# 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.
• I’ve refer­enced the Grothendieck quote in this post many times since it came out, and the quote it­self seems im­por­tant enough to be worth cu­rat­ing it.

I’ve also refer­enced this post a few times in a broader con­text around differ­ent math­e­mat­i­cal prac­tices, though definitely much less fre­quently than I’ve refer­enced the Grothendieck quote.

• This is a very valuable effort in out­lin­ing a hy­poth­e­sis, and us­ing the au­thor’s wide-rang­ing taste and knowl­edge to pull loads of sources to­gether. Definitely helped me a bit think about math­e­mat­ics and thought, and some of my friends too. I’ve es­pe­cially thought about that Grothendieck quote a lot.

• There’s an ob­vi­ous but cool meta-thing go­ing on with the num­ber 2 that might be use­ful to pick out. Some pieces of this thing:

All over the place, we speak in terms of di­chotomies and not tri­chotomies or more. The rea­son is ba­si­cally that each di­chotomy cor­re­sponds to do­ing PCA and pro­ject­ing space onto a sin­gle axis, and a one-di­men­sional line has two di­rec­tions. This sug­gests that much of the in­ter­est­ing con­ver­sa­tion about any given topic (i.e. axis) can be picked up by hav­ing ex­actly two peo­ple talk about it. Two peo­ple will always differ slightly on the axis. Ad­ding any ad­di­tional peo­ple to a con­ver­sa­tion has rapidly diminish­ing re­turns: you may have more to­tal dis­agree­ment, but rarely more to­tal di­men­sion­al­ity in the dis­agree­ment.

Duo Talks is the idea that most pro­duc­tive con­ver­sa­tions oc­cur be­tween two peo­ple. Even with Trio walks, the setup is for one per­son to stay on the sidelines and wait for a chance to ro­tate in. A sin­gle con­ver­sa­tion re­ally only hap­pens along one di­men­sion, and it re­quires only two dis­tinct peo­ple to de­tect.

I won­der if monogamy is some kind of at­trac­tor state be­cause in­ter­ac­tions be­tween two peo­ple are the most pro­duc­tive.

Ap­ply­ing the Soli­taire Prin­ci­ple, for all the same rea­sons it’s use­ful to have con­ver­sa­tions be­tween two peo­ple, it’s most use­ful to draw di­chotomies be­tween two pieces of the mind in­stead of more. This is why we have In­ter­nal Dou­ble Crux in­stead of Triple or more. A con­ver­sa­tion/​in­ter­nal con­flict is always about some­thing and should have a pur­pose, and that pur­pose pro­jects the en­tire con­ver­sa­tion onto the one rele­vant di­men­sion, so it’s re­ally only nec­es­sary to di­vide into two sides along this axis. Thus we get S1/​S2, Elephant/​Rider, Epi­sodic/​Di­achronic, etc.

• I like this a lot.

There are many rea­sons it’s so tempt­ing to pro­ject onto a sin­gle axis but maybe the foun­da­tional rea­son is the di­chotomy be­tween ap­proach­ing and avoid­ing, or if you pre­fer, be­tween pos­i­tive and nega­tive re­ward in re­in­force­ment learn­ing terms. This blows up into good vs. evil, friend vs. en­emy, and so forth.

Edit: Also this is why Venkatesh Rao is much more so­phis­ti­cated than we are; he does PCA but pro­jects onto 2 axes and makes a 2x2 square.

• Very pleased to see all of these di­chotomies col­lected in one place. The nat­u­ral ques­tion is whether these di­vides can be in­te­grated to a use­ful pic­ture with more pieces.

My take on the “Two Cul­tures” model of prob­lem-solvers and the­ory-builders: the­ory-build­ing fields of math­e­mat­ics like alge­braic topol­ogy (say) are those where the goal is to ar­tic­u­late grand meta-the­o­rems that are big­ger than any par­tic­u­lar ap­pli­ca­tion. This was the work of a Grothendieck.

Mean­while, con­crete prob­lem-solv­ing fields of math­e­mat­ics like com­bi­na­torics are those where the goal is to be­come the grand meta-the­o­rem that con­tains more un­der­stand­ing than any par­tic­u­lar the­o­rem you can prove. This was the style of an Er­dos. The inar­tic­u­late grand meta-the­o­rems lived in his cog­ni­tive strate­gies so that the the­o­rems he ac­tu­ally proved are in­di­vi­d­u­ally only faint im­pres­sions thereof.

• Yeah, there’s some­thing less leg­ible about com­bi­na­torics com­pared to most other fields of math­e­mat­ics. Peo­ple like Er­dos know lots of im­por­tant prin­ci­ples and meta-prin­ci­ples for solv­ing com­bi­na­to­rial prob­lems but it’s a tremen­dous chore to state those prin­ci­ples ex­plic­itly in terms of the­o­rems and no­body re­ally does it (the clos­est thing I’ve seen is Fla­jo­let and Sedgewick—by the way, amaz­ing book, highly recom­mended). A con­crete ex­am­ple here is the ex­po­nen­tial for­mula, which is or­ders of mag­ni­tude more com­pli­cated to state pre­cisely than it is to un­der­stand and use.

(Ray has been sug­gest­ing to me in per­son that an im­por­tant chunk of the cur­rent big LW de­bate is not about S1/​S2 but about illeg­i­bil­ity and that sounds right to me.)

I re­ally like the phras­ing “be­come the grand meta-the­o­rem.”

• I think I heard the “be­come grand meta-the­o­rem” phras­ing origi­nally from Alon & Spencer. I ac­tu­ally bought the Fla­jo­let and Sedgewick book a cou­ple months ago (only got through the first chap­ter), but it was mind-bog­gling that some­thing like this could be done for com­bi­na­torics.

Of course re­al­ity is self-similar, so it’s not sur­pris­ing that there’s cur­rently a big di­vide in com­bi­na­torics be­tween what I would call the “alge­braic/​enu­mer­a­tive” style of Richard Stan­ley con­tain­ing the Fla­jo­let and Sedgewick stuff, char­ac­ter­ized by fancy alge­bra/​ex­plicit for­mu­lae/​crys­tal­line struc­tures and the “an­a­lytic/​ex­tremal” style of Er­dos, char­ac­ter­ized by asymp­totic for­mu­lae and less leg­i­bil­ity. It’s sur­pris­ingly rare to see a com­bi­na­to­ri­al­ist bridge this gap.

• I went through most of the first half of Fla­jo­let and Sedgewick when I was 18 or so and was blown away, then re­cently went through the sec­ond half and was blown away in a com­pletely differ­ent way. It’s re­ally wild. Take a look. It’s where I learned the ar­gu­ment in this blog post about the asymp­totics of the par­ti­tion func­tion.

• Do you think that the­ory-build­ing and prob­lem-solv­ing maps at all to your ham­mers and nails di­chotomy? One would be about be­com­ing a ham­mer that can hit all the nails, and the other is more about re­ally un­der­stand­ing each par­tic­u­lar nail.

• I think it’s more about the na­ture of the ham­mers. The­ory-build­ing ham­mers are leg­ible: they’re big the­o­rems, or maybe big messes of defi­ni­tions and then the­o­rems (the term of art for this is “ma­chin­ery”). Prob­lem-solv­ing ham­mers are illeg­ible: they’re a bunch of tacit knowl­edge sit­ting in­side some math­e­mat­i­cian’s head.

• I mostly agree. By the end of the ham­mers and nails post I re­al­ized the real di­chotomy was be­tween sys­tem­atic (in­clud­ing both ham­mers and nails) and hap­haz­ard, and this is a differ­ent di­chotomy from all the oth­ers men­tioned in this thread be­cause I will ac­tu­ally make a value judg­ment that sys­tem­atic is just bet­ter.

Then again, these are just two stages of de­vel­op­ment and you can ex­trap­o­late there’s some third stage that’s even bet­ter than sys­tem­atic that looks like hap­haz­ard ge­nius from the out­side.

• Great! Th­ese are some of my fa­vorite es­says on math­e­mat­ics and I’m ex­cited to see what other ra­tio­nal­ists think of them.

I think many math­e­mat­i­ci­ans would ob­ject to lump­ing alge­bra and anal­y­sis to­gether; I know a lot of peo­ple who are great at alge­bra and ter­rible at anal­y­sis. My clusters here are not ter­ribly well-formed but my rough sense is that alge­bra, anal­y­sis, and ge­om­e­try are three fairly differ­ent things.

• Some­what weirdly, I have seen anal­y­sis usu­ally de­scribed as be­long­ing squarely into the in­tu­ition cluster. And I ac­tu­ally par­tially dis­cov­ered my love of anal­y­sis af­ter I read the corn-eat­ing post on anal­y­sis vs. alge­bra, re­al­ized that I eat corn like an an­a­lyst but thought of my­self as an alge­braist, and then re­al­ized that all the alge­bra per­spec­tives I like most are com­ing from the in­tu­ition/​ge­om­e­try per­spec­tive (Lin­ear Alge­bra Done Right and 3Blue1Brown’s videos be­ing two of my top 3 ed­u­ca­tional re­sources, and both be­ing heav­ily in­tu­ition-based as op­posed to alge­bra-based).

(I do not know whether the corn-eat­ing thing is real in any mean­ingful sense, but it did get me to re­con­sider my per­spec­tive on math­e­mat­ics)

• So, I was an un­der­grad math­e­mat­i­cian, and planned to be­come an aca­demic, but bailed out of my PhD and be­came a pro­gram­mer in­stead. I made notes as I was read­ing the var­i­ous ar­ti­cles.

My strong suit in maths was anal­y­sis. I just never ‘got’ alge­bra at all and didn’t touch it af­ter the first year.

Weirdly I was very good at lin­ear alge­bra/​ma­tri­ces/​spec­tral the­ory/​fourier anal­y­sis. But all that seemed like a ge­o­met­ri­cal, in­tu­itive the­ory about high-di­men­sional spaces to me. I had very strong re­li­able in­tu­ition there, but I never had any in­tu­ition, or idea about how one might go about ac­quiring one, for rings/​fields/​groups or math­e­mat­i­cal logic.

I never liked any sort of sym­bol-ma­nipu­la­tion. I felt I un­der­stood things if and only if I could make men­tal pic­tures of what was go­ing on that would im­ply the an­swers ‘as if by magic’.

M. Meray’s en­deav­ours seem un­ap­peal­ing. I ap­pre­ci­ate them in the ab­stract but can­not imag­ine get­ting in­ter­ested. Prof. Klein’s con­duct­ing sphere seems a fas­ci­nat­ing mas­ter­stroke.

Feyn­man/​Ein­stein are definitely ‘what I’d be if I was twenty times bet­ter’. I recog­nise their ways of think­ing, at least as they ex­plained them.

I agree whole­heart­edly with Arnold’s rant.

I’m am­biva­lent on the prob­lem-solver/​the­o­rizer dis­tinc­tion. I think I’m more of a the­o­rizer, but prob­lem-solv­ing is im­por­tant and they both mat­ter. I’d have been proud to have con­tributed in ei­ther way.

Maths is very vi­sual for me. The sym­bols mean noth­ing with­out the pic­tures.

As a pro­gram­mer, I:

loathe OO

love lisp, and found it mind blow­ing when I first found it. By de­fault I use a lisp var­i­ant called Clo­jure both per­son­ally and pro­fes­sion­ally, al­though I’ve tried al­most ev­ery­thing. I avoid java and c++ if I can.

have oc­ca­sion­ally tried Haskell, and feel that I ought to un­der­stand it, but it feels like pro­gram­ming with one hand tied be­hind my back. An awful lot of ex­tra effort for no gain.

am quite fond of python, al­though I use it as a wa­tered-down lisp and avoid all its OO fa­cil­ities.

adored “Why Arc isn’t es­pe­cially Ob­ject Ori­ented”

have never tried tem­plate metapro­gram­ming, C++ is just too dirty for me, al­though I love C it­self.

like both vi and emacs, and was origi­nally a vi user, but these days I use emacs al­most ex­clu­sively, and have done ever since I dis­cov­ered what a joy it is as a lisp ed­i­tor.

I think that all, with the ex­cep­tion of emacs, puts me strongly on the anal­y­sis/​in­tu­ition side of things and weakly con­firms the sug­gested di­chotomy and its re­la­tion­ship to pro­gram­ming styles.

But it’s been a long time since I ate corn-on-the-cob. When I try to vi­su­al­ise it I see my­self eat­ing it in rows rather than spirals. But I don’t want to go out and find some, be­cause then I’d bias the re­sult. Some­how I have to catch my­self in the act of eat­ing it un­con­sciously. Any sug­ges­tions?

• That post is hilar­i­ous, and fas­ci­nat­ing.

I eat corn like an an­a­lyst, vastly pre­fer Lisp to Haskell, use Vim, iden­tify much more strongly with the per­son­al­ity de­scrip­tion of the an­a­lyst, and while I haven’t done much higher math, have a deep and abid­ing love for the delta-ep­silon defi­ni­tion of a limit.

Very cu­ri­ous to hear other re­sults, ei­ther suc­cess­ful or not.

• I eat corn like an an­a­lyst, and

• [+] did my PhD in a fairly anal­y­sis-y field (but also fairly ge­o­met­ri­cal, con­trary to the anal­y­sis+alge­bra/​ge­om­e­try split kinda-im­plied by the OP here)

• [+] pre­fer Lisp to Haskell but [-] feel vaguely guilty about that from time to time and feel I re­ally “ought” to learn Haskell properly

• [+] use Vim but [-] only be­cause Emacs was bad for my wrists

• [-] don’t much care for fancy C++ tem­plate metapro­gram­ming but [+] also don’t much care for hard­core OO pro­gram­ming, though [-] I don’t by any means ob­ject to OO, “de­sign pat­terns”, etc.

• Am a com­puter sci­en­tist, work­ing on AI al­ign­ment the­ory.

• I’m prob­a­bly one of the peo­ple where I work who is more sym­pa­thetic to MIRI-style ways of think­ing about al­ign­ment.

• Leaned to­wards a type of think­ing that I la­bel­led “alge­braic” as a math un­der­grad.

• My best course in un­der­grad was in­tro to anal­y­sis, but it was taught by a PDEs guy. Our de­part­ment only had one real an­a­lyst, and was pre­dom­i­nantly com­posed of alge­bra peo­ple.

• My favourite take on lin­ear alge­bra in­volves a ‘ge­o­met­ric’ ap­proach, e.g. think­ing of lin­ear op­er­a­tors, not ma­tri­ces, and tak­ing this sort of view of the sin­gu­lar value de­com­po­si­tion.

• I wish that ev­ery­body would always de­note vec­tors with bra-ket no­ta­tion.

• My pri­mary aca­demic con­tri­bu­tion to CS was to take a bunch of proofs about one fam­ily of prob­a­bil­ity dis­tri­bu­tions, and see if they worked on a differ­ent fam­ily of prob­a­bil­ity dis­tri­bu­tions (if this doesn’t sound CS-y.… uh, I’m a fake CS boy).

• I re­ally like Haskell, and de­vi­a­tions from it re­ally bother me. In par­tic­u­lar, the bits I like are the fact that it’s func­tional and strictly typed.

• I mostly use Python be­cause it’s eas­ier.

• Ob­ject-ori­ented pro­gram­ming seems weird and creepy to me.

• I use emacs, and have briefly tried vi-type things but they never stuck.

• When eval­u­at­ing ar­gu­ments, I tend to ask ques­tions like “is this ar­gu­ment sym­met­ric in the ap­pro­pri­ate vari­ables”, “if you take this vari­able to 0 or in­finity, does the ar­gu­ment still work”, or “does this type check”. I could trans­late this into terms that make more sense for ver­bal/​non-math­e­mat­i­cal ar­gu­ments, but hon­estly this is how I think of it.

• When eat­ing corn on the cob, I think I do it in spirals.

• I only eat corn on the cob at my fam­ily home where I grew up, which is a differ­ent part of my life than the part that con­tains ev­ery­thing else on this list.

Look­ing over the post, I guess that I’m ba­si­cally an alge­braist ex­cept for the way I eat corn?

• Eat­ing corn on the cob is messy and gets stuff stuck in my teeth. It’s also slow. I always find a knife (even just a plas­tic but­ter knife), cut the corn off, and eat it with a fork or spoon. What cat­e­gory does that fit in? Un­til I started do­ing this, I think I kept ex­per­i­ment­ing with eat­ing in differ­ent pat­terns. I have no idea what it’s like to eat corn with­out try­ing to op­ti­mize the pro­cess.

• My feel­ing is that that’s prob­a­bly anal­y­sis-style rather than alge­bra-style. (Even though the ac­tual or­der of corn-ker­nel re­moval is more like that of alge­braists.) Are any of the other dis­tinc­tions that allegedly cor­re­late with it ones that you can match up with your life? Of course they won’t be if you’re not a math­e­mat­ics/​soft­ware type.

(It would be very in­ter­est­ing to know whether the alge­bra/​anal­y­sis di­vide among math­e­mat­i­ci­ans is a spe­cial case of some­thing that ap­plies to a much broader range of peo­ple, and corn-eat­ing might be a way to ex­plore that. But I don’t think cornol­ogy is far enough ad­vanced yet to make con­fi­dent con­jec­tures about what per­son­al­ity fea­tures might cor­re­late with differ­ent modes of corn-eat­ing.)

• I’m a soft­ware en­g­ineer and my de­gree in col­lege re­quired a good chunk of ad­vanced math. I am cur­rently in the pro­cess of try­ing to re­learn the math I’ve for­got­ten, plus some, so I’m think­ing that if this anal­y­sis/​alge­bra di­chotomy points at a real prefer­ence differ­ence, know­ing which I am might help me choose more effec­tive learn­ing sources.

But I find it hard to point to one cat­e­gory or an­other for most as­pects. Even the corn test is in­con­clu­sive! (I agree that it sounds more like an anal­y­sis thing to do.)

• I love the step-by-step bits of alge­bra and logic, but I also love ge­om­e­try.

• I think I do tend to form an “idiosyn­cratic men­tal model of spe­cific prob­lems.” As I come to un­der­stand prob­lems more, I feel like they have a qual­ity or char­ac­ter that makes them rec­og­niz­able to me. I did best in school when teach­ing my­self from out­side sources and then us­ing the teacher’s meth­ods to spot check and fill in gaps in my mod­els.

• I think ob­ject ori­ented pro­gram­ming is very use­ful, and func­tional pro­gram­ming is very ap­peal­ing.

• I use(d) vi/​vim be­cause that’s what I know well enough to func­tion in. I barely touched emacs a cou­ple times, was like, “dafuq is this?” and went back to vim. I never gave emacs a fair chance.

• I think I lean to­wards ‘build­ing up’ my un­der­stand­ing of things in chunks, filling in a big­ger pic­ture. But the skill of ‘break­ing down’ mas­sive con­cepts into bite-sized chunks seems like an im­por­tant way to do this!

My ten­ta­tive self di­ag­noses is that I have a weak prefer­ence for anal­y­sis. Read­ing more of the links in the OP might help me con­firm this.

• That is an amaz­ing post.

• I just start gnaw­ing on the corn cob some­where at ran­dom, like the hor­rible physi­cist I am :) But the ‘anal­y­sis’ style makes more sense to me of the two, it had never even oc­curred to me that you could eat corn in the ‘alge­bra’ style.

I also think about lin­ear alge­bra in a very vi­sual way. I’m miss­ing that for a lot of group the­ory, which was pre­sented to us in a very ‘mem­o­rise this ran­dom pile of defi­ni­tions’ way. Some time I want to go back and fix this… when I can get it to the top of the very large pile of things I want to learn.

• I’m on the bor­ing side of all di­chotomies in the OP, and the one with corn too. Fun­nily, my vi­sual imag­i­na­tion is pretty good (men­tal ro­ta­tion etc.) I just never seem to use it for math or pro­gram­ming, it’s step-by-step all the way.

• I hate eat­ing corn on the cob, I don’t re­mem­ber the last time I did it, and I can’t even re­ally in­ner sim do­ing it. Math­e­mat­i­cally I spend a lot of time talk­ing about alge­bra but am also, I think, bet­ter at anal­y­sis than other math­e­mat­i­ci­ans would pre­dict based on my rep­u­ta­tion.

• That’s one of the more use­ful posts I’ve read in a while since it gives me a way to con­soli­date a bunch of other loose thoughts that have been kick­ing around. Thanks.

• Ah, that prob­a­bly needs clar­ify­ing… I was us­ing ‘anal­y­sis’ in the sense of ‘op­posed to syn­the­sis’ as one of the di­chotomies, rather than the math­e­mat­i­cal sense of ‘anal­y­sis’. I.e. ‘break­ing into parts’ as op­posed to ‘build­ing up’. That’s pretty con­fus­ing when one of the other di­chotomies is alge­bra/​ge­om­e­try!

I agree that alge­bra and (math­e­mat­i­cal) anal­y­sis are pretty differ­ent and I wouldn’t par­tic­u­larly lump them to­gether. I’d per­son­ally prob­a­bly lump it with ge­om­e­try over alge­bra if I had to pick, but that’s likely to be a fea­ture of how I learn and re­ally it’s pretty differ­ent to ei­ther.

• I eat corn like an an an­a­lyst, and I am an an­a­lyst. I also use vim over emacs, like Lisp, and find ob­ject-ori­ented pro­gram­ming weirdly dis­taste­ful.

How­ever I don’t think anal­y­sis and alge­bra are usu­ally lumped to­gether and op­posed to ge­om­e­try; my un­der­stand­ing was that tra­di­tion­ally alge­bra, anal­y­sis, and ge­om­e­try were the three main fields of math.

I tend to think of the dis­tinc­tions within math as about how much we posit that we know about the ob­jects we work with. The ob­jects of study of math­e­mat­i­cal logic are very gen­eral and thus can be very “per­verse”; the ob­jects of study of alge­bra and topol­ogy are also quite gen­eral; the ob­jects of study of ge­om­e­try are more pinned down be­cause you have a met­ric; the ob­jects of study of anal­y­sis are the ”best be­haved” of all, be­cause they have smooth­ness and in­te­gra­bil­ity prop­er­ties.

I find anal­y­sis much eas­ier than alge­bra be­cause I rely a lot on the con­crete­ness of be­ing able to mea­sure, es­ti­mate, and (some­times) vi­su­al­ize. Peo­ple who are more alge­bra-ori­ented are more likely than me to be­come ir­ri­tated by do­ing fiddly com­pu­ta­tions, but they have more abil­ity to rea­son about very ab­stract ob­jects.

• No, I also definitely wouldn’t lump math­e­mat­i­cal anal­y­sis in with alge­bra… I’ve ed­ited the post now as that was con­fus­ing, also see this re­ply.

Your ‘how much we know about the ob­jects’ dis­tinc­tion is a good one and I’ll think about it.

Also vim over emacs for me, though I’m not ac­tu­ally great at ei­ther. I’ve never used Lisp or Haskell so can’t say. Ob­jects aren’t dis­taste­ful for me in them­selves, and I find Javascript-style pro­to­ty­pal in­her­i­tance fits my head well (it’s con­crete-to-ab­stract, ‘ex­am­ples first’), but I find Java-style ob­ject-ori­ented pro­gram­ming an­noy­ing to get my head around.

• Hav­ing been a ge­ome­ter that mi­grated to com­puter sci­ence via for­mal logic, I can tes­tify to this di­vi­sion—to some ex­tent.

When I first learnt for­mal logic and then ma­chine learn­ing, I had the same plod­ding, ‘alge­bra’ ap­proach. But now that I’ve grasped it bet­ter, I’ve started to de­velop an in­tu­ition in these ar­eas, that can short­cut most of the plod­ding ap­proach (and it’s so much more fun).

I think the differ­ence might be more in the way the ideas are com­mu­ni­cated. You can com­mu­ni­cate semi-rigor­ous ge­o­met­ric ideas in a (some­what) in­tu­itive way, and have other ge­ome­ters grasp them, at least enough that they can re-cre­ate them rigor­ously if needed. But alge­braic ideas have to be more ex­plicit if you want any­one be­yond your im­me­di­ate cir­cle to get them.

See for in­stance Bour­baki, where the in­ter­nal dis­cus­sions were filled with in­tu­ition and imagery, but where the writ­ten out­puts were fa­mously te­dious and rigor­ous.

• I’m a math/​econ un­der­grad, I’ve found that us­ing ge­om­e­try and imagery to con­tex­tu­al­ize all my classes is the eas­iest way for me to re­ally un­der­stand a sub­ject.

To use a small ex­am­ple: Learn­ing things like the chain rule or the product rule in calcu­lus be­came triv­ial once I learned via this method. How­ever, that is not a way of teach­ing that is pre­sent where I’m learn­ing. I’ve had lit­tle (but not zero) suc­cess in find­ing re­sources on my own that choose to com­mu­ni­cate ideas in this way. Or help me hone my vi­sual-math rea­son­ing skills (1 2). I feel like learn­ing other ways just re­quire too much mem­o­riza­tion and doesn’t eas­ily slot into my in­tu­ition. As a re­sult when­ever some­thing doesn’t in­tu­itively trans­late to imagery, I feel like I’m plod­ding along. Are there books, lec­tures, se­quences, or any­thing out there that I could use? Any­thing you could send my way would be re­ally ap­pre­ci­ated.

• Ok, here’s a 2x2 that cap­tures a lot of the vari­a­tion in OP:

ab­stract/​con­crete x in­tu­itive/​me­thod­i­cal.

In­tu­itive vs. Me­thod­i­cal is what Atiyah, Klein, and Poin­care are talk­ing about. Ab­stract vs Con­crete is what Gow­ers, Rota, and Dyson are talk­ing about.

Ab­stract and in­tu­itive is like Grothendieck.

Con­crete and in­tu­itive is like ge­om­e­try or com­bi­na­torics.

Con­crete and me­thod­i­cal is like anal­y­sis.

Ab­stract and me­thod­i­cal — I don’t know what goes in this space.

• This seems good. I was definitely get­ting the sense there were at least two axes, and these seem to cap­ture a lot of it.

Could Ab­stract/​Me­thod­i­cal be some­thing like Rus­sell and White­head’s Prin­cipia Math­e­mat­ica?

Also, I’m in­ter­ested that Con­crete/​Me­thod­i­cal is anal­y­sis, given the Corn post. I would have ex­pected it to be In­tu­itive? (I don’t ac­tu­ally do higher math, so I don’t know from per­sonal ex­pe­rience.)

• Pro­moted to front­page.

(You men­tion be­ing un­sure about whether it’s a good fit for the front­page. We’ve been hav­ing a de­bate about how in­tu­ition works lately on LW, and I think a post that brings in a bunch of solid data from a healthy field like math­e­mat­ics is ab­solutely ap­pro­pri­ate, and re­ally ap­pre­ci­ated.)

• This post is great, and is prob­a­bly one of my fa­vorite things on LessWrong 2.0 so far. Thank you a lot for writ­ing this, and I am look­ing for­ward to read­ing all the es­says in full when I find the time for that.

• In the chakra sys­tem, the 5th (throat) is as­so­ci­ated with alge­bra, while 6th (third eye) is as­so­ci­ated with ge­om­e­try. (Yet an­other place where this dis­tinc­tion has been noted.) I could prob­a­bly go off on an in­tu­ition-based rant about what this might im­ply, which should tell you which one I am.

I’d also pre­dict birds pre­fer games with some var­i­ance in them (card games), and frogs pre­fer de­ter­minis­tic games (chess, go).

Ka-kaw!

• The chakra thing sounds right; an­other way of putting it is that alge­bra is more ver­bal & ge­om­e­try is more vi­sual/​spa­tial. (IMO, anal­y­sis is vi­sual/​spa­tial too.)

• I’m not sure what my brain is do­ing when it thinks about anal­y­sis but I’m not con­vinced it’s vi­su­ospa­tial.

More con­cretely, let’s sup­pose I’m try­ing to an­a­lyze the asymp­totic be­hav­ior of some func­tion which is a sum of terms that have differ­ent growth rates, say . What I could be do­ing, if I were do­ing this vi­su­ally, is vi­su­al­iz­ing the the asymp­totic be­hav­ior of (“grows fast as gets big”) as a curve that curves up re­ally fast, and similarly for (“grows fast as gets small, goes to zero as gets big”) as a curve that starts big and gets small.

But I think that’s not what I’m ac­tu­ally do­ing first, al­though that is a mode of thought I can use and find helpful. I think I am ac­tu­ally work­ing with some­thing like a prim­i­tive felt sense of big­ness and smal­l­ness, which is not about vi­sual tal­l­ness (to tri­an­gu­late, an­other metaphor for big­ness that isn’t vi­sual is weight: gets “heavy” and gets “light”). Not sure though, be­cause the vi­sual thought also hap­pens pretty quickly af­ter this and ev­ery­thing is cor­re­lated (heavy things are large in my vi­sual field, etc.).

• Th­ese es­says had a pretty large im­pact on how I go about learn­ing math­e­mat­ics, I always had an eas­ier time when for­mu­las or ar­gu­ments could be mapped onto vi­sual struc­ture. In-fact, be­fore writ­ing this com­ment (and in gen­eral when con­struct­ing ar­gu­ments) I imag­ined a mind map con­tain­ing all the rele­vant ideas and re­la­tions I wanted to por­tray. I am now (some­what poorly) at­tempt­ing to trans­late my 3-D vi­sual ar­gu­ment into a lin­ear ver­bal one.

Some­thing else to be noted is vi­sual rea­son­ing and com­ple­men­tary cog­ni­tive ar­ti­facts seem to go hand in hand. Con­sider that learn­ing to use an aba­cus can al­low some­one to simu­late an aba­cus in their mind and pro­duce the out­puts of an aba­cus with­out need­ing to ac­tu­ally have one. A similar thing can be done with a slide rule. This prac­tice can also pro­duce other, pos­i­tive effects on cer­tain parts of cog­ni­tion*.

I would be sur­prised if the skill of be­ing able to con­struct com­ple­men­tary cog­ni­tive ar­ti­facts wasn’t po­ten­tially helpful in many do­mains. I don’t know how one would go about learn­ing this, but it seems like some­thing to con­sider as hav­ing pos­i­tive value if in­ves­ti­gated.

*Those pa­pers are the first things that came up with a google search. So I re­serve the right to be wrong about the ex­act con­se­quences.

• Late to com­ment­ing on this post, but where would Gri­gori Perel­man, the prover of the Poin­care con­jec­ture, fall then? I re­mem­ber this quote from his bi­og­ra­phy:

Golo­vanov, who stud­ied and oc­ca­sion­ally com­peted alongside Perel­man for more than ten years, tagged him as an un­am­bigu­ous ge­ome­ter: Perel­man had a ge­om­e­try prob­lem solved in the time it took Golo­vanov to grasp the ques­tion. This was be­cause Golo­vanov was an alge­braist. Su­dakov, who spent about six years study­ing and oc­ca­sion­ally com­pet­ing with Perel­man, claimed Perel­man re­duced ev­ery prob­lem to a for­mula. This, it ap­pears, was be­cause Su­dakov was a ge­ome­ter: his fa­vorite proof of the clas­sic the­o­rem above was an en­tirely graph­i­cal one, re­quiring no for­mu­las and no lan­guage to demon­strate. In other words, each of them was con­vinced Perel­man’s mind was profoundly differ­ent from his own. Nei­ther had any hard ev­i­dence. Perel­man did his think­ing al­most en­tirely in­side his head, nei­ther writ­ing nor sketch­ing on scrap pa­per. He did a lot of other things—he hummed, moaned, threw a Ping-Pong ball against the desk, rocked back and forth, knocked out a rhythm on the desk with his pen, rubbed his thighs un­til his pant legs shone, and then rubbed his hands to­gether—a sign that the solu­tion would now be writ­ten down, fully formed. For the rest of his ca­reer, even af­ter he chose to work with shapes, he never daz­zled col­leagues with his ge­o­met­ric imag­i­na­tion, but he al­most never failed to im­press them with the sin­gle-minded pre­ci­sion with which he plowed through prob­lems. His brain seemed to be a uni­ver­sal math com­pactor, ca­pa­ble of com­press­ing prob­lems to their essence. Club mates even­tu­ally dubbed what­ever it was he had in­side his head the “Perel­man stick”—a very large imag­i­nary in­stru­ment with which he sat quietly be­fore strik­ing an always-fatal blow.

from Perfect Ri­gor.