My reality tunnel
I got over the years two different perceptions of computers. What happened first, several years ago, was when we were computing the local index formula for foliations with Henri Moscovici. At that point we had a negative perception of computers. We were thinking it was much better to do computations by hand than to do them using a computer. So we spent separately 3 weeks 8 hours a day to compute some formula which we wanted to know. After that time of course we compared our results and we found there were minor mistakes. After these minor corrections our results agreed but we were supposed to get a cocycle and the formula didn’t give us a cocycle.
Then we got really worried that perhaps there was a mistake in the theory but we found by staring at the result that if we change the sign of 8 of the 36 terms appearing in the final formula then it became a cocycle. We first went back to check again these terms and realized after a while that the reason why their signs were wrong was quite subtle : we had made a conceptual mistake in doing the computation and had forgotten some crucial terms in the differential operators below the sub-principal symbol! At that point I convinced myself that we could never find this subtle correction without the intimate knowledge of all the terms of the formula which only the slow computation “by hand” could give us…
Several years later I changed my mind. Namely I met in my work with Michel Dubois-Violette a result that was expressed as the sum of 1440 integrals, each the integral over a period of an elliptic curve (with modulus q) of a rational function of high degree in theta functions and their derivatives. Even in the trigonometric limit it was difficult to compute this sum. So we used the computer in that limit case first and it produced a beautiful formula in terms of the parameters of the theory. Then it took us about 6 months to guess how to extend this formula in the elliptic case using Jacobi’s elliptic functions sn cn etc…but there was still an unknown function of the modulus q in front which we could not guess. We then found some simplifications in the theory which divided the computing time by a factor of fifty and then we got the first terms in the expansion in powers of q and recognized the first terms in the expansion of the ninth power of the Dedekind eta function. We could predict the next term but it was still beyond the computing power of a private computer and we had to use the computer system in ecole polytechnique. It produced exactly the predicted term. We were then sure that we had the correct formula. It then took a lot of hard work to understand its meaning but we eventually did it.
So then I cannot deny that computers can be incredibly useful because it was simply not possible to do things by hand in that case: each of the 1440 integrals, when expanded in powers of q up to the relevant power, was taking up to 200 pages of trigonometric formulas. You can’t do this by hand; its just impossible. Clearly computers make it possible to see much further in some circumstances! I view them as a great help, a bit like a slave doing without mistakes and complaints the most tedious tasks ever!
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- Gelfand: You mentioned computers. What has changed in mathematics since their appearance?
- Manin: What has changed in pure mathematics? The unique possibility of doing large-scale physical experiments in mental reality arose. We can try the most improbable things. More exactly, not the most improbable things, but things that Euler could do even without a computer. Gauss could also do them. But now, what Euler and Gauss could do, any mathematician can do, sitting at his desk. So if he doesn’t have the imagination to distinguish some features of this Platonic reality, he can experiment. If some bright idea occurs to him that something is equal to something else, he can sit and sit and compute a value, a second value, a third, a millionth. Not only that. People have now emerged who have mathematical minds, but are computer oriented. More precisely, these sorts of people were around earlier, but, without computers, somehow something was missing. In a sense, Euler was like that, to the extent that he was just a mathematician— he was much more than just a mathematician— but Euler the mathematician would have taken to computers passionately. And also Ramanujan, a person who didn’t even really know mathematics. Or, for instance, my colleague here at the institute, Don Zagier. He has a natural and great mathematical mind, which is at the same time ideally suited to work with computers. Computers help him study this Platonic reality, and, I might add, quite effectively. I myself am not this sort of person at all, but I understand what this is about and would be glad to have collaborators who might help me in this. So this is what computers have done for pure mathematics.
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