, I give myself a small problem, namely, to evaluate an infinite series of the form $1+1+......= ?$

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? On the other hand, sometimes research tool such as Google can be counterproductive as all the answers are on your fingertips and one link leads to another making me more scatterbrained.

But, how to get started then? How to discover mathematics on my own?

Answer 1

Well, Ramanujan did it to some extent, but I agree with @HagenvonEitzen in not recommending a clean room approach.

One (of the many) useful thing(s) one can do when studying, though, is to pause before a proof, and ask oneself "How would I do it?"

Or ask oneself questions like "Is the reverse implication true?", "Does this work if I relax the assumptions?" etc.

PS When I was given my first research problem (back in the seventies), my advisor rightly recommended a thorough search in the existing bibliography. All you had at that time was your library, and things like, say, forward searches were pretty difficult. So I am not complaining about having Google, ArXiv, MathSciNet, etc today.

Answer 2

I argue that it is possible; I have only learned high-school math but I have published a paper that used only vector geometry.

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 had a crazy amount of instinct. Besides instinct, of course it would help to avoid dead ends that others have pointed out. The interconnectivity 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.

As for "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).

If 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 (for brevity, see here for 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 PhD'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, hope this helps!

Answer 3

Plato was full of himself. We are surrounded by an immense number of our inventions, from controlled fire and domesticated animals and cooked food and clothing to cities and states and countries and governments and laws and airplanes and trains and streets and buildings and stores and plumbing and electricity and every one of our sciences and you name it. None of these already existed waiting to be discovered like the flora and fauna and geology and history etc. of the earth and the universe.

We INVENTED the objects of mathematics and DISCOVERED their properties. Once created, an object is a thing itself whose nature needs to be discovered. We domesticated dogs but discovered that they can be loyal, intelligent, tolerant, friendly (or not), capable of work, and so on. This is the case with everything we invent.

But "invent" does not mean create out of nothing. Every invention, including in mathematics, is purposeful. We named quantities when they became important to us. We made up symbols for them when we needed shorthand for dealing with many quantities. The same for the arithmetic operations negative numbers and various kinds of fractions and prime numbers and p-adic numbers and functions and everything else.

It took many thousands of years and brilliant minds to invent and discover our mathematics. None of even the most brilliant mathematicians started from scratch; but it might be interesting to try. If you do, keep us informed.