Is there a better way to read proofs?

Question

I'm finishing my undergraduate degree in 6 weeks and I'm pretty happy with how my education is coming along so far. I can write proofs, solve many different problems, and I even have some idea as to how mathematics ought to be conceptualized / founded / written.

One thing that's really letting me down, however, is my (in)ability to actually read proofs. I find myself losing track of what is assumed and/or what we're trying to prove pretty quickly, so I end up plodding through sentences word-by-word, symbol-by-symbol, not really understanding a lot of what is going on. So in the end, I can either:

1. Go on to the next section, having not really understood the proof.
2. Reread the proof many times, still not understanding.
3. Painstakingly try to reconstruct the proof myself.

So anyway, I'm currently using approaches 1 & 3, but its slow going and kind of tough.

Is there a better way to read proofs?

Answer 1

Before rewording my comment into an answer, I'd like to draw your attention to a fact people often forget about, and, if taken into account, may (hopefully) somehow reduce the frustration you are talking about. I do assume you are talking about proofs in textbooks (as opposed to research papers, which can be an entirely different story).

Especially when you are reading a book on a well-established topic (like all the standard topics in math: (linear) algebra, (nonlinear) (functional) analysis, measure theory, ...) you need to be aware of the fact that you are looking at a theory which has evolved over years and has been honed over years, often decades, by an army of brilliant minds. Not only the proofs were simplified and streamlined in this process, but also the objects the theories are about. Very often modern mathematical definitions incorporate years of research and insight. They have been, often in a very subtle way, boiled down to extremely efficient formulations which may have absolutely nontrivial results simply built-in.

An example of this is modern linear algebra, in which, as an easy first example, the proof of Pythagoras theorem is reduced to just calculate $||a-b||^2$ using the definitions. A maybe not so well known example is the fact that every odd-dimensional real division algebra is one dimensional (that is, isomorphic to $\mathbb{R}$). With modern linear algebra, every student can show that after one year of studying math. Just note that for a unit $u$ in the algebra left multiplication $x\mapsto ux$ is a vector space endomorphism, hence (cause of odd dimension) has a real eigenvalue. Do you see how to proceed? One more line is missing. Im sure that William Hamilton, who was a brilliant mathematician, would have, if he had known about linear algebra. He did not, instead he spent years of his life to find a three dimensional real division algebra.

There are consequences of this observation, which now, in fact, relate to your question.

First of all, sometimes things are really difficult, and it must not frustrate you too much if you don't understand them on first reading.

Second, and this is the point which I personnally think is the most important one, just reading the proofs will not do you any good. Almost any proof in a modern textbook will be streamlined in order to look nice and be efficient. The very moment you start and try to do yourself you will immediately stumble over those special cases, nasty sidetracks and hidden subtleties the author, in his streamlined proof, has optimized away. And stumbling is a good thing here: you will get to work with the entities the theorem is about! This is the only way to get acquaintet with them (assuming you are not one those very few giftet people who understand these things at first sight, think Will Hunting), which, in fact, is most often much more important than to understand that pesky proof. This is why I gave you the advice I already made in my comment, which might sound arrogant and cruel at first sight:

Try to understand the claim first. Try to figure out why it would be true, if possible, search for examples and counterexamples. Then try to sketch a proof. If it works, write down the details. If that works, too, skip the proof in the book, if not, try to understand what you cannot show. Then look at the proof to find out how they did.

In other words: try to do it yourself. And before you start, make sure you know what the statement is all about. Recap the definitions. Consciously understand the input, and the claimed output, then try to relate them. On the other hand you should always be aware of the complexity you are dealing with, and don't get frustrated if you don't succeed. One third important consequence of my remarks is that often you will not have a chance to find a proof on your own. The author might introduce relevant concepts only in the proof. Nonetheless, if you tried to do it yourself first, you will be able to recognize this and honor the ideas.

Another important aspect is time. You will often need time to realize the hidden implications of mathematical definitions and to familiarize your brain with the claim of a complicated theorem. If you notice that you don't make any progress, just leave it alone for some days. Never believe you can force yourself to understand a statement by sitting down for a certain amount of time and doing ... whatever.

Answer 2

The first question I usually try to answer when reading a proof is "Why does it work?", i.e. I try to figure out the reason the theorem in question is true, and how the proof uses that.

A well-written proof should emphasize that in some way. Though, unfortunately, not all proofs are well-written - in that case there isn't much choice except to try to extract that reason on your own. A good way to start is usually to check how and were the various prerequisits of the theorem are used.

Generally, reading and understanding a proof is easier if once you gain some intuition about the theorem in question. For that, it often helps to look at particular situations where the theorem hold, and particular situations where it fails. It also helps to have some context, i.e. to be familiar with theorems related to the theorem in question. So simply skipping a difficult proof at first isn't the worst choice. It might be that after you work through the parts that follows it, the proof will suddenly seem much simpler, because you have developed a better understanding of the concepts invovled.

Answer 3

I've been doing a lot of number systems and discrete math lately in school. I find it helpful when a proof is hard to understand completely to write out, next to each line, the axiom or previous proposition that allows that line to be written. Afterwards, you get a kind of flowchart that describes the proof. From here, you get an idea on which axioms were pertinent to the proof, which should help for future proofs of the same type.