Axiomathes

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Why Believe Infinite Sets Exist?

Original Paper
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Abstract

The axiom of infinity states that infinite sets exist. I will argue that this axiom lacks justification. I start by showing that the axiom is not self-evident, so it needs separate justification. Following Maddy’s (J Symb Log 53(2):481–511, 1988) distinction, I argue that the axiom of infinity lacks both intrinsic and extrinsic justification. Crucial to my project is Skolem’s (in: van Heijnoort (ed) From Frege to Gödel: a source book in mathematical logic, 1879–1931, Cambridge, Harvard University Press, pp. 290–301, 1922) distinction between a theory of real sets, and a theory of objects that theory calls “sets”. While Dedekind’s (in: Essays on the theory of numbers, pp. 14–58, 1888. http://www.gutenberg.org/ebooks/21016) argument fails, his approach was correct: the axiom of infinity needs a justification it currently lacks. This epistemic situation is at variance with everyday mathematical practice. A dilemma ensues: should we relax epistemic standards or insist, in a skeptical vein, that a foundational problem has been ignored?

Keywords

Justification of axioms The axiom of infinity Mathematical indispensability Richard Dedekind Thoralf Skolem Penelope Maddy Foundations of mathematics 

Notes

Acknowledgements

The paper is based on a talk given at the CLMPS in Helsinki (August 2015). I greatly benefited from conversations with Jim Cargile (UVa), Nora Grigore (UT Austin), Markus Pantsar (Helsinki), Gabriel Săndoiu (Sfântul Sava College), and Susan Vineberg (Wayne State).

Funding

This work was supported by the Jefferson Scholars Foundation through a John S. Lillard fellowship.

References

  1. Baker A (2005) Are there genuine mathematical explanations of physical phenomena? Mind 114(454):223–238CrossRefGoogle Scholar
  2. Boghossian P (2003) Blind reasoning. Proc Aristot Soc 77(1):225–248CrossRefGoogle Scholar
  3. Boolos G (1971) The iterative conception of set. J Philos 68(8):215–231CrossRefGoogle Scholar
  4. Brouwer LEJ (1913/1983) Intuitionism and formalism. In: Putnam H, Benacerraf P (eds) Philosophy of mathematics, 2nd edn. Cambridge University Press, Cambridge, pp 77–89Google Scholar
  5. Carnap R (1939/1983) The logicist foundations of mathematics. In: Putnam H, Benacerraf P (eds) Philosophy of mathematics, 2nd edn. Cambridge University Press, Cambridge, pp 41–51Google Scholar
  6. Cohen P (1966) Examples of formal systems. Primitive recursive functions. In: Set theory and the continuum hypothesis. W.A. Benjamin, New York, pp 20–32Google Scholar
  7. Dedekind R (1888/2007) The nature and meaning of numbers. In: Essays on the theory of numbers, 1st English ed. (1901) Chicago: Open Court; 2nd ed. ebook digitized by Project Gutenberg, pp 14–58. http://www.gutenberg.org/ebooks/21016
  8. Douven I, Meijs W (2007) Measuring coherence. Synthese 156(3):405–425CrossRefGoogle Scholar
  9. Hauser K, Woodin WH (2014) Strong axioms of infinity and the debate about realism. J Philos 111(8):397–419CrossRefGoogle Scholar
  10. Heil J (2003) Introduction. In: From an ontological point of view. OUP, Oxford, pp 1–14Google Scholar
  11. James W (1907/1981) Pragmatism. Hackett, LondonGoogle Scholar
  12. Kelly Th (2010) Peer disagreement and higher-order evidence. In: Feldman R, Warfield T (eds) Disagreement. OUP, Oxford, pp 111–174CrossRefGoogle Scholar
  13. Lavers G (2016) Carnap’s surprising views on the axiom of infinity. Metascience 25:37–41CrossRefGoogle Scholar
  14. Maddy P (1988) Believing the axioms—I. J Symbol Log 53(2):481–511CrossRefGoogle Scholar
  15. Maddy P (1992) Indispensability and practice. J Philos 89(6):275–289CrossRefGoogle Scholar
  16. Maddy P (2011) The problem. In: In defense of the axioms. OUP, OxfordGoogle Scholar
  17. Pantsar M (2014) An empirically feasible approach to the epistemology of arithmetic. Synthese 191(17):4201–4229CrossRefGoogle Scholar
  18. Potter M (2004) Set theory and its philosophy. Oxford University Press, ClarendonCrossRefGoogle Scholar
  19. Potter M, Smiley T (2002) Recarving content. Proc Aristot Soc 102:351–354CrossRefGoogle Scholar
  20. Quine WVO (1948) On what there is. In: From a logical point of view. Harvard University Press, Cambridge, pp 1–19Google Scholar
  21. Quine WVO (1951/1999) Two dogmas of empiricism. In: From a logical point of view. Harvard University Press, Cambridge, pp 20–46Google Scholar
  22. Quine WVO (1960) Ontic decision. In: Word and object. MIT, Cambridge, pp 233–276Google Scholar
  23. Resnik MD (1981) Mathematics as a science of patterns: ontology and reference. Noûs 15(4):529–550CrossRefGoogle Scholar
  24. Robinson A (1969) From a formalist’s point of view. Dialectica 23:45–49CrossRefGoogle Scholar
  25. Russell B (1908) Mathematical logic as based on the theory of types. Am J Math 30(3):222–262CrossRefGoogle Scholar
  26. Russell B (1919/2010) The axiom of infinity and logical types. In: Introduction to mathematical philosophy. George Allen & Unwin, London, pp 106–115Google Scholar
  27. Skolem Th (1922/1967) Some remarks on axiomatized set theory. In: van Heijnoort J (ed) From Frege to Gödel: a source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge, pp 290–301Google Scholar
  28. Sober E (1993) Mathematics and indispensability. Philos Rev 102(1):35–57CrossRefGoogle Scholar
  29. Vineberg S (1996) Confirmation and the indispensability of mathematics to science. Philos Sci 63(3):263Google Scholar
  30. von Neumann J (1925/1967) An axiomatization of set theory. In: van Heijnoort J (ed) From Frege to Gödel: a source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge, pp 393–413Google Scholar
  31. Zermelo E (1908/1967) Investigations in the foundations of set theory—I. In: van Heijnoort J (ed) From Frege to Gödel: a source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge, pp 199–215Google Scholar
  32. Zermelo E (1921/2010) Theses concerning the infinite in mathematics. In: Ebbinghaus HD, Fraser CG, Kanamori A (eds) Collected Works, vol. I. Springer, Berlin, p 307Google Scholar

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© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Corcoran Department of PhilosophyUniversity of VirginiaCharlottesvilleUSA

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