![]() ![]() It is with the later names that the terrain grows very difficult. David Foster Wallace starts straight in with Cantor but takes a look back at his predecessors when he thinks it necessary to do so. Brian Clegg begins at the beginning, introducing one great name after another until, with Newton and Leibniz in the 17th century and Cantor in the 19th, he has cleared the way for even better-informed great ones still to come. They are not unalike in tone both use a jaunty approach with the odd witticism to keep the class happy, but they differ in method. These books both tell the story of the development of mathematical thought from its beginnings to modern set theory, and Georg Cantor is the hero of both. ![]() Resisting the obvious was his profession, as it was Freud's. It is interesting that Georg Cantor, probably the greatest mathematician of the infinite, refused to believe in the Stratford Shakespeare. Our humanist may be tempted to regard the whole business as dealing in fictions which, as can sometimes happen, open up new ways of thinking, so that genius must of necessity prefer the freedom of fiction to the convention of fact. It is tempting to argue that when the notion of infinity is not related to God or the sublime but to the manipulation of infinitesimals and, say, of all the numbers between 0 and 1, something - but certainly not passion - has been forfeited. The humanist looks at these enterprises with some bewilderment. Paranoia is not unknown among mathematical geniuses. The matter of priority, of who first took the leaps, is important it is of small value to work out Fermat's last theorem if somebody has done it before and there was a classic row between Newton and Leibniz about which of them first identified what Newton called "fluxions" and Leibniz "calculus". They assume they must answer them logically, for the history of mathematics is one of logical development and amendment, sometimes in startling leaps and bounds. And sometimes even geniuses have enormous difficulty in answering what look like childlike questions. Moreover the child's discoveries resemble, if only a little, those of more recent mathematicians, which may often, though by no means always, be of no use. Yet without their work the modern world would be almost inconceivably different. They knew about pi of course, but geometrically, not as a number with endless decimal points, and couldn't get their heads round square roots, let alone infinity. What the Greeks discovered, and the mathematical universe they believed in, might also seem nowadays to be of little practical value. But the calculating child is emulating the Greek mathematicians. I am fairly sure I was not taught these things at school, perhaps because they are not really much use and don't lead anywhere. And incidentally, why is the sum of two odd numbers, like the sum of two even numbers, always an even number? Twos and fives need no help, and the only number that yields to no such tricks is seven. If a number is divisible by 11, the sums of its alternate digits will always be equal (121, 671, 2541. If a number is divisible by three, the sum of its digits is also divisible by three, eg 714, 1,002, 108,762. #Infinity and infinitesimals book fullEven those of us to whom calculus was a distant peak we had no prospect of climbing can remember a time of innocence when numbers were full of mysterious interest. ![]()
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