Abstract
There is at present a lively debate in cognitive psychology concerning the origin of natural number concepts. At the center of this debate is the system of mental magnitudes, an innately given cognitive mechanism that represents cardinality and that performs a variety of arithmetical operations. Most participants in the debate argue that this system cannot be the sole source of natural number concepts, because they take it to represent cardinality approximately while natural number concepts are precise. In this paper, I argue that the claim that mental magnitudes represent cardinality approximately overlooks the distinction between a magnitude and the increments that compose to form that magnitude. While magnitudes do indeed represent cardinality approximately, they are composed of a precise number of increments. I argue further that learning the number words and the counting routine may allow one to mark in memory the number of increments that composed to form a magnitude, thereby creating a precise representation of cardinality.
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Notes
This is not entirely accurate of all weak nativists. Susan Carey (2004, 2009a, 2009b) and Le Corre and Carey (2006, 2007) for example, argue that our acquisition of the first few natural number concepts depends on innate systems for representing objects and for understanding natural language quantifiers, that the system of mental magnitudes plays no role in this process, and that it only later plays a role in our acquisition of larger number concepts. I describe this position below.
The use of the terms strong and weak nativism is due to Laurence and Margolis (2007).
Saying that the present hypothesis appeals only to the system of mental magnitudes is somewhat inaccurate. In fact it will have to appeal to other resources, such as the ability to create one-to-one correspondences between representations. These other resources will be appealed to by other extant theories, however. Since my aim is to distinguish the present hypothesis from others, it is most useful to say, for example, that while some hypotheses appeal to object-files and mental magnitudes, and while others appeal to object-files and quantificational markers in natural language, the present hypothesis appeals only to mental magnitudes.
Rips et al. (2006, 2008a) and Rips et al. (2008b) discuss the origin of the general concept NATURAL NUMBER, and argue that it requires an at least implicit understanding of the axioms of arithmetic. But this is a separate problem from the problem of the origin of individual natural number concepts, and I will not take it up here. See Margolis and Laurence (2008) and Sarnecka (2008) for discussion.
This is important because, as Wynn (1992a) shows, children come to meet this requirement in stages, first developing the ability to distinguish groups of one from other groups, then groups of two, then groups of three, and then all at once groups of any size. I will describe this process in more detail below.
Some may worry that the requirement as stated implies that we do not possess concepts for very large numbers, since we would not be able to distinguish them from their nearest neighbors. For example, one could not distinguish one billion trillion objects from one billion trillion plus one objects. But note that this inability is due to a lack of sufficient time and memory resources. In contrast, small children are unable to distinguish numbers from their nearest neighbors, even for very small numbers, so their inability cannot be due to time and memory constraints. Thus, the caveat might be added that the requirement is in fact not necessary for the possession of the concept N, if distinguishing N from its nearest neighbors would outstrip the time and memory resources available to the subject. It would remain unclear then, exactly what the necessary and sufficient conditions on possession of N would be. But that is not problematic for the present account—we do not, after all, possess the concept ONE BILLION TRILLION in exactly the same way that we possess small number concepts, and the present account is primarily concerned with the latter.
See Carey and Spelke (1996) for a useful discussion of looking-time methods and results.
Feigenson et al. (2004) provide a useful review.
The numbers involved here are important. For some evidence suggests that cardinalities from one to about three or four are represented not by mental magnitudes but by an object-tracking system. I will describe this system below.
Some early experiments revealed this pattern too, as both Starkey and Cooper (1980) and Antell and Keating (1983) found that infants failed to discriminate four from six dots, and Strauss and Curtis (1981) found that infants were unable to distinguish between four and five dots. Strauss and Curtis (1981) also tested infants at three versus four, a discrimination which female infants were able to make and male infants were not. But again, these early studies failed to control for variables besides number (see footnote 14).
Notice, however, that this sensitivity does appear to improve with age. While Lipton and Spelke (2003) found that six-month-olds were unable to distinguish between eight and twelve sounds, they found that nine-month-olds were successful at discriminating eight from twelve sounds, though they failed to discriminate eight from ten.
For example, Wynn (1992b) used small numbers of toys.
They grow, in the sense of requiring more digits, (roughly) in proportion to the logarithm of the number represented.
Note that there are non-proportional systems of magnitude values. For example, suppose I have cups of water, each containing a number of ounces of water (from zero to nine ounces). But rather than use the empty cup to represent 0, the cup with one ounce to represent 1, and so on, I assign numbers to cups randomly so that e.g., cups with three ounces of water represent 7, cups with two ounces of water represent 9, etc.
As Le Corre and Carey (2007) put it, “Scalar variability holds the standard deviation of the estimate of some quantity is a linear function of its absolute value” (397).
Note that I assume scalar variability is caused by the compounding of noise in the size in the individual increments. Some may worry that this assumption is unfounded, as scalar variability could also be achieved with increments that were not noisy in size, but if there was a probability that for each object observed, the system produced between zero and two increments. But physical systems are inherently noisy, so there is already a reasonable explanation of scalar variability without needing to appeal to too many or too few increments. That hypothesis needlessly complicates matters.
A possible objection here that it is inappropriate to say that two magnitudes that are equal in size may represent different numbers, and that it would be more correct to say that some magnitudes represent a range of numbers. However, I take the representational content of a magnitude to be determined by the number of objects in response to which it was formed. A magnitude represents, for example, the number 5 just in case it was formed when the subject observed five objects or events.
As an illustration of the accumulator, many authors have described a model accumulator constructed using a supply of water and cups. Such a model provides an intuitive account of the variability inherent in accumulator representations, and indeed, of why the accumulator exhibits scalar variability. For instance, Dehaene (1997) writes,
A clear drawback of the accumulator is that numbers, although they form a discrete set, are represented by a continuous variable: water level. Given that all physical systems are inherently variable, the same number may be represented, at different times, by different amounts of water in the [basin]. Let us suppose, for instance, that water flow is not perfectly constant and varies randomly between 4 and 6 liters per second, with a mean of 5 liters per second. If [the user] diverts water for two-tenths of a second into the [basin], one liter on average will be transferred. However, this quantity will vary from 0.8 to 1.2 liters. Thus, if five items are counted, the final water level will vary between 4 and 6 liters. Given that the very same levels could have been reached if four or six items had been counted, [the] calculator is unable to reliably discriminate the numbers 4, 5, and 6. (29–30)
In the above description of the accumulator I have omitted three areas of disagreement. First, Meck and Church’s (1983) model enumerates objects serially. Other models (e.g., Dehaene and Changeux 1993) enumerate objects in parallel. Second, as described above there is variability inherently present in the model. Other descriptions (e.g., Gallistel et al. 2006) take the variability to be present in memory, not in the accumulator itself. Third, as explained above, according to this model the system’s inability to distinguish between numbers that are close together is due to scalar variability. In other models (e.g., Dehaene et al. 2008 and see also Cantlon et al. 2009b and Dehaene et al. 2009) this inability is due to a logarithmic compression of increment size. The second and third of these disagreements should have no effect on the arguments in the rest of this paper. The first, however, is relevant. The view I present below depends on the accumulator adding increments serially, as the subject slowly observes a group of objects (e.g., observes the objects one at a time). But the present view does not depend on the accumulator only operating in this manner; it allows that the accumulator may also operate in parallel. I will address this question in more detail in the concluding section. Note though that whether it operates serially or in parallel, or both, remains an open question. So it is worth exploring what sorts of hypotheses are available, assuming particular answers to that question. The present argument is that there is an important hypothesis that has been overlooked, assuming that mental magnitudes can be formed serially.
Gallistel et al. (2006) argue that some animals compare rates of return from foraging in different areas, and that positing an accumulator can explain this as well.
Gelman and Gallistel (1978) also describe the abstraction principle: that the symbols may be used to count heterogeneous groups of objects (e.g., toys together with cookies), and the order-irrelevance principle: that the order in which the objects are counted does not affect the cardinality of the group. Neither of these will affect the discussion here though, so I leave them out.
Gallistel et al. (2006) are not alone in this line of thinking. Dehaene (1997) describes the process in similar ways. He explains that to learn the meaning of the word “three,” for example, the child must
correlate [his] preverbal representation with the words he hears. After a few weeks or months, he should realize that the word ‘three’… is very often mentioned when his mental accumulator is in a particular state that accompanies the presence of three items. Thus, correlations between number words and his prior nonverbal numerical representations can help him determine that ‘three’ means 3. (107)
De Cruz (2008) writes that “Natural language is one among several tools that allow us to map exact cardinalities onto our approximate… mental number [representations]” though she also says that “External symbolic representations of natural numbers are not merely converted into an inner code; they remain an important and irreducible part of our numerical cognition” (487).
For instance, they suggest that the system may judge two magnitudes to be equivalent if it cannot reliably order them, or alternatively that the system may employ “shortcuts” such as assuming that when one increment is added to each of two equivalent magnitudes the results are equivalent, or finally, that “the discrete nature of the verbal representation… is the origin of our notion of exact equivalence” (Gallistel, et al. 2006, 266–7).
Carey (2001) argues on similar grounds that mental magnitudes cannot be the sole source innate source of natural number concepts.
See also Margolis and Laurence (2008).
Note the relationship here to the necessary condition on possession of individual number concepts I gave above. The requirement is that a person in possession of the concept N must be able to distinguish groups of objects with cardinality n from groups of objects with cardinality n+1 and from groups of objects with cardinality n-1. The intuition that 1-knowers possess the concept ONE, but no other number concepts, is the source of this requirement.
See Spelke (2003, 299–301).
Condry and Spelke (2008) put the hypothesis this way: “the counting routines of specific human cultures engender spontaneous, constructive processes within the child and that these processes build a unitary system of natural number concepts from a set of conceptual primitives delivered by distinct, core cognitive systems” (37).
See for example, Montemayor and Balci (2007).
Also see Laurence and Margolis (2005). Rips et al. (2006, 2008a) and Rips et al. (2008b) have argued against both forms of weak nativism described here, on grounds that the induction required cannot guarantee natural number concepts. The reason is that it cannot rule out other non-standard conceptual structures, such as loops. As Margolis and Laurence (2008) have pointed out though, this appears to be an instance of more general worries about induction (such as those discussed by Kripke 1982) and would seem to affect any account of the acquisition of number concepts, including strongly nativist accounts. The only exception would be an account according to which the entire set of natural number concepts is innate, but such an account would appear absurd on its face, and no one has offered such an account.
There are other weakly nativist accounts, in addition to Spelke’s and Carey’s. For instance, Decock (2008) discusses the possibility that number competence develops from the one-to-one principle, rather than from enumeration.
Though they note the possibility that the innate complement of natural number concepts could include the first few numbers.
Leslie et al. (2008) claim that any candidate for being the concept ONE must have this feature. But it is not clear that all features of the concept ONE must be innate. The idea that the number 1 is the (unique) multiplicative identity could be acquired as multiplication is learned. And this is true regardless of whether the concept ONE is innate or not.
The experimental evidence shows that when estimating the cardinality of a group of objects, adults are able to produce a number word, and adults’ rapid discrimination of Arabic numerals exhibits the well-known “distance” and “magnitude” effects, which are corollaries of Weber’s Law, and hence suggests the deployment of mental magnitudes. For further explanation and discussion, see for example Dehaene (1997) and Gallistel et al. (2006).
Also see Izard et al. (2008b), for discussion of speakers of Mundurucú, who possess numerical terms that “approximately” correspond with the numbers one through five.
The argument from the claim that some cultures do not possess natural number concepts to the claim that those concepts cannot be innate depends on a notion of innateness according to which if a concept is not present in all cultures then it is not innate. Such an account would have to be defended if the argument were to be successful. I am here noting that some have made the argument, but I am not relying on it, so I leave that discussion aside.
It is important to imagine the cement wall having been poured all at once. One should not imagine it being poured in separate forms that are then joined together.
Some might object to the example, since words can be individuated even when spaces are not left between them. This is true, but the example is intended to show that there are at least some systems in which representations are constructed from parts and the parts retain their individuality. Indeed, even when a written sentence has no spaces, there remains a canonical breakdown of the sentence into words.
It is important here to distinguish between two hypotheses. The first is that the accumulator encodes an increment for each object observed, and then combines these representations. In other words, it arrives at a completed representation of a group of objects by first forming a representation of each object in the group. The second is that the accumulator responds to individual objects, but does not encode separate representations for each – it creates the completed representation “all at once”. I am here suggesting the second, and not the first of these hypotheses, at least in cases in which the objects are observed rapidly. The hypothesis I offer below does, however, depend on the accumulator being able to add increments to a magnitude that has already been encoded, when for example, the objects are observed slowly. I will say more about this distinction below.
As I noted above, some evidence suggests that the accumulator does not in fact respond to cardinality at all (at least in the small number range), but rather to continuous variables such as total contour length of the objects in a display or the density of the objects. Thus, one should read the proposal here as hypothetical, depending on the assumption that the accumulator creates a representation by incrementing a magnitude n times for n objects.
Another way to make the point is that, while the literature on mental magnitudes has drawn a two-way distinction between cardinality represented and magnitude doing the representing, what is needed is a three-way distinction between cardinality represented, increments combined to form a total magnitude, and total magnitude.
Notice that this is not the same problem as somehow mapping number words to overlapping ranges of mental magnitudes. I take it that problem is unsolvable (as strong nativists seem to agree) because it requires mapping a precise system of representation directly to an approximate system. The present problem requires mapping a precise system of representation to another precise system (the number of increments combined in the formation of a mental magnitude).
There is also some evidence that learning a system of number words may also help to remove some of the variability inherent in magnitude representations. See for example, Piazza and Izard (2009).
It is true of course that Spelke (2003) argues that the length of time it takes children to learn the meaning of number words is evidence for her view—that the system of object-files must also play a role in that learning. There are two issues that point to the present hypothesis though, over Spelke’s. First, the present hypothesis is simpler, in that it does not require object-files, as does Spelke’s account. Second, notice that Spelke’s account does not predict that the number words must necessarily be learned in order. It allows that they are, and of course that is what one would expect. But her hypothesis is compatible with a child learning the number words out of order. The present hypothesis is not so compatible. Again, it requires that children first learn the meaning of “one”, then “two,” and so on. On the other hand however, this may also be a drawback for the present hypothesis. See the discussion of “Alex” the grey parrot, below.
Of course, after about “three” or “four” children learn the meanings of the rest of the number words all at once, which requires an inductive inference of some kind. None of the extant theories—including the present one— has an explanation of this part of the process. It must be acknowledged though, that the fact that that inductive inference happens after about “three” or “four” rather than, for example, at “two” or “five” is perhaps better explained by Spelke’s view, since that view relies on mapping number words to object-files, and that system has a set-size limit of about three or four.
See De Cruz (2008) for discussion.
See Nieder and Dehaene (2009), for example, for a review of neuroscientific studies on numerical cognition.
Note that in discussing the neuroscientific evidence, I am only arguing that it cannot distinguish between the present account and the others discussed here. I am not arguing that there is no evidence speaking against the theory that an accumulator mechanism is responsible for numerical competence in infants. As I noted above (footnote 24), there is much debate about whether an accumulator or some other kind of mechanism is responsible, and if it is an accumulator, exactly how it functions. But those debates remain unsettled, so it is worth exploring, on the assumption that an accumulator is responsible for infant numerical competence, the space of possible hypotheses concerning the acquisition of number concepts. The present argument is that one important hypothesis within that space has been overlooked.
And Rips et al. (2008a) account as well.
Famously, Frege (1884/1953) showed that integer concepts could be derived from one-to-one correspondence, which in turn could be reduced to logical relationships alone. I noted above that all of the accounts described here will have to rely on the ability to create one-to-one correspondences. But they all also rely on innate representational systems—mental magnitudes, object-files, or innate representations of natural numbers—and so are all distinct from Frege’s view.
Still, according the present hypothesis, instances in which the system operates in parallel cannot play a role in the acquisition of natural number concepts. Only instances in which the system operates serially can.
There are, however, explanations of how it would be possible to create magnitude representations in parallel even when objects are presented serially (see e.g., Church and Broadbent 1990).
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Acknowledgements
A version of this paper was presented at the conference The Number Concept: Axiomatization, Cognition, and Genesis. I am grateful to the audience for helpful discussion. I also want to thank Véronique Izard, Joshua Smith, three anonymous referees and the editor of this journal for providing comments and suggestions on earlier drafts that led to significant improvements in the paper.
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Katz, M. Mental Magnitudes and Increments of Mental Magnitudes. Rev.Phil.Psych. 4, 675–703 (2013). https://doi.org/10.1007/s13164-013-0158-z
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DOI: https://doi.org/10.1007/s13164-013-0158-z