Problems with Prolifigate Platonism

  • Colin Cheyne
Part of the The Western Ontario Series in Philosophy of Science book series (WONS, volume 67)


Variations on a new platonist epistemology have been offered recently. Mark Balaguer (1995, 1998a & 1998b) offers one version, and Bernard Linsky and Edward Zalta (1995) another. Although there are important differences between their proposals, what they have in common is the suggestion that if there is a plenitude of mathematical entities, then there is no problem about acquiring knowledge of them. I first discuss Balaguer’s proposal and then discuss the extent to which Linsky and Zalta’s account faces similar problems.1


Natural Number Abstract Object True Belief Mathematical Object Singular Term 
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  1. 1.
    My discussion of Balaguer’s proposal is based on Cheyne (1999).Google Scholar
  2. 2.
    This sketch should be seen as a rational reconstruction, gleaned from his (1995), (1998a) & (1998b). He may not agree with all details.Google Scholar
  3. 3.
    There is an analogy here with modal-realist talk. We may speak loosely of the possible world in which the Spice Girls sing Mozart opera, although there are infinitely many such worlds.Google Scholar
  4. 4.
    For detailed accounts of supervaluational semantics, see van Fraassen (1969) and Fine (1975).Google Scholar
  5. 5.
    Narrow causal theorists claim that if unicorn-like creatures were discovered, they would not be the referents of our word ‘unicorn’.Google Scholar
  6. 6.
    I assume that ‘slithy’ and ‘mimsy’ are nonsense words and disregard Humpty-Dumpty’s gloss on them.Google Scholar
  7. 7.
    I am not saying that the objects in question are bare particulars. (How, in consistency, could I?) Rather, I claim that for all we can say or know, they might as well be. See Balaguer (1998b, p. 87).Google Scholar
  8. 8.
    Burgess and Rosen (1997, Part I.A.2) survey this debate in detail and conclude that ‘in so many ways [debate over reference] is just a replay of debate over knowledge’ (p. 49), and that there is a ‘tendency [for] nominalist arguments and anti-nominalist counter-arguments to reach stalemate over burden of proof’ (p. 60).Google Scholar
  9. 9.
    From Hofstadter (1979), p. 216.Google Scholar
  10. 10.
    This ensures that we have a mathematical theory on our hands and that the theory can be combined with other mathematical theories.Google Scholar
  11. 11.
    I observe that if we can construe semantic consistency non-platonistically as the de dicto possibility of having a model, then it is an (apparently) small step from an AP account of mathematical truth to a modal account, whereby a mathematical theory is true just in case it is possible that it has a model.Google Scholar
  12. 12.
    The adventures of Alexander Selkirk only roughly match those of Robinson Crusoe whilst I am entertaining an exact match between novel and reality. Paul Griffiths suggests that a more realistic example would be Defoe’s Memoirs of a Cavalier, which, at the time of its publication, was widely believed to be a true and accurate record of actual events. I choose (and stick with) Robinson Crusoe because the remoteness of its island is somewhat analogous with the remoteness of the platonic realm (and the novel is better known).Google Scholar
  13. 13.
    Note that any object exemplifies the property of being such that p iff p. Hartley Slater complains that the suggestion that there are such properties is a grammatical nonsense. He demands to know what kind of an English sentence is ‘Mary is such that George loves Barbara’ (Slater 1997, p. 169). I shall let this pass. Linsky & Zalta do at least give truth conditions for the exemplification of such properties.Google Scholar
  14. 14.
    According to PP, there is an abstract object that encodes just the axioms of T, but the closure condition ensures that that object is not a theory.Google Scholar
  15. 15.
    Menzel (1993), Deutsch (1993), and Anderson (1993) make a number of criticisms to which Zalta (1993) replies. See also Slater (1997).Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • Colin Cheyne
    • 1
  1. 1.University of OtagoDunedinNew Zealand

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