Mastodawn

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.

Shae Erisson Dec 2

@dpiponi I'm feeling that as I slowly learn group theory with much help from @mjd
We gave up on two(?) documents before Mark found one with much better motivation and fewer of the surprises you describe.

Dan Piponi Dec 2

@shapr @mjd Yeah. I think proofs can play many roles. Sometimes they're pedagogical. But sometimes they're just slick because slick proofs are an art form. And sometimes you want a proof that is just a proof, ie. something to make you confident it's true.

Good luck with the Group Theory. I struggled with that for a while as an undergrad. Then one day I woke up and realised that conjugation, ie. forming a*b*inverse(a), was a technique I'd used for many years with Rubik's cube. A lot came together after that.

Mark Dominus Dec 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.

Pozorvlak Dec 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.

Dan Piponi Dec 2

@pozorvlak @mjd A good one is the "compact-open" topology about which I never developed any intuition at all but is conveniently exactly what you need to make the proofs work.

(I just had a look at wikipedia and it gives a characterization that is better than anything I've read before and I wish I'd seen it 40 years ago!)

Mark Dominus Dec 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.

Pozorvlak Dec 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.

Mark Dominus Dec 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.

Mark Dominus Dec 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

Dan Piponi Dec 2

@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.

Brent Yorgey Dec 2

@pozorvlak @mjd @dpiponi It sounds kind of like you're referring to my blog post: https://byorgey.github.io/blog/posts/2009/01/12/abstraction-intuition-and-the-monad-tutorial-fallacy.html , but I don't recall saying anything about groups, and (if you are indeed referring to my post) I think you've mischaracterized it. I was writing against tutorials that tried to *avoid* technical detail by just giving analogies; I was certainly NOT advocating for a purely formalist style of presentation! Indeed, I try to frontload my own teaching with motivation/intuition/synthesis, just as you all have been discussing.

blog :: Brent -> [String] - Abstraction, intuition, and the "monad tutorial fallacy"

Pozorvlak Dec 3

@byorgey nope, I'm definitely thinking of a post that talked about groups - I see nothing to disagree with in yours! Saying "groups are closed collections of symmetries" wouldn't be enough on its own either. @mjd @dpiponi

Martin Escardo Dec 2

@dpiponi writes "I don't feel I understand some mathematics until I think I could have invented it myself "

+1

The only way to understand something, at least in my case, is to reconstruct it. And often this involves doing it my own way.

@dpiponi I feel you! It's the same for me... I always feel kind of stupid if everyone during class nods along and I am just sitting there, questioning basically every other statement of a proof and wondering why that's supposed to be true.

Dan Piponi Dec 2

@climbingVectorspace I never understood lectures in mathematics as a student. If I ever went, I took notes, which were sometimes usable later, but it all seemed pointless except to find out what topics were covered.

Mark Dominus Dec 2

@climbingVectorspace @dpiponi If everyone else is nodding along, it's often that they don't understand either, and they are just too ashamed to speak up.

A big turning point in my life was when I decided to sit in on a graduate category theory seminar. I was baffled by everything and held everyone up by asking a lot of clueless questions. I was going to drop out, but one day after class, one of the grad students thanked me for asking the questions he had been to shy to ask, and then several others said the same.

And that's when I realized that that's part of my job in life, to be the person who admits to being baffled in those settings, because as a white man I am much less vulnerable to other people's contempt for it.

I keep the book from that seminar on the shelf, so I can take it down every few years and marvel at how it manages to make even the simplest things obscure.

@mjd @dpiponi That sounds like really good advice; thanks! Maybe I just have to start asking more questions!