[...]... The Art of the Infinite each of these square numbers was the sum of two triangular ones! 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... third of the pearls were collected by the maid-servant, one sixth fell on the bed—then half of 1 The Art of the Infinite what remained and half of what remained thereafter and again one half of what remained thereafter and so on, six times in all, fell scattered everywhere 1,161 pearls were still left on the string; how many pearls had there been in the necklace? Talking mostly to each other or themselves,... in the universe, would stand to one another as a ratio of two natural numbers Take the module of the way we count, the number 10 Is it a coincidence that it is the triangular sum of the first four counting numbers? 17 The Art of the Infinite The Pythagoreans didn’t think so: 10 must have seemed to them as compact of meaning as the genetic code, coiled within a cell, seems to us For not only did the. .. and taken together say something about how peculiar the art of mathematics is The same technique of merely going on adding 1 to itself shows you, on the one hand, how each of the counting numbers is built—hence where and what each one is; on the other, it tells you a dazzling truth about their totality that overrides the variety among them We slip from the immensely concrete to the mind-bogglingly abstract... once? The tension between these two points of view the potentially infinite of motion and the actual infinity of totality—continues today, unresolved, opening up fresh approaches to the nature of mathematics The uneasy status of the infinite will accompany us throughout this book as we explore, return with our trophies, and set out again Here is the next truth We can see that the sizable army of counting... each point in the array that stretches, like Banquo’s descendants, even to the crack of doom Every one of these counting numbers is just a sum of 1 with itself a finite number of times: 1 + 1 + 1 + 1 + 1 = 5, and with paper and patience enough, we could say that the same is true of 65,537 5 The Art of the Infinite These two truths—one about all the counting numbers, one about each of them—are very... future, their negative siblings recede toward the limitless past, with 0 forever in that middle we take to be the present It takes a real act of generosity, of course, to extend the franchise as we have, because we so strongly feel the birthright of the counting numbers “God created the natural numbers,” said the German mathematician Kronecker late in the nineteenth century, the rest is the work of man.”... explanations too will live in the middle distance: some in the appendix, others the more distant excursions—(along with notes to the text) in an on-line Annex, at www.oup-usa.org/artoftheinfinite Gradually, then, the music of mathematics will grow more distinct We will hear in it the endless tug between freedom and necessity as playful inventions turn into the only way things can be, and timeless laws are drafted—in... called the natural numbers, with N 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... extended to ratios in the guise of fractions, although uneasiness at splitting the atomic unit remained The fractions, preserving traces of their origin in their official name of Rational Numbers, were symbolized by the letter Q, for “quotient” Does this variety of names reflect the doubts about their legitimacy? To counteract these worries, notice that the integers now can be thought of as rationals too: . in the middle distance: some in the appendix, others the more distant excursions—(along with notes to the text) in an on-line Annex, at www.oup-usa.org/artoftheinfinite. Gradually, then, the. third of the pearls were col- lected by the maid-servant, one sixth fell on the bed—then half of what remained and half of what remained thereafter and again one half of what remained thereafter. about all the counting numbers, one about each of them—are very different in spirit, and taken together say some- thing about how peculiar the art of mathematics is. The same technique of merely