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Advice on math graduate school (my mistakes!)

After n years of pursuing a Phd in mathematics, where n is greater than or equal to 5, it appears that I may actually be finishing soon. I’m not done yet, but I’m starting to see the light at the end of the tunnel.

For a while, I’ve been wanting to write down some of my thoughts on how a person can go about learning mathematics, navigating mathematics graduate school in general, and eventually doing mathematical research. This last thing is something I am quite new to, so I don’t have too many thoughts in that regard. But I think I’ve learned quite a bit about the first two in these n years. At least, I’ve made quite a few mistakes, and I’d like to think I’ve learned from them.

I will present my advice in list form (I like lists).

1. Don’t be overly rigorous. But don’t be un-rigorous either.

I list this one first, because it was the lesson that took me longest to learn. There is an important balance to strike in learning  mathematics, and not grasping this balance can lead to disaster. Basically, there are two ways to go wrong in terms of how much we try to understand things: we can look at things too closely, or not closely enough.

I’ve always had absolute confidence in my own understanding of what constitutes a mathematical proof. Consequently, when I first started reading mathematical texts, and attending lectures, I insisted on abiding this standard. I would scrutinize every line of my notes; I would rewrite proofs in textbooks until the entire logic seemed clear to me.

As you might imagine, I got stuck a lot. For one thing, the fact is that many textbooks leave a lot out, and no math lecture ever contains all the details. What took me a long time to realize is that this is OK. You don’t always have to have all the details. Sometimes it is best not to fill in the gaps in proofs yourself–this can be horribly time-consuming, and is often less enlightening than you think it will be. What matters most is being able to understand what is going on, to have the right picture in your head so that you can really work with the objects in consideration.

It certainly matters a lot that you do understand what a proof is, and how to really make one yourself. Thus the opposite error, of being un-rigorous, can get you into a lot of trouble too. I get the feeling that some people read mathematical proofs and don’t actually realize that there are steps missing. Then they try to emulate this style of writing themselves–they copy the style without understanding the actual substance that underlies it. I think this is probably the origin of a lot of the nonsense mathematics out there.

So definitely you should know what a proof is, and especially know how to tell when you have proved something yourself. But, if you focus too much on the details of a particular textbook, you will never get far enough to understand why the authors are doing what they are doing anyway. Consequently:

2. Don’t read linearly.

Here is a great way to waste a few months of your life: decide that you are going to sit down with a giant tome (or even a thin little book) on a difficult subject, and read straight through it. I have embarked on this project several times, and have never been successful. Often the first few pages go well. Then it all falls apart.

What I have come to understand is that when people write a mathematics text, they aren’t usually thinking about the time constraints of the reader. They are just outlining the subject in the way that they find interesting. They put things together as they see them–all the examples and theorems that they think are important to developing the subject, or help them understand what is going on. This is a beautiful thing, and a lot can be gleaned from it if you know how to approach the book. But it can also lead you into trouble if you don’t.

I think it is helpful to look at things like this. In a given day, you only really have a finite amount of mental effort you can exert in trying to understand new things. What you are trying to do is maximize the insight gained with respect to this mental effort. So if I see a part of a proof that looks like its going to be a lot like something I’ve seen before (though one should be cautious in making this judgment), I skip it. There is no sense in banging my head on it; its far better to work out the parts that are new to you, that look like they have the potential to give you a new insight into what is going on. Sometimes I give up on trying to understand a proof altogether. I move on to something else. It is OK to do this! It is far better that to waste hours doing practically nothing, all for the sake of some false sense of completeness.

Rather than reading a text straight through, consider just looking at the definitions and major theorems first. Try reading a proof if you think it looks interesting–don’t worry too much about the previous results that it depends on. Take those for granted (but do understand what they say), and if you find yourself interested in them, you can always try to read the proofs later.

Then maybe try some exercises. This brings me to:

3. Exercises are more important for your understanding of a subject than are learning the proofs of major theorems.

Perhaps an analogy will help illustrate this one. Suppose you want to learn how to play golf. What you could do is get a video of Tiger Woods golfing in one of his classic performances. You could then find the exact golf course on which that performance took place, and try to replicate what Woods did: at each hole, you could try to swing exactly as he did in the video. When the ball doesn’t land in the same spot as in the video, you could pick it up, bring it back, and try again. You agree with yourself to move on as soon as you’ve basically replicated the shot that he took.

Of course, this is a ridiculous plan–assuming you aren’t already a great golfer, such a project would probably be endless. And here is the point: it wouldn’t be an effective way to learn how to golf. You’d be much better off by picking up a golf club, and asking someone for advice on how to swing. You could spend a few weeks just taking practice swings. Then you could incorporate putts, or whatever (I actually don’t know how to play golf). It would take a long time, and you’ll probably never get as good as Tiger Woods, but eventually you could get to the point where you are capable of something at least resembling a respectable round of golf. And you’d get to the point where, by watching Tiger Woods, you might actually learn something.

Math is similar. The problem is that, even if we learn what the great mathematicians have done, by going through it step by step, we don’t really learn how to think like them. We don’t learn a lot by examining the details of a difficult theorem if we aren’t yet accustomed to the ideas that go into it. It is good to know what has been done–this is why we should read theorems, and also get a general idea of where they came from. But it’s by doing exercises that we learn how to think like the people who came up with these theorems–it is by doing the exercises that we gain a feel and an intuition for the objects involved, and where we learn what it means to actually try to prove something.

I’d go so far as to say that it is far better to do all of the exercises in a book, and read none of the proofs, than to do the opposite–read each proof carefully and do none of the exercises. Another fun thing you can do is this: when you come to theorems that have a proof that looks short, try to work it out for yourself. Even if you get stuck you gain insight into what the challenge is.

4. Feel free to learn lots of theorems, ideas, and definitions–even if you don’t learn the details.

This is basically just a more general expansion on the advice above about reading textbooks. I think there are some important points to clarify here, and some nuances. There is always a danger of learning nothing but “what has been done”–of becoming an encyclopedia of famous results that, if it really came down to it, you couldn’t reproduce any of the details or even really say what the theorems you can easily quote are actually about. This should be avoided. I’m talking more about have a sense of what is going on in mathematics as a whole.

In my next post, I hope to talk about classes, seminars, and talking to other mathematicians. For now though, I think I should go work on my thesis…

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