Abstract
Neo-Fregeanism (NF) is a family of positions in the philosophy of mathematics that combines a certain type of platonism about mathematical abstracta with a certain type of logicism about the foundations and epistemology of mathematics. This paper addresses the following question: what sort of theory of reference can/should NF be committed to? The theory of reference I propose for NF comes in two parts. First, an alethic account of referential success: the fact that a term ‘a’ succeeds in referring to something depends on facts about truth. Second, a deflationary account of referential specification: given that ‘a’ refers to something, the fact that ‘a’ refers to b specifically follows from the disquotational schema for reference (‘a’ refers to x iff x = a) together with the fact that b = a. In the first section of the paper I argue that NF should be committed to the first part of this theory. This a point on which there is already (some) agreement. The bulk of the paper is therefore devoted to arguing that NF should be committed to the second part, given that it is committed to the first part. I close the paper by indicating some significant implications and a possible problem for this theory of reference.
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Notes
Øystein Linnebo's (2018) view could naturally count as neo-Fregean as well. But it is different enough from the views mentioned above that I will save discussion of it for another time.
Also known as “neo-logicism” or “abstractionism.”
You might think that, as a position in the philosophy of mathematics, the neo-Fregean does not need to tell us anything about reference at all. If there were a variety of theories of reference compatible with NF, then I agree that they wouldn’t have to commit themselves to any one in particular. But ultimately I am going to argue that there is only one such theory compatible with their position, and in that sense they are committed to such a theory of reference. Hence I will tend to focus on what theory they should (rather than can) be committed to.
The “appropriate type” of sentence is an extensional, atomic sentence. This won’t matter much in what follows, and so I will largely leave it tacit.
This paper is going to be full of claims about constitutive priority—about what constitutes what. I recognize that some are skeptical of any such relation, preferring to stick to claims about identity and modal covariance. I can’t come to a general defense of constitution here. But let me make three brief points. First, questions about constitution and relative fundamentality are, as a matter of fact, at the center of a lot of contemporary philosophy of language. See Burgess and Sherman (2014). Second, while I am perfectly happy to talk about what constitutes what, the point of the present paper is not so much that various constitutive claims are true, but rather that the neo-Fregean needs them to be true. General skepticism about constitutive priority should translate, therefore, not to skepticism about my thesis in this paper, but (if my thesis is correct) to skepticism about NF. Third, my arguments in this paper would work equally well with constitution replaced with any other relation of metaphysical priority (e.g., grounding).
The language of “referential success” and “referential specification” I take from Burgess (2012).
Two points. First, in §3.1 I will defend the idea that a complete theory of reference requires a theory of referential specification. Second, those familiar with the Julius Caesar objection to NF may see a connection between these issues and that objection. I will discuss the Caesar objection in §5.
The two parenthetical qualifications are obviously important to the sense in which NF classifies as a version of logicism, but this isn’t material in the present context.
The Canonical Argument is one version of a more general line of reasoning that runs throughout Hale and Wright’s work. The more general argument begins by observing that certain mathematical claims are true by “ordinary criteria” Wright (1983: p. 11), and then inferring from this that the constituent singular terms must refer. Hale and Wright (2001: p. 10) are clear that one way to establish the truth of ‘#Fs = #Gs’ in particular is to infer it from (HP) and the truth of ‘F \(\sim\) G’, as in the Canonical Argument.
There are two aspects of the Canonical Argument that I leave tacit here but which are otherwise very important to NF. The first is that ‘#Fs’ is a genuine singular term. The second is that the concept of an object is explained in terms of that of a singular term: what it is to be an object is to be the sort of thing that could be the referent of a singular term. A third aspect of the argument, relating truth and reference, I will come to presently.
Eklund's (2006, 2016) argument centers around (tr). Very briefly, the idea is this. (tr) is not a controversial principle; most theorists can be expected to accept it. But anyone who does accept (tr), but who is also antecedently skeptical of the existence of numbers, can then take the Canonical Argument and run it in reverse: numbers don’t exist; so ‘#Fs’ doesn’t refer; so—given (tr)—‘#Fs = #Gs’ is false (or at least not true). Which direction of reasoning you find persuasive, Eklund argues, will depend on how you view the constitutive relationship between truth and reference. Specifically, only if (TR) is true, so that truth is constitutive of reference, will it be appropriate to argue in the way that the neo-Fregean does. There’s much more to say about this argument, but in the interest of space I focus on just my own.
This also requires the disquotational properties of truth. Throughout I will assume these properties for convenience, so that we can move fluidly between formal and material modes. We could avoid all of this by speaking strictly in the formal mode, but that would be cumbersome.
From hereon I will assume that to stipulate is to successfully stipulate.
The principle does not claim that the “making it the case” relation is closed under necessitation, which would be implausible. That is one reason why it includes the condition that neither p nor q is true prior to making it the case that p. Moreover, my making it the case that q by doing \(\varphi\) might “go through” p in the sense that doing \(\varphi\) directly makes it the case that p which in turn grounds or constitutes q. This I count as a way of making it the case that q.
There may be other difficulties in stipulating the truth of (HP) and other abstraction principles (for instance bad company problems). My only point is that, if (TR) is true, the fact that truth requires reference does not all by itself pose any specific such difficulty.
Cf. Fine (2002: p. II.4).
The analogy is imperfect in various places, but I think it is illustrative nevertheless.
There’s a complication in this deflationary account. On one hand, facts about referential specification are just a matter of the meaning of the word ‘refers’, as defined by (R*). On the other hand, the definition of ‘refers’ given by (R*) is conditioned on independent facts about referential success, which are not (necessarily) just a matter of the disquotational meaning of ‘refers’—for instance, they may be constituted by facts about truth, as in (TR). Is this a problem for the general theory of reference that I am proposing for the neo-Fregean? It might well be. But it’s not something I’m going to sort out here on their behalf. Compare the issue here with the worry raised at the very end of the paper (§5): that there is a prima facie tension in combing a deflationary view of referential specification with any substantial account of referential success (such as (TR)). If at the end of the day it turns out that the theory of reference that, according to my arguments in this paper, underlies the neo-Fregean position is itself untenable, then I suppose so much the worse for neo-Fregeanism. But we are, at present, a long way off from that conclusion.
Technically it’s not this specific fact about truth that constitutes the referential success of ‘Barry’, but the fact that there is some such truth. For simplicity, I’ll ignore this here.
Variations on the idea that arbitrary referential indeterminacy should not be tolerated by a theory of reference have played center stage in a certain segment of philosophy of language over the past 40 years (Putnam 1981; Kripke 1982; Lewis 1984; Soames 1999; Sider 2014; Horwich 2010). I see (D) as keeping in that tradition.
Or least it doesn’t suffer these problems in the in the specific ways that the inflationary accounts do. At the very end of the paper, I will suggest, rather inconclusively, a very different way in which the deflationary account might be in tension with (TR) after all.
The suggestion is not that the neo-Fregean, as a position in the philosophy of mathematics, doesn’t have to tell us anything about referential specification. On that suggestion, see note 3. The suggestion is that, in general, a complete theory of reference need not say anything about referential specification—that the latter is not a phenomenon at all or at least not one that requires theorizing about.
I actually think that the deflationary account doesn’t so much avoid (D) as it does easily satisfy (D), as we’ll see below (§3.8). But I’ll put this aside.
For something in the ballpark of this sort of primitivism, however, see Boghossian (1990).
A Gödelian faculty of intuition is not irreducible in the sense of being non-natural, since it is usually taken to be a causal relation. But it is irreducible in the sense of being posited to explain something (knowledge) and yet being inexplicable itself: in order to account for how it is that we can have knowledge of mathematical objects, we posit a quasi-perceptual faculty, distinct from and irreducible to any other faculty of ours, say that via this faculty we can “intuit” mathematical objects, and then don’t say anything about what that faculty is or how it works. The parallels between this idea and an irreducible relation of reference are not perfect, but the spirit of the positions, and what many find objectionable about them, is similar.
We could also take \(\varSigma\) to be the set of sentences believed to be true by some agent. This complicates matters (since agents might have inconsistent beliefs) but not in any way that affects the arguments of this section.
The alethic account focused on in this section defined adequacy as the preservation of the truth-values of the members of \(\varSigma\). What about an account that took adequacy to be the preservation of their truth-conditions (i.e., functions from worlds to truth-values)? This wouldn’t help. Williams (2005: ch. 5.1.2) shows how the same sort of permuations problems arise for this type of account as well.
See Lewis (1984).
This requires that the space of objects and properties under the relation of relative naturalness be a metric space. That’s a substantive assumption. But it’s a fair assumption in the present context, since any relaxation of the structure of this space will put fewer constraints on the relation of reference, and will thereby make the arguments of this section all the easier.
Williams (2007) expresses worries that are similar in spirit to those I articulate here. But there are important differences between our lines of thinking.
The original interpretations from the first example are also both eligible in this second example. But the reduced interpretations demonstrate a different point: that there can be a tradeoff in where the naturalness is located (object or property) between interpretations.
The simplification is made to avoid quantification over sets. This does not affect my argument.
I don’t mean this in any technical sense. But note that, in the case where \(\kappa\) is infinite, \(\kappa ! = 2^{\kappa }\).
To get a total order in the example we need to further suppose (i) that a, b, F and G are totally ordered according to \(N\), and (ii) that all other objects and properties are themselves totally ordered. But none of this effects the analysis of the example, so long as in (i) we still maintain that the difference in naturalness between the objects is the same as the difference in naturalness between the properties.
My worries about (ERS) satisfying (D) run counter to what Lewis (1984) says about the matter. Lewis seems to think that something like an eligibility account is sufficient to rule out arbitrary indeterminacy of the sort that Putnam worries about in his model-theoretic argument. He says:
Ceteris paribus, an eligible interpretation is one that maximizes the eligibility of referents overall. Yet it may assign some fairly poor referents if there is good reason to…overall eligibility of referents is a matter of degree, making total theory come true is a matter of degree, the two desiderata trade off. The correct, ‘intended’ interpretations are the ones that strike the best balance. The terms of trade are vague; that will make for moderate indeterminacy of reference; but sensible realist won’t demand perfect determinacy Lewis (1984: pp. 228–229)
I agree that we shouldn’t demand (anything close to) perfect determinacy. But we should demand that any indeterminacy not be arbitrary. And I don’t see general reason to think that the residual indeterminacy that Lewis acknowledges won’t be of this sort.
Don’t read the quantifier on the right-hand-side as a second-order-quantifier; ‘P’ is a first-order variable ranging over linguistic expressions (specifically predicates) of the language \({\mathcal{L}}\).
Proof: Notice that the consequent of (1) entails the biconditional:
$$ \kern2pc \exists x\left[ {t\,{\text{refers}}\,{\text{to}}\,x} \right] \leftrightarrow \exists xC\left( {t, x} \right). $$And so from (1) we have:
$$ \kern2pc\exists x\left[ {t\,{\text{refers}}\,{\text{to}}\,x} \right] \to \left[ {\exists x\left[ {t\,{\text{refers}}\,{\text{to}}\,x} \right] \leftrightarrow \exists xC\left( {t, x} \right)]} \right], $$from which it follows, by propositional reasoning, that:
$$ \kern2pc \exists x\left[ {t\,{\text{refers}}\,{\text{to}}\,x} \right] \to \exists xC\left( {t, x} \right). $$Finally, by (2) we can substitute the antecedent of the preceding claim with its material equivalent to yield:
$$ \kern2pc\exists P[Pt\,{\text{is}}\,{\text{true]}} \to \exists xC\left( {t, x} \right). $$Might a singular term represent an object by denoting it rather than referring to it? Perhaps an attributive (use of a) definite description represents in this way. In response: Suppose D does not refer to a, but nevertheless represents a by denoting a. The latter is either a semantic feature of D or a pragmatic feature of (a use of) D. If pragmatic, then this feature is irrelevant in the present context, in which the operative sense of representation is (we are assuming) partially constitutive of sentential truth (where this is understood as a semantic rather than pragmatic feature of a sentence). But if denotation is semantic, then the hypothesized scenario looks incompatible with (tr), and so not something the neo-Fregean could accept. For if D does not refer to a, then, given (tr), either (i) D, despite representing a, cannot feature in any true (atomic, extensional) sentence, or (ii) D, despite representing a, refers to some object b (≠ a), or (iii) D is not a genuine singular term. Option (iii) is irrelevant. And neither (i) nor (ii) is tenable.
If I’m correct in thinking that NF must adopt a deflationary account of referential specification for singular terms, then the upshot of this subsection is that they must adopt a similarly deflationary account for predicates. There is an affinity between the latter type of view and a deflationary conception of properties, for instance as defended by Hale (2013, 2019). However, officially the views are distinct, one being a view about how predicates latch on to specific properties, the other about the nature of properties themselves.
The quantifier is not a second-order quantifier, however. ‘F’ is a variable that could replace an expression like ‘the property of a horse’. The latter is a definite description, and thus a singular (first-order) term, despite the fact that its referent is a property. The paradox of the concept of a horse is lurking here, but that issue crosscuts those we are presently concerned with.
The reasoning is similar to that contained in 39.
The exact relationship between the two sides of (HP)—e.g., the sense in which each “carves” the same content differently—is a source of much debate among proponents and critics of NF alike. See Hale and Wright (2001: Ch. 4) for discussion.
This is the sort of view endorsed, for example, by Sider (2014: Ch. 3).
Instead of (HP), Rayo (2013: p. 73) takes the following as a definition of number: For the number of the Fs to be n just is for there to be exactly n Fs.
Rayo focuses on directions* rather than directions so as to not presuppose anything about our actual direction terminology.
Personal correspondence.
In particular it’s worth noting that Wright himself would probably welcome this result, since he often seems to be thinking about meaning, not in terms of representation, but in terms of something like norms of assertion. See, for instance, Wright (1992).
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Acknowledgements
Some of the central ideas in this paper grew out of the many discussions of neo-Fregeanism that I had the pleasure of participating in over the past several years during meetings of the Foundations Interest Group at the Minnesota Center for Philosophy of Science. I thank the participants at those meetings for lively and stimulating discussion of these topics. An earlier version of this paper was presented at the Fifth Philosophy of Language and Mind Conference at the University of St Andrews in August of 2019. Thanks to the participants at that conference for their helpful questions and feedback. Finally, I’d like to pay special thanks to the following people for providing me with thoughtful comments on earlier drafts of this work: Samuel Asarnow, Roy Cook, Manuel García-Carpintero, Peter Hanks, Carlos Núñez, and two anonymous referees for this journal.
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Taylor, D.E. Reference for neo-Fregeans. Synthese 198, 11505–11536 (2021). https://doi.org/10.1007/s11229-020-02811-z
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DOI: https://doi.org/10.1007/s11229-020-02811-z