Mathematics as the Art of Abstraction

  • Richard L. Epstein
Part of the Logic, Epistemology, and the Unity of Science book series (LEUS, volume 30)


Mathematics, like the sciences, proceeds by a process of abstraction, so that mathematical claims like scientific claims are neither true nor false, but only true or false in an application of the theory to which they belong. A proof in mathematics is meant to show that a claim follows from the assumptions of a particular mathematical theory.


abstraction empiricism about mathematics if-thenism mathematical argument mathematical explanation mathematical proof. 



A previous version of this chapter was published as ‘On Mathematics’ in Richard L. Epstein, Logic as the Art of Reasoning Well, Advanced Reasoning Forum, 2008, 411–441. I am grateful to Fred Kroon, Charlie Silver, Carlo Cellucci, Jeremy Avigad, David Isles, Reuben Hersh, Ian Grant, Paul Livingston, and Andrew Aberdein for their comments on earlier versions.


  1. Aristotle (1930). Physics (R. P. Hardie & R. K. Gaye, Trans.). In W. D. Ross (Ed.), The works of Aristotle. Oxford: Clarendon.Google Scholar
  2. Bashmakova, I. G, & Smirnova, G. S. (2000). Geometry: The first universal language of mathematics. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 331–340). Dordrecht: Kluwer.CrossRefGoogle Scholar
  3. Bôcher, M. (1904). The fundamental conceptions and methods of mathematics. Bulletin of the American Mathematical Society, 11, 115–135.CrossRefGoogle Scholar
  4. Breger, H. (2000). Tacit knowledge and mathematical progress. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 221–230). Dordrecht: Kluwer.CrossRefGoogle Scholar
  5. Brouwer, L. E. J. ([1983] 1912). Intuitionism and formalism (Trans. by A. Dresden of ‘Intuitionisme en formalisme’, Inaugural Address at the University of Amsterdam). In P. Benacerraf & H. Putnam (Eds.), Philosophy of mathematics: Selected readings (pp. 77–89). Cambridge: Cambridge University Press.Google Scholar
  6. Bryant, P., & Nuñes, T. (2002). Children’s understanding of mathematics. In U. Goswami (Ed.), Blackwell handbook of childhood cognitive development (pp. 412–439). Oxford: Blackwell.CrossRefGoogle Scholar
  7. Cantor, G. (1883). Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Leipzig: Teubner.Google Scholar
  8. Dauben, J. W. (1979). Georg Cantor: His mathematics and philosophy of the infinite. Cambridge, MA: Harvard University Press.Google Scholar
  9. Davies, E. B. (2005). Whither mathematics? Notices of the American Mathematical Society, 52, 1350–1356.Google Scholar
  10. De Cruz, H., & Smedt, J. D. (2010). The innateness hypothesis and mathematical concepts. Topoi, 29, 3–13.CrossRefGoogle Scholar
  11. Detlefsen, M. (2005). Formalism. In S. Shapiro (Ed.), Oxford handbook of philosophy of mathematics and logic (pp. 236–317). Oxford: Oxford University Press.CrossRefGoogle Scholar
  12. Einstein, A. ([1961] 1915). Relativity: The special and general theory (R. Lawson, Trans.). New York: Crown.Google Scholar
  13. Epstein, R. L. (1988). A general framework for semantics for propositional logics. Text of invited address to the VII Latin American Symposium on Mathematical Logic. Contemporary Mathematics, 69, 149–168.Google Scholar
  14. Epstein, R. L. (1990). Propositional logics. Dordrecht: Kluwer. (2nd ed., Oxford: Oxford University Press, 1995. 2nd ed. with corrections, Belmont, CA: Wadsworth, 2000)Google Scholar
  15. Epstein, R. L. (1994). Predicate logic. Oxford, UK: Oxford University Press. (Reprinted, 2000, Belmont, CA: Wadsworth)Google Scholar
  16. Epstein, R. L. (1999). The metaphysical basis of logic. Manuscrito, 22(2), 133–148.Google Scholar
  17. Epstein, R. L. (2006). Classical mathematical logic. Princeton, NJ: Princeton University Press.Google Scholar
  18. Epstein, R. L. (2010). The internal structure of predicates and names with an analysis of reasoning about process. Typescript available at www.AdvancedReasoningForum.orgGoogle Scholar
  19. Epstein, R. L. (2012). Valid inferences. In J.-Y. Béziau & M. E. Coniglio (Eds.), Logic without frontiers: Festschrift for Walter Alexandre Carnielli on the occasion of his 60th birthday (pp. 105–112). London: College Publications.Google Scholar
  20. Epstein, R. L., & Carnielli, W. A. ([2008] 1989). Computability: Computable functions, logic, and the foundations of mathematics. Belmont, CA: Wadsworth & Brooks/Cole. (3rd ed., ARF, 2008)Google Scholar
  21. Fallis, D. (1998). Review of Hersh, 1993 et al. Journal of Symbolic Logic, 63, 1196–1200.CrossRefGoogle Scholar
  22. Ferreirós, J., & Gray, J. J. (Eds.). (2006). The architecture of modern mathematics. Oxford: Oxford University Press.Google Scholar
  23. Frege, G. (1980). Gottlob Frege: The philosophical and mathematical correspondence. Edited by G. Gabriel, H. Hermes, F. Kambartel, C. Thiel & A. Veraart. Chicago, IL: University of Chicago Press.Google Scholar
  24. Gardner, M. (1973). Mathematical games. Scientific American, 229(October), 114–118.CrossRefGoogle Scholar
  25. Hardy, G. H. (1928). Mathematical proof. Mind, 38, 11–25.Google Scholar
  26. Hersh, R. (1997). What is mathematics, really? London: Jonathan Cape.Google Scholar
  27. Hersh, R. (2006). Inner vision, outer truth. In R. Hersh (Ed.), 18 unconventional essays about the nature of mathematics (pp. 320–326). New York: Springer.CrossRefGoogle Scholar
  28. Isles, D. (1981). Remarks on the notion of standard non-isomorphic natural number series. In F. Richman (Ed.), Constructive mathematics, proceedings of the New Mexico State University Conference, Vol. 873 of Lecture notes in mathematics (pp. 111–134). Berlin: Springer.CrossRefGoogle Scholar
  29. Isles, D. (2004). Questioning articles of faith: A re-creation of the history and theology of arithmetic. Bulletin of Advanced Reasoning and Knowledge, 2, 51–59.Google Scholar
  30. Kitcher, P. (1975). Bolzano’s ideal of algebraic analysis. Studies in History and Philosophy of Science, 6, 229–269.CrossRefGoogle Scholar
  31. Kulpa, Z. (2009). Main problems of diagrammatic reasoning. Part I: The generalization problem. Foundations of Science, 14(1–2), 75–96.Google Scholar
  32. Lakoff, G., & Núñez, R. E. (2000). Where mathematics comes from: How the embodied mind brings mathematics into being. New York: Basic Books.Google Scholar
  33. Leibniz, G. W. ([1981] 1765). New essays on human understanding (Trans. and edited by Peter Remnant and Jonathan Bennett). Cambridge: Cambridge University Press.Google Scholar
  34. Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford: Oxford University Press.Google Scholar
  35. Mancosu, P. (2000). On mathematical explanation. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 103–119). Dordrecht: Kluwer.CrossRefGoogle Scholar
  36. Mill, J. S. (1874). A system of logic, ratiocinative and inductive, being a connected view of the principles of evidence and the methods of scientific investigation (8th ed.). New York: Harper & Brothers.Google Scholar
  37. Musgrave, A. (1977). Logicism revisted. British Journal for the Philosophy of Science, 28, 90–127.CrossRefGoogle Scholar
  38. Nagel, E. (1961). The structure of science. New York: Harcourt, Brace & World. (Reprinted by Hackett Publishing Company, Indianapolis, IN, 1979)Google Scholar
  39. Nelsen, R. B. (1993). Proofs without words. Washington, DC: Mathematical Association of America.Google Scholar
  40. Nidditch, P. H. (1960). Elementary logic of science and mathematics. Glencoe, IL: Free Press.Google Scholar
  41. Peirce, C. S. (1931). Collected papers (Vol. I). Cambridge, MA: Harvard University Press.Google Scholar
  42. Poincaré, H. (1921). The foundations of science (G. B. Halstead, Trans.). New York: The Science Press.Google Scholar
  43. Pólya, G. ([1977] 1963). Mathematical methods in science, Vol. IX of Studies in mathematics. Revised edition edited by Leon Bowden. Washington, DC: Mathematical Association of America.Google Scholar
  44. Popper, K. R. (1972). Objective knowledge. London: Oxford University Press.Google Scholar
  45. Putnam, H. (1967). Mathematics without foundations. Journal of Philosophy, 64(1), 5–22. (Reprinted in Putnam, Mathematics, matter, and method: Philosophical papers, (Vol. 1, pp. 43–59). Cambridge: Cambridge University Press).Google Scholar
  46. Putnam, H. (1975). What is mathematical truth? In Mathematics, matter and method: Philosophical papers (Vol. 1, pp. 60–78). Cambridge: Cambridge University Press.Google Scholar
  47. Resnik, M., & Kushner, D. (1987). Explanation, independence, and realism in mathematics. British Journal for the Philosophy of Science, 38, 141–158.CrossRefGoogle Scholar
  48. Russell, B. ([1901] 1986). Mathematics and the metaphysicians. In Mysticism and logic (pp. 75–95). London: Unwin.Google Scholar
  49. Sawyer, W. W. (1943). Mathematician’s delight. Harmondsworth: Penguin.Google Scholar
  50. Sherry, D. (2009). The role of diagrams in mathematical arguments. Foundations of Science, 14(1–2), 59–74.CrossRefGoogle Scholar
  51. Skorupski, J. (2005). Later empiricism and logical positivism. In S. Shapiro (Ed.), Oxford handbook of philosophy of mathematics and logic (pp. 51–80). Oxford: Oxford University Press.CrossRefGoogle Scholar
  52. Steiner, M. (2000). Penrose and platonism. In E. Grosholz & H. Breger (Eds.), The growth of mathematical knowledge (pp. 133–141). Dordrecht: Kluwer.CrossRefGoogle Scholar
  53. Van Dantzig, D. (1956). Is \(1{0}^{1{0}^{10} }\) a finite number? Dialectica, 9, 273–277.CrossRefGoogle Scholar
  54. Whorf, B. L. (1941). The relation of habitual thought to language. In L. Spier (Ed.), Language, culture, and personality: Essays in memory of Edward Sapir (pp. 75–93). Menasha, WI: Sapir Memorial Publication Fund. (Reprinted in Carroll J. B. (Ed.). (1956). Language, thought, and reality: Selected writings of Benjamin Lee Whorf (pp. 134–159). Cambridge, MA: MIT Press)Google Scholar
  55. Wigner, E. (1960). The unreasonable effectiveness of mathematics in the natural sciences. Communications on Pure and Applied Mathematics, 13(1), 1–14.CrossRefGoogle Scholar
  56. Wilder, R. L. (1944). The nature of mathematical proof. The American Mathematical Monthly, 51(6), 309–323.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2013

Authors and Affiliations

  1. 1.Advanced Reasoning ForumSocorroUSA

Personalised recommendations