Page 14 - Textos de Matemática Vol. 39

4 GRACIANO DE OLIVEIRA
the letter, and I immediately understood it was an exceptional result. I advised Sa´ to send it for publication. A little later I received another letter from him saying that there was a mistake in the first version (good mathematicians make mistakes too), and sending me the correction. I do not remember what the mistake was. I kept the two letters — I still remember where I had them, on a bookshelf in my home — until ... they disappeared. I regret this, as I would much like to read them again.
I would like to say something about the use of computers in research. Computers are amazing machines — I wonder what the first King of Portugal would do had he had one. But contrary to what many people think, they are not the philosopher’s stone. In the fields of creativity, imagination, ethics and esthetics, they are worth nothing when compared to the human brain. But they can be of help in research, as pencil and paper are as well. The difference is one of degree: the computer is like pencil and paper only with greater scope, much as these were preferable to stylus and clay tablet from millennia ago. When we do research in Mathematics, we often start, using pencil and pa- per (or stylus and tablet!), with simple cases, to see what happens after some computations... looking for inspiration to fly higher, to where only the human brain can go. With the computer, we can try to see what happens in more complicated cases, to proceed with the flight where the computer cannot even crawl. Two examples happened close to me. A few years ago, F. C. Silva sent me a paper for possible publication in Linear Algebra and its Applications. It contained an interesting detail: there was a proof done with the help of the computer. To complete the proof of a statement, the author needed, if I remember correctly, to show that an equation of a certain type, with coeffi- cients in a field, had a solution in that field. The author had a proof for that, but there remained some (small) finite fields for which the proof didn’t work, as is often — and annoyingly — the case. The exceptional cases, therefore, had to be dealt with one at a time. The problem was that their number was very large, I think in the hundreds or more. F. C. Silva wrote a program and put the computer to work, reaching the conclusion that his statement was true. The computer did the brute and repetitive work very fast and effectively, and with the customary lack of imagination.
A while later F. C. Silva found a proof in which the computer was not needed, and the paper came to be accepted and published with the computer aid reduced to word processing.
There is another example involving Marques de Sa´ and his interlacing inequalities, where the role of the computer was different: someone more intel- ligent than the machine was led to imagine new things. When I returned to Coimbra in the fall of 1975, Marques de Sa´, to break the monotony of life in the capital, was playing with a new piece of equipment in the Instituto de F´ısica e Matema´tica, where he then worked. The thing included a plotter. Since I had