‘every Cauchy sequence with a convergent subsequence converges’.

And this I think is part of a general one in mathematics where a ‘smaller’ subproblem determines the larger one. It also helps educate one into looking for similar kinds of thinking. One might think of this as learning a kind of aesthetics in mathematics.

]]>IMHO the idea about pairing Proof and Discovery is a great idea on its own. Pairing same is the way memorization should be structured too.

Prof Gowers, would it be ok for me to email you privately the one time? Reason being I have sort of tracked you down hoping to talk about a proof. I’m too embarrassed to talk about it public?

]]>That way way punching over my weight. Then when my daughter was born she had some learning difficulties. They marked her IQ 70 at one point. But by the time she was 9 she made it pretty clear that if life was going to one long coming last and being invisible, then she basically didn’t want it. So I quite work, and coached her, throwing some incantations and running. Two years later she banned me from her memorization files. Passed 10 GCSE’s and is currently on Math AS Level of all things.

All true. Al Hibbs KCL 94-97 mech eng

p.s. kind of the reason I wanted to bragg all that, is because that memorization tutorial and the generally positive, or at least if they’re doing it anyway, let’s make work better. I was really moved actually. I’ve never seen a math person or science person not totally pejorative about techniques. I’ve tried to raise the matter so often, that it’s not rote when methodical. But people throw out counter intuitive Popperian ways learning must be.

]]>But how we memorize proofs is itself closely bound up with how we *discover* proofs. As I am trying to convey in the post, we train ourselves to do certain steps almost automatically, and once we have done that we no longer have to memorize those steps — we just remember that the obvious thing works and then do the obvious thing.

Finally, I believe that a hard science of how we discover proofs is possible, at least for a wide range of proofs. For example, in my work with Mohan Ganesalingam, I have found that there is a very strong correlation between the technical difficulties that arise when we try to think how a computer could discover such-and-such a proof, and the difficulties that we see undergraduates encountering when we supervise them. So we have become sensitive to a very fine gradation of difficulty, which seems to be an objective one and amenable to rigorous study.

]]>I mean between the kinds of methods that do it right for memorization, and maths itself. Not memorization.

Do you think a hard science theory is possible for the nature of science. A proof?

Only if you have time.

]]>I think the last occurrence of “Cauchy” should say “convergent”?

*Indeed it should. Corrected now — thanks.*

A typical request that I receive is that I should select a small subset of the proofs and declare those as the ones that might turn up in the exam, because otherwise there are too many to memorize. (I am, however, only willing to declare a very small number of proofs to be non-examinable.)

I have made my own efforts to counter this based on the idea that the important thing is understanding the material and developing general fluency in at least many of the “routine” portions of “doing proofs”. However a significant number of students still appear to abandon understanding in favour of memorization. In the anonymous feedback, some students have explicitly told me that the abstract material and reasoning/doing proofs is too big a jump from A level to cope with, and that is why they have resorted to memorization.

I shall continue to refer my students to this blog, in the hope that it will help!

Joel ]]>