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.
This is great, and something I thought was missing from the SSC discussion (pity I’m too slow to ever get round to commenting). I only really got a tumblr to read some stuff, but I am fascinated by this topic so here is some related crap…
I think there’s a lot to this ‘different ways of understanding maths’ idea, sometimes it seems that you can pretty much give a mathematician a pen and they will start writing an essay on two types of mathematician. The clusters seem to be roughly ‘algebra/problem-solving/analysis/logic/precision’ vs. geometry/theorising/synthesis/intuition/hand-waving’ but there is plenty of variation.
I keep meaning to collect together a set of all of these I can find, so this has motivated me to make a first attempt:
- The earliest one I know is Poincare’s ‘Intuition and Logic in Mathematics’, 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.”
- 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)
- Freeman Dyson calls his groups ‘Birds and Frogs’ (this one’s more physics-focussed)
- Vladimir Arnold turns the whole thing into a massive ideological war in his wonderful rant ‘On Teaching Mathematics’
- 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.’
- 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’. There is also a wonderful list of ways of understanding the concept of a derivative in Section 2.
OK hopefully I know how to write a post now! Hope this is interesting to someone.