A brief history of infinity

EBSCOhost Full Display Page 1 of7 8 Resul3o3 gg CRD CES QUES GUAM [So To Full Text] [Tips] Title: A brief history of infinity. Subject(s): INFINITE; ARCHIMEDES; ARISTOTLE; CANTOR sets Source:Scientific American, Apr95, Vol. 272 Issue 4, p112, 5p,1 chart, diagrams. 2 Ic, 1bw Author(s): Moore, A.W. Abstract: Discusses the history of the concept of infinity. Eudoxus' method of exhaustion; Archimedes’ use of infinityin estimating the area of a circle; Effect on the Pythagoreans ' belief that everything can be explainedin terms of positive integers; Aristotle's actual and potential infinite; Equinumerosity paradoxes; Georg Cantor's continuum hypothesis. INSETS: Archimedes and the area of the circle; Diagonalization and Godel's theorem. AN: 9504040512 ISSN: 0036-8733 ¿de A Database: Academic Search Elite Print: M Click here to mark for print. I [Go To Citation] Best Part 4 A BRIEF HISTORY OF INFINITY The infinite has always been a slippery concept. Even the commonly accepted mathematical view, developed by Georg Cantor, may not have truly placed infinity on a rigorous foundation For more than two millennia, mathematicians, like most people, were unsure what to make of the infinite. Several paradoxes devised by Greek and medieval thinkers had convinced them that the infinite could not be pondered with impunity. Then, in the 1870s, the German mathematician Georg Cantor unveiled transfinite mathematics, a branch of mathematics that seemingly resolved all the puzzles the infinite had posed. In his work Cantor showed that infinite numbers existed, that they came in different sizes and that they could be used to measure the extent of infinite sets. But did he really dispel all doubt about mathematical dealings with infinity? Most people now believe he did, but I shall suggest that in fact he may have reinforced that doubt. vNeErM‚oAorwe, The hostility of mathematicians toward infinity began in the fifth century B.C., when Zeno of Elea, a student of Parmenides, formulated the well-known paradox of Achilles and the tortoise [see "Resolving Zeno's Paradoxes,” by William I. McLaughlin; SCIENTIFIC AMERICAN, November 1994]. In this conundrum the swift demigod challenges the slow tortoise to a race and grants her a head start. Before he can overtake her, he must reach the point at which she began, by which time she will have advanceda little. Achilles must now make up the new distance separating them, but by the time he does so, she will have advanced again. And so on, ad infinitum. It seems that Achilles can never overtake the tortoise. In like manner Zeno argued that it is impossible to complete a racecourse. To do so, it is necessary to reach the halfway point, then the three-quarters point, then the seven-eighths point, and so on. Zeno concluded not only that motion is impossible but that we do best not to think in terms of the infinite. The mathematician Eudoxus, similarly wary of the infinite, developed the so-called method of exhaustion to circumvent it in certain geometric contexts. Archimedes exploited that method some in 100 years later to find the exact area of a circle. How did he proceed? In the box on page 114, I y” present not his actual derivation but a corruption of it. Part of Archimedes’ own procedure was to y” consider the formula for the area of a polygon with n equal sides--call it P[sub n]--inscribed inside a ~ circle C. According to the distortion of his argument, this formula can be applied to the circle itself, which is just a polygon with infinitely many, infinitely small sides. The perversion of Archimedes’ argument has some intuitive appeal, but it would not have satisfied Archimedes. We cannot uncritically make use of the infinite as though it were just some unusually big integer. Part of what is going on here is that the larger n is, the more nearly P[sub n] matches C. But it is also true that the larger n is, the more nearly P[sub n] approximates a circle with a bulge-- lFalltavt acnDracnltCatlA—DNANNANNN1 E LhitMNim—2 LrhnnlannTarmnurthanaranna Lrfheren

A Brief History of InÞnity The infinite has always been a slippery concept. Even the commonly accepted mathematical view, developed by Georg Cantor, may not have truly placed infinity on a rigorous foundation by A. W. Moore A. W. MOORE is a tutorial fellow in philosophy at St. HughÕs College of the University of Oxford. He studied for his Ph.D. in the philosophy of language at Balliol College at Oxford. His main academic interests are in logic, metaphysics and the philosophies of Immanuel Kant and Ludwig Wittgenstein, all of which have informed his work on the inÞnite. He is at work on a book about the metaphysics of objectivity and subjectivity. 112 SCIENTIFIC AMERICAN April 1995 Zeno argued that it is impossible to complete a racecourse. To do so, it is necessary to reach the halfway point, then the three-quarters point, then the seven-eighths point, and so on. Zeno concluded not only that motion is impossible but that we do best not to think in terms of the inÞnite. The mathematician Eudoxus, similarly wary of the inÞnite, developed the socalled method of exhaustion to circumvent it in certain geometric contexts. Archimedes exploited that method some 100 years later to Þnd the exact area of a circle. How did he proceed? In the box on page 114, I present not his actual derivation but a corruption of it. Part of ArchimedesÕ own procedure was to consider the formula for the area of a polygon with n equal sidesÑcall it PnÑ inscribed inside a circle C. According to the distortion of his argument, this formula can be applied to the circle itself, which is just a polygon with inÞnitely many, inÞnitely small sides. The perversion of ArchimedesÕ argument has some intuitive appeal, but it would not have satisÞed Archimedes. We cannot uncritically make use of the inÞnite as though it were just some unusually big integer. Part of what is going on here is that the larger n is, the more nearly Pn matches C. But it is also true that the larger n is, the more nearly Pn approximates a circle with a bulgeÑ call it C*. The key point intuitively is that C, unlike its deformed counterpart C*, is the limit of the polygonsÑor what they are tending toward. Still, it is very hard to see any way of capturing this intuition without, once again, thinking of C as an ÒinÞnigon.Ó Archimedes provided a way. He pinpointed the crucial diÝerence between C and C* by proving the following point: no matter how small an area you consider, call it ε (the Greek letter epsilon), there exists an integer n that is large enough for the area of Pn to be within ε of the area of C. The same is not true of C*. This fact, combined with a similar result for circumscribed polygons and supplemented with a reÞned version of the logic contained in that argument, Þnally enabled Archimedes to show, without ever invoking the inÞnite, that the area of a circle equals π r 2. The Actual and Potential InÞnite A lthough Archimedes successfully ducked the inÞnite in this particular exercise, the Pythagoreans (a religious society founded by Pythagoras) happened on a case in which the inÞnite was truly inescapable. This Þnd shattered their belief in two fundamental cosmological principles: Peras (the limit), which subsumed all that was good, and Apeiron (the unlimited or inÞnite), which encompassed all that was bad. They had insisted that the whole of creation could be understood in terms of, and indeed was ultimately constituted by, the positive integers, each of which is Þnite. This reduction was made possible, they maintained, by the fact that Peras was ever subjugating Apeiron. Pythagoras had discovered, however, that the square of the hypotenuse (the longest side) of a right-angled triangle is equal to the sum of the squares of the other two sides. Given this theorem, the ratio of a squareÕs diagonal to each side is √ 2 to 1, since 12 + 12 = (√ 2) 2. Were Peras impervious, this ratio should be expressible in the form p to q, where p and q are both positive integers. Yet this is impossible. Imagine two positive integers, p and q, such that the ratio of p to q, or p divided by q, is equivalent to √ 2. We can assume that p and q have no common factor greater than 1 (we could, if necessary, UNLIMITED EXPANSE is conveyed by the baked ßoor of CaliforniaÕs Death Valley. To grasp the inÞnite, mathematicians have confronted several paradoxes, concluding that inÞnities come in diÝerent types and that some are bigger than others. Copyright 1995 Scientific American, Inc. GARY YEOWELL Tony Stone Images F or more than two millennia, mathematicians, like most people, were unsure what to make of the inÞnite. Several paradoxes devised by Greek and medieval thinkers had convinced them that the inÞnite could not be pondered with impunity. Then, in the 1870s, the German mathematician Georg Cantor unveiled transÞnite mathematics, a branch of mathematics that seemingly resolved all the puzzles the inÞnite had posed. In his work Cantor showed that inÞnite numbers existed, that they came in diÝerent sizes and that they could be used to measure the extent of inÞnite sets. But did he really dispel all doubt about mathematical dealings with inÞnity? Most people now believe he did, but I shall suggest that in fact he may have reinforced that doubt. The hostility of mathematicians toward inÞnity began in the Þfth century B.C., when Zeno of Elea, a student of Parmenides, formulated the well-known paradox of Achilles and the tortoise [see ÒResolving ZenoÕs Paradoxes,Ó by William I. McLaughlin; SCIENTIFIC AMERICAN, November 1994]. In this conundrum the swift demigod challenges the slow tortoise to a race and grants her a head start. Before he can overtake her, he must reach the point at which she began, by which time she will have advanced a little. Achilles must now make up the new distance separating them, but by the time he does so, she will have advanced again. And so on, ad inÞnitum. It seems that Achilles can never overtake the tortoise. In like manner