Abstract
This essay uses a mental files theory of singular thought—a theory saying that singular thought about and reference to a particular object requires possession of a mental store of information taken to be about that object—to explain how we could have such thoughts about abstract mathematical objects. After showing why we should want an explanation of this I argue that none of three main contemporary mental files theories of singular thought—acquaintance theory, semantic instrumentalism, and semantic cognitivism—can give it. I argue for two claims intended to advance our understanding of singular thought about mathematical abstracta. First, that the conditions for possession of a file for an abstract mathematical object are the same as the conditions for possessing a file for an object perceived in the past—namely, that the agent retains information about the object. Thus insofar as we are able to have memory-based files for objects perceived in the past, we ought to be able to have files for abstract mathematical objects too. Second, at least one recently articulated condition on a file’s being a device for singular thought—that it be capable of surviving a certain kind of change in the information it contains—can be satisfied by files for abstract mathematical objects.
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Notes
I am using Kaplan’s (1989) convention of using square brackets to describe physical gestures made when uttering statements or thinking thoughts.
The expression “thinks (1)” here abbreviates “thinks the thought that would be standardly expressed by making statement (1)”. Thus “thinks either (1) or (2)” abbreviates “thinks either the thought that would be standardly expressed by making statement (1) or the thought that would be standardly expressed by making statement (2)”.
I am using “logical form” in Davidson’s sense, where “to give the logical form of a sentence is to give its location in the totality of sentences, to describe it in a way that explicitly determines what sentences it entails and what sentences are entailed by it” (Davidson 2006, p. 64). This is not necessarily the same as the level of mental representation of the syntactic form postulated by certain Chomskyan linguistic theories known as LF.
\({}^\lceil {\upalpha }\) is \(\Phi ^{\rceil }\) abbreviates “the thought expressed by the sentence resulting from inserting the value of \(\upalpha \) into the argument place of the value of \(\Phi \)”.
This is François Recanati’s congruence principle: “in literal communication ... the same state of affairs is represented by the speaker’s thought and by the utterance which (literally) expresses that thought” (1993, pp. 54/5).
Obviously there are other candidates—and plausibly, different agents could use “e” to abbreviate different descriptions. Here I set aside any complications arising from this.
One could appeal here to the fact that (4) conjoined with the axioms of real analysis entails “\(\exists x \sim (x\) can be expressed as a fraction)”. But using this fact to explain ordinary speaker intuition requires crediting all ordinary speakers with belief in the axioms of real analysis, which doesn’t seem very plausible.
Cf. Dehaene (2011), pp. 55–8.
While the mental files view of singular thought is popular, it is not without its detractors. See in particular Dickie (2015) and Goodman (2016a, b). Dickie argues that claims about mental files can be systematically replaced by claims about mental processes. Thus I conjecture that if my arguments go through on a mental files view of singular thought, they ought to go through on Dickie’s ‘process’ view too (though it would take us too far afield to pursue this fully). I address the pertinent details of Goodman’s view in due course.
I stay officially neutral on whether the information comprising a mental file must only be conceptual or propositional, or whether it can include pictorial, auditory, or otherwise non-conceptual information. Recanati (2012) appears to allow only propositional information to comprise mental files. (pp. 37/8.)
E.g. “[T]here are also non-descriptive senses or modes of presentation, and these, I claim, are mental files.” (Ibid., p. 40.) And: “mental files are ‘about objects’: like singular terms in the language, they refer, or are supposed to refer. They are, indeed, the mental counterparts of singular terms” (Ibid., p. 35).
See ibid., p. vii and the references mentioned therein.
Cf. Russell (1911): “We have acquaintance with sense-data, with many universals, and possibly with ourselves, but not with physical objects or other minds” (p. 127).
“Mental files are based on what Lewis calls ‘acquaintance relations’ [begin footnote 5] The paradigm is, of course, perceptual acquaintance, but the notion of acquaintance can be generalized ‘in virtue of the analogy between relations of perceptual acquaintance and other, more tenuous, relations of epistemic rapport. [...] In each case there are causal chains from him [the referent] to me of a sort which permit the flow of information.” (My emphasis; Recanati quotes from Lewis 1999, pp. 380–1.) See also Hansen and Rey’s remark that “it’s a shame that Recanati relegated this expansion of (ER) relations to a footnote” (Hansen and Rey 2016, p. 427).
“Mental files are ‘about objects’: like singular terms in the language, they refer, or are supposed to refer. What they refer to is [...] the individual we are acquainted with (in the appropriate way), not the individual which best ‘fits’ the information” (Recanati 2012, p. 35). Note the intellectual debt to Evans (1973). There is a question of interpretation here: is reference a function of the information itself, or the relation through which the information flowed? The latter allows files containing no information to refer, whilst the former does not. I address this in footnote 37.
Kaplan (1989), p. 536: “There is nothing inaccessible to the mind about the semantics of direct reference, even when the reference is to that which we know only by description. What allows us to take various propositional attitudes toward singular propositions is not the form of our acquaintance with the object but is rather our ability to manipulate the conceptual apparatus of direct reference.” See also Harman (1977), p. 174: “If Mary believes there is a certain unique thing satisfying certain conditions \(C_{1}\), \(C_{2}\), \(C_{3}\), she can introduce a new mental name a into her system by forming the beliefs that a is \(C_{1}\), that a is \(C_{2}\), and that a is \(C_{3}\). This name functions as a name of the unique thing satisfying these conditions if there is one; otherwise it does not name anything. Moreover, the name continues to name this thing, as long as Mary uses it, even if nothing or something different should be[come] the unique thing satisfying those of her beliefs involving the name a.”
“Dthat” is an expression that when prefixed to a definite description yields a singular term referring to the satisfier of that description (if it has exactly one) (Kaplan 1989, pp. 521/2).
“Almost all theorists think that Semantic Instrumentalism is false—indeed, wildly off” (Jeshion 2010, pp. 106–7).
Another way of putting this argument against semantic instrumentalism is Evans’ (1982, p. 50) invocation of Grice’s (1969) dictum that we cannot gain new beliefs ‘at the stroke of a pen’. Note also that there is a nearby argument that semantic instrumentalism entails voluntarism about new knowledge—the view that one can gain new knowledge by deciding to do so (and being competent with the relevant semantic apparatus)—and this is implausible.
“Semantic Instrumentalism supposes that we can will a singular intention. But how? By thinking harder, with more intensity, with feeling? This lacks plausibility” (Jeshion 2010, p. 107).
Readers disliking the eternalism Kaplan’s example apparently presupposes can exchange it for the perhaps more palatable “let “Oldman 1” refer to the first child born in the sixteenth century” (This example is modified from Jeshion 2002, p. 72).
I’m placing ‘cognition’ in scare-quotes to indicate that I am using that term to mean whatever Jeshion uses it to mean. From now on the phrase “significant to S” abbreviates “significant to S’s sub-agential ‘cognition”’.
“An agent making a judgment “this is significant” is not sufficient for engendering the significance needed for singular thought” (2010, p. 136).
Rachel Goodman has claimed that we should read Jeshion as taking the significance condition as not applying to perceptual demonstrative thought (2016b, pp. 246–7). While there may be reasons for this—in particular, that Jeshion discusses perceptual demonstrative thought and, so it seems, can’t plausibly have been unaware of the problem—given that this contradicts what Jeshion herself says, and that she responds to the objection in the passage quoted by asserting the automatic significance of perceived objects, I will continue to read her significance condition as applying to all singular thought.
She also gives the ‘aesthetically motivated collector’ thought experiment in Goodman (2016b), p. 249, which is intended to motivate the same conclusion.
Cf. chapters 6 and 7 in particular. Note that while I will speak of perceptual links rather than informational links more generally, this shouldn’t affect my larger conclusions.
“A demonstrative file exists only within a limited context: it exists only as long as the subject bears the demonstrative relation (whatever that relation is exactly) to some object x—a relation which makes it possible for the subject to focus his or her attention on x. If x disappears from view for sufficiently long, a change of context takes place and the file comes out of existence” (2012, p. 68; my emphasis.)
“When the contextual relation to the object is severed, the temporary file based on it disappears, but the information stored in the file does not disappear: it is transferred into the new file” (2012, pp. 62/3). And: “Conversion is the process through which information stored in a file is transferred into a successor file when the ER relation which sustains the initial file comes to an end.” (Ibid., p. 81).
I say ‘almost nothing’ because Recanati does say that “when an object is encountered and some information about it is gained, that information is typically preserved in memory and made available when the object is encountered again and recognized...” (2012, p. 81; my emphasis). However saying that demonstrative files are typically converted into memory files does nothing to explain why this is so. Likewise, chapter 5 of Mental Files in Flux (Recanati 2016), is an extended discussion of conversion that does not mention the conditions under which it takes place.
Note that this is not the only feature of the information contained in a file that could fix that file’s reference. Another is that the file refers to the satisfier of that information (if there is one), as in Goodman’s descriptive files. Yet another is that a file refers to the satisfier, if there is one, of the description from which that information was inferred, rather than the satisfier of the information itself. These inferences may be invalid (we are fallible reasoners), in which case the satisfier of the original description – i.e. the referent of the file—need not be the satisfier of the inferred information contained in the file.
If the referent of a file is a function of the information it contains. However there are more purely causal views according to which reference is fixed not by any feature of information contained in the file, but by causal relations between the agent and the referent. One main problem with purely causal views is how to account for reference change over time (as in Evans’ 1973 ‘Madagascar’ case). The mental files theory I am describing deals nicely with reference change over time, and so I won’t consider more purely causal views here; but see section 5.4 of Devitt (1981) for an attempt to account for reference change over time within a purely causal framework (not though that Devitt’s account requires allowing that both reference and truth can come in degrees, whereas as mine does not. I leave it to the reader to decide whether or not this counts against Devitt’s approach).
This is where the question of interpretation mentioned in footnote 18 becomes relevant. An alternative interpretation of the Evans/Recanati ‘dominant causal source’ model is that reference is a function not of the information contained in the file, but the information channel through which that information was obtained. Thus a file refers to whatever object was in fact at the other end of that information channel (if there was one). This allows that a file could refer in spite of being empty of information. I do not have space to go into a full argument against this version of the ‘dominant causal source’ view here, but I will mention another reason why empty files cannot be used to refer, which this alternative is also susceptible to. The main idea is that S’s a-file refers to o only if S can use that a-file to have beliefs about o. If S uses their a-file to have the belief \({}^\lceil a\) is \(F^{\rceil }\), then their a-file contains the information \({}^{\lceil }\) is \(F^{\rceil }\), whence the file is not empty after all. Thus a file that is empty of information cannot be used to have beliefs. And we might think (but I won’t argue here) that if a file cannot be used to have beliefs, then there is little reason to regard it as a device for reference.
Thanks to an anonymous reviewer for bringing this case to my attention.
The uniqueness of the channel transmitting information to S is not a feature of Sutton’s case, but I have added it because it strengthens the point.
Kripke’s ‘Paderewski’ case has S believe distinct things about two homonymously named individuals (that are really one) (Kripke 1979, pp. 265/6).
Thanks to an anonymous reviewer for bringing this point to my attention.
Cf. Hansen and Rey (2016), p. 433: “Specific causal relations are no doubt relevant in many cases; but why think they are present in all? Why think there’s any general solution to puzzles of this sort beyond occasion-relative pragmatics and forensics? Indeed, pace the recent resurgence of interest in traditional “metaphysics,” why think that there’s a general satisfactory account of all the multitude of “things” that we are able to think about? It’s hard not to suspect that the majority of such issues are really just matters of pragmatics and forensics” (Emphasis original).
Modulo concerns about reference change over time, which I leave aside.
This is not the only difference between descriptive and singular reference for Goodman (Cf. Goodman 2016a, pp. 445–6). However it would take us too far afield to address the other differences and their connections with the difference in terms of persistence conditions here.
Modulo concerns regarding sortalism about reference—the claim that to think about an entity requires not being mistaken about what kind of entity it is, in some sense of ‘kind’. I here set aside such concerns—they shouldn’t have any impact on the arguments I’m going to give.
“[H]olistically descriptive files are such that, in principle, they allow for complete information overhaul—that is, they allow for ‘ship of Theseus-style’ changes.” (Goodman 2016a, p. 458; my emphasis.)
The use of NFU is solely to avoid paradox in the example. NFU is both consistent (relative to Peano arithmetic) and also countenances a universal set. Nothing (else) of substance hangs on this choice.
Recanati claims that what accounts for the difference in content between “Hesperus \(=\) Hesperus” and “Hesperus \(=\) Phosphorus” is that one file is deployed twice when thinking the former, but two files are deployed once each in the latter (Recanati 2012, p. 42).
“[M]ost philosophers of mind accept the ... thesis that you have transparent access to the content of your own thoughts: provided you’re minimally rational, you simply cannot mistake one conceptual content for another” (Schroeter 2007, p. 597; quoted in Recanati 2012, p. 117.) Recall also Sutton’s (2004) intuition discussed in Sect. 4.
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Acknowledgements
I thank Imogen Dickie, Dominic Alford-Duguid, and two anonymous referees for discussions of previous versions of this essay.
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Davies, J.E. Towards a theory of singular thought about abstract mathematical objects. Synthese 196, 4113–4136 (2019). https://doi.org/10.1007/s11229-017-1644-0
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DOI: https://doi.org/10.1007/s11229-017-1644-0