Written: June 11, 2002
I just finished reading Sir Michael Atiyah's beautifully-written, extremely lucid and intentionally provocative article ``Mathematics in the 20th Century'' (Bull. of the London Math. Soc. 34 (2002) 1-15). Since provocative statements are meant to provoke, Sir Michael might be pleased to learn that he succeeded spectacularly, at least in my case. I got really MAD.
One approach at rebuttal is defense. Try to respond and defend, sentence by sentence, all the statements that one disagrees with. Such an approach was broadly taken by Tim Gowers, in his beautifully-argued `two-culture' masterpiece (available from Tim Gowers's Papers site). But another, equally effective way is offense: fight provocation by counter-provocation. So here goes.
20th century mathematics (with the exception of the great events that Atiyah explicitly excluded from his discussion, associated with the names of Hilbert, Gödel and Turing, and I would add Chomsky) will very soon be completely trivialized and superseded by computer-generated algorithmic mathematics. And much sooner than you think!
In particular, K-theory, the Atiyah-Singer Index Theorem, and all the rest of mainstream 20th-century mathematics, will be very soon only of historical interest, and if studied at all, it would only be as theology.
It is ironic that Atiyah does not mention, even once, the single greatest contribution of the 20th century, next to which all else is dwarfed: the COMPUTER. He does mention once "logic and computing" (as something he will NOT talk about), but by this he means, at best `theoretical computer science', not the computer itself. Perhaps it is also implicit in the word `application', which he decided not to talk about, because they are `so numerous and need special treatment'. Be that as it may, don't you think that the eight-letter string: COMPUTER, deserves to show up at least ONCE?
The one paragraph that outraged me the most (quoted by Gert-Martin Gruel in his ECCAD 2002 talk) is the one on top of page 7, saying that using algebra (and, by implication, computer-algebra) is a `Faustian deal' selling out the `geometrical intuition' soul in return for mere mathematical facts.
You got it all backwards, Sir Michael! `Geometrical Intuition' (whatever it is), is the DEVIL, making humans too braggy, while ALGEBRA, especially in its manifestation as COMPUTER ALGEBRA, is the MESSIAH, that will free us from our pre-historic humanistic, `naked-brain' `conceptual' fixations and hang-ups, and enable us to jump-start the first genuinely non-trivial mathematical revolution.
The few accomplishments that will be ultimately remembered from the mathematics of the 20h century are those that Atiyah disdains the most. One major achievement, mentioned by Atiyah, as `not exciting', since `it closed, rather than opened, a field', is the Classification of Finite Simple Groups. Why open new fields? Don't we have enough interesting open problems to last us for ever? The proof of the Classification of Finite Groups is very EXCITING because it showed how many people can work together to solve a particularly deep and difficult problem. And it is this model of problem-solving, rather than `working alone in the attic for seven years' that will prevail. Of course, in the future it would be done mostly, and perhaps soley, by computers, but the design and `division of labor' will be the great challenges to the human mathematical engineer. So even if the proof of the Classification theorem did not open any new field (which by the way, is not true, even Atiyah concedes that the Monster did), it did something far more EXCITING. It started a new METHODOLOGY for problem-solving, that can be used and extended for other major open problems in the future.
My second 20th-century favorite was not considered worthy of mention by Atiyah, even not in a derogatory way. It is Ken Appel and Wolfgang Haken's GORGEOUS proof of the Four-Color Theorem. Few people are aware that it is really a ONE-LINE Proof: `The following finite set of reducible configurations, let's call it S, is unavoidable'. The set S itself does not have to be actually examined by human eyes, and perhaps should not. The computer would be much more reliable than any human in checking its claim. The human challenge was to design algorithms for constructing the set S. Right now we still need (human) IDEAS to find such algorithms, but not for `explaining' `why' four colors suffice. FOUR COLORS SUFFICE BECAUSE THE COMPUTER SAID SO!. Four Colors suffice for the same reason that the speed of light is what it is, and the 1001005th digit of Pi is what it is, it is contingent rather than necessary knowledge. So The Ugly Duckling of the Appel-Haken proof was just ahead of its time, and very soon will turn into a Beautiful Swan, as our tastes and priorities will change, and we will get rid of the narrow-minded, micro-managing human trait of trying to `understand all the details', and our obsession with `elegant proof', that is but a euphemism for `trivial proof'.
But the MOST significant contribution by 20th-Century human mathematicians and computer scientists was the creation of COMPUTER ALGEBRA. It is still in a very preliminary stage, but is already revolutionizing the way we DO and THINK about mathematics. Even computer foes often `cheat' and use Maple or Mathematica on the side. It is a scandal that, so far, the pioneers of computer algebra did not get their due recognition by the mathematical establishment. One glaring oversight, reminiscent of Gödel only becoming a professor at the age of 47, is that Bruno BUCHBERGER, the great pioneer of computer algebra, whose algorithm revolutionized all the spectrum of mathematics, from robotics to very abstract algebraic geometry, does not get a tiny fraction of the recognition granted to the Atiyahs and the Botts. He is not even a member of the Academy of Science of his native Austria! But one should not fault this or that specific Academy. After all, at least today, national academies consist of members of that narrow-minded, short-sighted species called humans. Bruno Buchberger's heritage, and his Gröbner Bases will survive long after most of the current members of the Austrian Academy of Science will be forgotten.