Examples in Mathematics

After read­ing Luke’s in­ter­view with Scott Aaron­son, I’ve de­cided to come back to an is­sue that’s been bug­ging me.

Speci­fi­cally, in the an­swer to Luke’s ques­tion about ob­ject-level tac­tics, Scott says (un­der 3):

Some­times, when you set out to prove some math­e­mat­i­cal con­jec­ture, your first in­stinct is just to throw an ar­se­nal of the­ory at it. (..) Rather than look­ing for “gen­eral frame­works,” I look for easy spe­cial cases and sim­ple san­ity checks, for stuff I can try out us­ing high-school alge­bra or maybe a five-line com­puter pro­gram, just to get a feel for the prob­lem.

In a similar vein, there’s the Hal­mos quote which has been heav­ily up­voted in the Novem­ber Ra­tion­al­ity Quotes:

A good stack of ex­am­ples, as large as pos­si­ble, is in­dis­pens­able for a thor­ough un­der­stand­ing of any con­cept, and when I want to learn some­thing new, I make it my first job to build one.

Every time I see an opinion ex­press­ing a similar sen­ti­ment, I can’t help but con­trast it with the opinions and prac­tices of two wildly suc­cess­ful (very) the­o­ret­i­cal math­e­mat­i­ci­ans:

Alexan­der Grothendieck

One strik­ing char­ac­ter­is­tic of Grothendieck’s mode of think­ing is that it seemed to rely so lit­tle on ex­am­ples. This can be seen in the leg­end of the so-called “Grothendieck prime”. In a math­e­mat­i­cal con­ver­sa­tion, some­one sug­gested to Grothendieck that they should con­sider a par­tic­u­lar prime num­ber. “You mean an ac­tual num­ber?” Grothendieck asked. The other per­son replied, yes, an ac­tual prime num­ber. Grothendieck sug­gested, “All right, take 57.” But Grothendieck must have known that 57 is not prime, right? Ab­solutely not, said David Mum­ford of Brown Univer­sity. “He doesn’t think con­cretely.” Con­sider by con­trast the In­dian math­e­mat­i­cian Ra­manu­jan, who was in­ti­mately fa­mil­iar with prop­er­ties of many num­bers, some of them huge. That way of think­ing rep­re­sents a world an­tipo­dal to that of Grothendieck. “He never re­ally worked on ex­am­ples,” Mum­ford ob­served. “I only un­der­stand things through ex­am­ples and then grad­u­ally make them more ab­stract. I don’t think it helped Grothendieck in the least to look at an ex­am­ple. He re­ally got con­trol of the situ­a­tion by think­ing of it in ab­solutely the most ab­stract way pos­si­ble. It’s just very strange. That’s the way his mind worked.”

(from Allyn Jack­son’s ac­count of Grothendieck’s life).

Maxim Kontsevich

Saito: This is one typ­i­cal point of your work. But I find that in much of your work, by hear­ing one symp­tom you cap­ture the cen­tral point of the prob­lem and then give some gen­eral big frame­work. That’s my gen­eral im­pres­sion of what you are do­ing.

Kont­se­vich: Yeah, I re­ally don’t work on ex­am­ples at such a level.

Saito: How can you work in that way?

Kont­se­vich: For my­self some­times I work on one or two ex­am­ples, but...

Saito: You already keep some ex­am­ples in mind, but still you con­struct the­ory.

Kont­se­vich: Yes. And gen­er­ally I find ex­am­ples some­times to be mis­lead­ing. [Laugh­ter]. Be­cause of­ten the prop­er­ties of ex­am­ples are too spe­cial, you can­not see gen­eral prop­er­ties if you con­stantly work too much on con­crete ex­am­ples.

(from the IPMU in­ter­view).

Are they fool­ing them­selves, or is there some­thing to be learned? Per­haps it’s pos­si­ble to men­tion Gow­ers’ Two Cul­tures in the an­swer.

P.S. First con­tent post here, I would ap­pre­ci­ate feed­back.