Are the exercises necessary to understand the subject of a mathematical textbook?

Question

When I am reading a mathematical textbook, I tend to skip most of the exercises. Generally I don't like exercises, particularly artificial ones. Instead, I concentrate on understanding proofs of theorems, propositions, lemmas, etc..

Sometimes I try to prove a theorem before reading the proof. Sometimes I try to find a different proof. Sometimes I try to find an example or a counter-example. Sometimes I try to generalize a theorem. Sometimes I come up with a question and I try to answer it.

I think those are good "exercises" for me.

EDIT What I think is a very good "excercise" is as follows:

(1) Try to prove a theorem before reading the proof.

(2) If you have no idea to prove it, take a look a bit at the proof.

(3) Continue to try to prove it.

(4) When you are stuck, take a look a bit at the proof.

(5) Repeat (3) and (4) until you come up with a proof.

EDIT Another method I recommend rather than doing "homework type" exercises: Try to write a "textbook" on the subject. You don't have to write a real one. I tried to do this on Galois theory. Actually I posted "lecture notes" on Galois theory on an internet mathematics forum. I believe my knowledge and skill on the subject greatly increased.

For example, I found this while I was writing "lecture notes" on Galois theory. I could also prove that any profinite group is a Galois group. This fact was mentioned in Neukirch's algebraic number theory. I found later that Bourbaki had this problem as an exercise. I don't understand its hint, though. Later I found someone wrote a paper on this problem. I made other small "discoveries" during the course. I was planning to write a "lecture note" on Grothendieck's Galois theory. This is an attractive plan, but has not yet been started.

EDIT If you want to have exercises, why not produce them yourself? When you are learning a subject, you naturally come up with questions. Some of these can be good exercises. At least you have the motivation not given by others. It is not homework. For example, I came up with the following question when I was learning algebraic geometry. I found that this was a good problem.

Let $k$ be a field. Let $A$ be a finitely generated commutative algebra over $k$. Let $\mathbb{P}^n = Proj(k[X_0, ... X_n])$. Determine $Hom_k(Spec(A), \mathbb{P}^n)$.

As I wrote, trying to find examples or counter-examples can be good exercises, too. For example, this is a good exercise in the theory of division algebras.

EDIT Let me show you another example of self-exercises. I encountered the following problem when I was writing a "lecture note" on Galois theory.

Let $K$ be a field. Let $K_{sep}$ be a separable algebraic closure of $K$. Let $G$ be the Galois group of $K_{sep}/K$.

Let $A$ be a finite dimensional algebra over $K$. If $A$ is isomorphic to a product of fields each of which is separable over $K$, $A$ is called a finite etale algebra. Let $FinEt(K)$ be the category of finite etale algebra over $K$.

Let $X$ be a finite set. Suppose $G$ acts on $X$ continuously. $X$ is called a finite $G$-set. Let $FinSets(G)$ be the category of finite $G$-sets.

Then $FinEt(K)$ is anti-equivalent to $FinSets(G)$.

This is a zero-dimensional version of the main theorem of Grothendieck's Galois theory. You can find the proof elsewhere, but I recommend you to prove it yourself. It's not difficult and it's a good exercise of Galois theory. Hint: Reduce it to the the case that $A$ is a finite separable extension of $K$ and X is a finite transitive $G$-set.

EDIT If you think this is too broad a question, you are free to add suitable conditions. This is a soft question.

Answer 1

If your goal is to become a research mathematician, then doing exercises is important. Of course, there will be the rare person who can skip exercises with no detriment to their development, but (and I speak from the experience of roughly twenty years of involvement in training for research mathematics) such people are genuinely rare.

The other kinds of exercises that you describe are also good, and you should do them too!

The point of doing set exercises is to practice using particular techniques, so that you can recognize how and when to use them when you are confronted with technical obstacles in your research.

In my own field, two books whose exercises I routinely recommend to my students are Hartshorne's Algebraic geometry text and Silverman's Elliptic curves text. The exercises at the end of Cassels and Frolich are also good.

Atiyah and MacDonald also is known for its exercises.

One possible approach (not recommended for everyone, though) is to postpone doing exercises if you find them too difficult (or too time-consuming, but this is usually equivalent to too difficult), but to return to them later when you feel that you understand the subject better. However, if upon return, you still can't fairly easily solve standard exercises on a topic you think you know well, you probably don't know the topic as well as you think you do.

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If your goal is not to become a research mathematician, then understanding probably has a different meaning and purpose, and your question will then possibly have a different answer, which I am not the right person to give.

Answer 2

Depends on the textbook, I suppose. Some textbooks introduce a lot of material in the exercises that isn't developed in the main text.

Answer 3

I think that the most important point in mathematics is to think about the subject for long periods of time. If you think about mathematics, then you will often develop intuition which is very important. Of course, if you think about something for long periods of time, then your memory of the material is better as well.

Ultimately, the point is that people generally learn more by doing (compare active learning to passive learning). Of course, there are exceptions to every rule and you are the person who best understands your own strengths and weaknesses. The important point is to identify your weaknesses and work hard on them through a combination of active thinking and problem solving.

Answer 4

Of course this is entirely subjective and depends on your intelligence and memory. I'd suspect most all people on this site are high in both areas, at least in logic/mathematics.

Understanding theorems and working through them like you explained is a very good way to understand material, especially if you can keep it in mind when needed. Exercises are usually repetitious, but also can give you some context that you can/should apply the theorems in.

So, no going through exercises isn't a requirement, but going through some is a good idea to both confirm you understand the material and to help commit that material to memory. That said I wouldn't suggest doing the first few exercises (in most textbooks they're the easiest) rather pick a few in the middle or some that the answer or how to solve them isn't obvious to you. Usually the ones toward the end of a section are either tough, long-winded or both. Doing some of those might be worth it, but some might just be long and strewn out and ultimately not worth the time.

This is just my experience with Math Textbooks, but I've only gotten to the undergraduate level, so how true this is at higher levels I don't know.

Answer 5

absolutely you can't truly say you understand something until you work through it

Answer 6

I'm with the OP on this, I skip exercises too.

Here's the logic: A true understanding of maths is about being creative in its applications and not just the material initself.

Exercises, by definition, stifle creativity by presenting a sandbox in which to think.

It's a bit like Rocky 3, do you learn your craft by working out in a gym or by lifting logs down in the forest?

...and another thing, I reckon following examples in books leads to degenerate mathematics. They perpetuate particular styles of thinking about problems.