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Abstract
Carey has argued that there is a system of core numerical cognition – the analog magnitude (AM) system – in which (approximate) cardinal numbers are explicitly represented in iconic format. While the existence of this system is beyond doubt, this paper aims to show that its representations cannot have the combination of features attributed to them by Carey. According to the argument from abstractness, the representation of the (approximate) cardinal number of a collection of individuals as such requires the representation of individuals as such, and this in turn requires non-iconic format, from which it is concluded that the explicit representation of the (approximate) cardinal number of some individuals requires non-iconic representational format. In support of the first premise, an account is given of what approximate cardinal numbers might be (namely, quantifiers), and in support of the second, a direct argument is articulated and defended. Finally, in response to an objection, a second argument (from parts) for the central thesis is provided. While the discussion is couched in the terms of Carey’s work, the considerations it adduces are perfectly general, and the conclusion should therefore be taken into consideration by all those aiming to characterize the AM system.
Notes
1. That is, at least since Locke argued that number is a primary quality, and so “utterly inseparable” (Citation1975, book II, Chapter 8, Section 9) from that which objectively has it. Locke also thereby incurred a commitment to the claim that our ideas of numbers resemble the numbers of which they are ideas.
2. And vice versa. For instance, Berkeley long ago noted that number is relative to a concept: “we say one book, one page, one line … yet the book contains many pages and the page contains many lines” (Citation1998, Section 12). He concluded, contrary to Locke (see note 1), that number is subjective. Frege (Citation1980) accepted Berkeley’s first point, but argued that concepts are objective, so that number is too. As we shall see, my argument below relies on this Berkeley–Frege point, which is also noted by Burge (Citation2010).
3. See, for example, Beck (Citation2012, 2014), Burge (Citation2010), Carey (Citation2009, 2011), Dehaene (Citation2011), and references contained therein for both further details of the empirical results and some theoretical attempts at accounting for them.
4. In the book, Carey claims that “later-developing explicit knowledge differs from core cognition in every single one of these six properties” (Citation2009, p. 68). However, in the Precis, she corrects this error (Carey, Citation2011, p. 114), explicitly recognizing that the first feature is shared with conception.
5. See Shea (Citation2011, p. 131) for a similar reading of Carey’s notion of core cognition.
6. Burge (Citation2010) uses the term “explicit representation” in a manner that is closer to Carey’s second suggestion, namely for any representation that is accessible to consciousness. (If a representation were personally or centrally accessible if and only if it were accessible to consciousness – which, however, I suspect it isn’t – these uses would coincide.) While I have some sympathy with Burge’s criticism of the contrast I will make between implicit and explicit representation, drawing the distinction in roughly the way Carey (first) does allows me to more directly engage with her work.
7. This is roughly what Burge (Citation2010) calls a body.
8. I find some support for this way of thinking in Burge (Citation2010). In particular, I agree with Burge that representing an object, or a number, as such – that is, as an object or as a number – is necessary for that object or number to figure in the content of the representation (Fodor, Citation2007 appears to disagree). I also agree with him that to represent something as an F it is necessary to respond differentially to Fs and “representationally relevant alternatives” (Citation2010, p. 466) to Fs. However, I here call this “implicitly representing” the property of being F, whereas he calls it “perceptually attributing” that property; my terminological decision allows me to more closely follow Carey’s usage.
9. It is widely held that the natural numbers are the finite cardinal numbers. But then, how can we reconcile attributing to Carey the view that the latter are explicitly represented in the AM system with her claim that “none of the [three] representational systems that underlie infants’ or animals’ behavior on nonlinguistic number tasks [including the AM system] represent number in the sense of natural number or positive integer” (Citation2009, p. 296)? Part of the answer, I think, comes from the recognition (above) that to explicitly represent the cardinals, one must represent them, at least implicitly, as cardinals. (A further part of the resolution of the worry involves noting the qualification Carey makes about the contents of AM system representations – see below.) Carey thinks that the AM system does not even implicitly represent the concept natural number, since it does not implicitly represent the concept successor; thus, it does not represent the natural numbers as such. By contrast, it does implicitly represent the concept cardinal number, and so is able to explicitly represent cardinal numbers.
10. In Dehaene and Changeux’s (Citation1993) model of the AM system (discussed in Section 10 below), there are quite clearly no representations obeying the picture principle in the fourth and final, behavior driving, output layer of the numerosity detection module. Yet Carey (Citation2009, pp. 132–134) argues on empirical grounds that this model is preferable to accumulator models such as those of Meck and Church (Citation1983), so she might be thought to be misrepresenting her own preferred model of the AM system, or at least representing it in a manner that is likely to mislead. (Thanks to an anonymous referee for stressing this point.) It is less clear, however, that there are no representations of number obeying the picture principle at the third, “summation” layer of the model. Nevertheless, I will argue for this claim below and I believe my conclusion also holds for accumulator models in which something akin to the third layer “summation” representations discussed here govern behavior.
11. As we shall see below, Carey qualifies the claim made in the content passage to suggest that the contents may be approximate cardinals.
12. Following Berkeley (Citation1998) – see note 2.
13. If we take properties to be intensions (i.e., functions from possible worlds to individuals), then their identity conditions are sensitive to the individuals to which they actually and possibly apply; and even if we don’t endorse this identification, nevertheless it is clear that the properties in question are properties of individuals (e.g., glasses of water) and not of stuff (e.g., water itself).
14. That is, at least insofar as it concerns the exact cardinals; the case of approximate cardinals will be considered below.
15. In fact, the abstractness of number representations appears to be established by Frege’s considerations even more directly than the abstractness of numbers themselves; it is the attribution of number which is relative to a way of representing a phenomenon (as a collection of individuals).
16. By contrast, the representation of greenness does not require the representation of individuals as such: I can represent an area (e.g., of a surface) as green without individuating that area.
17. Burge says, “the representational capacity [of the AM system] seems to be analog or continuous rather than discrete” (Citation2010, p. 477), and he takes this to count against the idea that the contents of AM representations are numbers, perhaps because the higher order character of numbers implies their discreteness. At the same time, however, he claims that “the integers could be represented by analog representations” (Citation2010, p. 477). Yet nowhere does he say that analog representations are distinguished by being iconic in Carey’s sense; and in fact, the above quote suggests he thinks they are differentiated by continuity, not their adherence to the parts principle. Burge does claim that “the representational form of numerosity estimates is analogous” (Citation2010, p. 477) to that of number attributions in being higher order; but he also suggests that these representations, in effect, lose track of the individuality of the members of the collections whose numerosity is being estimated. So I see nothing in Burge’s work that conflicts with the soundness of my argument from abstractness and some points which appear to support it.
18. Roughly, Fregean objects are Aristotelian substances of which something can be predicated, but which cannot themselves be predicated; that is, they are individuals.
19. See Williamson (Citation1994) for this kind of epistemicist treatment of vagueness.
20. Perhaps I have fully conceptual knowledge of the object(s) in question, or perhaps perception does not represent by virtue of employing pictures. Either is consistent with the line of thought espoused here. (See Spelke (Citation1988) for the claim that objects are not represented perceptually, but only conceptually. Obviously I am not committed to that claim here, though what I say is consistent with it.).
21. Suppose we add as a further premise the claim (D0): If an individual is explicitly represented, then it is (at least implicitly) represented as an individual. Then we can get to the stronger conclusion (D4): If an individual is explicitly represented, then the format of representation is not iconic. Moreover, (D0) was defended above, albeit briefly. But (D4) is not absolutely required for our purposes here; accordingly, I focus here on the reasoning from (D1) to (D3).
22. Fodor (Citation2007) explicitly embraces compositionality for icons obeying the picture principle.
23. Note that I don’t say predicated of the object; perhaps there can be non-predicative (e.g., perceptual) attribution, as on Burge’s (Citation2010) view.
24. Either p is represented as an individual by this part of the representation or it isn’t. If it isn’t, then the problem suggested in the main text arises. If it is, then the part of the representation in question is semantically privileged in the sense that is currently at issue; we should accordingly choose some other part of o and a corresponding part of the representation and run the argument again.
25. Compare Shea’s claim that ultimately, even on Carey’s view, “objecthood … is [not] being represented iconically” (Citation2011, p. 131).
26. Let me stress at this point that my direct argument in general, and its second premise in particular, is not so strong as to make all representation in accordance with the picture principle impossible. I am arguing that representing an object as such involves semantic privilege, and that this involves syntactic privilege for some part(s) of the representation at the expense of others. But this is perfectly consistent with the existence of representations with no syntactically privileged parts – they simply can’t represent objects as such, but must instead have a more minimal content. Indeed, as Kulvicki (Citation2014, p. 101) has argued, a photo of a photo might be indiscernible from the photo of which it is a photo: the “bare bones” content of the latter – as Kulvicki calls it, following Haugeland (Citation1991) – should therefore be thin enough as to be compatible with the veridicality conditions of the former. And since there is no object boundary around any part of the photographed photo for the photographing photo to represent, the photographed photo does not represent anything as an object (with boundaries) either. To return to Carey’s example: although a picture may be caused by a dog (and so in that sense be a representation of a dog), and its parts caused by the parts of that dog (so that they are, in that sense, representations of its parts), as far as the picture itself is concerned (as opposed to our interpretation of it), the dog is (as it were) camouflaged (it is not represented as a dog, or other agent, or indeed, as an object). Nevertheless, the visual scene depicted by the whole picture will be determined (compositionally) by those depicted by its parts and the way they are put together.
27. In particular, in this latter case, parts that cut across the boundaries between privileged and non-privileged parts will not be semantically significant.
28. Of course, we will represent the object (i.e., the dog) as an object when we look at the picture; but it doesn’t follow that the picture itself does.
29. Perhaps another example will help. Consider a color photograph of a wall on which the word “red” is painted in green. The photo represents the token of the word on the wall iconically, in accordance with the picture principle (for instance, green parts of the photo represent green parts of the wall); but it is itself a token of the word “red” and represents the color red symbolically. One and the same thing, the piece of photographic paper, “has” both iconic and symbolic format, but it is simultaneously an instance of each of two distinct representations with distinct contents.
30. A similar account can be given of bounded continuity for three-dimensional objects: just substitute “surface” for “line” and “volume” for “area.”
31. I use the phrase “proper part” in the standard mereological sense of a part which is not identical with the whole. Crucially, I do not use it to mean a syntactically privileged constituent.
32. This is perhaps especially obvious in the case of the scattered parts of the dots, which are not continuous and therefore have no boundaries; but it is equally true of continuous, bounded proper parts – just think of those other parts of the dot that are outside this part’s boundary.
33. This argument, like the direct argument, is inspired by a remark of Shea’s. “It is hard to see,” he says, “how there are parts of the property of being an agent, so that parts of a representation could represent parts of the property” (Citation2011, p. 131). But Shea is somewhat equivocal when it comes to the representations of the analog magnitude system: he denies that “numerosity is being represented iconically” (Citation2011, p. 131); yet he also says that “if analog magnitude representations are realized by some quantity in the brain, then they will indeed be iconic” (Citation2011, p. 131).
34. In the current context, the number representations in question are those at the third level of the Dehaene and Changeux numerosity detector module; I suppress this qualification throughout.
35. In fact, even if this analysis is not correct and these representations have no parts, the first premise on its own will pose problems for the view that they are non-trivially iconic in format (where a non-trivial icon is one that has proper parts that represent parts of what the whole represents).
36. Frege’s own view is that numbers are equivalence classes of extensions of concepts; but as his theory of extensions is inconsistent, neo-Fregeans, following Wright (Citation1983), reject the idea that numbers are constructed from other entities. It is this that I have in mind by calling them sui generis individuals.
37. It might be thought that numbers are to be identified with certain set-theoretic constructs other than those suggested by the Russellian quantifier view: we might, for instance, take zero to be the empty set and take each successor number to be the singleton set containing just its immediate predecessor, or we might take successors to be the sets containing all and only their predecessors (immediate or otherwise). I don’t consider such views in the main text because I take it that Benacerraf (Citation1965) has shown them to be implausible: how could the (finite cardinal) numbers really be one, rather than the other, of these constructions? (The existence of these constructions shows only that arithmetic is consistent relative to set theory; it tells us nothing about the metaphysics of numbers.) But suppose we take them seriously. If we accept Lewis’ claim that the parts of a set are its subsets, then on the first view the only parts of a number n are 0 and n itself (so 2, for instance, isn’t a part of 3); and on the second, although each smaller number is a part of n (so that 2 is a part of 3), so are lots of things that aren’t numbers – for instance, the set whose only members are 1 and 3 is not a number on this view but is a part of 4. So neither proposal vindicates what we might think of as the natural generalization of Carey’s suggestion that 2 is a part of 3, namely that the parts of a given number are precisely its predecessors (which might give pause to those aiming to identify the numbers with either construction). It is, of course, implausible that anything in the AM system, or in the third layer of Dehaene and Changeux’s model, represents any such non-numerical parts of numbers. But the picture principle doesn’t require that every part of what’s represented itself be represented; it only requires that every part of the representation represents some part of the what the whole representation represents. Nevertheless, we can mount a variation on the argument from parts – the argument from too few parts – against the second view of numbers considered here (or indeed, against the view that embraces the second set-theoretic identification of numbers but insists, contra Lewis, that the members of a set are its parts, so that all and only the predecessors of a number are its parts – as, one suspects, Carey intended). The idea is that numbers have too few parts to be represented in the third layer of the Dehaene and Changeux model in accordance with the picture principle. (Having no, or no relevant, parts, as discussed in the main text, is then just a special case.) The argument is as follows: the representations of number (on the third layer in Dehaene and Changeux’s model) have more parts than the numbers represented have; therefore, not every part of the representation represents a part of what the whole represents (contra the picture principle). To see that the sole premise of the argument is true, take for example the activity in the third layer that is correlated with four objects being presented. In the model, ten units on the third layer are (typically) activated in such cases; but even counting 0 and 4 itself, 4 only has five numbers as parts (on the views under consideration). But then, just counting the active units, we have more parts in the representation than in the number represented – so not every part represents a part of the number; and, of course, things look even worse for the picture principle if we count the continuously varying quantity of activity on the third layer instead of the number of active units, for then the representation has uncountably many parts (and so clearly more than five). The picture principle for number representations can’t be salvaged by appealing to implausible views of the metaphysics of numbers.
38. If it is, though, it does not represent the exact cardinals, since they are not “semantically undifferentiated in the extreme” (Goodman, Citation1968, p. 160).