The Argument of Mathematics pp 11-29

Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 30) | Cite as

Non-deductive Logic in Mathematics: The Probability of Conjectures

Chapter

Abstract

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case.

References

  1. Aschbacher, M. (1980). The finite simple groups and their classification. New Haven, CT: Yale University Press.Google Scholar
  2. Aschbacher, M. (2001). The status of the classification of the finite simple groups. Notices of the American Mathematical Society, 51, 736–740.Google Scholar
  3. Baker, A. (2007). Is there a problem of induction for mathematics? In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical knowledge (pp. 59–73). Oxford: Oxford University Press.Google Scholar
  4. Baker, A. (2008). Experimental mathematics. Erkenntnis, 68, 331–344.CrossRefGoogle Scholar
  5. Baker, A. (2009). Non-deductive methods in mathematics. Stanford Encyclopedia of Philosophy. Accessed May 2013. http://plato.stanford.edu/entries/mathematics-nondeductive/
  6. Bender, H. (1970). On the uniqueness theorem. Illinois Journal of Mathematics, 14, 376–384.Google Scholar
  7. Borwein, J. M., & Bailey, D. H. (2004). Mathematics by experiment: Plausible reasoning in the 21st century. Natick, MA: A. K. Peters.Google Scholar
  8. Borwein, J. M., Bailey, D. H., & Girgensohn, R. (2004). Experimentation in mathematics: Computational paths to discovery. Natick, MA: A. K. Peters.Google Scholar
  9. Brent, R., van de Lune, J., te Riele, H., & Winter, D. (1982). On the zeros of the Riemann Zeta Function in the critical strip. II. Mathematics of Computation, 39, 681–688.Google Scholar
  10. Brown, J. R. (1999). Philosophy of mathematics: An introduction to the world of proofs and pictures. London: Routledge.Google Scholar
  11. Chandler, C. (1999). Hello, I must be going: Groucho Marx and his friends. Garden City, NY: Doubleday.Google Scholar
  12. Collins, M. J. (1980). Finite simple groups, II. London: Academic.Google Scholar
  13. Corfield, D. (2003). Towards a philosophy of real mathematics. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  14. Derbyshire, J. (2003). Prime obsession: Bernhard Riemann and the greatest unsolved problem in mathematics. Washington, DC: Joseph Henry Press.Google Scholar
  15. Dove, I. J. (2009). Towards a theory of mathematical argument. Foundations of Science, 14(1–2), 137–152.CrossRefGoogle Scholar
  16. du Sautoy, M. (2003). The music of the primes: Searching to solve the greatest mystery in mathematics. New York: Harper Collins.Google Scholar
  17. Echeverría, J. (1996). Empirical methods in mathematics. In G. Munévar (Ed.), Spanish studies in the philosophy of science, volume 86 of Boston studies in the philosophy of science (pp. 19–55). Dordrecht: Kluwer.Google Scholar
  18. Edwards, H. M. (1974). Riemann’s zeta function. New York: Academic.Google Scholar
  19. Epstein, D., Levy, S., & de la Llave, R. (1992). About this journal. Experimental Mathematics,1, 1–13.Google Scholar
  20. Fadiman, C. (1955). The American treasury. New York: Harper.Google Scholar
  21. Fallis, D. (1997). The epistemic status of probabilistic proof. Journal of Philosophy, 94, 165–186.CrossRefGoogle Scholar
  22. Fallis, D. (2000). The reliability of randomized algorithms. British Journal for the Philosophy of Science, 51, 255–271.CrossRefGoogle Scholar
  23. Feit, W., & Thompson, J. G. (1963). Solvability of groups of odd order. Pacific Journal of Mathematics, 13, 775–1029.CrossRefGoogle Scholar
  24. Franklin, J. (1987). Non-deductive logic in mathematics. British Journal for the Philosophy of Science, 38(1), 1–18.CrossRefGoogle Scholar
  25. Franklin, J. (2001). Resurrecting logical probability. Erkenntnis, 55, 277–305.CrossRefGoogle Scholar
  26. Franklin, J. (2009). What science knows and how it knows it. New York: Encounter Books.Google Scholar
  27. Franklin, J. (2011). The objective Bayesian conceptualisation of proof and reference class problems. Sydney Law Review, 33, 545–561.Google Scholar
  28. Gorenstein, D. (1979). The classification of finite simple groups (I). Bulletin of the American Mathematical Society, 1, 43–199 (New Series).Google Scholar
  29. Gorenstein, D. (1980). An outline of the classification of finite simple groups. In B. Cooperstein & G. Mason (Eds.), The Santa Cruz conference on finite groups, volume 37 of proceedings of symposia in pure mathematics (pp. 3–28). Providence, RI: American Mathematical Society.Google Scholar
  30. Gorenstein, D. (1982). Finite simple groups. New York: Plenum.Google Scholar
  31. Gorenstein, D., Lyons, R., & Solomon, R. (1994–2005). The classification of the finite simple groups (6 Vols.). Providence, RI: American Mathematical Society.Google Scholar
  32. Gourdon, X. (2004). The 1013 first zeros of the Riemann Zeta Function, and zeros computation at very large height. Accessed May 2013. http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e1%3-1e24.pdfGoogle Scholar
  33. Hunt, D. (1980). A computer-based atlas of finite simple groups. In B. Cooperstein & G. Mason (Eds.), The Santa Cruz conference on finite groups, volume 37 of proceedings of symposia in pure mathematics (pp. 507–510). Providence, RI: American Mathematical Society.Google Scholar
  34. Ivić, A. (2003). On some reasons for doubting the Riemann Hypothesis. Accessed May 2013. http://arxiv.org/abs/math/0311162
  35. Janko, Z. (1966). A new finite simple group with abelian 2-Sylow subgroups and its characterization. Journal of Algebra, 3, 147–186.CrossRefGoogle Scholar
  36. Jaynes, E. T. (2003). Probability theory: The logic of science. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  37. Keynes, J. M. (1921). A treatise on probability. London: Macmillan.Google Scholar
  38. Kolata, G. B. (1976). Mathematical proofs: The genesis of reasonable doubt. Science, 192, 989–990.CrossRefGoogle Scholar
  39. Lehrer Dive, L. (2003). An epistemic structuralist account of mathematical knowledge. PhD thesis, University of Sydney.Google Scholar
  40. Linnik, Y. V. (1952). Some conditional theorems concerning the binary Goldbach problem. Izvestiya Akademii Nauk SSSR, 16, 503–520.Google Scholar
  41. Lyons, R. (1972). Evidence for a new finite simple group. Journal of Algebra, 20, 540–569.CrossRefGoogle Scholar
  42. Marsaglia, G. (2005). On the randomness of pi and other decimal expansions. Accessed May 2013. http://www.yaroslavvb.com/papers/marsaglia-on.pdf
  43. Mason, G. (1980). Preface. In B. Cooperstein & G. Mason (Eds.), The Santa Cruz conference on finite groups, volume 37 of proceedings of symposia in pure mathematics (p. xiii). Providence, RI: American Mathematical Society.Google Scholar
  44. Müller, J., & Neunhöffer, M. (1987). Some computations regarding Foulkes’ conjecture. Experimental Mathematics, 14, 277–283.CrossRefGoogle Scholar
  45. O’Nan, M. (1976). Some evidence for the existence of a new finite simple group. Proceedings of the London Mathematical Society, 32, 421–479.CrossRefGoogle Scholar
  46. Pólya, G. (1954). Mathematics and plausible reasoning (2 Vols.). Princeton, NJ: Princeton University Press.Google Scholar
  47. Renyi, A. (1962). On the representation of an even number as the sum of a prime and an almost prime. American Mathematical Society Translations, 2nd series, 19, 299–321.Google Scholar
  48. Riemann, B. (1859 [1974]). On the number of primes less than a given magnitude. In H. Edwards (Ed.), Riemann’s zeta function (pp. 299–305). New York: Academic.Google Scholar
  49. Ruhkin, A. (2001). Testing randomness: A suite of statistical procedures. Theory of Probability and its Applications, 45, 111–132.CrossRefGoogle Scholar
  50. Sabbagh, K. (2002). Dr Riemann’s zeros. London: Atlantic Books.Google Scholar
  51. Solomon, R. (2001). A brief history of the classification of the finite simple groups. Bulletin of the American Mathematical Society, 38, 315–352.CrossRefGoogle Scholar
  52. Tenenbaum, G. (1995). Introduction to analytic and probabilistic number theory. Cambridge: Cambridge University Press.Google Scholar
  53. Tits, J. (1971). Groupes finis simples sporadiques. In A. Dold & B. Eckmann (Eds.), Séminaire Bourbaki, volume 180 of Springer Lecture Notes in Mathematics (pp. 187–211). New York: Springer.Google Scholar
  54. Van Kerkhove, B., & Van Bendegem, J. P. (2008). Pi on earth, or mathematics in the real world. Erkenntnis, 68, 421–435.CrossRefGoogle Scholar
  55. Wang, Y. (Ed.). (2002). Goldbach conjecture. River Edge, NJ: World Scientific.Google Scholar
  56. Weil, A. (1948). Variétés abéliennes et courbes algébriques. Paris: Hermann.Google Scholar
  57. Williamson, J. (2010). In defence of objective Bayesianism. Oxford: Oxford University Press.CrossRefGoogle Scholar
  58. Zeilberger, D. (1993). Theorems for a price: Tomorrow’s semi-rigorous mathematical culture. Notices of the American Mathematical Society, 46, 978–981.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia

Personalised recommendations