Indispensability and Platonic Knowledge

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


Willard Quine and Hilary Putnam argue that the methods by which we confirm scientific theories are the means by which we acquire knowledge of platonic objects. In outline, they argue as follows.1 Our best theories about the world postulate entities that we cannot observe (for example, electrons) in order to make sense of our experiences. But those same theories postulate platonic objects (for example, numbers and sets). Mathematical objects are just as indispensable to science as theoretical entities like electrons. Electron theory quantifies over numbers and other platonic entities, just as it quantifies over electrons. So we have the same reason for thinking that numbers exist as we do electrons. Our knowledge of platonic objects is obtained in the same way as our knowledge of physical objects, from sense experience. The process by which we obtain platonic knowledge is the same as the process by which we obtain non-platonic scientific knowledge. Indeed, platonic knowledge is an indispensable part of scientific knowledge.


Scientific Theory Mathematical Object Causal Power Ontological Commitment Mathematical Entity 
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  1. 1.
    Quine (1961, ch. 1 & 1966, ch. 20) and elsewhere; Putnam (1979).Google Scholar
  2. 2.
    Sometimes, in physics, there have been competing theories that differ in their mathematical commitments. But the mathematics to which they are respectively committed can be reduced to set theories, which have in common a hierarchy of sets at least up to cardinality N0.Google Scholar
  3. 3.
    For example, Shapiro (1983 & 1993, pp. 458–65), Liston (1993), Chihara (1990, ch. 8), and many others. But see also Mundy (1992) and Balaguer (1998b, ch. 6) for some recent attempts to nominalise quantum mechanics.Google Scholar
  4. 4.
    See also Burgess (1990) p. 7.Google Scholar
  5. 5.
    Davis & Hersh attribute the percentages to J.D. Monk but I have been unable to track down the source. For an example of confused attempts by a highly reputable mathematician to sort out the ontological difficulties, see Mac Lane (1986).Google Scholar
  6. 6.
    The following argument draws on Maddy (1992, esp. pp. 280–82).Google Scholar
  7. 7.
    First philosophy is the a priori, armchair approach to philosophy, which sees philosophy as prior to empirically based knowledge.Google Scholar
  8. 8.
    My arguments in this and the following section draw on Cheyne & Pigden (1996).Google Scholar
  9. 9.
    For example, see Shapiro (1983) and Field’s reply (1989, ch. 4).Google Scholar
  10. 10.
    For recent attempts, see Mundy (1992) and Balaguer (1998b, ch. 6).Google Scholar
  11. 11.
    It isn’t, of course. Frege has shown how we can say that there are three butts in the ashtray without reference to the number three. But if we want an example in which indispensability is more likely, we shall need to delve into the realms of General Relativity or quantum mechanics. If platonists believe that they can strengthen their case with such an example, I look forward to seeing it.Google Scholar
  12. 12.
    See for example Hale (1987 & 1990), Hale & Wright (1992), Resnik (1990) and Shapiro (1983).Google Scholar
  13. 13.
    Cf. Stove (1991, p. 53) on the synthetic a priori. 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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