Too "young" for advanced mathematics?

Question

Is it harmful to try to learn an advanced topic that is way beyond your mathematical maturity, even if you're really interested in it? Should I focus only on standard material for undergraduates and leave the rest to grad-school?

Answer 1

It would only be harmful if you found it frustrating and dry.

Mathematical maturity is important. But there are a few kinds of it. You gain maturity by becoming familiar with subjects. By learning problem solving techniques. By coming to grips with "unmotivated abstraction" -- learning to figure out what a mathematical construct does without somebody giving you the "magic words". You'll need all of these, at the right level, to get a good grade in a class.

But you can gain -- especially in dealing with unmotivated abstraction -- if you study beyond your means. Learning to ask the right questions is a very good thing! Also, a lot of mathematics comes down to memorization -- memorization of definitions, theorems, and techniques. You might not be able to memorize many techniques if you can't try them out, but you can get a big head start on the other two.

But, like I said, stop if it gets boring or too frustrating.

Answer 2

I know I'm late to this party, and I more or less agree with what nomen said. However, it happens that this is related to pretty much the only advice I've ever found useful about studying math, so I thought I'd share.

Ravi Vakil [who happens to be an algebraic geometer] has an advice page which answers your second question strongly in the negative. However, he doesn't recommend "studying" way-over-your-head material in the same way that you are used to "studying" for a class.

Here's a phenomenon I was surprised to find: you'll go to talks, and hear various words, whose definitions you're not so sure about. At some point you'll be able to make a sentence using those words; you won't know what the words mean, but you'll know the sentence is correct. You'll also be able to ask a question using those words. You still won't know what the words mean, but you'll know the question is interesting, and you'll want to know the answer.

Then later on, you'll learn what the words mean more precisely, and your sense of how they fit together will make that learning much easier. The reason for this phenomenon is that mathematics is so rich and infinite that it is impossible to learn it systematically, and if you wait to master one topic before moving on to the next, you'll never get anywhere. Instead, you'll have tendrils of knowledge extending far from your comfort zone. Then you can later backfill from these tendrils, and extend your comfort zone; this is much easier to do than learning "forwards".

(Caution: this backfilling is necessary. There can be a temptation to learn lots of fancy words and to use them in fancy sentences without being able to say precisely what you mean. You should feel free to do that, but you should always feel a pang of guilt when you do.)

Having learned a lot of math this way myself, I'm fully in favor of this advice. I also think it can stave off the boredom or frustration that nomen talks about, so it is somewhat practical to learn this way.