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.
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.