Abstract
In his book Philosophy of Logic, Putnam (1971) presents a short argument which reads like—and indeed, can be reconstructed as—a formal proof that a nominalistic physics is impossible. The aim of this paper is to examine Putnam’s proof and show that it is not compelling. The precise way in which the proof fails yields insight into the relation that a nominalistic physics should bear to standard physics and into Putnam’s indispensability argument.
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Notes
The assumption that r is rational has no real bearing on the proof, other than the fact that it allows for infinitely many distinct statements of the required form. Putnam does not allow r to be an arbitrary real number because he worries that we cannot have names for uncountably many objects.
Putnam expresses a similar idea about nominalism later, writing that “[n]ominalists must at heart be materialists, or so it seems to me: otherwise their scruples are unintelligible” (Putnam (1971), p. 36).
Note that we are following standard model theoretic practice and thinking of the languages \(L_n\) and \(L_p\) as just containing the basic predicates of the language. Infinitely many sentences can be formulated in \(L_n\), although, as we will shortly see, many of them are logically equivalent to one another.
If the signatures contain function or constant symbols, there are two more conditions that F must satisfy. For further details on reconstruals and translations see Barrett and Halvorson (2016a).
Putnam (1983) is his most detailed discussion of equivalence, but the topic comes up in a number of other papers as well.
He did not call this theory nominalistic because it included appeal to modalities, which he believed the nominalist must reject. Furthermore, he does not discuss whether “mixed predicates” like “the mass of object A is r” might be dispensable, which is his main concern in Putnam (1971). See (Burgess and Rosen (1997), III.B.2.d) for a brief discussion of how to square Putnam (1967) with Putnam (1971), and Burgess (2018) for a critical discussion of Putnam (1967).
See Weatherall (2019) for a survey of recent work.
The situation in (Putnam 1983) is slightly more complicated. Putnam (1983, p. 40) suggests that “one expects some type of translation” to exist between two theories if they are to be considered equivalent. This is the case, as we will shortly discuss, with definitional equivalence, but Putnam here explicitly mentions a standard of equivalence called mutual relative interpretability, which is weaker than definitional equivalence (Barrett and Halvorson 2019). He claims that mutual relative interpretability plus the “informal requirement” that the interpretations preserve explanations will suffice for full equivalence of theories. The problem with this idea is that mutual relative interpretability is known to be a poor formal standard of equivalence, in that it considers too many theories to be equivalent; for an example see Barrett and Halvorson (2019). One therefore suspects that he would be happy instead endorsing definitional equivalence, which is a much more reasonable standard, as the formal requirement. In either case, the same argument goes through to establish C2 from C1 and P4.
Note that a nominalist might instead respond to Putnam’s proof by denying that the kind of translation that they are aiming for is the kind defined above. If they are aiming for a sufficiently weak kind of translation from \(T_p\) to \(T_n\), then Putnam’s proof may no longer go through. But the burden is then on them to make precise the kind of weak translation that they desire. The nominalist who does aim for \(T_n\) to be equivalent to \(T_p\) must also find a way to argue that \(T_n\) does not commit to abstract objects, despite the fact that it is equivalent to a theory \(T_p\) that does appear to make that commitment.
This has been a contentious feature of Field’s theory. The issue is whether or not Field should count as a ‘genuine’ nominalist given his willingness to quantify over spacetime points. It has been argued that these objects are not sufficiently concrete for the nominalist to admit them into their physics. But this debate is beyond the scope of this paper. P2 would also be violated if every model of \(T_n\) were finite, but some had a different finite number of elements than others. One suspects that Putnam’s proof would still go through with a weakened version of P2 that stated only that \(T_n\) has only finite models. Even with a weakened P2, however, it is not clear that nominalists would be obligated to accept it, and Field’s theory certainly would not satisfy it.
See Barrett and Halvorson (2016b) for a precise definition.
See Weatherall (2019) for a review of recent work on equivalence.
The concept of a “natural transformation” is often used to define when two categories are equivalent. C and D are equivalent if there are functors \(F:C\rightarrow D\) and \(G:D\rightarrow C\) such that FG is naturally isomorphic to the identity functor \(1_D\) and GF is naturally isomorphic to \(1_C\). See Mac Lane (1971) for the definition of a natural transformation and for proof that these two characterizations of equivalence are the same.
Instead of appealing to a single preferred coordinate system as AG does, one might formulate analytic geometry by laying down a family of ‘generalized coordinates’ (Burgess and Rosen 1997, II.A.3). This kind of formulation will affirm a wider class of symmetries than AG does, making it so that the argument in proof of Proposition 5 does not go through. Indeed, one suspects that this formulation of analytic geometry will be equivalent to \(E_2\). It is natural to think, however, that moving to \(E_2\) in this case does not actually dispense with anything. Rather, it simply elucidates what the actual commitments of this formulation of analytic geometry were in the first place. By affirming this wider class of symmetries there is a sense in which we ‘take back’ or ‘weasel away’, to use a phrase of Melia (2000), some of AG’s commitments. For example, if we were to put forward the Newtonian theory of space, with its standard of absolute rest, while at the same time affirming that ‘Galilean boost’ symmetries preserved all of the spacetime structures structures that we took to be significant, then there would be a strong sense in which we were not actually committing to there being an absolute standard of rest. A full discussion of this idea is unfortunately beyond the scope of this paper. See Weatherall (2016) for arguments about classical electromagnetism and Newtonian gravitation that are closely related.
It is worth considering whether or not the requirement that T and \(T^-\) are empirically equivalent is robust enough to capture what is going on in some cases of dispensability. In many cases it actually seems that the two theories bear a much closer relationship to one another than mere empirical equivalence. Indeed, since neither of the theories AG or \(E_2\) have any empirical content, this suggests that perhaps the standard definition of dispensability merits revision. As we will see below, if we require that T and \(T^-\) bear too close a relationship to one another, the notion of dispensability we end up with is implausible. So the question is: exactly what relationship should we require \(T^-\) to bear to T? We will make a brief remark about this in the conclusion. Note also that some theory \(T^-\) might not postulate X but make new or better empirical predictions than the original theory T. While there is no doubt a sense in which \(T^-\) dispenses with X, this is not the kind of dispensability that Colyvan is after. He is trying to pin down the kind of dispensability at play in Field-style attempts to dispense with abstract objects, where one is trying to capture the same empirical content as the original theory without appeal to X.
This basic idea shares much in common with Alston (1958), whose target was the Quinean practice of ‘paraphrasing away’ ontological commitments. He pointed out that if the paraphrases preserve the content of the original statements, then they must have precisely the same commitments.
If one has constants denoting two different points, then one can use the techniques of Tarski (1959) to define a real-closed field using \(E_2\). (A similar result is true of the system of Field (2016). His discussion in chapter 4.1 is closely related to the following point.) One might worry that this means that \(E_2\) is still committed to numbers. This thought is, however, a bit misleading. The definition of such a real-closed field relies on those two constant symbols, which label the ‘0’ and ‘1’ points. It is more appropriate to say, therefore, that the extension of \(E_2\) to a signature containing these two additional constant symbols (with an axiom saying they are non-equal)—and not there mere theory \(E_2\)—can define a real-closed field. And it is natural to think that this extension of \(E_2\) has more structure than \(E_2\) itself did, since \(E_2\) cannot itself define those two constant symbols.
This result is closely related to a well known fact about ‘intertranslatability’ and definitional equivalence. See Barrett and Halvorson (2016a).
As mentioned at the beginning of Sect. 3, one can imagine a nominalist—for example, the hermeneutic nominalist of Burgess and Rosen (1997)—who wants to show that \(T_p\) and \(T_n\) are equivalent in order to show that \(T_p\) was not actually committing to abstract objects in the first place. Such a nominalist is not proposing \(T_n\) with the aim of dispensing with something from \(T_p\), but rather in order to clarify the content of \(T_p\). See, for example, the earlier discussion in footnote 19. This kind of nominalist would be fine with there being a translation from \(T_p\) to \(T_n\) and would have to deny P2 or P3.
It is worth mentioning that one expects a similar argument to go through even if one moves to a more general notion of translation. Given some of the claims he made in later years about the equivalence of geometry with points and geometry with lines, one can imagine Putnam endorsing a more liberal standard of equivalence (like Morita equivalence), which corresponds to a ‘looser’ notion of translation than the one we have discussed here (Barrett and Halvorson 2017). The important point, however, is that if a good translation exists from \(T_p\) to \(T_n\)—regardless of how one makes it formally precise—that will allow one to express in \(T_n\) all of the statements about numbers that \(T_p\) is capable of formulating.
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Acknowledgements
Thanks to Jeff Barrett, Alex LeBrun, Hannes Leitgeb, and two anonymous referees for their extremely thoughtful and helpful comments.
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Barrett, T.W. On Putnam’s Proof of the Impossibility of a Nominalistic Physics. Erkenn 88, 67–94 (2023). https://doi.org/10.1007/s10670-020-00340-2
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DOI: https://doi.org/10.1007/s10670-020-00340-2