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7. Solutions and Disillusionment
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In mathematics there is an elitist moment that corresponds to the correct intuition of the solution to a problem and is reserved for the enlightened few, and a second, genuinely democratic moment when that solution is revealed to one and all through a proof. On closer inspection, a mathematical proof is a succession of small, logical steps, connected to one another so that anyone may examine the links as thoroughly as possible. Ideally, each one of the steps should be so simple that any person possessing even the most basic acquaintance with symbols could check it almost automatically, verifying each connection in a “local” way, just as a computer traces innocent little lines in minuscule squares on a screen without knowing that they will ultimately form a portrait of the Mona Lisa. This combination of imagination and freedom to conjecture solutions , and of transparency and rigor in proofs, might well be the key to the depths that mathematical thought has reached, as compared to the relatively horizontal accumulation of knowledge found in other disciplines. Nevertheless, the complexity of certain problems and 7 Solutions and Disillusionment1| 87 | | 88 | BORGES AND MATHEMATICS the use of computers can dramatically change the concept of “solution ” and the nature of proofs. One of the most important problems in algebra—how to classify certain mathematical objects known as finite groups—required a Herculean effort involving a team of dozens of mathematicians. It’s very likely that only the director was able to perceive the outlines of the larger picture in the puzzle being assembled: no single mathematician trying to convince himself could have reproduced all the details in a human lifetime. For many years Russian mathematicians in the former Soviet Union would put an asterisk of warning on their work whenever they found themselves obliged to use this theorem. They considered it to be more an act of faith on the part of their Western colleagues than an admissible piece of mathematical reasoning . Similarly, it is interesting to note the shiver of anxiety produced throughout the mathematical world when Andrew Wiles announced his solution to Fermat’s last conjecture, an open wound for over three centuries. His original proof contained an error that only three or four specialists could detect; they were the same three of four specialists that certify that the error has now been corrected. I don’t mean to suggest that there’s any doubt that the theorem has been proved at last. But the proof covers one hundred pages that refer to one hundred algebra books and three centuries of the history of mathematics. This naturally alters the democratic character of the proof. If Fermat could come back to life, he would surely protest. He believed he had a brief, basic, admirable argument—a good, oldfashioned proof. Things can get worse when computers come into play. One of the most famous problems in mathematics is that of the four colors: given a map of any arbitrary countries, what is the minimum number of colors necessary to paint the map so that neighboring countries have different shades? It was known that five colors were sufficient and three were not enough. For many years, people tried to prove that the minimum number was four. Finally, a “demonstration” was produced: it’s a book of programs that, once run, exhaust thousands [3.237.65.102] Project MUSE (2024-03-28 20:32 GMT) SOLUTIONS AND DISILLUSIONMENT | 89 | of ramifications of a classification that is as detailed as it is discouraging . No mathematician would be willing to accept something like that as a demonstration strictly from the standpoint of aesthetics or mathematical need. It wins, but it doesn’t convince, just like Deep Blue, the computer that was able to defeat Garry Kasparov at chess, but which didn’t really play the same game. Without a doubt, an acute aesthetic problem emerges into focus here. I’ve read that in the United States they’re offering a million dollars to anyone that can solve any of the seven pending mathematical problems. Perhaps they should add that the solution must be verifiable in human time. Deep Thought, the supercomputer imagined by Douglas Adams in The Hitchhiker’s Guide to the Galaxy, completes its calculations and prints the final answer, “42,” in a future so distant that no one can remember the question. Note 1. Published in “Radar,” Página/12, 20 Jan. 2002. ...