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:
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.”
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.