Understanding mathematical texts

Question

Please could you comment on following:

I always wanted to know what mathematicians mean by "understanding a piece of mathematics". For example, I have just finished the second chapter from Rudin's Real Analysis. I can recite all theorems and their proofs in my head. But maybe not for long. Does this mean I understand the chapter? There are not many articles on the web on this, and I think the most interesting opinion on the subject is by H. Poincar\'e who in his Essay On Mathematical Creation writes:

... Why then does it not fail me in a difficult piece of mathematical reasoning where most chess-players would lose themselves? Evidently because it is guided by the general march of the reasoning. A mathematical demonstration is not a simple juxtaposition of syllogisms, it is syllogisms placed in a certain order, and the order in which these elements are placed is much more important than the elements themselves. If I have the feeling, the intuition, so to speak, of this order, so as to perceive at a glance the reasoning as a whole, I need no longer fear lest I forget one of the elements, for each of them will take its allotted place in the array, and that without any effort of memory on my part...

Evidently the "certain order" in particular piece of mathematical text is what one should be reaching for in order to be able to claim understanding of the material.

How do I reach it? Like I said, I doubt that this kind of "understanding" is equivalent to working out all proofs to theorems and even solving all problems at the end of the chapter.

Answer 1

There are many ways to build better understanding.

One way that is important is to work through many examples of concepts, and as you do so, to connect them to the theory.

Just learning the theorems, and the various syllogisms etc. that connect them, is not sufficient (for most students) to yield a real understanding.

Similar, just working problems in (mental) isolation, is not sufficient. You have to think of the problems as a guide to developing your understanding of the theory, and at the same time, use the theory as a guide to understanding the various examples provided in the problems. When you see a phenomenon in an example, see if you can relate it to one of the concepts defined in the theory. If you make a computation, see if you can abstract it --- perhaps the conclusion of the computation has a more theoretical explanation, in terms of some of the concepts involved in the theory you are studying.

(E.g. in an analysis book, in an exercise you may prove various inequalities in some context. You should try to think whether there is some underlying phenomenon that explains them all; e.g. maybe there is a linear operator in the background, and the various inequalities are a consequence of it being bounded, or maybe bounded with bounded inverse, or maybe compact, or ... .)

When you read or listen to a lecture on the subject you are trying to understand, say analysis, and the text/lecturer says "Consider a measure space $X$", you don't want the letter $X$ to appear in your head. What you want in your head is all your cumulative experience with measure spaces, all the different measure spaces you know, the properties each of them do and don't satisfy, and so on. As the lecture/text continues, the examples you are holding in your mind should change in a fluid way as various hypotheses are imposed or removed. In this way, you should be following the argument in a conceptual manner, and so hopefully understand the various steps, what their meaning and consequences are, why they are being applied, and how they are furthering the argument.

This doesn't have to happen literally; for those with a lot of mathematical experience it's almost subconscious; until the lecturer they are listening to states something false, at which point the experienced members of the audience can immediately call out a counterexample --- because they have an intimate grasp of the properties and key example of the objects being discussed, and so can immediately recognize if the lecturer oversteps the bounds of reality, and produce an example illustrating this.

When you are beginner, you need to do this in a more conscious, and probably slower, way. As you read the statements of theorems, write down examples you know of that satisfy the hypothesis. Do your best to check, using the tools you know, that they satisfy the conclusion. Already this may help give you a sense of the difficulty of the statement. Sometimes your check will actually not use the particular properties of your example, and you will discover the general proof; other times this won't be the case, and then you will realize that there are logical connections out there that you can't see yet, and so (as you begin to study the proof) you can try to look for where these connections are being made. Then you can go back to your example: is the theorem easier to verify for your example than in the general case? Why? What properties does your example have that aren't hypotheses of the theorem? Can you state an easier, or maybe different, theorem that is more tailored to your example?

The more you do this, the more intuitive it becomes, until eventually it becomes an automatic process, indeed probably the dominant process, in your reading and listening to mathematics. At that point as you listen to or read a proof, you understand in a deep way how the pieces of the argument fit together and what their role is, which is the kind of understanding Poincare is describing.

Answer 2

This isn't a complete answer by any means.

A couple of weeks ago there was a conference based on the work of William Thurston. There were several references made to an idea he used (I believe) of having levels of understanding. When you first meet something, you can read the theorems, and get a first level of understanding. But as you come back to it, in different contexts, seeing it from different points of view etc, you gain more and more insight.

Answer 3

In my experience learning mathematics is a lot like learning a language. You need that basic vocabulary, but in order to really have a conversation you need a deep understanding of what all the words really mean and how they fit together and interact, and all the subtleties therein.

Once we are proficient at a language we no longer worry about what each individual word means, rather we understand sentences as a whole. So now when someone says "How's it going" we take that at one piece of language, rather than three separate words.

For me mathematics has been the same. While in undergraduate studies I would always need to look at all the pieces of something to know what's going on. But in my MSc year I started noticing that I was starting to think about the areas that I studied in a much broader sense. Rather than having to drudge through all the details in my mind I could call up the relevant parts and leave the other bits alone, confident that they were there somewhere, but ok with not having them at the front of my mind because I knew how it all fitted together. This I guess is the type of understanding that you are asking about here.

So here is my answer to your question: This understanding comes from good old-fashioned hard work, repetition and lots and lots and lots of working through examples and approaching the ideas from different viewpoints and directions. It is only in my fifth year of university mathematics that I have even started to feel I have started to have this type of understanding, mathematicians spend their whole lives sharpening it and getting better. But there is no great secret, just like learning a language you start with the basics and then practise, practise, practise.... and one day you may find yourself writing poetry in this new language.

Answer 4

I think what Poincare calls "certain order" can also be called the mathematical idea (behind the subject you are studying). To understand a mathematical idea, the following items are important:

1. A mathematical idea is a dynamic creature and it continually evolves according to its applications. Take for example "continuity". It started as a notion for functions from $\mathbb{R}$ into $\mathbb{R}$ and its definition has extended to every function between two topological spaces. In fact, the main motivation for development of topology is to address the idea of "continuity" in its more general sense.

2. One should look for the preceding challenge that has raised the necessity for certain theoretical explanation and subsequently gave birth to the new idea. For example, in Euclidean geometry all curves (and lines) are implicitly assumed to be continuous curves. For instance, when we consider two arcs meeting each other at an intersection point we implicitly assume continuity. To see how continuity is used, note that the equation $x^2 = 2$ has no solution in rational numbers (irrational number did not exists at the time), but it is assumed that the curve $y=x^2 -2$ intersects the $x$-axis in two points. This conceptual challenge and its solution (or theoretical explanation) gave rise to several fundamental ideas of mathematical analysis such as continuity, real numbers, etc.

3. For understanding a mathematical idea, sometimes it is helpful to clarify the relationship between the idea at hand and other mathematical ideas. For example, to understand why groups are defined the way they are defined, considering groups as the symmetries of a geometric object is extremely helpful. surprisingly, this realization of groups as symmetries of geometric objects not only helps beginners to understand groups easily, but it is also a fundamental idea used by experts in group theory. In fact, the subject is called geometric group theory.

4. Never expect you understand a mathematical idea fully at the first encounter. It takes time to get a feeling about an idea. Therefore you should consider the learning of the formal definition and basic properties of a mathematical notion as the first step and you should develop and further your understanding gradually. I remember, first time I read the definition of a category, I memorized it but I had no feeling about it. However after a while, the definition of a category was so natural for me that I could not imagine another definition for it.

5. A useful practice to understand a mathematical idea (or subject) is to discuss it with others or even better give an informal (or semi-formal) presentation about it. Some people even teach a course to extend and deepen their understanding of a subject.

These are fundamental questions that how we understand something and how we can facilitate our understanding of a subject (in mathematics or other fields). So the above items are just some basic comments.

Answer 5

Not an answer but more of an extended comment: I tell my students to forget the textbook at first, just try to solve the exercises/problems and then if you get stuck at a problem then go read the textbook. It is a deliberate exaggeration but the point is that reading a book or article will give one a false sense of understanding. Only when we get stuck and discover we don't know how to use the book results then and only then we start real understanding.