{"product_id":"infinite-ascent-isbn-9780812978711","title":"Infinite Ascent","description":"In \u003cb\u003eInfinite Ascent\u003c\/b\u003e, David Berlinski, the acclaimed author of \u003ci\u003eThe Advent of the Algorithm, A Tour of the Calculus\u003c\/i\u003e, and \u003ci\u003eNewton’s Gift\u003c\/i\u003e, tells the story of mathematics, bringing to life with wit, elegance, and deep insight a 2,500-year-long intellectual adventure.\u003cbr\u003e\u003cbr\u003eBerlinski focuses on the ten most important breakthroughs in mathematical history–and the men behind them. Here are Pythagoras, intoxicated by the mystical significance of numbers; Euclid, who gave the world the very idea of a proof; Leibniz and Newton, co-discoverers of the calculus; Cantor, master of the infinite; and Gödel, who in one magnificent proof placed everything in doubt.\u003cbr\u003e \u003cbr\u003eThe elaboration of mathematical knowledge has meant nothing less than the unfolding of human consciousness itself. With his unmatched ability to make abstract ideas concrete and approachable, Berlinski both tells an engrossing tale and introduces us to the full power of what surely ranks as one of the greatest of all human endeavors.“For mathematically challenged folk like me, David Berlinski comes again to help with a thin volume that, like his A Tour of the Calculus, renders mathematics not easy, but accessible and absorbing. He portrays through history how mathematical thought evolved, from the genius of the few to its application by the many. Personalities, times, cultures, and opportunities all play their dramatic roles and Berlinski, knowing how they interacted, brings them vividly to life. You’ll enjoy yourself.” \u003cbr\u003e–\u003cb\u003ePaul McHugh\u003c\/b\u003e, Distinguished Service Professor of Psychiatry, Johns Hopkins University\u003cbr\u003e\u003cbr\u003e“This is literary science at its best. I was charmed by this top-down and introspective presentation of the subject of mathematics. It is not just highly readable; because it is one step above the subject, it can even inspire the professional.” \u003cbr\u003e–\u003cb\u003eNassim Nicholas Taleb\u003c\/b\u003e, Dean’s Professor in the Sciences of Uncertainty, University of Massachusetts at Amherst, author of Fooled by Randomness\u003cbr\u003e\u003cbr\u003e“A humorous and graceful short history of mathematics, quite deceptively easy to read. Berlinski is actually a sophisticated insider, and every page of this book glows with his love of mathematics and with his sardonic appreciation for humanity’s foibles.”\u003cbr\u003e–\u003cb\u003eGregory Chaitin\u003c\/b\u003e, IBM Thomas J. Watson Research Center, author of Meta Math! The Quest for OmegaDavid Berlinski received his Ph.D. from Princeton University and has taught mathematics, philosophy, and English at Stanford, Rutgers, the University of Puget Sound, and the Université de Paris at Jussieu. He has been a research fellow at both the International Institute for Applied Systems Analysis in Austria and the Institut des Hautes Études Scientifiques in France. His many books have been translated into more than a dozen European and Asian languages. His essays in Commentary have become famous. A senior fellow at the Discovery Institute in Seattle, he lives and works in Paris.Chapter 1\u003cbr\u003e\u003cbr\u003e    Number\u003cbr\u003e\u003cbr\u003e    The history of mathematics begins in 532 BC, the date marking the birth   of the Greek mathematician Pythagoras. Having fled the island of Samos   in order to escape the tyranny of Polycrates, Pythagoras traveled to   Egypt, where, like so many impressionable young Greek men, he “learned   number and measure from Egyptians [and] was astonished at the wisdom of   the priests.” Thereafter, he settled in southern Italy; he began   teaching and quickly attracted disciples. Very little is known directly   of his life, except that his contemporaries considered him admirable.   Nothing from his own hand remains: He has been preserved against the   worm of time by the amber of various literary artifacts. Admission to   the Pythagorean sect was evidently based on mathematical ability.   Secrecy was enforced and dietary restrictions against beans maintained.   New members were required to keep silent for a number of years, a   policy that even today many teachers will find admirable, and they were   expected during this time to meditate and reflect. Some members of the   Pythagorean sect regarded the external world as a prison, a cave filled   with flickering shadows and dull brutish shapes. Let me add to this   confused but static scene the heat lightning of superb mathematical   intuition.\u003cbr\u003e\u003cbr\u003e    Until the mid-twentieth century, the thesis that in mathematics as in   almost everything else, the Greeks were there at first light, did not   require an elaborate defense. With their forearms draped in friendship   over any number of toga-clad shoulders, classicists who had spent years   mastering infernal Greek declensions naturally assumed that the “Greeks   were fellows of another college.” The history of the Ancient Near East   has come into sharper focus over the past century, great scholars   poring over cuneiform tablets and re-creating the life of ancient   empires that had until their work been swallowed up as the impenetrable   before. They have found remarkable things, a history before classical   history, evidence that men and women have used and loved mathematics in   the time before time began. Neolithic ax-marks have even suggested that   the origins of mathematics lie impossibly far in the past, and that men   living in caves, their hairy torsos covered by vile-smelling furs,   chipped the names of the numbers onto their ax handles as bison grease   spattered over an open fire. And why not? Like language itself,   mathematics is an inheritance of the race.\u003cbr\u003e\u003cbr\u003e    The burden of those impossibly distant centuries now disappears. It is   roughly six centuries before the birth of Christ. The Greeks are just   about to elbow their way into all the corridors of culture. They give   every indication of knowing everything and having known it all along.   Yet the Babylonians already possessed a remarkably sophisticated body   of mathematical knowledge. They were matchless observational   astronomers, and they had brought a number of celestial phenomena under   the control of precise mathematical techniques. They were immensely   clever. “I found a stone, but did not weigh it,” one scribe wrote. “I   then weighed out six times its weight, added two gin, and then added   one third of one seventh, multiplied by twenty-four. I weighed it. [The   result came to] one ma-na.” “What,” the scribe now asks his oil-haired   students, “was the original weight of the stone?” Mathematicians are   apt to see an all-too-familiar face peeping through the problems of a   Babylonian scribe—their face, of course, ubiquitous and always the   same.\u003cbr\u003e\u003cbr\u003e    But those classicists sipping sherry in the common room of time had   been right all along. The Greeks were there at first light.\u003cbr\u003e\u003cbr\u003e    The natural numbers 1, 2, 3, . . . begin at one and they go on forever,   the mathematician’s dainty dots signifying an endless progression. As   soon as anyone attempts to cap the natural numbers, anyone can find a   way to cap the cap, say by adding one to the last natural number   capped. If the numbers are infinite, they are also wonderfully various.   When the great Indian prodigy Srinivasa Ramanujan lay dying in a London   hospital, the cold English winters eating his lungs away now ending his   life, his friend, the mathematician G. H. Hardy, paid him a visit.   Paralyzed by his own reticence, Hardy could think only to blurt out the   number of the taxi that had brought him to the hospital—1729, as it   happens.\u003cbr\u003e\u003cbr\u003e    “I don’t suppose it is a very interesting number,” he added.\u003cbr\u003e\u003cbr\u003e    “Oh, no, Hardy,” Ramanujan replied at once, “it is the smallest number   expressible as the sum of two cubes in two different ways.”\u003cbr\u003e\u003cbr\u003e    And so it is: 1729 = 13 + 123 = 93 + 103. No smaller number has this   property. The story has become famous. No one quite knows what it   means, but every mathematician understands why it is told.\u003cbr\u003e\u003cbr\u003e    Like Ramanujan, the Pythagoreans were taken with the inexhaustible   variety of the natural numbers, their personalities. They were   fascinated by 1, 3, 6, and 10, because these numbers could be expressed   geometrically as triangles composed of dots. They quite understood the   importance of numbers that are divisible only by themselves and 1—the   prime numbers such as 2, 3, 5, 7, and 11; and they may well have   understood that the prime numbers are fundamental, lying like dark   rubies amid the pale panoply of the ordinary numbers. They discovered   that certain numbers such as 6, 28, and 496 could be expressed as the   sum of their divisors. They lived in caves—I mean such is the   legend—and squatting there, a pile of smooth pebbles in their laps,   they saw that there are square numbers as well as triangular numbers,   and amicable relationships between numbers, as when each of two numbers   is the sum of the other’s divisors, or when the sum of two consecutive   triangular numbers such as 3 and 6 is a square number, and progressions   from one series of numbers to another; and in all this, as the tallow   dripped from their candles, they treated the natural numbers as if they   were themselves men at play, serious but never solemn, their endless   curiosity amounting at times to a form of intellectual rapture and so   entirely alien to the beetle-browed scribes and accountants of the   Ancient Near East, men forever plodding along the severe utilitarian   axis of a commercial culture.\u003cbr\u003e\u003cbr\u003e    What did the Pythagoreans care for some pharaoh’s monstrous pyramid or   staring one-eyed sphinx? They were mathematicians.\u003cbr\u003e\u003cbr\u003e    Superstitious? Of course they were, but Pythagoras and the Pythagoreans   were devoted to a higher spookiness. It is their distinction. With his   vein-ruined hands describing circles in the smoky air, Pythagoras has   come to believe in numbers, their unearthly harmonies and strange   symmetries. “Number is the first principle,” he affirmed, “a thing   which is undefined, incomprehensible, [and] having in itself all   numbers.” The number one, the Pythagoreans termed the monad, and at   times they seemed to suggest that the natural numbers might be   subordinated to a dull grunting process by which all of the numbers   could be generated from the monad, number creation monstrous and   pullulating. “And the first principle of numbers is in substance the   first monad, which is a male monad, begetting as a father all other   numbers.” The numbers two, three, and four enter into Pythagorean   thought scent-marked from the first, the number two, because it is   squat and feminine, and three, because it marks a return to the   masculine, its three-tipped triangle when inverted (base up, apex down)   looking very much as if a pair of wide-spread shoulders were descending   toward a manly groin. The number four merits celebration—but I really   have no idea why it does, except for the fact that one, two, three, and   four sum to ten, at which point the number series topples back to one,   with eleven expressed as the sum of one and ten. It is the number ten   that served the Pythagoreans as the object of a sacred oath, one   offered at night in the owl-hooted landscape and dedicated to “him that   transmitted to our soul the tetraktys, which has the spring and root of   ever-flowing nature.”\u003cbr\u003e\u003cbr\u003e    Half-mad, I suppose, and ecstatic, Pythagorean thought offers us the   chance to peer downward into the deep unconscious place where   mathematics has its origins, the natural numbers seen as they must have   been seen for the very first time, and that is as some powerful erotic   aspect of creation itself. “Number,” the Pythagoreans wrote, “is the   essence of all things.” Time has long scattered the Pythagoreans and   canceled their sense of play, and yet the declaration that number is   the essence of all things has lost none of its thrilling intellectual   power. Number? And the essence of all things? Of all things? The Greeks   heard those unearthly and mysterious words and tried to give them   sense, but sand needed to sift over the monuments of antiquity before   they would again enter into the mathematician’s self-confident   self-consciousness. When Galileo initiated the great scientific   revolution of the West, writing that the Book of Nature is written in   the language of mathematics, he was reconveying that Pythagorean note,   those Pythagorean words.\u003cbr\u003e\u003cbr\u003e    The Pythagoreans never succeeded in explaining what they meant by   claiming that number is the essence of all things. Early in the life of   the sect, they conjectured that numbers might be the essence of all   things because quite literally “the elements of numbers were the   elements of all things.” In this way, Aristotle remarks, “they   constructed the whole heaven out of numbers.” This view they could not   sustain. Aristotle notes dryly that “it is impossible that [physical]   bodies should consist of numbers,” if only because physical bodies are   in motion and numbers are not. At some time, the intellectual   allegiances of the sect changed and the Pythagoreans began to draw a   most Platonic distinction between the world revealed by the senses and   the world revealed by the intellect. The literal aspect of the   Pythagorean doctrine gives way. Numbers are one thing, the world of   sensory objects another. But numbers still remain the essence of all   things, the Pythagoreans groping their way toward the remarkable   doctrine that the harmony between numbers offers a guide to the harmony   between things.\u003cbr\u003e\u003cbr\u003e    “To give an example of my meaning,” Aristotle remarked in describing   the Pythagoreans, “inasmuch as ten seemed to be the perfect number and   to embrace the whole nature of numbers, they asserted that the number   of bodies moving through the heavens were ten, and when only nine were   visible, for the reason just given they postulated the counter-earth as   the tenth.” This is neither muddied nor mystical: The inference on   which the Pythagoreans relied has been championed by physicists from   the seventeenth to the twenty-first centuries. It is the rock of their   faith.\u003cbr\u003e\u003cbr\u003e    In the 1920s, for example, the French mathematician Paul Dirac set   himself the problem of extending the Klein-Gordon field equations so   that they encompassed relativistic solutions for the electron. The   details of Dirac’s project may safely be subordinated; what is at issue   is a risky navigational maneuver in which a man sets off on a road he   cannot see trusting for guidance in a road he has already seen. Dirac   quickly encountered difficulties. Equations needed to be factored, as   when x2 + 11x + 10 is resolved into x + 10 and x + 1, and new   mathematical objects were needed to accomplish this. Groping now and   guessing like mad, Dirac succeeded in solving the Klein-Gordon   equations, the relativistic electron appearing as a physical correlate   to a mathematical object. And then Dirac noticed something odd. The   solutions to the Klein-Gordon equations were split like the devil’s   tail. One solution corresponded to the expected electron, the   solution’s negative sign matching the electron’s negative charge; but   another and opposite solution seemed to correspond to the electron in   all of its properties except charge. Lesser mathematicians might have   discretely discarded this anomalous solution and carried on. Dirac   ignored their advice and affirmed the existence of the positron.\u003cbr\u003e\u003cbr\u003e    He had seen the fork on the devil’s tail. Some years later—not many,   mind you—experimental physicists confirmed the existence of the   positron.\u003cbr\u003e\u003cbr\u003e    Let us by all means cast out what is unwholesome in Pythagorean   lore—the beans, the number mysticism, and the mumbo jumbo. There   remains this. The doctrine that number is the essence of all things,   passing through the prism of a thousand philosophical tracts, remains   the central insight of Western science, the indispensable key of   coordination. And this, too: The fact that this key opens so many locks   has often been celebrated, but it has never been explained.\u003cbr\u003e\u003cbr\u003e    Greek historians tell an odd little story. A ship is sailing across the   Aegean Sea, the sound of the waves slapping against its wooden hull,   the rowers chanting. On board are a number of mathematicians,   Pythagoreans all, although why a group of mathematicians should have   decided on an ocean voyage, I do not know.\u003cbr\u003e\u003cbr\u003e    Now Pythagoras is best known in the official history of mathematics for   the theorem that bears his name. I am going to need that theorem close   at hand, where it can do some good. Consider a right triangle whose   tips are labeled A, B, and C. Distances between distant points are what   they seem—distances, and so numbers. If distances cannot be correlated   with numbers, very little remains of the grand Pythagorean proposition   that number is the essence of all things. Although very simple, a right   triangle—any old right triangle—is an object quivering with unsuspected   numerical relationships; and in particular, Pythagoras discovered that   when it comes to any old right triangle, the distances between A and B,   and again between B and C, are coordinated with the distance between A   and C. They are, in fact, coordinated by the simple formula: (A - B)2 +   (B - C)2 = (A - C)2. The proof that Pythagoras offered is all grunt and   shove. A number squared suggests an area in prospect. The area of a   square is, after all, the product of its length and width, and since,   by definition, they are the same so far as squares go, it is the   product of its length or its width with itself. Pythagoras thought to   construct squares along each of the triangle’s sides, and then by a   series of geometrical adjustments—the grunt now follows—which involved   shoving those squares around, he showed that the square . . .\u003cbr\u003e\u003cbr\u003e    But now that I have given away the key, the readers may follow   Pythagoras through that open door. In mathematics, it is always the key   that counts.\u003cbr\u003e\u003cbr\u003e    The Pythagorean theorem dooms any naive version of the Pythagorean   program, the denouement taking place on board that sailing vessel just   recently seen leaving port. A mathematician named Hippasus of   Metapontum has just drawn a right triangle whose sides are one unit in   length on the dusty surface of a ship’s plank; throat cleared wetly to   draw attention, he observes that by the Pythagorean theorem, the length   of its diameter must correspond to the square root of two.\u003cbr\u003e\u003cbr\u003e    Now suppose, Hippasus continued, that the square root of two is a   number or that it may be represented as the ratio of two numbers. In   that case, S2w = m\/n. The steps that follow have a concision suggesting   the taps of a telegraphic key:\u003cbr\u003e\u003cbr\u003e    Tap. Suppose that m\/n has been reduced to its lowest common form by   division.\u003cbr\u003e\u003cbr\u003e    It follows that either m and n are both odd, or that m is even and n   odd, or, finally, that m is odd and n even.\u003cbr\u003e\u003cbr\u003e    Nods all around. It is a fine thing to be on board a ship.\u003cbr\u003e\u003cbr\u003e    Tap. Squaring both sides of S2w = m\/n, it follows again that\u003cbr\u003e\u003cbr\u003e    2 = m2\/n2.\u003cbr\u003e\u003cbr\u003e    Tap. Then 2n2 = m2, so that m2 is even.\u003cbr\u003e\u003cbr\u003e    If so, then m = 2x, where x is now some number. This is, after all,   what it means to say that m is even.\u003cbr\u003e\u003cbr\u003e    Tap. Squaring things lavishly, it follows that m2 = 4x2 = 2n2. . . .\u003cbr\u003e\u003cbr\u003e    My telegraphic taps now end just before the final tap; but like a   newspaper announcing a great victory in headlines with details to   follow on subsequent pages, this message is really complete. To get to   those subsequent pages, the reader need only see that n2 = 2x2 so that   . . .\u003cbr\u003e\u003cbr\u003e    But if my taps have come to an end, Hippasus kept right on tapping,   pointing out with evident satisfaction that a contradiction had been   reached, and that—tap, tap, tap—it consequently made no sense to   suppose that the square root of two corresponds to the ratio of two   numbers, and that—tap, tap, tap—it follows that certain distances   cannot be measured by the natural numbers at all, and that—tap, tap,   tap—\u003cbr\u003e\u003cbr\u003e    But here the story really ends. The Pythagoreans pitched Hippasus   overboard where, still tapping, he perished ignominiously.\u003cbr\u003e\u003cbr\u003e    It is said that at some point in his mathematical career, Pythagoras   proclaimed himself a god.\u003cbr\u003e\u003cbr\u003e    He was right to do so.A Modern Library Chronicles Book [22]; Bestselling author of A Tour of the Calculus","brand":"Modern Library","offers":[{"title":"Default Title","offer_id":46304671334629,"sku":"NP9780812978711","price":20.0,"currency_code":"USD","in_stock":false}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/1842\/7735\/files\/9780812978711.jpg?v=1767730038","url":"https:\/\/k12savings.com\/products\/infinite-ascent-isbn-9780812978711","provider":"K12savings","version":"1.0","type":"link"}