[spekcular]

This is not bad advice, especially for people new to proof-based mathematics. (I've noticed more mathematically experienced people do something like the linked method intuitively, without writing things down.)

But, it's only half the story. After you learn the definitions and theorems, you have to learn how to apply them to do computations and solve problems. This means working at least a few "easy" problems to learn how to crank through rote computations, and a few harder ones to learn how to think through novel applications.

If you can't solve problems with the material you've supposedly learned, you haven't actually learned it. (Otherwise, what did reading all that stuff really accomplish? You picked up some cool vocabulary words?)

[vymague]

> This is not bad advice

I was going to say this article was a bad advice.

I second the solving problems approach. It's pretty mainstream opinion really, at least among math/phys students. You can try to follow the article and feel you "understand" the topic. And still unable to solve any of the homework problems, let alone exams.

[verytrivial]

Reading is necessary but insufficient, but you can at least make it efficient -- that's what the article is covering I think.

[Trex_Egg]

I second this

[jvvw]

One thing that is hard is that not all areas of mathematics have obvious easy problems to work through or rote computations, although you should clearly do that if those exist. I do think problem-solving and 'getting your hands dirty' is important, but you are also unlikely to manage to reinvent hundreds of years of mathematics by yourself so there is value in reading too. You do also need to learn the vocabulary that other mathematicians use.

When reading textbooks, I would often try and prove theorems myself before looking at the proof - and when I looked at it because I was stuck I would just try and see what the next insight or step and go from there.

I suppose I am thinking somewhat of reading maths papers here which don't come with a nice set of exercises, which probably isn't what we are talking about - but mathematics does move gradually in that direction from areas where the routine computations are laid out to areas where you have to work out for yourself how to make the abstract seem less abstract and where there are assumptions that you will fill in lots of details yourself.

[GuB-42]

It seems like proof-based maths and applied maths are different fields. The latter is more closely related to engineering.

I am the kind of person who likes to solve more practical problems but it is more of a hindrance in advanced maths, where you manipulate concepts that are too abstract for practical use, like infinities.

[spekcular]

Especially in abstract mathematics, you must practice computations with basic examples (e.g. computing some homotopy groups of spheres). If you cannot do this, you haven't actually grasped the mathematical content of the reading. The abstractions exist precisely because they are concise, powerful ways to deal with various examples.

[eru]

I agree with you, but I think you might misunderstand what GuB-42 means by 'practical'?

Computing some homotopy groups of spheres is a good simple exercise for the right kind of abstract math. But probably not 'practical' by GuB-42's standards?

[spekcular]

It is a little hard for me to interpret their comment. I took it as meaning that the learning strategies should differ for pure and applied mathematics, hence my response.

Maybe another way to say this is: pure math is just a kind of applied math where the applications are resolving theoretical problems. Essentially all of the big mathematical programs/fields/whatever were created to solve or understand some Big Central Theoretical Problem(s), and prove their worth by continuing to be useful in solving other problems. And generally these problems can be understood in terms of concrete examples.

Even Grothendieck, perhaps the canonical example of a "theory builder," had resolving the Weil conjectures firmly in mind while writing his famous texts (and then got annoyed at Deligne for doing it the "wrong way"; see https://webusers.imj-prg.fr/~leila.schneps/grothendieckcircl...).

[garmaine]

That's an explanation that only makes sense from the perspective of pure math. Many of us engineers are wholly incapable of caring about the resolution of theoretical problems. Learning how to take a real world problem and map it into the domain of theory is a special skill that is not required for pure math, because it has an intrinsically messy interface into the real world.

I suspect that a pure mathematician would look scornfully at that as a waste of time, as the whole point is the math doesn't depend on the particular instance. An applied mathematician / engineer on the other hand sees no point in the mathematics unless it is manifested in the form of physical problems.

[spekcular]

I don't think mathematicians look scornfully upon people doing the mapping between applied and theoretical domains. On the contrary, I think there's a lot of respect (and perhaps some envy). It's an entirely different set of skills, and a lot of great math has been inspired by applications, which would never have been possible if pure mathematicians worked in isolation. (The respect/envy comes from the fact that most mathematicians don't have those skills but recognize their value.)

I'm thinking in particular of a lot of stuff in mathematical physics and PDEs.

[mkl]

As an applied mathematician, I disagree. Proofs are often useful in applied maths, and most applied maths is much more closely related to pure maths than to engineering. Infinities show up in practical applied maths, but there are more abstract things that don't (yet).

[eru]

Yes.

Though just like libraries in programming mean that you don't have to worry about how to implement a hash table when using Python's dicts, advances in math mean that you don't have to worry about infinities and infinitesimals when you are constructing a bridge using differential equations.

That's progress!