The relevance of knowing "just facts"

Question

When you are new to a research area, whether as a PhD student, a young postdoc or even a more experienced researcher, you have to absorb a lot of information. Of course you have to learn the math behind your new area, but also you have to know the main problems, the main results and the relevant history/literature.

This is something that always puzzled me a bit. When you see a publication or you go to a lecture of a senior researcher, the lecturer usually contextualizes his/her research by citing a lot of other works and authors. At least to me, the feeling is that the lecturer knows in great detail every such cited work: what was done, what was not done, the proofs etc. However, as a young researcher, I sometimes feel a bit overwhelmed with such a huge amount of information in the field. What I mean is that I can oftentimes memorize who proved what and when, but almost in an encyclopedic way. However, it seems almost impossible to keep track of much more than that. In fact, sometimes a single problem was studied a lot of times under different techniques or slightly different hypothesis, and each paper is many pages long. Even if I want to upload every such paper and skim over it, it is beyond me to understand all the ideas and still have time to do my own research. On the other hand, memorizing results seems almost meaningless if I don't know more than what was proved and by who/when.

So, my question is: Is it common for mathematicians to know about important results in their own field only in a superficial way, at least at the beginning of their careers? Is this also relevant knowledge? And are there good practices that allow one to learn a bit more than only facts and still be able to do their research?

Answer 1

The very short answer is yes. :-) For many people learning is not linear, you pick up facts and with time you fill in the gaps, especially if you use them. But sometimes not, for example, no one (more or less) knows the full proof of the classification of finite simple groups, but everyone uses it. In my particular case, I have a terrible patchy memory and health problems made it even worse. However, I work with collaborators that complement my weakness, so this would be my advice to you.

Answer 2

I would suggest the following perspective: Your goal in research is to prove something new. Learning what other people have done is just a means toward that end. In particular:

• Obviously, a result isn't new if someone has already proved it, so before you can claim novelty, you need to check that your result isn't already known.
• It's usually—though not always!—much easier to prove something new if you stand on the shoulders of giants.

So it's not really necessary to know in great detail everything that has already been done. To ensure that the question that you're trying to answer is new, you only need to know what has already been proved, and not how it was proved. As for standing on the shoulders of giants, if you pick a random paper in your field and see what previous work it really relies on (as opposed to citations that just provide context and motivation), you'll typically find that it relies only on a small amount of the existing literature. Of course, this will vary according to the field, because some fields are more technical than others and require lots of background knowledge, but even then, no paper relies on everything that has been done before. It's common practice to cite a lot of prior work, but this is done mostly to provide context and motivation, and to avoid offending people by inadvertently omitting their name, not because the author knows all the prior work in detail.

I typically take something like the following approach. I pick some unsolved problem that I find interesting. Then I try to prove a partial result, using just what I already know. Typically, I will find that what I've proved is already known. In that case, I'll look closely at the paper that anticipated my result, and study it in detail to see how they proved it. Did they prove it the same way? Is my approach new even if the result isn't? Can I extend either my approach or their approach to push the frontier forward? Iterating this process, you should eventually be able to prove something new, and along the way, you'll master some techniques which should serve you well in the future.

Answer 3

Acquisition of knowledge progresses approximately the same for everyone:

1. Encounter new subject, be completely ignorant.
2. Study new subject, become less ignorant.
3. After many years, become one of the least ignorant people on that subject.

Congratulations, you're now an expert. We're all ignorant, just some less than others.