my old tribe

There was one more thing I meant to port over from the old tumblr and forgot: a list of what I loved about my old research group. It was never under the ‘mathbucket’ tag, but goes a long way to explaining what I care about in maths and physics, what I missed horribly when I left and what I’m working towards finding again.

  • Everyone is interested in a wide variety of things – other areas of maths and physics, other academic subjects, various sports and arts and hobbies. Nobody expects you to just be narrowly focussed on learning about your specialism.

  • Getting better at these things is valued. A little bit of bragging is alright as long as you don’t get too obnoxious about it.

  • Helping other people get better at these things is valued. Being able to explain your work in plain language is valued. Writing a clear paper, giving an entertaining talk or writing for a nonspecialist audience are all considered worthwhile, as well as technical competence in your own research area.

  • (This one’s important) An almost total absence of that competitive one-upping thing where everyone spends their time proving how much smarter they are than everyone else, or looking down on other subdisciplines as less important/fundamental/difficult/rigorous than their own. This is all over the place in physics, I hate it, and I was very very lucky to avoid most of it.

  • Playfulness, silliness, gurning, stupid repetitive injokes, awful songs played over and over again, pointless fun distracting projects with absolutely no relevance to anyone’s research.

  • A kind of glorying in being stubbornly independent-minded and prepared to defend your own stupid opinion. ‘That is bollocks and I will tell you why.’ But always grinning as you say it, and sometimes you discover that it isn’t bollocks and admit you’ve changed your mind.

(Disclaimer I added a bit later: I’m not saying it was a perfect fit for me. It was more undisciplined and structureless and anarchic than I really knew how to deal with, and I was very lazy and unfocussed a lot of the time. This stored up problems for me in the long run, and finishing on time was a miserable ordeal. But there was a lot of good there.)

Advertisements

I’m a bricoleur scientist

I’ve just read a fascinating paper, ‘Epistemological Pluralism and the Revaluation of the Concrete’ by Sherry Turkle and Seymour Papert. I’m lucky that I only found the paper recently: I love Papert but I’m not sure I’d have been able to stomach it even two years ago. The very first paragraph manages to combine a couple of ideas I’m seriously allergic to:

The concerns that fuel the discussion of women and computers are best served by talking about more than women and more than computers. Women’s access to science and engineering has historically been blocked by prejudice and discrimination. Here we address sources of exclusion determined not by rules that keep women out, but by ways of thinking that make them reluctant to join in. Our central thesis is that equal access to even the most basic elements of computation requires an epistemological pluralism, accepting the validity of multiple ways of knowing and thinking.

So, first of all, this is a paper about Women In STEM, considered capitalised as an Important Social Issue. Being lumped in with my gender automatically puts me on edge, as I tend to assume that I’m not going to fit in very well.

Then we have the phrase ‘ways of knowing’, which I’ve sort of unfairly come to associate with the worst of pomo nonsense. Like that anthropology course my flatmate did, where literally any bullshit explanation of anything ever advanced by some isolated tribe had to be taken seriously as an ‘equally valid’ way of understanding the world.

Put these two together and this article threatens to be about, er, ‘women’s ways of knowing in STEM’, a phrase which is literally making me cringe as I type it out. A couple of years I would have stopped here, unable to cope with the kind of associations this gave me with the awful gender-essentialist woo stuff that some women inexplicably find inspiring and not horrific. Like, stuff of the form ‘women have special kinds of intuition, which are probably to do with being really in touch with the earth or something, and also lots of feelings are going to be involved’.

Anyway I’ve calmed down about this a bit recently, to the point where I could possibly even extract something worthwhile from a full-fat gender-essentialist-woo piece of writing. And of course this paper is not like that.

Even so, this paper pretty much is about ‘women’s ways of knowing in STEM’ (in broad statistical strokes, rather than an essentialist claim that This Is How All Women Feel). And, um, it actually fits me rather well? Some of it is off, but it also includes the best description of my particular learning style that I have ever come across anywhere.


The basic setup here is one of those ‘two types of mathematician’ divisions I love. Except here there are two types of programmer. There’s this standard (straw? I don’t think so, but it’s hard for me to tell) idea of a programmer:

For some people, what is exciting about computers is working within a rule-driven system that can be mastered in a top-down, divide-and-conquer way. Their structured “planner’s” approach, the approach being taught in the Harvard programming course, is validated by industry and the academy. It decrees that the “right way” to solve a programming problem is to dissect it into separate parts and design a set of modular solutions that will fit the parts into an intended whole. Some programmers work this way because their teachers or employers insist that they do. But for others, it is a preferred approach; to them, it seems natural to make a plan, divide the task, use modules and subprocedures.

Then there’s ‘a very different style’:

They are not drawn to structured programming; their work at the computer is marked by a desire to play with the elements of the program, to move them around almost as though they were material elements — the words in a sentence, the notes on a keyboard, the elements of a collage.

Turkle and Papert call this ‘bricolage’, a term they got from Levi-Strauss. I know nothing about Levi-Strauss so can’t really say what he meant by it. The Wikipedia article on bricolage describes it as ‘the construction or creation of a work from a diverse range of things that happen to be available, or a work created by such a process’, which seems close enough to the usage in the paper.

Bricoleur scientists, apparently, work in the following way:

The bricoleur scientist does not move abstractly and hierarchically from axiom to theorem to corollary. Bricoleurs construct theories by arranging and rearranging, by negotiating and renegotiating with a set of well-known materials.

To which all I can say is:

!!!

This is the thing! This is a perfect description of the thing!


My favourite sort of problem is something that could probably be labelled ‘synthesis’, but at ground level looks like this: you have a bunch of concepts you don’t understand very well, but for some reason you’re convinced they can be combined. Sometimes this is a pointless exercise in making patterns out of noise, like staring at the Easyjet seat pattern for too long. Other times you have valid intellectual reasons for why they would fit together.

This is a bit vague, so here are some examples. There are some ideas in maths and physics that have this particular quality for me. They aren’t ones where I’m making much useful progress, and at least one is probably outright bad. They’re just examples of the kind of thing where once it’s in my head, it’s really in my head.

  • There’s a variant form of general relativity called teleparallel gravity. GR takes place in curved spacetime, and one way of thinking of this mathematically is that as you move from place to place, your frame of reference rotates in a manner described by an object called the connection. The GR connection has nonzero curvature, but there’s also some other geometrical property it could have called torsion, that’s set to zero in GR.

It turns out that you can also make a perfectly good connection with zero curvature (it’s ‘teleparallel’ – parallel lines stay parallel). Instead, it has nonzero torsion. And if you choose some coefficients right in some Lagrangian, you can reproduce GR in some sense. Buh? The formulation is pretty opaque, so what’s really going on?

  • Pedalling back a bit because we quite clearly need to, what are these curvature and torsion thingies? You can calculate quite well with limited understanding of what’s going on geometrically. GR people love to do this in a very opaque way with lots of shuffling little superscripts and subscripts around (it’s fast once you’ve learned it). In an intro course this is normally connected back to geometry at a specific ritual point, which involves shoving a vector round a loop and observing that it rotates a bit. This is not especially satisfying. It’s obviously possible to get a far better understanding, and people in the field manage this, but at least for me that’s involved extracting it painfully one piece at a time from many different sources.

  • A subquestion of this that wasted hours and hours of my time over several years (this is the ‘probably outright bad’ one): there’s curvature and torsion of a connection, but there’s also the simpler idea of curvature and torsion of a a curve in 3D space. I’d convinced myself that there was some sort of analogy between them that had to do with taking a curve off a manifold and developing it in flat Euclidean space. In fact I even got it into my head that I’d read this one of Cartan’s own books! But a lot about the idea didn’t fit so well.

I eventually couldn’t stand it any more and risked asking about it on Mathoverflow, where I feel massively underqualified. Robert Bryant answered me, which was pretty amazing. There is probably nobody better placed in the world to answer questions about Cartan – he’s apparently read the whole lot. He very politely explained that he thinks it’s a red herring, and that Cartan had a different picture in mind when he introduced the torsion of a connection. And I can’t find anything about my brilliant idea in the Cartan book I read.

So it looks like there’s probably nothing there, but I can’t quite say it’s fully out of my head yet. It’s the Easyjet seat pattern of maths questions.

  • A current one: what’s going on in QFT that makes it different to classical perturbation theory? Suddenly the diagrams have loops; why? OK, so some propagator’s nonzero at some point. What does that mean? Why can’t I get that out of a classical theory?

There’s two main parts to all these questions. One, how do the things fit together? And in order to answer this: two, what are these things really? Where ‘really’ is poorly defined, but just being able to reproduce a formal calculation definitely won’t cut it.

And the process for working on them? It’s exactly as in the quote: you do it ‘by arranging and rearranging, by negotiating and renegotiating with a set of well-known materials’. ‘Well-known’, because you’ve spent hours thinking about specific concrete instantiations, in the process of trying to understand what they ‘really’ are. Particular connections, particular propagators. ‘Negotiating and renegotiating’, because they’re your friends by now and you want them to get on. Maybe one side of your explanation meshes poorly with another side. Maybe there’s a reframing that can combine them.

If this is bricolage, then sign me up.


Doing maths and physics in this style requires a certain stubbornness in the face of never getting taught that way. I lost confidence eventually, but I seem to have it back now. I’m convinced that it absolutely can work. It’s not some kind of second-prize way to flail around the curriculum, inferior to a more structured approach. It has its own distinctive methods and produces its own distinctive questions, which I think are often good questions.

It could work even better if it was supported better.

I’m a bricoleur scientist.

Crackpot time!

I’m pretty serious now about becoming a crackpot physicist. I even have an expandable bullet-pointed plan:

crackpot

Actually I’ve had this list for a while now, but it’s becoming less of a joke. And a couple of weeks ago I went to the most amazing workshop, organised by early-career physicists who are pretty unimpressed with the state of a lot of academic physics themselves, and they were very understanding and welcoming. So now it looks like I have allies on the inside!

The aim is to be a sort of high-class crackpot, rather than the sort who spams mailing lists with EINSTEIN WAS WRONG screeds. I want to engage with the actual physics community, and learn more physics.

Learning is going well already. I have a full time job so have to work around it, but I’ve got fairly disciplined now and have learned that you can do a lot with half-hour focussed blocks of time. In some ways it’s more freeing than the PhD situation, because you never have to worry that you should be working on your stupid thesis instead.

Research will be a harder one to crack. I’m pretty sure it just needs longish stretches of uninterrupted time, and I don’t have many of those. So long term I also need to work out a better employment situation. I’ve punted those decisions to October at the earliest, though, and for now I’ll focus on learning.

This is the plan:

  • QFT first pass: focus on scalar phi^4 theory

This is pretty much the simplest option, and I’m sticking to it pedantically in the spirit of ‘examples first’. I’ve had this ‘sort out my embarrassing total lack of QFT background’ plan going for a while now, so I’m already a good way through this one. The aim is just to know roughly how to calculate some shit without getting too hung up on technical details.

I lucked out by buying the book Quantum Field Theory for the Gifted Amateur, by Blundell and Lancaster. It’s a bit of a weird name but turns out to be brilliant – it’s pitched at exactly the level I want, giving a reasonable overview of how to actually calculate things with out being immediately bogged down in the sort of practitioner-level detail that the likes of Peskin and Schroeder need to go into. I’m starting to think I should ask the authors for some kind of sales commission as I keep plugging it to all the physicists I meet. I’ve got a few chapters on renormalisation to go and then I think I’m done with the phi^4 material.

I don’t want to go completely mad on conceptual questions either, but I do want to clear up some of my ‘quantum without quantum’ confusions and figure out what’s actually novel and what’s a bunch of methods that could just as easily be used in classical physics.

I think the key question is still: where are the loops coming from? You don’t get those in classical perturbation theory. I’m suspicious that the weirdo Feynman propagator is part of the issue, and I’m curious what happens when you introduce it in the classical case. I should stop vaguely wondering about this and actually go and do some maths.

  • Quantum foundations grab bag

Now I’m a proper crackpot I can think about the foundations of quantum mechanics as much I as want 😀

The main part of this is going through Matt Leifer’s Perimeter course video lectures, which I’d already started on. But now I also have a big old supplementary reading list from the workshop. Also I might get hold of Bell’s Speakable and Unspeakable, which was passed around there like some sort of holy relic.

  • QFT second pass: nonzero spin

Not sure of the exact plan as this is some way off, but it may evolve into some giant detour where I learn geometric algebra too.

I always thought that was one of those weird things with the worryingly zealous true believers, but at the workshop a couple of true believers did a good job of convincing me it was worth bothering with. Apparently they find it to be a really useful tool for going up and down the ladder of abstraction, with the same notation helping them both with getting a clean geometric understanding of a problem and with cranking actual numbers out. That’s exactly the kind of thing I’m always looking for! So maybe if I’m going to be messing about with spinors this’ll help.

Also now I’m a crackpot I can learn whatever I like. Being a crackpot is fun!

  • Looking for the fine structure constant in the Bible

… or maybe not. I’m not that far gone yet! (Growth mindset?)

Cognitive dancing! Cognitive style!

(Pictured: the inside of my head. This has been a pretty bad earworm recently.)

Figure out what your own cognitive style is. Embrace and develop it as your secret weapon; but try to learn and appreciate other styles as well.

I keep thinking about this quote (from How To Think Real Good). And specifically about what my continuing adventures in being a terrible programmer are adding to my own toolkit, as it’s definitely quite a lot.

Of course everyone and their cat is now a programmer so these are not in any way hard-to-find insights. Actually most of what I read seems to be saturated with them. But my brain is quite resistant to learning any of this, so it’s only just starting to go in.

The ultimate aim would be to finally become passable at some of the tricks the current culture throws at me, and also hang on to my own weirdo cognitive style. Cognitive bilingualism!

Anyway let’s be specific. Here’s three related things that I’m starting to get a grip on.

‘Seeing the skull beneath the skin’.

As in, being able to isolate the structural skeleton of a problem, the bits that actually matter in this application, and mapping it on to some kind of data structure.

You’d think a maths degree would have taught me this already, but no. I was able to rely on intuition and good old rote memorisation for when that failed, and it worked just about OK. Now there’s nothing I can yet grasp well enough to visualise and not much to memorise either, so I suppose I have to learn how to do the thing.

This’ll be a slow one and I don’t yet have any strategies for getting better. I’m still in the process of ‘falling in love with the gears’, attempting to attach any sort of ‘positive affective tone’ to the idea so that I can be bothered to learn it at all. In that respect it’s interesting to listen to people who can do the thing. I joined my current job on some sort of graduate scheme thingy and there are some good CS graduates who were starting at the same time. Sometimes just in the pub or whatever they’ll mention the kinds of problems they idly think about for fun and they have this sort of quality. ‘How would I code this up?’ ‘What structure can I fit it to?’

I like these people so maybe I can like the thing? That always sounds dumb written out but it’s worked before.

(The ‘skull beneath the skin’ quote comes from here but even more I’m thinking of this bit from Watership Down:

“When we think of the downs, we think of the downs in daylight, as we think of a rabbit with its fur on. Stubbs may have envisaged the skeleton inside the horse, but most of us do not: and we do not usually envisage the downs without daylight, even though the light is not a part of the down itself as the hide is part of the horse itself.”

Seems like there are increasing numbers of people who are able to look for the skeleton inside the horse, which is interesting but also weird to me.)

‘Is there a process for that?’

I’m actually getting my head round this one quite nicely, as part of a general project of getting better at structure and discipline and organisation. (This has been way more successful than I would have expected — I actually get up at six and learn physics before going to work these days, which I’d never have imagined would work even a couple of years ago.)

The trick here is that if you’ve got something you need to do, you set up some kind of system so it happens automatically. My boss and his boss are both good at this and take it seriously – it’s worth having an in-depth dull conversation once for, say, how to keep track of emails in a shared inbox, because once you’ve got a good process it takes care of itself and you never need to think about it again.

Most productivity advice is of this form, but it took me a long time to start engaging with it. It just sounded so tedious and bean-county and I kind of liked the image of myself as an messy eccentric with piles of paper everywhere. But it turns out that getting shit done efficiently is extremely useful, and now I want to get more shit done efficiently.

I seem to pass as an organised person quite often these days, which is funny. My boss asked me if I had any thoughts on Trello, for example – I know nothing much about Trello, but it was nice to be considered as the sort of person who could plausibly have something useful to say.

Automate everything.

The general version of ‘is there a process for that?’ is the classic programmer tendency to ‘automate everything’, and it’s actually really hard for me to learn outside of a few special cases. I love repetition. I’m pretty sure there’s a gigantic lead flywheel in my brain which takes forever to spin up, but once it gets there I have enormous cognitive inertia and will happily do the same thing for ages. This is probably obvious from the contents of this blog, and even more obvious from my youtube music listening history. I’ve done many low-level menial admin temp jobs in the past because I have an unusually high tolerance for doing a rote task over and over again.

Whereas the programmer ideal seems to be to pick up a new task lightly, identify the abstract structure as cleanly as possible, and then automate the hell of the bastard so you never have to see it again. And then move on to the next one.

This is a massive nuisance for me! There’s no such thing picking up a task lightly when you have a lead flywheel to spin up, and once it’s spun up you’re properly invested. And then once you’re into that enjoyable flow state of knowing how to do something you’re supposed to rip it all up and make a computer have all the fun instead? Programmers are crazy!

I’m not sure it’s worth tackling this one upfront. I also need to work on ways to get my existing abilities to help me out and route around obstacles, and I’m thinking about that too, but that will be another post sometime.

sleep deprivation, part 2

After the flight, I got a four hour coach back to Bristol, and the fun really started. While waiting in the airport I read a bit more of Keith Johnstone’s Impro, and got on to some pretty crazy-sounding free association exercises (the earlier part of this is what inspired my post about the pastebin of utter crap). Most of them required a partner, or at least a situation where you could talk out loud without sounding like a total nutter, but I thought I could at maybe try improvising stories in my head, allowing myself no pauses to think about what should come next.

I’ve forgotten most of what I came up with, and it was probably mostly crap anyway, but there were a few segments that were a) strange and memorable; b) somewhat in the style of those in the book; and c) nothing at all like what I would have expected to come up with, given what I normally think is in my head:

1. Plovers make their winter migration to the pole star, where all lines intersect. The surface of the pole star is covered in bulrushes, but the inner core is ice. Each plover brings a small amount of warmth to the star, and melts a bit of the ice. When the ice is fully melted, the universe ends and infinity returns.

2. Wolves are streaming down off the tundra in lines. In each line, each wolf carries the tail of the preceding wolf in its jaws. The air smells of lichen and wet feet.

3. There’s an internment camp for people who have forgotten the names of the four seasons. Most of them have internalised that they are stupid and deserve to die.

A man there gets angry and kicks his chair to bits. He makes a fire from the pieces. Staring into the ashes, he intuits new and better names for the seasons. He shouts them at the guards and they drop down dead. The inmates run out through the gates and all four seasons happen at once.

(These are sort of tidied up, but I think only slightly; the verbal content was almost exactly as I wrote it down, but there was a funny mix of visual imagery in there with it which I obviously can’t reproduce. This was interesting to me because I normally have a rather weak visual imagination. The reincorporation of the seasons at the end of the third one is probably because I’d just read the section on reincorporation. I’m pretty sure there were wolves in one of the Impro stories too.)

It’s fascinating to me that this stuff is just there. As Johnstone says, it all turns up when you stop fussing about whether the thoughts belong to or mesh well with you in any sense, and just let them come out with no particular owner. As ever, I wonder about whether mathematical intuition is made out of similar stuff.

After a while of this, I finally fell asleep and napped until Bristol.

sleep deprivation, part 1

Here’s an image of the seat pattern on Easyjet aircraft, which I stared at for an hour or so while sleep deprived yesterday:

easyjet-big

[image source]

It’s been carefully designed to be maximally annoying to whatever kind of stupid brain I have, in that pretty much every element of the pattern almost but doesn’t quite repeat. Actually the segment in the photo isn’t too bad, but the bit I had in front of me had the ‘L’ shape appearing in three of the possible orientations but not the fourth one, and no ways to make larger squares using the grey squares as the corners, and some other incredibly frustrating feature that I have thankfully now forgotten.

Anyway this pattern ruined any chance I may have had of a good nap, but it did make me hazily realise that whatever this mental mode is is one of the key elements of my cognitive style, for better or worse. I normally label it as ‘synthesis’, which sounds kind of grand, but what I really mean is just this thing of idly mashing little bits of conceptual pattern together in a semi-conscious, automatic way, over and over and over until I maybe get something I can use.

It has a pretty similar subjective flavour to having an annoying tune or a bunch of odd word fragments stuck in your head, which is also something that I have going on all the sodding time.

OK, I’m maybe not selling this skill as an enjoyable one here, but actually I have a lot of affection for it. Whatever this skill is, it somehow manages to balance out my terrible ability at conscious rule-following and gives me a way to survive in mathematics.

long pastebin of utter crap

it’s been abstracted too far for my tastes and stinks of algebra all over
that putrid stench? that would be the dead rotten hand of algebra

I dug out an interesting-ish writing experiment from a few years ago, where I just wrote out rubbish that came into my head as fast as possible over several evenings. I guess I should be horribly embarrassed by the result, but for some reason I like it, so I’m posting the whole lot to a pastebin here.

It starts out stilted and self-conscious and pretentious and derivative and crap, and then eventually relaxes a bit into being, well, still very pretentious but vaguely worth reading. And every so often there’s an odd little section I’d never have imagined I could write, based on my normal output.

This was a year or so before I started reading SSC, got a tumblr and got fascinated with the whole odd rationalist-adjacent world, so in a lot of ways the prevailing weather in my head was pretty different. I mean, some things never change – as well as the algebra bit there’s another maths rant 500 lines or so in, this time about terminology in differential geometry. But I was finishing up my thesis and was utterly sick of it, and winter was hanging around a lot longer than usual, and the main loops in my head were these sort of miserable ones linked up somehow with M. John Harrison’s Viriconium series and the Anatomy of Norbiton blog. The Pastel City and the Ideal City. And of course underneath it The Waste Land and the Unreal City, my longest-lived and most faithful source of brain noise (that and some Empson stuff).

There was some kind of depressing economics-related news around too, whatever this Harrison post was soaked in. I honestly can’t remember what particular thing it was now that the whole of the news is such a shitshow, but it may have been something about automation/’the post-work future’? “Outside it’s minus ten & you have no idea what’s happening on the old housing estates by the river.” That image of boiling an egg in the same water is completely lodged anyway, even if the specifics have gone out of my head.

Anyway it’s been interesting to read back. I’m a lot more cheerful right now despite the news, and my brain is currently working around some new loops, so maybe I should try this exercise again.

Examples First highlights

Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.

Rereading a bit, that blog post comment section is probably the original source of my minor obsession with the role-of-intuition-in-maths literature.

(This isn’t actually a particularly major real-life obsession, just a secondary one that I like writing about/have worked out how to write about/have bored everyone I actually know with so I need to go talk about it somewhere else where people can opt out easily)

The only thing I can’t find in there is the Vladimir Arnold rant, which I must have picked up somewhere else. And actually Rota doesn’t make an appearance either. But there’s a load of the standard lore, the usual quotes from Thurston and Poincaré (it’s always the same quotes because this subfield is tiny). 

I’d kind of forgotten that, because I think of it more as my starting point in figuring out how to learn differential geometry. Following the references in the comments taught me more than any maths course I took in undergrad. 

Anyway, some highlights:

  • “I have a colleague (in CS, not math) who reads papers as follows: First he skims the paper by skipping all English and reading only formulas, then he reads the introduction, and then he reads again forcing himself to read some of the English too.”

  • that story about Grothendieck thinking that 57 was prime

  • “One day I realized it was all a lot clearer if I specialized the arguments. As a simple example, a theorem about differentiable real-valued functions on an interval might reduce to the case of the behavior, at 0, of a differentiable function f satisfying f(0) = 0 and f'(0) = 0. Cosmetic assumptions like these simplify the difference quotient and make the key issues clearer (to a novice anyway). The “general case” of such a theorem is often the result of composing the specific proof with an affine transformation. The symbols implementing this transformation play no essential role in the argument.”

  • Fields Medallist admits they never really understood what all that Sylow subgroup stuff was on about

  • ”My undergraduate days left me afraid of many subjects: complex analysis, measure theory, most of algebra and almost all geometry, for example”

  • link to John Baez on normal subgroups. I gave up with trying to understand group theory when I didn’t understand what the definition of a normal subgroup was on about. Unfortunately this is like week 3 of an intro to group theory course, so that was kind of it for me and abstract algebra. I clicked on the link but never read it properly and still don’t really get what a normal subgroup is. Even so, after reading this I felt better for realising it wasn’t some completely obvious thing I ought to ‘just get’ and didn’t.

  • “Dualization is a rather simple idea but I think it is perhaps one of, if not the, most powerful tools in mathematics, especially in the modern era. There is, I’m sure, a good story about why. Perhaps someone can explain or tell me where to find an explanation?”

  • someone asks how to start learning differential geometry and a student of Chern turns up to answer

  • ‘I like to call differential geometry “nonlinear linear algebra”.’

  • a long involved interesting argument about whether you should identify the tangent and cotangent space when you can to save bothering to keep track of a distinction you don’t need right now, or whether you’ll confuse yourself more in the long run

  • anecdote about helping a six-year-old who “could do 3+2 with no problem whatsoever. In fact, she had no trouble with addition. She just couldn’t get her head around all these wretched apples, cakes, monkeys etc that were being used to “explain” the concept of addition to her.”

  • “I was talking to two students about conjugation and talked about how gfg^{-1} is the function that takes g(x) to g(y) if f takes x to y. I then asked them to come up with a function from the reals to the reals that takes x^3 to (x+1)^3 for every x. After a while, one of them had the idea of taking the cube root, adding one, and cubing. But it was clear that he did that by forgetting all about my discussion of conjugation and just looking at the example. Only afterwards, when I pointed it out, did he realize that he had just done a conjugation.”

  • “Kazhdan’s advice to my friend: You should know everything in this book but don’t read it.”

deep paths of mathematics

Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.

That education thing made me remember I wrote something about my maths teacher a few months ago as setup for a complex point that I never wrote down and have now completely forgotten. I’m probably not going to remember so let’s post it as is:


So I’m digging through some differential equation stuff trying to fill a few gaps in my knowledge, mostly arsing around ‘doing some prerequisites’ for quantum field theory instead of just jumping in. This time it’s the Fredholm Alternative. It looks like one of those bits of arcane mathematical methods textbook lore with the silly names, like the bilinear concomitant or Rayleigh’s quotient or ‘the method of undetermined coefficients’, which I always thought was just called ‘guessing’. This looks pretty useful and general though, looking at some inner product to see whether boundary value problems have one solution or no solutions or infinitely many solutions.

Actually it looks a bit like… oh, yeah, look, there’s even a matrix version. In fact,…


I’m in a classroom with the other Further Maths nerds. It doesn’t fit on the timetable so we’re stuck in there again after school, eating vending machine sweets and solving systems of simultaneous equations using Gaussian elimination. As always, our teacher has gone beyond the rote work of the syllabus and is making sure we understand what’s going on geometrically, using three planes as an example. We’re systematically working through the possibilities: all three planes parallel, two parallel and one crosses them (‘British Rail logo’), all three intersect along a single line, all the rest. We can end up with no solutions, or a unique solution at a single point, or a whole line or plane of solutions.

Then someone looks out the window. It’s someone’s brother in Year 8, mucking about on the flat roof across the playground. He thinks everyone’s gone home.

Our teacher opens the window, still caught up in his system of equations. “GET OFF THAT PLANE!!! NOW! YOU’RE IN DETENTION TOMORROW!”

And afterwards, as the kid complies: “…did I just say ‘plane’ instead of ‘roof’?”

We go back to the example. I get plenty more linear algebra next year at university, but however abstractly they dress it up, in my head it’s just the same old intersecting planes.


I’ve seen the words ‘Fredholm Alternative’ somewhere else, though, too, on one of my unmotivated afternoons in the library pulling books off the shelves. Ah yes, googling around it must have been Booss and Bleecker’s Index Theory, which aims to drag even applied mathematicians up to the heights:

Index Theory with Applications to Mathematics and Physics describes, explains, and explores the Index Theorem of Atiyah and Singer, one of the truly great accomplishments of twentieth-century mathematics whose influence continues to grow, fifty years after its discovery. The Index Theorem has given birth to many mathematical research areas and exposed profound connections between analysis, geometry, topology, algebra, and mathematical physics. Hardly any topic of modern mathematics stands independent of its influence.

And there’s the Fredholm Alternative in Chapter 2, one of the steps on the path.

Maybe I’m not just digging out random crap from the textbook. It looks like I accidentally found one of the Old Ways of mathematics, linking my A Level classes with some great confluence up in the stratosphere. With like a million steps above me still, but sometimes it’s nice just to know you’re on the path.

tastes in the head

Note: these posts are copied over from the ‘mathbucket’ section of my old tumblr blog and I haven’t put much effort into this, so there is likely to be context or formatting missing.

[I don’t really know where I’m going with this post, but my brain seems to be fixated on writing it. It never comes out right, so this time I’m just going to push it out the door in whatever confused form I can manage, and then hopefully my brain will shut up.]

Tastes in the head is some idiosyncratic piece of mental furniture I have cluttering up the place. I’m not even sure where the best place to look is if I want to find a better vocabulary for this sort of thing (phenomenology? meditation practice? do I have to drag my way through Heidegger or something? ugh). I first got it from Empson’s Seven Kinds of Ambiguity:

“… what the poet has conveyed is no assembly of grammatical meanings, capable of analysis, but a ‘mood’, an ‘atmosphere’, a ‘personality’, an attitude to life, an undifferentiated mode of being. Probably it is in this way, as a sort of taste in the head, that one remembers one’s own past experiences, including the experience of reading a particular poet.”

He obviously liked the phrase, because he uses it again a bit later to snark about the Romantic poets:

“They admired the poetry of previous generations, very rightly, for the taste it left in the head, and, failing to realize that the process of putting such a taste into a reader’s head involves a great deal of work which does not feel like a taste in the head while it is being done, attempting, therefore, to conceive a taste in the head and put it straight on to their paper, they produced tastes in the head which were in fact blurred, complacent, and unpleasing.”

I don’t know if I’m that close to what he meant, but I interpret his ‘tastes in the head’ as that kind of preverbal emotional tone ideas have – once you have some description in language you can manipulate that and catch some of the meaning, but there is also some qualitative essence that is harder to get at. I’m not going to give any examples now, because 1. that ‘preverbal’ bit makes it really hard, 2. the ones I tried were really distracting and ended up taking over the post, but I will quote a couple of sources further along that hopefully clarify it a bit.


This is a bit of an aside, but I’m really not aiming for clear structure in this one: while I’m talking about the the New Critics, there’s an interesting link here to the idea of the ‘objective correlative’ popularised by T.S. Eliot. I often see this used to mean just, like, imagery in literature that helps to convey the emotional tone of the piece, but iirc he originally used it to make a much stronger, wrong-but-interesting claim: that each image in literature maps to a distinct ‘taste in the head’ – a specific nonverbal emotional tone – that is ‘objective’, i.e. the same for each person (provided they have cultivated the Correct level of literary sensitivity, is the disclaimer at this point.)

This would be amazing if true: we could generate a load of poetic imagery, discover exactly what impossible-to-convey-with-normal-language emotional tone it mapped to, and then produce a giant lookup table that people could use to reliably convey the background texture of their thoughts, and then nobody would ever misunderstand each other ever again. E.g., taking the latest SSC post as an example, Scott digs up the image that exactly corresponds to lived-experience-sympathy-for-the-plight-of-overworked-junior-doctors (it’s probably some kind of whale), sticks it in the post, and we all understand it on an intuitive level and none of us ever need to argue about it again. This sounds like the ultimate rationalist project, but unfortunately we’re going to fail given the obvious problem that the same image provokes very different responses in different people. (And even in the same person at different times.) It’s actually hard for me to believe that this idea was even on the table, but, well, behaviourism was popular at one point, and this is a good deal less reductionist.


Normally I approach this stuff through my endless droning on about the role of intuition in maths. In that case I sometimes also think of the feeling as ‘falling in love with the gears’. This comes from the intro to Seymour Papert’s Mindstorms (pdf) where he talks about his early fascination with cars, and how playing with gears helped his early intuition for maths:

“I believe that working with differentials did more for my mathematical development than anything I was taught in elementary school. Gears, serving as models, carried many otherwise abstract ideas into my head. I clearly remember two examples from school math. I saw multiplication tables as gears, and my first brush with equations in two variables (e.g., 3x + 4y = 10) immediately evoked the differential. By the time I had made a mental gear model of the relation between x and y, figuring how many teeth each gear needed, the equation had become a comfortable friend.

“Assimilating equations to gears certainly is a powerful way to bring old knowledge to bear on a new object. But it does more as well. I am sure that such assimilations helped to endow mathematics, for me, with a positive affective tone that can be traced back to my infantile experiences with cars.

“A modern-day Montessori might propose, if convinced by my story, to create a gear set for children. Thus every child might have the experience I had. But to hope for this would be to miss the essence of the story. I fell in love with the gears.”

That ‘positive affective tone’ is what I mean by ‘taste in the head’, and for me learning maths is all about the process of finding that kind of tone in new areas:

As an example I’ve been going on about a lot, for me at the moment differential geometry is strongly associated with a positive tone, and abstract algebra most definitely isn’t. (This hasn’t always been true, so hopefully one day I can like both!) Lie groups provide one natural bridge, as an algebraic object that is also a differentiable manifold. Using that I can start to see a path to more “unpleasantly algebraic” components, like, I don’t know, classifying Lie algebras or something, and hopefully later I can move further along it.


So this has mostly just been a lot of quoting various links. I was going to try and make some rambling ill-defined claims at this point, but I don’t think I’m going to be able to manage that this time so I’ll just wave vaguely at one of them.

I must be unusual in that one part of Less Wrong that I actually like is all the stuff about ‘akrasia’ and reducing procrastination – I don’t get a whole lot of practical use out of this, but the vocabulary of ‘ugh fields’ works very well at communicating the pre-verbal feel of what’s going on. Some of Alicorn’s ‘Living Luminously’ posts are even closer to what I’m interested in, including a much more developed idea of ‘hacking yourself’ to like things (like my Lie group example, but she seems to manage a greater level of control.)

When I read this stuff (also the parts about cognitive biases, Kahneman’s System One, etc., but there I haven’t read so much of the LW content), LW seems really good at taking all this preverbal substructure seriously. And then suddenly, bam!, I’ll be reading something else and it’s all about fitting everything into some incredibly restrictive language-based formal structure. I don’t know, I am possibly just missing the part of my brain that can get anything out of philosophical discussions of ethics, but this part of LW in particular gives me this strong feeling of “how the hell have have you reduced this gigantic mess of tastes in the head to a clean conceptual system, which in certain versions is clean enough to actually take values in the real numbers? This is worse than the T. S. Eliot Emotional Lookup Table! And more importantly, what the hell have you left out by doing this?”

I don’t know, sometimes formal systematizing does work, really well. I would be very surprised if this is one of those times.

OK, I’m going to finish here, and push it out the door like I said I would. Hopefully this has got some of what I wanted out of my head.