pp 1–16 | Cite as

Benacerraf, Field, and the agreement of mathematicians

  • Eileen S. Nutting


Hartry Field’s epistemological challenge to the mathematical platonist is often cast as an improvement on Paul Benacerraf’s original epistemological challenge. I disagree. While Field’s challenge is more difficult for the platonist to address than Benacerraf’s, I argue that this is because Field’s version is a special case of what I call the ‘sociological challenge’. The sociological challenge applies equally to platonists and fictionalists, and addressing it requires a serious examination of mathematical practice. I argue that the non-sociological part of Field’s challenge amounts to a minor reformulation of Benacerraf’s original challenge. So, I contend, Field’s challenge is not an improvement on Benacerraf’s. What is new to Field’s challenge is as much a problem for the fictionalist as it is for the platonist.


Benacerraf Field Platonism Epistemological challenge 



Thanks to Corinne Bloch-Mullins, Ben Caplan, Sam Cowling, Lina Janssen, Kathryn Lindeman, May Mei, Reed Solomon, Audrey Yap, three anonymous referees, my colleagues at the University of Kansas, and an audience at Denison University. Thanks also to audiences that heard an ancestor of this paper at the University of Missouri, the 2014 Midwest Epistemology Workshop, and a workshop at the University of Umeå.


  1. Armstrong, D. M. (1973). Belief, truth, and knowledge. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  2. Azzouni, J. (2004). Deflating existential consequence: A case for nominalism. Oxford: Oxford University Press.CrossRefGoogle Scholar
  3. Benacerraf, P. (1973). Mathematical truth. The Journal of Philosophy, 70(8), 661–680.CrossRefGoogle Scholar
  4. Burgess, J., & Rosen, G. (1997). A subject with no object. Oxford: Oxford University Press.Google Scholar
  5. Casullo, A. (1992). Causality, reliabilism, and mathematical knowledge. Philosophy and Phenomenological Research, 52(3), 557–584.CrossRefGoogle Scholar
  6. Cheyne, C. (1998). Existence claims and causality. Australasian Journal of Philosophy, 76(1), 34–47.CrossRefGoogle Scholar
  7. Clarke-Doane, J. (2013). What is absolute undecidability? Noûs, 47(3), 467–481.CrossRefGoogle Scholar
  8. Clarke-Doane, J. (2014). Moral epistemology: the mathematics analogy. Noûs, 48(2), 238–255.CrossRefGoogle Scholar
  9. Clarke-Doane, J. (2017). What is the Benacerraf problem? In F. Pataut (Ed.), Truth, objects, infinity: New perspectives on the philosophy of Paul Benacerraf (pp. 17–44). Dordrecht: Springer.Google Scholar
  10. Elgin, C. Z. (2004). True enough. Philosophical Issues, 14(1), 113–131.CrossRefGoogle Scholar
  11. Field, H. (1984a). Is mathematical knowledge just logical knowledge? Philosophical Review, 93(4), 509–552.CrossRefGoogle Scholar
  12. Field, H. (1984b). Critical notice of crispin Wright’s Frege’s conception of numbers as objects. Canadian Journal of Philosophy, 14(4), 637–662.CrossRefGoogle Scholar
  13. Field, H. (1986). The deflationary conception of truth. In G. MacDonald & C. Wright (Eds.), Fact, science and morality (pp. 55–117). Oxford: Blackwell.Google Scholar
  14. Field, H. (1988). Realism, mathematics, and modality. Philosophical Topics, 16(1), 57–107.CrossRefGoogle Scholar
  15. Field, H. (1989). Realism, mathematics, and modality. Oxford: Basil Blackwell.Google Scholar
  16. Goldman, A. (1979). What is justified belief? (Reprinted from Epistemology: An anthology, pp. 340–353, by E. Sosa, Ed., 2008, Oxford: Blackwell).Google Scholar
  17. Hart, W. D. (1991). Benacerraf's Dilemma. Crítica: Revista Hispanoamericana de Filosophía, 23(6), 87–103.Google Scholar
  18. Hodes, H. T. (1984). Logicism and the ontological commitments of arithmetic. The Journal of Philosophy, 81(3), 123–149.CrossRefGoogle Scholar
  19. Hofweber, T. (2005). Number determiners, numbers, and arithmetic. Philosophical Review, 114(2), 179–225.CrossRefGoogle Scholar
  20. Kasa, I. (2010). On field’s epistemological argument against platonism. Studia Logica, 96(2), 141–147.CrossRefGoogle Scholar
  21. Lewis, D. (1986). On the plurality of worlds. Malden: Blackwell.Google Scholar
  22. Liggins, D. (2006). Is there a good epistemological argument against platonism? Analysis, 66(2), 135–141.CrossRefGoogle Scholar
  23. Liggins, D. (2010). Epistemological objections to platonism. Philosophy Compass, 5(1), 67–77.CrossRefGoogle Scholar
  24. Liggins, D. (2017). The reality of field’s epistemological challenge to platonism. Erkenntnis. Scholar
  25. Linnebo, Ø. (2006). Epistemological challenges to mathematical platonism. Philosophical Studies, 129(3), 545–574.CrossRefGoogle Scholar
  26. Linnebo, Ø. (2012). Reference by abstraction. Proceedings of the Aristotelian Society, 112(1), 45–71.CrossRefGoogle Scholar
  27. Maddy, P. (1984). Mathematical epistemology: What is the question? The Monist, 67(1), 46–55.CrossRefGoogle Scholar
  28. Maddy, P. (1988a). Believing the Axioms I. The Journal of Symbolic Logic, 53(2), 481–511.CrossRefGoogle Scholar
  29. Maddy, P. (1988b). Believing the axioms II. The Journal of Symbolic Logic, 53(3), 736–764.CrossRefGoogle Scholar
  30. Nutting, E. S. (2016). To bridge Gödel’s gap. Philosophical Studies, 173(8), 2133–2150.CrossRefGoogle Scholar
  31. Nutting, E. S. (2017). Ontological realism and sentential form. Synthese. Scholar
  32. Potter, M. (2007). What is the problem of mathematical knowledge? In M. Leng, A. Paseau, & M. Potter (Eds.), Mathematical Knowledge (pp. 16–32). Oxford: Oxford University Press.Google Scholar
  33. Wright, C. (1983). Frege’s conception of numbers as objects. Aberdeen: Aberdeen University Press.Google Scholar

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Authors and Affiliations

  1. 1.University of KansasLawrenceUSA

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