I’ve recently been browsing through Season 1 of Venkatesh Rao’s Breaking Smart newsletter. I didn’t sign up for this originally because I assumed it was some kind of business thing I wouldn’t care about, but I should have realised it wouldn’t stray far from the central Ribbonfarm obsessions. In particular, there’s an emphasis on my favourite one: figuring out how to make progress in domains where the questions you are asking are still fuzzy and ambiguous.
‘Is there a there there? You’ll know when you find it’ is explicitly about this, and even better, it links to an interesting article that ties in to one of my central obsessions, the perennial ‘two types of mathematician’ question. It’s just a short Wired article without a lot of detail, but the authors have also written a pop science book it’s based on, The Eureka Factor. From the blurb it looks very poppy, but also extremely close to my interests, so I plan to read it. If I had any sense I’d do this before I started writing about it, but this braindump somehow just appeared anyway.
The book is not focussed on maths – it’s a general interest book about problem solving and creativity in any domain. But it looks like it has a very similar way of splitting problem solvers into two groups, ‘insightfuls’ and ‘analysts’. ‘Analysts’ follow a linear, methodical approach to work through a problem step by step. Importantly, they also have cognitive access to those steps – if they’re asked what they did to solve the problem, they can reconstruct the argument.
‘Insightfuls’ have no such access to the way they solved the problem. Instead, a solution just ‘pops into their heads’.
Of course, nobody is really a pure ‘insightful’ or ‘analyst’. And most significant problems demand a mixed strategy. But it does seem like many people have a tendency towards one or the other.
A nice toy problem for thinking about how this works in maths is the one Seymour Papert discusses in a fascinating epilogue to his Mindstorms book. I’ve written about this before but I’m likely to want to return to it a lot, so it’s probably worth writing out in a more coherent form that the tumblr post.
Papert considers two proofs of the irrationality of the square root of two, “which differ along a dimension one might call ‘gestalt versus atomistic’ or ‘aha-single-flash-insight versus step-by-step reasoning’.” Both start with the usual proof by contradiction: let , a fraction expressed in its lowest terms, and rearrange it to get
The standard proof I learnt as a first year maths student does the job. You notice that must be even, so you write it as , sub it back in and notice that is going to have to be even too. But you started with the fraction expressed in its lowest terms, so the factors shouldn’t be there and you have a contradiction. Done.
This is a classic ‘analytical’ step-by-step proof, and it’s short and neat enough that it’s actually reasonably satisfying. But I much prefer Papert’s ‘aha-single-flash-insight’ proof.
Think of as a product of its prime factors, e.g. . Then will have an even number of each prime factor, e.g. .
But then our equation is saying that an even set of prime factors equals another even set multiplied by a 2 on its own, which makes no sense at all.
This proof still has some step-by-step analytical setup. You follow the same proof by contradiction method to start off with, and the idea of viewing and as prime factors still has to be preloaded into your head in a more-or-less logical way. But once you’ve done that, the core step is insight-based. You don’t need to think about why the original equation is wrong any more. You can just see it’s wrong by looking at it. In fact, I’m now surprised that it didn’t look wrong before!
For me, all of the fascination of maths is in this kind of insight step. And also most of the frustration… you can’t see into the black box properly, so what exactly is going on?
My real, selfish reason for being obsessed with this question is that my ability to do any form of explicit step-by-step reasoning in my head is rubbish. I would guess it’s probably bad compared to the average person; it’s definitely bad compared to most people who do maths.
This is a major problem in a few very narrow situations, such as trying to play a strategy game. I’m honestly not sure if I could remember how to draw at noughts and crosses, so trying to play anything with a higher level of sophistication is embarrassing.
Strategy games are pretty easy to avoid most of the time. (Though not as easy to avoid as I’d like, because most STEM people seem to love this crap 😦 ). But you’d think that this would be a serious issue in learning maths as well. It does slow me down a lot, sometimes, when trying to pick up a new idea. But it doesn’t seem to stop me making progress in the long run; somehow I’m managing to route round it. So what I’m trying to understand when I think about this question is how I’m doing this.
It’s hard to figure it out, but I think I use several skills. One is simply that I can follow the same chains of reasoning as everyone else, given enough time and a piece of paper. It’s not some sort of generalised ‘inability to think logically’, or then I suppose I really would be in the shit. Subjectively at least, it feels more like the bit of my brain that I have access to is extremely noisy and unfocussed, and has to be goaded through the steps in a very slow, explicit way.
Another skill I enjoy is building fluency, getting subtasks like bits of algebraic manipulation ‘under my fingers’ so I don’t have to think about them at all. This is the same as practising a musical instrument and I’m familiar with how to do it.
But the fun one is definitely insight. Whatever’s going on in Papert’s ‘aha-single-flash-insight’ is the whole reason why I do maths and physics, and I wish I understood it better. I also wish there were more resources for learning how to work with it, as I’m pretty sure it’s my main trick for working round my poor explicit reasoning skills.
My workflow for trying to understand a new concept is something like:
- search John Baez’s website in the hope that he’s written about it;
- google something like ‘[X] intuitively’ and pick out any fragments of insight I can find from blog posts, StackExchange answers and lecture notes;
- (back when I had easy access to an academic library) pull a load of vaguely relevant books off the shelf and skim them;
- resign myself to actually having to think for myself, and work through the simplest example I can find.
The aim is always to find something like Papert’s ‘set of prime factors’ insight, some key idea that makes the point of the concept pop out. For example, suppose I want to know about the Maurer-Cartan form in differential geometry, which has this fairly unilluminating definition on Wikipedia:
Then I’m done at step 1, because John Baez has this to say:
Let’s start with the Maurer-Cartan form. This is a gadget that shows up in the study of Lie groups. It works like this. Suppose you have a Lie group G with Lie algebra Lie(G). Suppose you have a tangent vector at any point of the group G. Then you can translate it to the identity element of G and get a tangent vector at the identity of G. But, this is nothing but an element of Lie(G)!
So, we have a god-given linear map from tangent vectors on G to the Lie algebra Lie(G). This is called a “Lie(G)-valued 1-form” on G, since an ordinary 1-form eats tangent vectors and spits out numbers, while this spits out elements of Lie(G). This particular god-given Lie(G)-valued 1-form on G is called the “Maurer-Cartan form”, and denoted ω.
This requires a lot more knowledge going in than the square root of two example, because I need to know what a Lie group and a Lie algebra and a 1-form are to get any use out of it. But if I’ve already struggled through getting the necessary insights for those things, I now have exactly the further insight I need: if you can translate your tangent vector back to the identity it’ll magically turn into a Lie algebra element, so then you’ve got yourself a map between the two sorts of things. And if I don’t know what a Lie group and a Lie algebra and a 1-form are, it’s pointless me trying to learn about the Maurer-Cartan form anyway.
Unfortunately, nobody has locked John Baez in a room and made him write about every topic in mathematics, so normally I have to go further down my algorithm, and that’s where things get difficult. There’s surprisingly poor support for an insight-based route through maths. If you want insights you have to dig for them, one piece at a time.
Presumably this is at least partially a hangover of the twentieth century’s obsession with formalism. Insights don’t look like proper logical maths with all the steps written out. You just sort of look at them, and the work’s mostly being done by a black box in your head. So this is definitely not a workflow I was taught by anyone during my maths degree; it’s one I improvised over time so that I could get through it anyway, when presented with definitions as opaque as the one from the Wikipedia article.
I’m confident that we can do better. And also that we will, as there seems to be an increasing interest in developing better conceptual explanations. I think Google’s Distill project and their idea of ‘research debt’ is especially promising. But that article’s interesting enough that it should really be a separate post sometime.