a healthy perspective on "knowing everything"

Question

This is a question about attitude, but related to math studies. I have trouble with two things: 1. making "normal" progress in my learning and 2. having the satisfaction that I understand what is going on.

My experience in university (where "normal" progress occurs) suggests that at least 50% of the math learning is just "remembering". My feeling is that nobody spent any significant amount of time trying to prove the main theorems in the book on their own. The main theorems were read, remembered, and then used in exercises. But doesn't this kind of education violate the ethos of the activity? If I don't know why a main theorem works, is it kosher to keep reading and developing more theory on top? This seems to be "what people do" and how learning is done.

When I approach my math education, I want to hold myself to the standard where I fully understand the ins and outs of the main ideas before I proceed. But this slows progress significantly. I get the feeling that I should just accept that there are things that I won't have time to understand that I should nevertheless believe. But "believing" without "understanding" makes me uncomfortable...

I would appreciate input on this.

Answer 1

It depends on what level of math you're in. In a course, you should know why a math theorem works, which can mean anything from reproducing its full proof to remembering the main idea of it to remembering whatever clever tricks it involves, depending on its importance and complexity. If you're doing math research and publishing results, you should have a better understanding of what results you're invoking (which doesn't necessarily require building them up from scratch yourself). The point of math is to understand the subject and prove more results, not merely to snap other people's results together like Legos and treat the subject as a black box to solve equations or analyze experiments with. In a course, you're definitely responsible for understanding the proofs of results, rather than just being able to regurgitate and apply them.

That having been said, I freely use the Feit-Thompson theorem, for example, despite unable to tell you much about it besides the final result. Some theorems are quite involved but manageable in a class: the $h$-cobordism theorem and Mostow rigidity, for example, are usually covered in classes on their respective subjects, although they take many sessions to prove. But that's fine; the point of those classes is to understand more about the subject than just the packages results themselves, and knowing the proof allows you to prove related things. The Whitney trick, for example, involved in the proof of the $h$-cobordism theorem pops up everywhere in high-dimensional topology.

Answer 2

We are always "chunking" knowledge—grouping small results into a unified, memorable result while often forgetting the pieces themselves. I learned how to invert a matrix by hand many years ago, but for the last several decades have used software to invert matrices, never performed it by hand.

Think of driving a car. Do you really need to know how a catalytic converter works to drive a car? Of course not. But if you're designing and engine, well then certainly.

So the level at which you "chunk" or group your knowledge depends upon the tasks at hand. If you're an applied mathematician, you will more likely profit from having great breadth of knowledge, i.e., knowing what theorems and techniques might be useful in your problem. If you're a theoretical or pure mathematician, then you will more likely profit from having great depth of knowledge. (Of course the greatest mathematicians have both.)

Moreover, you may find that just using your mathematical knowledge will lead to greater understanding. I studied multi-variate calculus in high school and did fine, but (like you) there were several results that I "merely memorized" and couldn't fully explain. However, when I used this mathematics in physics class, then I understood the math.

So... don't worry that you don't understand every element of all the math you learn. Some of that math you'll never use, but the math that you do use, you'll gain deeper understanding from using it.