Kitcher, mathematics, and naturalism

Abstract

This paper argues that Philip Kitcher's epistemology of mathematics, codified in his Naturalistic Constructivism, is not naturalistic on Kitcher's own conception of naturalism. Kitcher's conception of naturalism is committed to (i) explaining the correctness of belief-regulating norms and (ii) a realist notion of truth. Naturalistic Constructivism is unable to simultaneously meet both of these commitments.

Notes

1 Those who have qualms about the metaphysical status of propositions can take ‘proposition’ and its cognates in this paper as shorthand for contents of beliefs, whatever one thinks such contents are.

2 This conception of justification sometimes goes by ‘warrant’[Plantinga 1993b; Plantinga 1993a; Kitcher 1984].

3 I have given considerable attention to this issue elsewhere [manuscript a].

4 There is evidence that Kitcher widely individuates processes relevant to knowing [2000: especially §III].

5 Assuming, of course, that we aim to have true beliefs.

6 Kitcher defends a correspondence theory of truth in [1993: 128–33].

7 Thanks to an anonymous referee for raising this issue.

8 Here P_t is mathematical practice at time t. The careful reader will note that I blur the distinction between beliefs and (accepted) statements with respect to the members of K. This is unproblematic for present purposes.

9 Here ‘⊂’ means proper subset.

10 This terminology is Kitcher's.

11 Recall that K^P is the set of statements accepted by the practitioners of P.

12 I use ‘regulative norms’ as a synonym for ‘belief-regulating norms’.

13 I treat ‘acceptance’ and ‘belief’ as synonyms.

14 For the sake of argument, I grant that no problems arise for the ontological status of these operations as the process of taking operations on operations on operations, etc. is iterated.

15 Cf. the discussion of truth and the ideal gas theory at the end of this section.

16 The compatibility at issue here requires at least logical or conceptual consistency. I bracket the question of what it may additionally require.

17 Intuitively, this says that adding one object to a finite collection C yields a collection equinumerous with C. The relevant definitions can be found in Kitcher [1984: chap. 6, §III].

18 For ‘means’, see the block quotation at the beginning of this section.

19 One might be tempted to help Naturalistic Constructivism simultaneously honour both (KC₁) and (KC₂) by modifying it to accommodate the claim that mathematics is a priori. However, such a move would conflict with Kitcher's express rejection of mathematical apriorism [1984].

20 This says that pressure times volume is equal to the so-called ‘ideal gas constant’ times temperature.

21 For present purposes it doesn't matter what ‘M’ is.

22 I ignore the other axiom that is not a prenexed universal quantification (axiom (10)), which is effectively the induction schema.

23 Note that since (2UP3) is part of currently accepted arithmetic practice, Kitcher will not object to this.

24 This problem arises from Kitcher's use of the material conditional. So one might try to help Kitcher by instead using a subjunctive conditional, though Kitcher himself self-consciously rejects this [Chihara 1990: 231–2, esp. n. 12]. The challenge for one who advocates this move is to provide an account of truth conditions for subjunctive conditionals that are neither realist nor violate Kitcher's empiricist scruples (e.g., by countenancing possible worlds).

25 Notice that the problems raised for the ideal theory of truth here don't obviously derail Kitcher's epistemology of science more generally. They do present a challenge to his account of knowledge where ideal theories are concerned, but I'm aware of no reason to think that the bulk of theoretical knowledge on Kitcher's view concerns ideal theories.

26 Thanks to Richard Boyd, Jon Cogburn, Eric Gilbertson, Harold Hodes, Brendan Jackson, Richard A. Shore, Zoltán Szabó, Jessica Wilson, and anonymous referees for this journal for valuable discussion and comments.