Written: Oct. 25, 2002
Just because we do mathematics everyday, and make a living out of it, does not mean that we know what mathematics is.
If you want to find out, then read Tim Gowers's recent 140-page little paperback (published by Oxford University Press). In a captivating and lucid style, he explains his view of the nature of mathematics. You don't have to agree with everything (or even anything) he says, but it will get you thinking.
I particularly liked his slogan `a mathematical object is what it does' (p. 18), and his description of the abstract method, especially his belief that `if one learns to think abstractly, then many philosophical difficulties disappear'.
The final chapter 8 makes a good dessert because of its meta-mathematical themes. Gowers's explanation why very few mathematicians do their first breakthrough after 30 is a very convincing Bayesian argument, analogous to why an `old-maid', or `confirmed old-bachelor', are unlikely to marry. It is not that she (or he) are not attractive any more, but if they were marrying types, they most likely would have already gotten married. It is also nice to know that math professors are not as nerdy and weird as math students (p. 127), because if you are too weird it is unlikely that you'll make it to tenured-professorship. But, then again, perhaps all it means that being mathematically talented does not suffice to climb the academic ladder, and one needs a modicum of political skills.
It was also heart-warming to know that Andrew Wiles is `extremely clever' , but not a genius, and that `the most profound contributions to mathematics are often made by tortoises rather than hares'.
Of course, in such an opinionated piece, everybody would find something to disagree with. I will now list the few, but important, points that I take issue with.
His current description of how computers are used today in mathematical research (p. 134) does not account for such major breakthroughs as the proof of the Four Color Theorem and the Kepler Conjecture, that should have been mentioned anyway. It is nice that Gowers holds the `minority view' that in the future computers will take over, but they are already starting to do a great job, over and above `routine verification', and `checking'.
The subsection `Why do people positively dislike math?' (p. 131), states the obvious reason that people just hate the experience of learning math. His remedy is to find better teachers. I have even a better solution. Don't make mathematics a requirement at all! Most mathematicians don't like this solution, either for `moral reasons', being mathematical fanatics who believe that everybody must take math, or for parochial `special-interest reasons', since we need students to make a living. But this does not justify the inhumane (proverbial) rape of mathematically-challenged people, that are made to take math whether they like it or not, and whether they are good at it or not. I am sure that their contribution to society will not diminish, and perhaps will even increase, if math will stop being compulsory. If everybody who wants to go to college would have to learn how to play the violin, and made to practice four hours a day, regardless of musical ability, I am sure that there would be as many music-haters as there are math-haters.
Finally, it bothered me that Gowers uses the word ``mainstream'' in a neutral-to-positive sense, where for me it is one of the dirtiest and most obscene words in our meta-mathematical language, used by the mathematical establishment to maintain the status-quo. Today's mainstream will soon be tomorrow's obsolete anthropocentric superstition.
But, whether you agree or disagree, this amazing little book will make you think! Order your copy right now!
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