Dan PiponiDec 2

It's a blessing and a curse.

I don't feel I understand some mathematics until I think I could have invented it myself and I often work hard to reach that stage. Well that's an illusion so I should qualify it a lot - I need to feel like if I was suitably "primed" I could have invented it. There's no way I could have explicitly invented monads, say, without lots of clues from all the papers I tried to read.

I feel like a good textbook or paper should lead you to a point where you can see what the main "trick" is going to be just before the big reveal. It allows you to develop pattern recognition for the type of problem.

But it does mean I waste a lot of time on stuff people may think is trivial. Machine learning papers are full of derivations like "log(A) = log(A/B) + log(B), now apply Jensen's equality" where B has magically appeared out of a hat. I can easily follow the argument but unless I know why this B was chosen I haven't learnt a reusable skill.

Show threadMark DominusDec 2

@dpiponi I've been complaining for years that mathematical exposition concentrates so heavily on analysis and so little on synthesis. So many expositions have the form:

THEOREM 23: There exists a semispatulated space.

PROOF: Let S be defined as follows: … Then …. Therefore S is semispatulated.

As I wrote that, I couldn't stop thinking of the Munkres presentation of the one-point compactification. Munkres defines a complicated -seeming construction, then defines a one-point compactification and then proves that the complicated-seeming construction is one, leaving many people baffled.

But it doesn't have to be like that! The one-point compactification is very concrete and intuitive!

I have never been a fan of Munkres.

Does he say that the mysterious-seeming product topology is precisely the simplest (that is, coarsest) topology that makes projection mappings continuous? I bet he doesn't. But that's kind of the whole point.

Category theory can be really abstract, but at least when you see a universal property you know immediately what is important about it.

I sympathize with your feeling about the conjugation operation. I _did_ do well in undergraduate group theory, I could prove all the theorems about conjugation, but I didn't understand what it was actually about until much later. And it's so simple! It's just “turn your head”!

Or normal subgroups. I was thunderstruck when, years out of school, someone pointed out to me that a subgroup is normal if it and only if it is a union of conjugacy classes. Of _course_ it is! And I had thought I knew what a normal subgroup was, but I hadn't known that.

Show threadPozorvlakDec 2
@mjd @dpiponi if I knew that characterisation of normal subgroups, I'd forgotten it - but of course it's obvious from the "closed under conjugation" formulation, which in turn you can get from "the map g -> gN is a group homomorphism" without too much difficulty. I remember finding the definition of ideal very puzzling, but again it boils down to "a thing you can take quotients by". And I also found the definition of product topology weird, until (much later) I saw it was a special case of the categorical definition, which generalised the more obvious product operations for algebraic structures.
Show threadMark DominusDec 2

@pozorvlak @dpiponi Right. I think there is way too much emphasis on algebraic definitions (“A normal subgroup H is one where g^{-1}Hg = H for all g\in G”) rather than on what the definition is *for*. When I taught group theory at a math camp in 1987, we taught normal subgroups exactly as you said: “a thing you can take quotients by”.

We introduced the idea of a group quotient _first_—it's intuitive and there are lots of examples—and then said:

“And when can you do this? Look, it doesn't always work, the subgroup needs to have this property: … So when a subgroup does that, we call it a “normal” subgroup.”

Again the point is synthesis, not analysis: Here's an interesting constructions, let's formalize it, now let's see what conditions are required to make it work, and that motivates the definition. Rather than: Here's the definition, now prove some properties and then let's see how that leads to an interesting construction.

Show threadPozorvlakDec 2
@mjd @dpiponi somewhere out there there's a blog post by some Haskell person saying "you can't say what groups *are* other than by pointing at the definition! You can give examples, but ultimately you have to stare at the equations until you've absorbed them". (I think he then went on to say that the same was true of monads, and therefore all burrito analogies were misguided). But that's nonsense! It's very easy to say what groups are! They're *closed collections of symmetries*, both because that's the motivation behind the definition, and because you can realise any group as a one-object category in which that's obviously true. Grrrrrr.
Show threadMark DominusDec 2

@pozorvlak @dpiponi

100%.

Even the name! “Groups” literally started out as “groups of symmetries”, and the important property was _closure_, because the operation was always composition of symmetries, composition is always associative, of course there is an identity symmetry, and of course inverses of symmetries are symmetries. So what makes something a *group* of symmetries? It's closed.

Talking about this somewhere, V.I. Arnol'd said that sometimes people answer this by claiming that the abstract, algebraic definition is more general, but they're wrong… because that's exactly Cayley's theorem, that the abstract definition is _not_ more general. It's just more abstract.

Show threadMark DominusDec 2
@pozorvlak @dpiponi I found the Arnol'd thing about Cayley's theorem: https://archive-dsweb.siam.org/The-Magazine/All-Issues/vi-arnold-on-teaching-mathematics.html
Show threadDan Piponi
@mjd @pozorvlak Oooh...now I have an authority to cite when I say mathematics is a branch of physics even though many mathematicians look at me askance for saying it.
Dec 2 at 11:14pmWeb