Putnam, Gödel, and Mathematical Realism Revisited

ABSTRACT

I revisit my 1993 paper on Putnam and mathematical realism focusing on the indispensability argument and how it has fared over the years. This argument starts from the claim that mathematics is an indispensable part of science and draws the conclusion, from holistic considerations about confirmation, that the ontology of science includes abstract objects as well as the physical entities science deals with.

KEYWORDS:

• platonism
• Putnam
• Gödel
• Quine
• indispensability

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1. For an overall view and references to the extensive literature, see Colyvan in the Stanford Encyclopedia Colyvan 2023; see also his monograph Colyvan (2001).

2. Quine (1948, 1951, 1976b, 1981b, 1981c); for Putnam’s much less positivistic version see especially Putnam 1975b.

3. For more detail see Weir, (2013a) §7.

4. It should be noted, however, that such holism was already to be found in Carnap and especially Neurath.

5. For discussion see Weir, (2006a) §3 and 2013a §§6–7.

6. It is sometimes said that Quine derives, via his verificationism, a form of semantic holism; I am calling this holistic verificationism but note that, on my account, it does not apply to observation sentences, not in a fully coherent version of Quine’s position anyway.

7. The argument is to be found in (Quine 1969a, 80–81), but the argument is already pre-figured in section V of ‘Two Dogmas’, Quine (1951). Føllesdal (1973, 290–1) seems to give the same reading. (Quine 1986, 155–6) endorses Føllesdal’s account.

8. Quine, trying to hold on to vestiges of realism, at one point proposes adopting a ‘sectarian’ line on such issues, see (Weir 2006a, 246).

9. The earliest occurrence appears to be in ‘Designation and Existence, Quine (1939), 708, a paper containing the ‘bulk’, Quine says, of a paper read at Fifth International Congress for the Unity of Science, Cambridge, Mass., September 9, 1939, under the title ‘A Logistical Approach to the Ontological Problem.’ The conference proceedings never appeared, owing to the German invasion of the Netherlands. Quine published it Quine (1939/76) in Ways of Paradox, in 1976a although there is very little literal overlap with ‘Designation and Existence’.

10. Which could be the universal set, supposing there is such a thing, as Quine did in his NF set theory. Lacking the power set axiom, however, NF cannot form a basis, even if it is consistent, for standard semantics. Some philosophers reject ‘domain semantics’ route and branch, for some discussion of this and the related notion of ‘absolute generality’, see Weir (2006b).

11. Here I am in agreement with (Quine 1969b, 100), that there is (there exists) no important metaphysical difference between Being and Existence.

12. A sceptic about propositions, actual Quine as well as Viennese Quine for example, would need to re-fashion this account in terms of interpreted sentences.

13. For a critique, see Weir (1991).

14. Maddy (2005), 445–457, and for comments Weir (2005) §III.

15. Although theory has practical benefits. We amend and change theories in the light of experimental evidence and new ideas, often guided by a model suggested by the theory; this we cannot do with the program. Still, this is surely too pragmatic a justification for theory to satisfy a scientific realist.

16. As a first attempt at explanatory theory; more complex and decontextualised contents will be more explanatory. The idea is analogous to that in two-dimensional semantics but not the same. There is no appeal, even as a façon de parler, to worlds as points of evaluation for example.

17. Or of acceptance or whatever cognitive attitude play the role of determining what theories a person holds.

18. Perhaps we can give more content to ‘direct detection’ by using Hacking’s idea, in slogan form, that ‘if you can spray it, it’s real’, i.e. causal interaction such as the spraying of a niobium ball using an electron gun, is required before scientists come to believe in the existence of newly proposed entities, such as (polarized) electrons and niobium atoms (Hacking 1983, 274.).

19. Most believe. But not, it seems, Gödel. I have a brief discussion of his apparent belief in something like perception of numbers in 1993, p. 256.

20. Some unpacking of the informational content here might be reasonable given how people understand concepts such as ‘even’ and ‘prime’ but it is still clear that it is not part of the Fregean sense that there exists a concrete proof token of the sentence, the Fregean cognitive test: one can believe(disbelieve) the sentence but not one expressing its metaphysical content, or vice versa, shows that.

21. Only some, various conditions have to apply. See Milne (2007).

22. Roughly, there is, for any standard formal system S, no fixed polynomial function f such that for every theorem, its shortest proof is no more than f(x) long where x is the length of the theorem.

23. There is no inconsistency in the ‘neo-formalist’ as I baptised my version of formalism, talking freely about numbers since the neo-formalist believes standard mathematics is true; the anti-platonism consists in giving an anti-realist, ontologically reductionist, at the metatheoretic level, account of mathematical truth.

24. Here ‘⇑’ represents tetration or ‘superexponenation’, so that’s a very large number: n⇑0 = 1, n⇑k+1 = $n^{n^{\Uparrow }\mathit{k}}$ so it’s an exponential stack of 10,000 tens.

25. Actually I require a little more, that certain ‘primality’ properties hold of the system. See Weir (2010), 10 and passim.

26. To be found in papers on neo-logicism such as Weir (2003) and (Shapiro and Weir 1999, 2000).

27. With an appeal in effect to free logic to defuse the paradoxical reasoning and give the appearance of fully naïve comprehension.

28. See (Weir 1998a) for my argument that naïve comprehension ‘trumps’ classical structural rules.

29. For a general account see Restall (2000); for some among many papers in the expanding area of non-transitive and non-contractive logic see Ripley (2012), Zardini (2013, 2019), Petrukhin and Shangin (2023).