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

170 some considerations on infinity in physicsnot mean that the picture is inconsistent. It only means that our common intuitionabout space can be incorrect, and that Euclidean geometry may not be the correctmathematical tool for describing real physical space, below a certain scale.It would not be the first time that science contradicted our common intuition.According to our intuition, after all, the earth is flat, and it does not move! In conclusion,contemporary research in fundamental physics is seriously considering the possibilitythat space is not infinitely divisible. There is no necessarily infinite toward the small.7.2 Infinite Extension of Space: The Archytas Problem andthe Solution of Dante Alighieri and Albert EinsteinArchytas of Tarentum was a fifth-century Greek thinker. An argument by Archytashas reached us, thanks to a fragment of Eudemus preserved by Simplicius. In thisargument, Archytas claims that the universe must be infinitely extended because ofthe following: “If I arrived at the outermost edge of the heaven, could I extend myhand or staff into what is outside or not? It would be paradoxical not to be able toextend it.” More in detail: if there is nothing blocking my hand, space continues, but ifthere is a “wall” that blocks my hand from being further extended, the wall is beyondthe boundary; hence, space continues there. Archytas can advance to the new limit,the other side of the wall, and ask the same question again, so that there will alwaysbe something into which his hand can be extended, beyond the supposed limit. Hence,space is clearly unlimited. This argument was considered by Aristotle to be the “mostimportant” reason why people believe in the existence of the infinite (Aristotle 1952,p. 203b22).A solution of this problem was given by Albert Einstein (1996) in a celebrated paperwritten in 1917. This very solution was foreseen by the greatest of the Middle Agepoets, Dante Alighieri. I do not know if Dante got this idea from somebody else or ifit is his own idea. The idea, as expressed by Dante, is the following:In his poem, Dante describes his great trip across the universe and gives a grandiose visionof the entire universe. The universe described by Dante has no boundary: there is no placewhere space ends, and Archytas could attempt to push his hand ahead. Nevertheless, theuniverse described by Dante has a finite extension. It is not infinitely extended. This seemsparadoxical at first, but it is not.Let’s see how this works. According to Dante, who followed Aristotelian cosmologyto some extent, the universe has a spherical structure centered on the earth. The earth isa sphere. (The common idea that in the Middle Ages people thought that the earth wasflat is completely false. Around the Mediterranean, the earth was universally knownto be spherical since about five centuries before Christ.) Around the earth, accordingto Dante and Aristotle, there are several larger concentric spheres: the spheres of theMoon, Venus, and so on, all the way until the sphere of the fixed stars, or the “primusmovens,” the “first mover.” Each of these spheres surrounds the previous one.What next? What is around the sphere of the fixed stars? Here Dante departsfrom Aristotle. For Aristotle, the universe ends there, with all the difficulties raisedby Archytas trying to push his hand further. For Dante, things are not so simple. In