Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

56 the mathematical infinityfrom the tortoise, the tortoise will have moved forward a little more). Zeno’s paradoxis not a joke. It leads to very meaningful questions, among them whether we acceptinfinite divisibility and the concept of a continuum, or not. (In Zeno’s case, we dealwith space and time, which I will leave to the physicist, but in mathematics it is againa fundamental question.) Every mathematical concept had to be described in finite,precise terms. A straight line was a good geometrical concept because of its constancyof direction, and so were circles, squares, triangles, and polygons built of triangles.Ellipses, parabolas, and hyperbolas could be considered part of geometry, because theywere conic sections (the intersection of a plane and a circular cone in three-dimensionalspace), but they were not part of the Euclidean geometry of circles and lines.The first limitations of such an ideal view of geometry appeared with the discoveryof the irrational. How was it possible that the diagonal of a square, obviously a goodgeometric object, could not be commensurable with the side of the square? The proofof such a statement is very simple. Suppose that the diagonal and side of a squarewere in a rational ratio m/n in whole numbers. Then Pythagoras’s theorem would givem 2 = 2n 2 . Hence, m would be even and we could write m = 2p, with p an integer.Therefore, 4p 2 = 2n 2 and n 2 = 2p 2 . This process could be repeated without end,leading to an impossible “infinite descent” of positive integers. Fermat’s technique of“infinite descent” is today an important tool of number theory.2.2 Three Famous Problems of AntiquityAlthough this paradox could be resolved by accepting the fact that geometric constructionsbelong to a mathematical world ampler than pure arithmetic, new difficultiesarose. The problem of constructing a segment equal to the side of a cube with volumetwice the volume of a given cube could not be solved within the scope of an Euclideangeometry that allowed only lines and circles. The problem of trisecting an arbitraryangle suffered the same fate, although in both cases solutions could be found by meansof three-dimensional geometry or by mechanical constructions. Archimedes himselffound a simple mechanical way of trisecting the angle using only ruler and compass.On the other hand, the quadrature of the circle, namely, the geometric constructionof a square with the same area as a given circle, could not be solved in this way. Theproblem became famous, and even Dante Alighieri (1966–1967), mentions it in hisCommedia when confronted with the impossibility of understanding the divine mysteryof the Trinity:Dante, Paradiso, Canto XXXIII:Qual è ’l geomètra che tutto s’affigeper misurar lo cerchio, e non ritrova,pensando, quel principio ond’ elliindige,tal era io a quella vista nova:veder voleva come si convennel’imago al cerchio e come vis’indova;ma non eran da ciò le proprie penne:The Longfellow translation:As the geometrician, who endeavoursTo square the circle, and discovers not,By taking thought, the principle he wants,Even such was I at that new apparition;I wished to see how the image to the circleConformed itself, and how it there findsplace;But my own wings were not enough forthis,