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The Art of the Infinite: The Pleasures of Mathematics by Robert M. Kaplan
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The Art of the Infinite: The Pleasures of Mathematics
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The Art of the Infinite : The Pleasures of Mathematics
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Subject to credit approval. See terms - opens in a new window or tab. Zero was only a notational convenience, but this nothing, which yet somehow is, gave a new depth to our sense of number, a new dimension—as the invention of a vanishing point suddenly deepened the pictorial plane of Renaissance art a subject to which we shall return in Chapter Eight.
The Pleasures of Mathematics
But is zero a number at all? It took centuries to free it from sweeping the hearth, a humble punctuation mark, and find that the glass slipper fitted it perfectly. The deep principle at work here—which we will encounter again and again—is that something must not only act like a number but interact companionably with other numbers in their republic, if you are to extend the franchise to it.
This was difficult in the case of zero, for it behaved badly in company. You do what the French did with Tom Paine and make him an honorary citizen. So zero joined the republic of numbers, where it has stirred up trouble ever since.
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Our primary mathematical experience, individually as well as collectively, is counting—in which zero plays no part, since counting always starts with one. The counting numbers take 17 as a random example , parthenogenetic offspring of that solitary Adam, 1, came in time to be called the natural as their symbol. Think of them strolling there in that boundless garden, innocent under the trees. For all that we have now found a way to organize them by tens and hundreds, they seem at first sight as much like one another as such offspring would have to be.
Yet look closer, as the Greeks once did, to see the beginnings of startling patterns among them. Are they patterns we playfully make in the ductile material of numbers, as a sculptor prods and pinches shapes from clay? Or patterns only laid bare by such probing, as Michelangelo thought of the statue which waited in the stone? Of all the arts, mathematics most puts into question the distinction between creation and discovery. Each is bigger than the previous one by its bottom row, which is the next natural number. This pattern clearly undulates endlessly on.
The first six square numbers, each gotten by adding a right angle of dots to the last,. What light does it shed on the nature of things, what use could it possibly be? Light precedes use, as Sir Francis Bacon once pointed out. If there was, it was probably well hidden. Perhaps he recalled what the Greek philosopher Heraclitus had said: A hidden connection is stronger than one we can see. Hidden how? Poking his holes again in the sand, looking at them from one angle and another, he suddenly saw:.
Then the leap from seeing with the outer to the inner eye, which is the leap of mathematics to the infinite: this must always be so. Our insight sharpens: the second square number is the sum of the first two triangular numbers; the third square of the second and third triangulars, and so on. You might feel the need now for a more graceful vessel in which to carry this insight—the need for symbols—and make up these:.