Two types of mathematician

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.

“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.

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.