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

74 the mathematical infinityof a proposed solution could be done in polynomial time, the same would be true forthe apparently much harder problem of producing an actual solution. For example, theproblem of finding a clique of size k in a graph (an arrangement of points, called vertices,some of which are joined by segments, called edges) is NP-complete. Obviously, it isin P: the oracle simply points out to the viewer the k vertices of a clique, and the viewerchecks the presence of the k(k − 1)/2 edges forming the clique. However, finding theclique without the help of the oracle is very hard, as our experience with the Ramseynumber R(6, 6) shows (at least, according to Erdős).By now, thousands of classical problems in finite combinatorics have been shown tobe NP-complete, and most mathematicians believe that NP is not equal to P. Notwithstandingvery deep studies of the question, the problem remains open. Is it possible thatthe solution to this basic problem, with its finitistic formulation, will require a daringexcursion into the realm of infinity? Only time will tell. Certainly, the computer hasshown us, in a dramatic way, the distinction between the “finite” in real life and the“finite” beyond our grasp, as in George Gamow’s One,Two,Three...Infinity.AcknowledgmentsThe author ackowledges the following additional source, besides those quoted in theReferences, used for the compilation of this chapter: Wikipedia, the Free Encyclopedia,in http:/en.wikipedia.org/wiki, for technical information and the Erdős quote. Figures2.1 and 2.2 were drawn using the Mathematica software program. The picture of theHydra is by permission from the J. Paul Getty Museum, Villa Collection, Malibu,California.ReferencesAx, J., and Kochen, S. 1965. Diophantine problems over local fields. I. American Journal of Mathematics87: 605–30.Bartocci, U., and Vipera, M.C. 2004. Variazioni sul problema dei buoi di Archimede, ovvero, allaricerca di soluzioni “possibili” . . . http://www.cartesio-episteme.net/mat/cattle-engl.htm.Busard, H. L. L. 2001. Johannes de Tinemue’s Redaction of Euclid’s Elements, the So-called AdelardIII version. vol. 1. Stuttgart: Franz Steiner Verlag.Cohen, P. J. 1969. Decision procedures for real and p-adic fields. Communications on Pure andApplied Mathematics 22: 131–51.Cook, S. 1971. The complexity of theorem proving procedures. In Proceedings, Third Annual ACMSymposium on the Theory of Computing, ACM, New York, pp. 151–58.Correspondence d’Hermite et de Stieltjes. 1905. T.II. Paris: Gauthier-Villars.Dante Alighieri. 1966–1967. La Commedia secondo l’antica vulgata a cura di Giorgio Petrocchi.Edizione Nazionale a cura della Società Dantesca Italiana. Milano: Arnoldo Mondadori Editore.Davis, M. (ed.) 1965. The Undecidable. Hewlett, NY: Raven Press. http://cs.nyu.edu/pipermail/fom/2007-October/012156.html. http://forum.wolframscience.com/showthread.php?s=&threadid=1472.Dickson, L. E. 1952. History of the Theory of Numbers, vol. 2. Reprint New York: Chelsea.De Morgan, A. 1872. A Budget of Paradoxes. London: Longmans, Green.