[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.

[eru]

> 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).

Proper abstract mathematicians view applications not with scorn, but as a challenge to come up with even more abstract math. :)