Yesterday Doron Zeilberger talked at the British Combinatorial Conference.
As always it was a stimulating performance, challenging the audience’s assumptions (not least about whether it is permissible to use electronic equipment during a lecture). I won’t discuss the whole thing, but the idea seems worth commenting on.
Suppose you ask your students to prove that the sum of squares of the first n natural numbers is n(n+1)(2n+1)/6. Probably they will say: this is true for n = 0 (the empty sum is equal to (0·1·1)/6), and then verify that if it is true for a value n then it is true for n+1 (by adding (n+1)2 to both sides of the equation). There is also, as he said, quite a good chance that they will not really understand what they are doing, but simply try to parrot what their professor told them.
On the other hand, a student who wrote
- empty sum = 0 = (0·1·1)/6
- 12 = 1 = (1·2·3)/6
- 12+22 = 5 = (2·3·5)/6
- 12+22+32 = 14 = (3·4·7)/6
would get 9 out of 10 from Doron, because all you have to do is mutter an incantation and this is a perfectly valid proof.
In this case, the incantation [my word, he just muttered “mutter something”] is: “both sides are obviously polynomials of degree 3; if they agree at four points then they are the same”.
In a slightly more elevated form, this principle applies to a lot of mathematics. He claims that many papers are published proving results which could be done just be a finite calculation. In his words, every theorem is trivial in God’s eyes, so no paper should be rejected on the grounds of triviality; but some papers, like cigarette packets, should carry warnings that the theorem can be proved by calculation.
His opinion of the Riemann hypothesis is: Odlyzko proved that the first billion Riemann zeros have real part 1/2; sometime in the next fifty years someone will come up with the appropriate incantation to show that only a finite amount of calculation is needed, maybe as few as 50 zeros, so Odlyzko did far more than was necessary.
As usual with Doron’s lectures, I was entertained but not convinced.