*(I posted this on Less Wrong back in April and forgot to cross post here. It’s worth reading over there for the comments, which are great.)*

This is an expansion of a linkdump I made a while ago with examples of mathematicians splitting other mathematicians into two groups, which may be of wider interest in the context of the recent elephant/rider discussion. (Though probably not *especially* wide interest, so I’m posting this to my personal page.)

The two clusters vary a bit, but there’s some pattern to what goes in each – it tends to be roughly ‘algebra/problem-solving/analysis/logic/step-by-step/precision/explicit’ vs. ‘geometry/theorising/synthesis/intuition/all-at-once/hand-waving/implicit’.

*(Edit to add: ‘analysis’ in the first cluster is meant to be analysis as opposed to ‘synthesis’ in the second cluster, i.e. ‘breaking down’ as opposed to ‘building up’. It’s not referring to the mathematical subject of analysis, which is hard to place!)*

These seem to have a family resemblance to the S2/S1 division, but there’s a lot lumped under each one that could helpfully be split out, which is where some of the confusion in the comments to the elephant/rider post is probably coming in. (I haven’t read *The Elephant in the Brain* yet, but from the sound of it that is using something of a different distinction again, which is also adding to the confusion). Sarah Constantin and Owen Shen have both split out some of these distinctions in a more useful way.

I wanted to chuck these into the discussion because: a) it’s a pet topic of mine that I’ll happily shoehorn into anything; b) it shows that a similar split has been present in mathematical folk wisdom for at least a century; c) these are all really good essays by some of the most impressive mathematicians and physicists of the 20th century, and are well worth reading on their own account.

- The earliest one I know (and one of the best) is Poincare’s ‘Intuition and Logic in Mathematics’ from 1905, which starts:

“It is impossible to study the works of the great mathematicians, or even those of the lesser, without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic; to read their works, one is tempted to believe they have advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance.

The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard.”

- Felix Klein’s ‘Elementary Mathematics from an Advanced Standpoint’ in 1908 has ‘Plan A’ (‘the formal theory of equations’) and ‘Plan B’ (‘a fusion of the perception of number with that of space’). He also separates out ‘ordered formal calculation’ into a Plan C.
- Gian-Carlo Rota made a division into ‘problem solvers and theorizers’ (in ‘Indiscrete Thoughts’, excerpt here).
- Timothy Gowers makes a very similar division in his ‘Two Cultures of Mathematics’ (discussion and link to pdf here).
- Vladimir Arnold’s ‘On Teaching Mathematics’ is an incredibly entertaining rant from a partisan of the geometry/intuition side – it’s over-the-top but was 100% what I needed to read when I first found it.
- Michael Atiyah makes the distinction in ‘What is Geometry?’:

Broadly speaking I want to suggest that geometry is that part of mathematics in which visual thought is dominant whereas algebra is that part in which sequential thought is dominant. This dichotomy is perhaps better conveyed by the words “insight” versus “rigour” and both play an essential role in real mathematical problems.

There’s also his famous quote:

Algebra is the offer made by the devil to the mathematician. The devil says: `I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.’

- Grothendieck was seriously weird, and may not fit well to either category, but I love this quote from
*Récoltes et semailles*too much to not include it:

Since then I’ve had the chance in the world of mathematics that bid me welcome, to meet quite a number of people, both among my “elders” and among young people in my general age group who were more brilliant, much more ‘gifted’ than I was. I admired the facility with which they picked up, as if at play, new ideas, juggling them as if familiar with them from the cradle – while for myself I felt clumsy, even oafish, wandering painfully up an arduous track, like a dumb ox faced with an amorphous mountain of things I had to learn (so I was assured), things I felt incapable of understanding the essentials or following through to the end. Indeed, there was little about me that identified the kind of bright student who wins at prestigious competitions or assimilates almost by sleight of hand, the most forbidding subjects.

In fact, most of these comrades who I gauged to be more brilliant than I have gone on to become distinguished mathematicians. Still from the perspective or thirty or thirty five years, I can state that their imprint upon the mathematics of our time has not been very profound. They’ve done all things, often beautiful things, in a context that was already set out before them, which they had no inclination to disturb. Without being aware of it, they’ve remained prisoners of those invisible and despotic circles which delimit the universe of a certain milieu in a given era. To have broken these bounds they would have to rediscover in themselves that capability which was their birthright, as it was mine: The capacity to be alone.

- Freeman Dyson calls his groups ‘Birds and Frogs’ (this one’s more physics-focussed).
- This may be too much partisanship from me for the geometry/implicit cluster, but I think the Mark Kac ‘magician’ quote is also connected to this:

There are two kinds of geniuses: the ‘ordinary’ and the ‘magicians.’ an ordinary genius is a fellow whom you and I would be just as good as, if we were only many times better. There is no mystery as to how his mind works. Once we understand what they’ve done, we feel certain that we, too, could have done it. It is different with the magicians… Feynman is a magician of the highest caliber.

The algebra/explicit cluster is more ‘public’ in some sense, in that its main product is a chain of step-by-step formal reasoning that can be written down and is fairly communicable between people. (This is probably also the main reason that formal education loves it.) The geometry/implicit cluster relies on lots of pieces of hard-to-transfer intuition, and these tend to stay ‘stuck in people’s heads’ even if they write a legitimising chain of reasoning down, so it can look like ‘magic’ on the outside.

- Finally, I think something similar is at the heart of William Thurston’s debate with Jaffe and Quinn over the necessity of rigour in mathematics – see Thurston’s ‘On proof and progress in mathematics’.

**Edit to add:** Seo Sanghyeon contributed the following example by email, from Weinberg’s *Dreams of a Final Theory*:

Theoretical physicists in their most successful work tend to play one of two roles: they are either sages or magicians… It is possible to teach general relativity today by following pretty much the same line of reasoning that Einstein used when he finally wrote up his work in 1915. Then there are magician-physicists, who do not seem to be reasoning at all but who jump over all intermediate steps to a new insight about nature. The authors of physics textbook are usually compelled to redo the work of the magicians so they seem like sages; otherwise no reader would understand the physics.