How do mathematicians conduct research?

Question

I am curious as to how mathematicians conduct research. I hope some of you can help me solve this little mystery.

To me, mathematics is a branch where you either get it or you don't. If you see the solution, then you've solved the problem, otherwise you will have to tackle it bit by bit. Exactly how this is done is elusive to me.

Unlike physicists, chemists, engineers or even sociologists, I can't see where a mathematician (other than statisticians) gather their data from. Also, unlike the other professions mentioned above, it is not apparent that mathematicians perform any experiments.

Additionally, a huge amount of work has already been laid down by other mathematicians, I wonder if there is a lot of "copy and pasting" as we see in software engineering (think of using other people's code)

So my question is, where do mathematicians get their research topics from and how do they go about conducting research? What is considered acceptable progress in mathematics?

Answer 1

As far as pure mathematics, you are quite right: there are neither data nor experiments.

Drastically oversimplified, a mathematics research project goes like this:

1. Develop, or select from the existing literature, a mathematical statement ("conjecture") that you think will be of interest to other mathematicians, and whose truth or falsity is not known. (For example, "There are infinitely many pairs of prime numbers that differ by 2.") This is your problem.

2. Construct a mathematical proof (or disproof) of this statement. See below. This is the solution of the problem.

3. Write a paper explaining your proof, and submit it to a journal. Peer reviewers will decide whether your problem is interesting and whether your solution is logically correct. If so, it can be published, and the conjecture is now a theorem.

The following discussion will make much more sense to anyone who has tried to write mathematical proofs at any level, but I'll try an analogy. A mathematical proof is often described as a chain of logical deductions, starting from something that is known (or generally agreed) to be true, and ending with the statement you are trying to prove. Each link must be a logical consequence of the one before it.

For a very simple problem, a proof might have only one link: in that case one can often see the solution immediately. This would normally not be interesting enough to publish on its own, though mathematics papers typically contain several such results ("lemmas") used as intermediate steps on the way to something more interesting.

So one is left to, as you say, "tackle it bit by bit". You construct the chain a link at a time. Maybe you start at the beginning (something that is already known to be true) and try to build toward the statement you want to prove. Maybe you go the other way: from the desired statement, work backward toward something that is known. Maybe you try to build free-standing lengths of chain in the middle and hope that you will later manage to link them together. You need a certain amount of experience and intuition to guess which direction you should direct your chain to eventually get it where it needs to go. There are generally lots of false starts and dead ends before you complete the chain. (If, indeed, you ever do. Maybe you just get completely stuck, abandon the project, and find a new one to work on. I suspect this happens to the vast majority of mathematics research projects that are ever started.)

Of course, you want to take advantage of work already done by other people: using their theorems to justify steps in your proof. In an abstract sense, you are taking their chain and splicing it into your own. But in mathematics, as in software design, copy-and-paste is a poor methodology for code reuse. You don't repeat their proof; you just cite their paper and use their theorem. In the software analogy, you link your program against their library.

You might also find a published theorem that doesn't prove exactly the piece you need, but whose proof can be adapted. So this sometimes turns into the equivalent of copying and pasting someone else's code (giving them due credit, of course) but changing a few lines where needed. More often the changes are more extensive and your version ends up looking like a reimplementation from scratch, which now supports the necessary extra features.

"Acceptable progress" is quite subjective and usually based on how interesting or useful your theorem is, compared to the existing body of knowledge. In some cases, a theorem that looks like a very slight improvement on something previously known can be a huge breakthrough. In other cases, a theorem could have all sorts of new results, but maybe they are not useful for proving further theorems that anyone finds interesting, and so nobody cares.

Now, through this whole process, here is what an outside observer actually sees you doing:

• Search for books and papers.

• Read them.

• Stare into space for a while.

• Scribble inscrutable symbols on a chalkboard. (The symbols themselves are usually meaningful to other mathematicians, but at any given moment, the context in which they make sense may exist only in your head.)

• Scribble similar inscrutable symbols on paper.

• Use LaTeX to produce beautifully-typeset inscrutable symbols interspersed with incomprehensible technical terms, connected by lots of "therefore"s and "hence"s.

• Loop until done.

• Submit said beautifully-typeset gibberish to a journal.

• Apply for funding.

• Attend a conference, where you speak unintelligibly about your gibberish, and listen to others do the same about theirs.

• Loop until emeritus, or perhaps until dead (in the sense of Erdős).

Answer 2

Actually, even in pure mathematics, it very often is possible to do experiments of a sort.

It's very common to come up with a hypothesis that seems plausible but you're not sure if it's true or not. If it's true, proving that is probably quite a lot of work; if it's false, proving that could be quite a lot of work, too. But, if it's true, trying to prove that it's false is a huge amount of work! Before you invest a lot of effort into trying to prove the wrong direction, it's good to gain some intuition about the situation and whether the statement seems more likely to be true or to be false. Computers can be very useful for this kind of thing: you can generate lots of examples and see if they satisfy your hypothesis. If they do, you might try to prove your hypothesis is true; if they don't, you might try to refine your hypothesis by adding more conditions to it.

See also Oswald Veblen's answer which talks about doing similar "experiments" by hand.

Answer 3

Unlike physicists, chemists, engineers or even sociologists, I can't see where a mathematician (other than statisticians) gather their data from. Also, unlike the other professions mentioned above, it is not apparent that mathematicians perform any experiments.

I "gather data" and perform experiments" by working out my conjectures in the context of specific examples. If the conjecture works out in several examples, that makes me more confident that it may be true in general.

For example, suppose that I think that every topological space of a certain form has a particular property. I will start by looking at some "simple" spaces, like the real line, and see if they have the property. If they do, I may look at some more complicated space. Often, when I look at what specific attributes of the examples were necessary to show they had the property in question, it tells me what hypotheses I need to add to make my conjecture into a theorem.

This is not the same as scientific experimentation, nor the same as computer experimentation, which is also important in various areas of mathematics. But it is its own form of experimentation, nevertheless.

Answer 4

One point to note is that, for some questions, it is possible to do experiments to get data. Certain questions are things we now have computer programs to generate, and previously they could have been done on a far more limited scale by hand. So in some cases mathematicians do work more like experimental scientists. On the other hand, once they've found what seems to be a pattern, they change approach. Gathering further examples isn't much use (unless you then find a counter-example, but it can be encouraging) - you need to find an actual proof.

More generally, nearly every big result will come from some 'experiments': you try special cases, cases with more hypotheses, extreme cases that might result in failures...

On the 'copy-and-paste' point, mathematicians do use a lot of what other people have done (generally they must), but whereas you might copy someone's code to use it, when you cite a theorem you don't need to copy out the proof. So in terms of written space in a paper, the 'copied' section is very small. There are (fairly large) exceptions to this: fairly often a proof someone has given is very close to what you need, but not quite good enough, because you want to use it for something different to what they did. So you may end up writing out something very similar, but with your own subtle tweaks. I guess you could see this as like adjusting someone else's machine (we call things machines too, but here I mean a physical one). The difference is that generally in order to do this sort of thing you must completely understand what the machine does. Another big reason for 'copying' is that you may need (for actual theoretical reasons or for expositional ones) to build on the actual workings of the machine, not just on the output it gives.

More to the point of the question: As a mathematician, you generally read, and aim to understand, what other people have done. That gives you a bank of tools you can use - results (which you may or may or may not be completely able to prove yourself), and methods that have worked in the past. You build up an idea of things that tend to work, and how to adapt things slightly to work in similar situations. You do a fair amount of trial and error - you try something, but realise you get stuck at some point. Then you try and understand why you are stuck, and if there's a way round. You try proving the opposite to what you want, and see where you get stuck (or don't!).

Once you have a working proof, you see whether there are closely related things you can/can't prove. What happens if you remove/change a hypothesis? Also, does the reverse statement hold? If not entirely, are there some cases in which it does? Can you give examples to show your result is as good as possible? Can you combine it with other things you know about?

Another source of questions is what other people are interested in. Sometimes you know how to do something they want doing, but you didn't think of it until they asked.

One more point I'd like to make in the 'methods of proof category' is that, for me at least, there's a degree to which I work by 'feel'. You know those puzzles where all the pieces seem to be jammed in place but you're meant to take them apart (and put them back together again)? You sort of play around until you feel a bit that's looser than the rest, right? Sometimes proofs are a bit like that. When you understand something well, you can 'feel' where things are wedged tight and where they are looser.

Sometimes you also hope that lightning (inspiration) will strike. Occasionally it does.

(All of this may not exactly answer the question, but hopefully it gives some insight.)