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

68 the mathematical infinityThis purist view has changed today. Indeed, nowhere differentiable functions doappear quite often as solutions of very natural problems ranging from number theoryto dynamical systems. Thus, they have lost their early negative connotation as uselesspathological examples.In geometry, one of the very first concepts that appear is the notion of dimension.A point has dimension 0, a curve has dimension 1, a surface has dimension 2, andso on. The dimension represents the degrees of freedom for motion on the geometricobject. So, we live in three-dimensional space, four-dimensional if we take into accounttime (a different type of freedom there, because the motion in time is only forward).Algebraic geometers (they study geometric objects defined by systems of polynomialequations) routinely talked about curves having ∞ 1 points, surfaces having ∞ 2 points,and three-dimensional space having ∞ 3 points. All this was put into discussion byPeano’s example of a curve filling a square, which indicated that it was a “curve” ofdimension 2. How could this happen? What was the correct notion of dimension?It turns out that Peano’s curve has self-intersections, and, hence, the correspondingmap it gives from [0, 1] is not injective. In fact, any continuous surjective map likePeano’s curve cannot be injective. The geometric notion of dimension, by now classic,is that the dimension of a space S is the smallest integer d such that any open coverof the space can be refined to a cover in which no point of S is covered more thand + 1 times. With this definition, the dimension turns out to be invariant by continuousinjective mappings, thereby conforming to the naive intuition that, for example, a planehas “more points” than a line. This does not contradict the theorem of Cantor that thereis a biunivocal map between the points of an interval and the points of a square; actually,it clarifies that any such map must be far from continuous. In retrospect, this shouldnot be surprising, because continuity is a notion of ordering, while sets are unorderedaggregates of elements. The peculiarity of the Peano curve is now explained.Although until recently the Peano curve was considered by many to be a strangepathological object, it is of interest that such curves have found practical applications.For example, we may imagine data given by points in a square and an idea for sorting aset of particular points that consists in sorting them in the order by which they appear onthe Peano curve. Sorting methods of this type are actually used in several sophisticatedsoftware applications for handling very large databases.Another aspect of infinity appears in the notion of self-similarity. If one looksat a geographical map, one finds that the general shape of coastlines appears to beindependent of the scale of the map. This phenomenon of self-similarity occurs quiteoften in mathematical models of dynamical systems (depending on parameters) inwhich the evolution of the system is determined by the repeated, in the limit infinite,iteration of simple rules. For certain values of the parameters the behavior of thedynamical system may be chaotic, and it becomes important to study the structureof the boundary between regular and chaotic behavior. It turns out that very oftenthis boundary has a self-similarity property, namely, after a change of scale it looksessentially the same, and more and more so by iterating the change of scale ad infinitum,an echo of Voltaire’s Micromégas universe.The study of such sets, called fractals, has become of considerable importance formodeling chaotic phenomena. Today, fractals are no longer considered as “pathological”objects.