2.3.1 Approximate Number System

The ANS is a primitive, prelinguistic cognitive system, also sometimes called the number sense (Reference DehaeneDehaene [2011]). It is distinguished from other cognitive systems by two characteristic features: quoting Reference Clarke and BeckClarke and Beck (2021, p. 2), the ANS’s “numerical discriminations are imprecise and conform to Weber’s Law.” Intuitively, the ANS’s discriminations are “imprecise” in that instead of tracking, say, exactly 20 objects, it instead tracks approximately 20 objects.Footnote ⁶ Moreover, it conforms to a psychophysical principle known as Weber’s Law – its ability to discriminate magnitudes decreases as the ratio between those magnitudes approaches 1:1. Consequently, it is just as easy to distinguish 5 objects from 10 as it is 10 objects from 20, since the ratios between the two pairs are the same. In contrast, it is easier to discriminate 2 from 4 objects than 8 from 10, since the ratios between the two pairs are different, even though the absolute difference is the same. (This is known as the magnitude effect.) Similarly, it is easier to distinguish four from eight objects than four from seven, since the ratio is greater. (This is called the distance effect.) Figure 1 illustrates typical stimuli testing for such effects.

Figure 1 Illustration of typical stimuli for ANS experiments.

As ordinarily construed, the ANS is a kind of analog magnitude system, roughly in that states of the system represent quantities, and also that states of the ANS involve magnitudes which are approximately proportional to what they represent. According to one influential view, for example, ANS states represent the approximate cardinalities of collections in virtue of instantiating physical magnitudes of some sort – e.g. amounts of electrical charge – roughly proportional to the cardinalities of the collections represented. Reference CareyCarey (2009a) usefully illustrates this feature in terms of a system in which lines of differing lengths represent different cardinalities.

$\begin{array}{cc}\text{Number}& \text{Length}\\ 1& \underset{\_}{}\\ 2& \underset{\_}{}\\ 3& \underset{\_}{}\\ 4& \underset{\_}{}\\ 5& \underset{\_}{}\\ 6& \underset{\_}{}\\ 7& \underset{\_}{}\\ 8& \underset{\_}{}\end{array}$

The important point is that within this representational system, “number is represented by a physical magnitude that is [merely] roughly proportional to the number of individuals in the set being enumerated” (Reference CareyCarey [2009a, p. 118], our emphasis). Precisely how best to characterize the numerical contents of these analog representations remains a point of ongoing debate. For instance, on some views, such states represent real numbers, on others rationals, and on some only approximate cardinalities (Section 3.2.1).

There is plenty of empirical evidence for the existence of the ANS in both humans and nonhuman animals, which we will not review here.Footnote ⁷ And while different models of the ANS exist,Footnote ⁸ the one most frequently cited by Mainstream theories is the accumulator model of Reference Meck and ChurchMeck and Church (1983), the crucial features of which Reference Laurence and MargolisLaurence and Margolis (2005, pp. 218–219) capture with the following metaphor:

Figure 2, taken from Reference Gallistel and GelmanGallistel and Gelman (2000), illustrates the metaphor, where lines on the beaker represent cardinalities, which map one-to-one with natural numbers.

Figure 2 Theoretical model of the ANS.

Of course, no-one is seriously suggesting that the ANS uses beakers of water to represent cardinalities. Instead, the presumption is that electrochemical events and properties – e.g. pulses of electricity and magnitudes of stored electrical charge – are doing the relevant work. However, this metaphor preserves the crucial features of how the ANS is assumed to represent. Moreover, it provides some sense of why theorists have disagreed on what the representational capacities of the ANS might be. If we focus on the fact that the ANS discriminates cardinalities in a manner that exhibits magnitude and distance effects, then it is natural to suppose, along with Carey, that it represents approximate cardinalities. However, if one instead focuses on the continuous nature of the states of the accumulator, following Gelman and Gallistel, then one may be led to suppose that states of the ANS structurally resemble and represent real numbers. We will return to the question of whether the ANS plausibly represents numbers of any kind in Section 3.

Finally, we emphasize that the accumulator is widely assumed to conform to the same principles ensuring that count words in natural language represent cardinality. Quoting Reference CareyCarey (2009a, p. 132):

As we’ll see, it is thanks to this supposed correspondence between ANS states and count words that, according to some Mainstream accounts, count words inherit their denotations from such states.

2.3.2 The Small Number System

In addition to the ANS, a second core system is widely assumed to be relevant to number cognition, what we’ll call the “Small Number System” (SNS). However, there is considerable disagreement concerning how precisely to characterize it. Here are three prominent characterizations:

• The Object Tracking System: A system employed by the visual system for focusing attention on up to three or four objects at a time. It does so, thanks to the object-indexing system [Reference Leslie, Xu, Tremoulet and SchollLeslie et al. (1998); (Reference Scholl, Leslie, Lepore and PylyshynScholl and Leslie (1999)], which incorporates up to four symbols, or indexes, acting like pointers to the objects attended to, based on their spatial and temporal properties.

• The Parallel Individuation System: A domain-general capacity to construct and manipulate mental models of objects [Reference SimonSimon (1997); Reference Le Corre and CareyLe Corre and Carey (2007); Reference SpelkeSpelke et al. (2022)]. Mental models consist of distinct symbols, one for each object represented, held briefly in working memory. Consequently, only up to three or four objects are represented at any time.

• The Subitizing Module: An innate, domain-specific system responsible for precisely representing cardinality properties – oneness, twoness, threeness, and possibly fourness – without counting [Reference HurfordHurford (1987), Reference Margolis and LaurenceMargolis and Laurence (2008); (Reference MargolisMargolis (2020)]. There is a small stock of representations which are causally responsive to “numerical quantities” [Reference Margolis and LaurenceMargolis and Laurence (2008); (Reference MargolisMargolis (2020)]. For example, in the presence of a pair of shoes, this system immediately registers the twoness of the shoes.

On all these proposals, the hypothesized system tracks small cardinalities – its states causally covary with the number of items presented, up to three or four. However, there is significant disagreement regarding whether cardinality is represented. According to the first two proposals, the number of symbols or indices merely covaries with the number of objects detected. Only the last, subitizing module hypothesis, assumes that the SNS also represents numerical properties as such. Regardless, on any of these views, the SNS is implicated in the possession of at most the first three or four number concepts. Indeed, as we’ll see, possessing number concepts beyond these appears to mark a significant developmental milestone, one which may require transcending the repertoire provided by the SNS or the ANS.

3.2.1 The Representation Requirement

According to the Representation Requirement, count words inherit their denotations from ANS states only if both denote the same things. On two of the major extant alternatives, these denotations are numbers, in the form of the reals or rationals. Thus, a significant problem with either suggestion is simply that numbers are not plausible denotations for count words. To see why, consider (3a,b).

1. 1. a. There are two Elmos on the table.

   2. b. Two is an even number.

These different uses of ‘two’ not only plausibly belong to different morphosyntactic categories, they also clearly serve different semantic functions (Section 4.3). Specifically, whereas ‘two’ in (3a) is an adjective or determiner enumerating Elmos, ‘two’ in (3b) is a numeral, i.e. a name of a number, in this case two. Thus, of these two uses, only ‘two’ in (3a) deserves the moniker “count word,” as only it is plausibly in the business of enumerating, semantically speaking. Furthermore, as we elaborate further in Section 4, contrastive pairs like (4a,b) strongly suggest that count words, unlike numerals or complex phrases like ‘the number two’, do not denote numbers.

1. 1. a. How many Elmos are on the table? {Two/??The (real/rational) number two}.

   2. b. Which number is Mary writing about? {Two/The (real/rational) number two}.

According to the Inheritance Thesis, count words inherit their denotations from ANS states. So, if the latter denote numbers, in the form of rationals or reals, then so should the former. In that case, the count word ‘two’ in (4a) would be coreferential with the numeral ‘two’ in (4b), and so we would expect (4a) and (4b) to be equally acceptable, contrary to fact. Consequently, views on which ANS states denote real or rational numbers are seemingly incompatible with the Inheritance Thesis.

Indeed, in this respect, views on which states of the ANS denote approximate cardinalities come closer to respecting the semantic evidence. On many extant analyses, ‘two’ in (4a) denotes the sort of thing answering ‘how many’-questions, i.e. a cardinality [Reference MoltmannMoltmann (2013); Reference ScontrasScontras (2014); Reference SnyderSnyder (2017)]. Furthermore, it has been suggested, notably by Reference Kennedy and SyrettKennedy and Syrett (2022), that children assign these same entities as the denotations of count words at the earliest stages of numerical development. Thus, if ANS states denote cardinalities, albeit approximate ones, they would plausibly denote entities of the right sort, given the Inheritance Thesis.

But what is an approximate cardinality? Quoting Reference Laurence and MargolisLaurence and Margolis (2005, p. 221): “Instead of picking out 17 (and just 17), an Accumulator-based representation indeterminately represents a range of numbers in the general vicinity of 17.” Fleshing this out somewhat, consider (5a–c):

1. 1. a. Exactly seventeen women attended the conference.

   2. b. Approximately seventeen women attended the conference.

   3. c. Seventeen women attended the conference.

Intuitively, (5a) is true if the number of women who attended the conference is precisely seventeen – no more, no less. In contrast, (5b) could be true if sixteen or eighteen women attended the conference, since these fall within a contextually given range which counts as close enough to seventeen. Now, within the context of performing the count routine, an utterance of (5c) is far more plausibly equivalent to (5a) than (5b). Indeed, if eighteen women actually attended the conference, then an utterance of (5c), like (5a), would be false, even though eighteen would be close enough to seventeen to count as approximately seventeen.

More precisely, suppose ‘exactly’ has something like the denotation in (6a), where ‘ $〚\alpha 〛$ ’ represents the denotation of ‘ $\alpha$ ’, ‘ $n$ ’ ranges over natural numbers, ‘ $S$ ’ ranges over sets, and ‘ $|S|$ ’ denotes the cardinality of $S$ .

1. 1. a. $〚$ exactly $〛=\lambda n.\lambda S.|S|=n$

   2. b. $〚$ exactly seventeen $〛=\lambda S.|S|=17$

According to (6b), ‘exactly seventeen’ is true of those sets whose cardinality is strictly identical to 17. In contrast, suppose ‘approximately’ has the denotation in (7a), where ‘ $\simeq$ ’ is a context-sensitive relation holding between numbers $n$ and $n^{\prime }$ just in case $n$ is among a range of numbers which are, in a given context, close enough to $n^{\prime }$ .

1. 1. a. $〚$ approximately $〛=\lambda n.\lambda S.|S|\simeq n$

   2. b. $〚$ approximately seventeen $〛=\lambda S.|S|\simeq 17$

According to (7b), ‘approximately seventeen’ denotes sets whose cardinality is among a range of numbers which are, in a given context, close enough to 17, say {16, 17, 18}.Footnote ¹¹

Seen this way, views on which states of the ANS denote approximate cardinalities are incompatible with the Inheritance Thesis simply because count words do not plausibly denote approximate cardinalities. As (5a–c) demonstrate, there is little empirical plausibility to the claim that ‘seventeen’ is synonymous with ‘approximately seventeen’. Rather, it is far more plausible that in the context of performing the count routine, ‘seventeen’ in (5c) denotes collections of exactly seventeen things [Reference HornHorn (1972)]. In which case, ‘seventeen’ could not inherit its actual denotation from some ANS state, assuming such states denote approximate cardinalities.

Something similar can be said for views on which ANS states denote what Reference BurgeBurge (2010, p. 482) calls pure magnitudes: “A pure magnitude is a magnitude not specific to any further type of magnitude such as spatial extent or size, temporal duration, weight, and so forth.” That is, pure magnitudes are magnitudes divorced from any specific dimension of measurement, but which nevertheless permit basic arithmetic operations, such as addition, subtraction, multiplication, and division. They can also fall into ratios, in virtue of being subject to the Eudoxus definition. Let $a$ and $b$ be magnitudes of the same kind, e.g. two intervals of time, and let $c$ and $d$ be magnitudes of the same kind, e.g. two cardinalities. Then according to the Eudoxus definition, for any natural numbers $n$ and $m$ , $a:b::c:d$ if and only if three facts obtain:

1. (ED1) $ma>nb$ if and only if $mc>nd$

2. (ED2) $ma=nb$ if and only if $mc=nd$

3. (ED3) $ma<nb$ if and only if $mc<nd$

Since pure magnitudes are divorced from specific dimensions of measurement, the Eudoxus definition also applies to them, only there is no distinction to be drawn between the kinds of magnitudes forming both sides of the proportion.

Reference BurgeBurge (2010, p. 482) speculates that ANS states represent pure magnitudes:

The important contention is that ANS states represent magnitudes which fail to distinguish between different dimensions of measurement, such as cardinality and length. Yet, as noted earlier, ‘two’ in (3a) most plausibly denotes a cardinality. Moreover, on standard accounts within linguistic semantics, what distinguishes cardinalities, as entities, from other kinds of magnitudes is precisely that the former encodes a particular, discrete dimension of measurement, namely cardinality [Reference ScontrasScontras (2014); Reference SnyderSnyder (2021a)]. As such, on the present proposal, count words could not denote pure magnitudes, and so could not inherit their denotations from ANS states.

3.2.2 The Feasibility Requirement

According to the Feasibility Requirement, count words inherit their denotations from ANS states only if there is a feasible mapping from count words to ANS states with an appropriate denotation. Here, ANS states are being individuated semantically, i.e. by their semantic contents. Thus, whether the Feasibility Requirement can be met will depend importantly on the presumed contents of ANS states. If, for instance, each state denotes a positive integer [Reference Clarke and BeckClarke and Beck (2021)], then a feasible mapping could presumably exist: just map each count word to a corresponding positive integer representation. However, if ANS states denote real numbers [Reference Gallistel and GelmanGallistel and Gelman (2000); Reference Gallistel, Gelman, Cordes, Levinson and JaissonGallistel et al. (2006)], then it it is hard to see how such a mapping could exist, in principle.

The problem is evident: because there are infinitely many reals between any two naturals, the prospect of establishing a mapping between a given count word and its corresponding integer value will be infinitesimally small. Indeed, as Galistel and Gellman themselves later note:

In light of this problem, why would G&G propose that ANS states denote real numbers in the first place? The answer, suggested by Reference Gallistel and GelmanGallistel and Gelman (2000), appears to be that this assumption is required to explain how the ANS represents duration as a continuous magnitude, as Reference Clarke and BeckClarke and Beck (2021, p. 12) explain:

However, this line of reasoning is committed to a pair of related, problematic assumptions.Footnote ¹² The first concerns the representational vehicles which may denote continuous magnitudes. Specifically:

• Denotation-to-Vehicle: If a representational vehicle $R$ denotes a value for some continuous magnitude, e.g. duration, then $R$ is itself continuous.

This appears necessary to transition from the premise that duration – what’s purportedly denoted by neural magnitudes – is continuous to the conclusion that the neural magnitudes representing duration – the vehicles themselves – are also continuous. However, the assumption is subject to clear counterexamples [Reference Laurence and MargolisLaurence and Margolis (2005)]. For example, digital clocks, digital thermometers, and symbols like ‘ $\pi$ ’ and ‘ $\sqrt{2}$ ’ each involve discrete symbols representing continuous magnitudes – time, temperature, and irrational numbers, respectively.

The second problematic assumption is the converse of the first:

• Vehicle-to-Denotation: If a representational vehicle $R$ is a continuous magnitude, e.g. a spike of electrochemical activity, then what $R$ denotes is continuous.

This appears to be required to transition from the empirical thesis that numbers (of some sort) are represented by continuous neural magnitudes, to the conclusion that the numbers so represented are reals. However, this assumption is also subject to obvious counterexamples [Reference Laurence and MargolisLaurence and Margolis (2005)]. For example, digital computers routinely use continuous magnitudes, such as voltage, to represent discrete values.

In summary, not only are there good reasons to deny that count words inherit their denotations from real-valued ANS representations, there were never especially good reasons to suppose that ANS states denote real numbers to begin with. We conclude that the Inheritance Thesis is implausible: given extant accounts of what ANS states denote, it is empirically implausible that count words inherit their denotations from such states.

3.4.1 Spelke’s Hybrid View

Reference SpelkeSpelke (2017, p. 148) summarizes her view as follows:

This requires unpacking. Specifically, how does Spelke conceive of the ANS and the SNS, and what role is language assumed to play in furnishing children with count concepts? We’ll consider these issues in turn.

According to Spelke, the ANS is a system for “representing sets and their approximate numerical magnitudes” – presumably their approximate cardinalities. In contrast, Spelke construes the SNS as a parallel individuation system, one which does not represent numerical properties, such as the number of entities in a visual array, but each individuated entity attended to – e.g. this cat and that dog – up to a limit of four objects. Each entity represented is associated with a distinct mental symbol [Reference Hyde, Simon, Berteletti and MouHyde et al. (2017)]. More specifically, the system generates mental indices, or object files, corresponding to each entity represented.

As Spelke sees it, both the ANS and SNS are involved in the acquisition of count concepts. However, it is language that seems most central to her proposal. Specifically, according to Reference SpelkeSpelke (2017, p. 156), “children’s discovery of the natural numbers depends on their mastery of the generative rules of their language,” which they discover in four steps. The first three are intended to explain the child’s slow and laborious mastery of the first three count words; the last is intended to explain how children expand their numerical vocabulary beyond ‘three’ (or its equivalent).

At the first step, children “construct a prolific and productive system of representations of object kinds, by mastering noun phrases that, in English, are composed of determiners and singular count nouns $\dots$ (e.g. a cup, the cat, your hand)” [Reference SpelkeSpelke (2017, p. 157)]. The idea appears to be that children first master singular noun phrases (NPs), which specify entities of a given kind, by “linking each phrase to representations from three core systems: systems for representing cohesive, bounded objects, for representing animate agents and their actions, and for representing visual forms.”

At the second step, “children learn natural language expressions that contain terms for individuals of different kinds (the dog and the cat) or individuals with different identities (Evelyn and Simon; my cup and your cup; this dog, that dog, and the other one).” Thus, by conjoining singular NPs previously mastered, children are able to form conjunctive NPs denoting pluralities of entities. These may be of the same kind, as with ‘my cup and your cup’, or different, as with ‘the dog and the cat’. Importantly, Spelke assumes that conjunctive NPs are co-denoting with numerically modified NPs – e.g. ‘my cup and your cup’ is co-denoting with ‘two cups’, and ‘this dog, that dog, and the other one’ is co-denoting with ‘three dogs’. Specifically, both kinds of phrases “refer to sets of two or three distinct individuals.” In this way, Spelke proposes that children “learn part of the meanings of the first number words.”

At the third step, children establish mappings from NPs involving the first three count words to two sorts of mental representations: (i) “representations of numerical magnitudes, delivered by the ANS,” and (ii) “representations of arrays of 1–3 individual objects,” delivered by the SNS. The mapping to ANS representations is intended to enable children to “discover that expressions like three dogs and Rover, Fido, and Lassie refer to sets of a larger numerical magnitude than the sets picked out by expressions like two dogs or Rover and Fido.” Thus, Spelke seemingly assumes that the ANS is capable of comparing the cardinal sizes of the sets denoted by such expressions, at least for those denoting sets of two or three objects. The explanatory role of the mapping from NPs to “representations of arrays of 1–3 individual objects” is less clear. It appears intended to accommodate the numerical exactness of NPs, such as ‘three dogs’ – a kind of exactness that is not readily explained by the ANSs approximate representations. Hence, by virtue of establishing this pair of mappings, Spelke proposes that:

However, limits on the parallel individuation system and the ANS prevent similar comparisons for larger sets. Thus, more is required to explain how children expand their numerical vocabulary beyond ‘three’.

The final step in Spelke’s account proposes that children expand their numerical vocabulary by applying “the grammatical rules for forming expressions that refer to two or three individuals (this dog and that one; two dogs) so as to form expressions that refer to two or three sets of individuals (three cows and one more, two geese and three ducks, two groups of three puppets).” Specifically, Spelke suggests three ways in which children may “construct representations of larger numbers from representations of smaller ones.” The first is through appending ‘and one more’ to a numerically modified NP, as with ‘three cows and one more’, thus forming a phrase apparently co-denoting with ‘four cows’. The second is through conjoining numerically modified NPs, as with ‘three cows and two horses’, which, according to Spelke, “designates a set composed of a specific number of animals: five animals.” Finally, we may use group nouns, such as ‘group’ or ‘collection’, to collect pluralities of things into individuated, and thus countable, entities [Reference LandmanLandman (1989); Reference Snyder and ShapiroSnyder and Shapiro (2022)]. To use Spelke’s example, whereas we can apparently use ‘three horses and three cows’ to denote the same set denoted by ‘six animals’, we can alternatively use ‘two groups of three animals’ to apparently represent this same set.

This last example illustrates two ways of apparently referencing sets – by conjoining numerically modified NPs or by using group nouns. According to Spelke, these encode different arithmetic operations – addition and multiplication, respectively. In fact, Spelke intimates that in virtue of encoding such operations, speakers may come to recognize that the natural numbers are infinite. Quoting Reference SpelkeSpelke (2017, p. 158):

In summary, “the productive rules of natural languages, together with the resources of core knowledge, allow children to gain two critical insights”: (i) numerical expressions “can be composed to express new numbers,” and (ii) by forming complex, numerically modified NPs, “numbers can be added and multiplied to form other numbers.”

3.4.2 Troubles for Spelke

In our view, the critical problem with Reference SpelkeSpelke’s (Reference Spelke2017) proposal is that it mischaracterizes the developmental target – count words that children come to understand. To succeed, Spelke’s proposal should explain how children learn meanings that count words actually have. However, the meanings Spelke assigns to numerically modified NPs, such as ‘two dogs’, are simply not plausible, empirically speaking. So, while Spelke’s account could perhaps explain how agents learn the meanings of expressions in some possible language, we maintain that it fails to explain how children learn numerical expressions in any extant natural language, such as English.

To appreciate the point, we need to say a bit more about Spelke’s semantic assumptions. Though these are largely implicit in her presentation, she appears to suppose something like Reference Chierchia and RothsteinChierchia’s (1998a,Reference Chierchiab) semantics for nouns. This is illustrated in Figure 3, where ‘ $a$ ’, ‘ $b$ ’, and ‘ $c$ ’ denote urelements, and arrows represent the subset relation ( $\subseteq$ ).

Figure 3 A set-based semantics for nouns.

According to this analysis, whereas the singular noun ‘dog’ denotes urelements, in the form of dogs, the plural noun ‘dogs’ denotes sets resulting from taking the union of their unit sets.Footnote ¹⁵ The question here is how this analysis may be extended to provide a semantics similar to that plausibly assumed by Spelke.

First, consider definite singular NPs, such as ‘Rover’, ‘the dog’, or ‘this dog’. Since these denote particular objects of a given kind, according to Spelke, such expressions are singular terms, referring directly to those entities. This is reflected in (8), where ‘ $r$ ’ represents Rover.

1. (8) $〚$ Rover/the dog/this dog $〛=r$

Second, consider conjunctive NPs, such as ‘Rover and Snoopy’ or ‘this dog and that dog’. These denote sets of entities referenced by each conjunct, according to Spelke, as suggested in (9).

1. (9) $〚$ Rover and Snoopy $〛=\{r,s\}$

Hence, the most straightforward compositional analysis would have it that ‘and’ in the expressions of interest denotes set-union, as suggested in (10).

1. (10) $〚$ and $〛=\lambda S.\lambda S^{\prime }.S\cup S^{\prime }$

Third, consider numerically modified NPs, such as ‘two dogs’. These are co-denoting with conjunctive noun phrases, according to Spelke. Hence, ‘two dogs’ should share a denotation with, e.g. ‘Rover and Snoopy’.

1. (11) $〚$ two dogs $〛=\{r,s\}$

Fourth, consider elliptical phrases like ‘one more (dog)’, as in ‘two dogs and one more (dog)’. Since the latter denotes a set consisting of three dogs, according to Spelke, the elliptical phrase presumably denotes a unit set of the same kind denoted by the preceding NP. Combining this with the previous denotations for ‘and’ and ‘two dogs’ thus results in (12), where ‘ $t$ ’ represents a third dog.

1. (12) $〚$ two dogs and one more (dog) $〛=\{r,s,t\}$

Similarly, (10) and (11) jointly predict that the conjoined numerically modified NP ‘three dogs and three cats’ denotes a set containing three dogs and three cats – the same set denoted by ‘six animals’, according to Spelke.

1. (13) $〚$ three dogs and three cats $〛=〚$ six animals $〛=\{r,s,t,a,b,c\}$

In this respect, conjoined numerically modified NPs can perhaps be seen as encoding addition, as Spelke suggests. Finally, consider group NPs, such as ‘group of animals’. Since ‘two groups of three animals’ is co-denoting with ‘three dogs and three cats’, according to Spelke, group NPs presumably denote sets of things described by the accompanying noun. For example, ‘group of animals’ presumably denotes a set of animals. This, combined with Spelke’s assumption that conjunctive phrases of the form ‘ $a$ and $b$ ’ are co-denoting with ‘two $F$ s’ whenever $a$ and $b$ are both $F$ s, may imply that ‘two groups of three animals’ denotes the union of two sets consisting of three animals, thus leading to the same denotation given in (13). In this way, numerically modified group NPs can perhaps be similarly understood as codifying multiplication, as Spelke suggests.

Though we think that the preceding analysis is the one most likely intended by Spelke, it should be evident that it cannot be correct as an account of the actual denotations of English expressions. The most significant problem concerns the assumed denotations of numerically modified NPs, such as ‘two dogs’. Contra Spelke, these cannot generally be co-denoting with conjunctive NPs, such as ‘Rover and Snoopy’. Consider (14), for instance, where each name refers to a different dog.

1. (14) Snoopy and Rover are two dogs, and so are Toby and Ulyses.

Spelke’s analysis wrongly predicts that (14) should be false in this context, since the two conjunctive NPs would refer to different sets.

What’s gone wrong? Conceptually speaking, the problem with (11) appears to be that it involves what Reference GeachGeach (1972) calls quantificatious thinking: that all expressions occupying subject position serve the same kind of semantic function, namely referring, and that the referents of such expressions are all of the same kind, namely sets. Yet as Frege made clear long ago, there is little empirical plausibility to the claim that obviously quantificational expressions, such as ‘nothing’ and ‘everything’, function as singular terms [Reference Kratzer and HeimKratzer and Heim (1998, Ch. 6)]. In terms of semantic types, the problem is that ‘two dogs’, when occurring in subject position, has the wrong type to function this way.

1. (15) {Two/All/No} dogs are on the porch.

Within contemporary semantic theory, the phrases underlined in (15) are standardly assumed to have the type of generalized quantifiers, denoting sets of sets [Reference Barwise and CooperBarwise and Cooper (1981)]. For example, ‘two dogs’ denotes the set of pairs of dogs. Thus, whereas one such pair may include, say, Rover and Snoopy, it may also include others, say, Toby and Ulyses.Footnote ¹⁶

Though the above objections principally target Spelke’s assumptions regarding the semantics of numerically modified NPs, they also ramify for her ontogenetic story. Most obviously, they undermine Steps 2 and 4 of the proposal. At Step 2, children are supposed to “learn part of the meanings of the first number words,” by recognizing that numerically modified NPs, such as ‘two dogs’, are co-denoting with conjoined NPs, such as ‘this dog and that dog’. But since such expressions are not co-denoting, a child who underwent Step 2 would, in fact, seriously misunderstand numerically modified NPs.

A similar problem afflicts Step 4. Here, children are presumed to expand their numerical vocabulary by applying “the grammatical rules for forming expressions that refer to two or three individuals.” Importantly, this vocabulary expansion succeeds in virtue of these newly formed complex NPs combining denotations from previously understood, component NPs: “The sets to which these noun phrases refer inherit the properties of the sets that are the referents of noun phrases containing the first three number words.” Clearly, then, no vocabulary expansion could take place unless these component expressions are already understood. Yet for each kind of complex phrase Spelke deems relevant to Step 4, the component NPs are numerically modified NPs whose meanings are supposed to be grasped at Step 2. Again, however, numerically modified NPs are not plausibly synonymous with conjunctive NPs. Thus, without some further, independent account of how understanding the latter could lead to understanding the former, we have little reason to think that Step 4 is achievable.

To summarize: though Spelke’s proposal offers an intriguing account of how children come to understand count words, we maintain that it suffers substantial theoretical difficulties. In Section 4.2.3, we critically assess a more fundamental assumption that, while widely adopted within the Mainstream, is prominently illustrated by Spelke’s proposal, namely whether the denotations of number words and number concepts are best construed in set-theoretic terms. For the moment, however, we turn our attention to SNS-dominant models of the sort most notably defended by Susan Carey.

3.5.1 Conceptual Continuity and Quinean Bootstrapping

As a preliminary, it is worth locating Carey’s proposal within the context of a broader debate that has not figured prominently in our discussion so far, but which is central to developmental psychology, at least since Piaget – issues regarding the possibility of conceptual discontinuities. A long-standing anxiety, both in philosophy and psychology, concerns how it is so much as possible for an individual to learn novel concepts – to learn concepts they did not possess at some prior time. Prima facie, we routinely perform this feat. For example, when learning science or mathematics, students seemingly learn such concepts as ELECTRON, QUARK, PHONEME or POLYNOMIAL.

Some nativists – most famously, in recent times, Jerry Fodor – have maintained that appearances are misleading – that such apparently novel concepts are not, in fact, learned, but rather innate [Reference Fodor and FodorFodor (1981)]. Yet few developmentalists are much inclined to accept this sort of “mad dog nativism” [Reference CowieCowie (1998)]. Rather, a specific model of concept learning has tended to prevail: one in which we learn new concepts via processes which combine antecedently available representations. To illustrate, consider the widespread suggestion that concepts are acquired via a process of inductive extrapolation involving hypothesis formation and testing [Reference Fodor and FodorFodor (1981)]. On such a view, for example, acquiring BACHELOR would result from forming and then confirming a hypothesis of something like the following form,

where the right-hand side of the hypothesis specifies conditions under which BACHELOR is correctly applied. Though the details of this proposal needn’t detain us, three related demands of this view should be appreciated:

The Correctness Requirement:

Learning BACHELOR requires that one already possesses whatever concepts are needed to formulate a correct hypothesis regarding the application of BACHELOR. For if one did not, the hypothesis could not be formulated.

The Non-Circularity Requirement:

On pain of explanatory circularity, the concept being acquired cannot also be used on the right-hand side of the hypothesis. For example, the hypothesis could not be:

The concept BACHELOR applies to x if and only if (and because) x is a BACHELOR.

For this would presuppose that the agent already possesses the concept whose acquisition we seek to explain.

The Basing Requirement:

The previous two requirements imply that learning a concept $C$ requires the prior possession of whatever other concepts are required to accurately formulate $C$ ’s application conditions. Suppose, for example, that the relevant hypothesis is:

The concept BACHELOR applies to x if and only if (and because) x is UNMARRIED and x is MALE, and x is an ADULT, and x is HUMAN.

Then, the learner must antecedently possess the concepts UNMARRIED, MALE, ADULT, and HUMAN. That is, the learner must antecedently possess a relevant conceptual base.

1. (CC) CS1 and CS2 are conceptually continuous if and only if all the thoughts expressible in CS2 are expressible using only the concepts of CS1. That is, the expressive power of CS2 is no greater than that of CS1.

Moreover, if we further suppose that concepts are only learned via the aforementioned method, then we appear committed to the following theses:

Primitive Basing:

There is a base of primitive conceptual representations – representations that do not have other representations as proper parts.

Continuity Thesis:

All acquired conceptual systems are conceptually continuous with the initial, primitive base system.

Though such a view has been defended by some very influential figures [e.g. Reference MacnamaraMacnamara (1986)], it has striking implications that many have found implausible. With regard to mathematics, specifically, Jean Piaget famously observes:Footnote ¹⁷

In light of this, many developmentalists have denied the Continuity Thesis, seeking instead to provide accounts of conceptual change that allow for discontinuities – changes where some of the thoughts expressible in CS2 are not expressible using only the concepts of CS1. Carey herself seems to be developing a view of this sort by characterizing a kind of learning she calls Quinean bootstrapping – “Quinean” because she claims to detect the general idea in the work of W.V.O. Quine, “bootstrapping” because, as with the well-worn metaphor of pulling oneself up by one’s bootstraps, the proposed process enables progress in a manner which, at first glance, appears impossible. As Reference CareyCarey (2000, p. 20) puts it:

Specifically, according to Carey, bootstrapping permits the acquisition of a more powerful conceptual system on the basis of less powerful ones.

So what is Quinean bootstrapping, exactly? Though Carey’s characterization is impressionistic, the following presumed features are clearly central. First, it is intended to effect conceptual discontinuities. Second, it is supposed to be a species of learning process. In particular, it is a computational process which involves operations on mental representations and preserves rational relations between inputs and outputs. Moreover, it recruits a wide array of other cognitive processes combined in various ways, including analogical mapping, deduction, and hypothesis formation and testing.

Third, Quinean Bootstrapping is assumed to rely on the initial availability of what Carey calls a placeholder system: a system of explicit symbols, such as a list of words in a natural language, which initially is “at most only partially interpreted with respect to antecedent concepts.” More specifically, the meanings initially attributed to the symbols in the placeholder system, by the learner, are exhausted by their (presumably nonsemantic) relations to each other, as opposed to being “interpreted in terms of already existing representations” [Reference CareyCarey, (2009a, p. 246)].

Finally, according to Carey, Quinean bootstrapping is a kind of “modeling process,” in that as a result of deploying the sorts of processes mentioned earlier, the learner is ultimately able to assign an interpretation to symbols in the placeholder system. According to Carey, it is by assigning such interpretations that the learner comes to possess new concepts.

3.5.2 Bootstrapping the Count List

Broadly speaking, on Carey’s account, children acquire number concepts in the course of learning how to count. Her proposal may be divided into three stages, each corresponding to periods in the developmental timetable outlined in Section 2.4.

Stage 1, occurring around two years of age, consists in the child learning the initial placeholder system, i.e. the count routine. During this period, the child learns to count intransitively and to satisfy the One-to-One Principle. According to Carey, children do not use count words to express any specific concepts at this point. Rather, count words are merely uninterpreted, perhaps phonologically individuated, tokens. To the extent that they have any meanings for the child, these are initially exhausted by their relations to each other. Specifically, words in the count list “get initial interpretations as part of a placeholder structure, the count list itself, in which meaning is exhausted by the fact that the list is ordered” [Reference CareyCarey (2009a, p. 19)].

Stage 2, corresponding to the subset-knower stage, consists in acquiring an understanding of the meanings of the first four count words. According to Carey, this results from mapping each word to a “long-term memory model” – a representation created by the parallel individuation system and stored in long term memory. Carey speculates that such models represent sets while also containing symbols representing individuals – “{i}, {j k}, {m n o}, {w x y z}” [Reference CareyCarey (2009a, p. 19)]. Moreover, she proposes that the first four count words have their meanings in virtue of a computational procedure for assigning count words to sets:

Thus, in the mouths of subset-knowers, the meanings of the first four number words are determined by a computational procedure in which they participate: a procedure ensuring that each word can be applied to any set in 1-1 correspondence with an appropriate set-theoretic representation stored in long-term memory.

Stage 3, which centers on the transition to becoming a CP-knower, involves mastery of the Cardinal Principle (Section 2.4). To appreciate Carey’s account, it is important to distinguish this from an initially similar principle, namely what [Reference FusonFuson (2012)] calls “The Last Word Rule”

LWR:

The last count word used in the counting procedure answers the relevant ‘how many’-question posed when performing that procedure.

We find an analogy to Reference SearleSearle’s (1982) Chinese room helpful here. In this familiar thought experiment, a non-Chinese speaker is handed cards containing Chinese characters through a slot in a door. There are instructions in the person’s native language for processing each card, producing further cards with Chinese characters, and passing these new cards through another slot. We may imagine that the person becomes so good at following these instructions that they process cards in a time comparable to a native Chinese speaker, even without the assistance of instructions. In that case, the two speakers might appear behaviorally equivalent: for each input, the two speakers return the same output, in a comparable time. Yet unlike the Chinese speaker, the non-Chinese speaker does not understand what is on the cards. Rather, they are merely following a procedure, albeit very well.

The analogy is this: In the context of counting, mastering LWR merely enables the child to deliver responses that are appropriate. Indeed, a child grasping LWR could be very good at performing the transitive counting routine without grasping the meanings of count words or possessing count concepts. In contrast, according to Reference CareyCarey (2009a), CP-knowers are like Searle’s Chinese speakers – when grasping the Cardinal Principle, they don’t merely respond appropriately, they also understand the meanings of the count words, and so possess count concepts. In making this claim, Carey appears to be applying similar considerations to those militating in favor of attributing number concepts to subset-knowers. For instance, the standard reason for attributing the concepts ONE, TWO, and THREE to three-knowers is that they manifest successful performance on Give-N and related tasks, up to the word ‘three’. Analogously, since CP-knowers successfully perform such tasks for all count words on their count list, similar considerations suggest the possession of concepts for all those count words.

Suppose with Reference CareyCarey (2009a) that on becoming CP-knowers, children tacitly recognize that count words denote what Carey calls cardinal values, or collections of equinumerous sets. The challenge is to explain how they come to grasp the meanings of all words in their count list, and not just the first four. To this end, Carey invokes bootstrapping. Specifically, she proposes that children recognize a structural analogy between count words and the sets they denote. Count lists are linearly ordered.Footnote ¹⁸ That is, just as ordering the natural numbers by $<$ results in the sequence

$1<2<3<\dots$ ordering count words by the relation of coming next in the count list, represented here as ‘ $\prec$ ’, results in a similar sequence:

‘one’ $\prec$ ‘two’ $\prec$ ‘three’ $\prec \dots$ Now, consider the following relation on sets, which we call cardinal successor, represented as ‘ $\ll$ ’.

1. (16) $S\ll S^{\prime }$ if and only if for some $x\notin S$ , there’s a one-to-one correspondence between $S\cup \left\{x\right\}$ and $S^{\prime }$

According to (16), $S^{\prime }$ is the cardinal successor of $S$ just in case adding a new object to $S$ results in a set which is in one-to-one correspondence with $S^{\prime }$ .Footnote ¹⁹ Clearly, ordering sets by cardinal successor results in a linear ordering. For example, we have

$\left\{a\right\}\ll \{a,b\}\ll \{a,b,c\}\ll \dots$ This appears to be the presumed structural analogy CP-knowers tacitly recognize: count words ordered by $\prec$ and the sets they are assumed to denote, ordered by $\ll$ , form isomorphic structures [Reference Cheung, Rubenson and BarnerCheung et al. (2017)].

Importantly, recognizing this analogy licenses what Reference CareyCarey (2009a, p. 289) calls “the crucial induction”: “For any symbol in the numeral list that represents cardinal value n, the next symbol on the list represents cardinal value $n+1$ .” More precisely, Carey’s comments suggest the following metasemantic principle governing the presumed denotations of count words.

1. (MP) For every count word $\alpha$ , there’s a unique next count word $\beta$ such for any set $S$ in the extension of $\alpha$ , if an object $x$ is not in $S$ , then adding $x$ to $S$ results in a set $S^{\prime }$ in the extension of $\beta$ .

In plain English, (MP) implies the conjunction of two claims:

The Morphosyntactic Condition:

Every count word is followed by a unique next count word.

The Semantic Condition:

If a count word is true of sets containing $n$ -many things, the next count word is true of sets that result from adding an extra object.

The Successor Principle:

For each natural number, there is a unique natural number immediately following it.

Under the aforementioned assumptions, then, if children become CP-Knowers as a result of learning (MP), then they should also come to tacitly know, or at any rate be in a position to infer, the Successor Principle. Accordingly, Reference CareyCarey (2009b, p. 251) describes the key cognitive difference between subset-knowers and CP-knowers as follows:

The suggestion appears to be that in virtue of bootstrapping (MP), CP-knowers tacitly possess knowledge of successor, a powerful consequence of which is an ability to represent every natural number, in principle.

A similar idea is echoed in Reference vanMarle and BanguvanMarle (2018, p. 132):

Again, the presumption is that an ability to represent the full system of natural numbers, at least in principle, is concomitant with becoming a CP-knower, something which only makes sense if bootstrapping (MP) implies tacit knowledge of the Successor Principle.

Reference CareyCarey (2009a, p. 328) summarizes her account as follows: “I have argued here that the numeral list representation of number is a representational resource with power that transcends any single representational system available to prelinguistic infants.” In other words, bootstrapping (MP) allows CP-knowers to have thoughts which are in principle inaccessible to pre-CP-knowers, namely those involving count concepts beyond FOUR. As a result, tacitly grasping (MP) constitutes a kind of conceptual discontinuity.

3.5.3 Troubles for Carey

Carey’s rich, provocative account of number concept acquisition has been the focus of much discussion. Though, in Section 4, we critically assess some of its more general assumptions, e.g. that count words denote sets, and that natural numbers are collections of sets, in this section, we focus on a pair of issues. The first concerns the extent to which early CP-knowers understand the Successor Principle. The second concerns whether Carey’s account delivers a conceptual discontinuity in the course of bootstrapping (MP).

Bootstrapping and Discontinuity

Our second objection targets the programmatic aspirations of Carey’s account. Recall that major motivation for bootstrapping number concepts is that such learning requires a conceptual discontinuity. In what follows, however, we argue for three claims:

1. i. The notion of a conceptual discontinuity, as Carey uses it, is equivocal between two readings: a cumulative reading and a distributive reading.

2. ii. Even on the assumption that she accurately characterizes both what the child learns, and the core systems involved in this learning process, there are good reasons to suppose that her account fails to deliver conceptual discontinuities in either sense.

3. iii. Given (i) and (ii), there is no reason to suppose that the transition to CP-knowing involves a conceptual discontinuity.

Two notions of conceptual discontinuity.

Earlier we characterized the notion of conceptual continuity as a relation between conceptual systems:

1. (CC) CS1 and CS2 are conceptually continuous if and only if all the thoughts expressible in CS2 are expressible using only the concepts of CS1. That is, the expressive power of CS2 is no greater than that of CS1.

Conversely, conceptual discontinuity between two systems is the failure of continuity – i.e. where the expressive power of CS2 is greater than that of CS1. Although such relations may obtain synchronically – e.g. between mine and your conceptual systems at a time – diachronic relations of conceptual continuity and discontinuity are of special interest to developmentalists, since they concern ways in which conceptual systems may evolve over time. It is important to see, however, that we can individuate conceptual systems in two quite different ways that lead to importantly different sorts of diachronic (dis)continuities.

According to the first, cumulative conception, a conceptual system is an agent’s total conceptual resources at a time, considered jointly. Crudely, CS1 consists of all conceptual structures the agent possesses at time ${}_1$ , while CS2 consists of all conceptual structures possessed at time ${}_2$ . Thus, to say that CS2 is discontinuous with CS1 implies that the total, collective conceptual resources of the agent at time ${}_1$ are insufficiently powerful to permit thoughts the agent could have at time ${}_2$ . Plausibly, it is this notion of discontinuity figuring most prominently in, say, the disputes between Piaget and Chomsky. For example, when Jerry Fodor asserts the ‘impossibility of acquiring “more powerful” structures’, he is denying this sort of discontinuity. Call this the cumulative conception of discontinuity.

However, on an alternative conception, rather than considering CS1 as a single system consisting of all conceptual structures jointly possesses at time ${}_1$ , we instead think of CS1 as partitioned into multiple, distinct conceptual or representational systems – parallel individuation, approximate magnitude, theory of mind, language, and so on – at a time. To assert the existence of a conceptual discontinuity then implies that each of the agent’s representational systems at time ${}_1$ is such that it is insufficiently powerful to permit thoughts that the agent could have at time ${}_2$ . For example, the parallel individuation system might be insufficiently powerful to represent natural number, as might the ANS. Call this the distributive conception of discontinuity.

Below, we assess Carey’s arguments for such a discontinuity in the case of number. For the present, however, we clarify the logical relations between these two notions of discontinuity. First, the distributive notion is logically weaker than the cumulative one: a cumulative discontinuity entails a distributive discontinuity, but not vice versa. That is, if all representational systems collectively comprising CS1 cannot jointly express what CS2 can, then none of those systems individually can, but not necessarily vice versa. Conversely, the absence of a distributive discontinuity implies the absence of a cumulative discontinuity as well. Specifically, CS1 will fail to be distributively discontinuous with CS2 only if at least one of the individual systems that comprise CS1 can express what CS2 can. Yet if a proper part of CS1 can express what CS2 can, then so can the whole CS1, of course.

Which sort of conceptual discontinuity does Carey’s account secure?

The answer, we suggest, is “Neither.” First, as evidenced from the following passage, Reference CareyCarey (2011, p. 118) appears only to be arguing for a distributive discontinuity with respect to the representation of exact cardinality:

CS2 is qualitatively different from each of the CS1s because none of the CS1s has the capacity to represent the integers. Parallel individuation includes no symbols for number at all, and has an upper limit of 3 or 4 on the size of sets it represents. The set-based quantificational machinery of natural language includes symbols for quantity (plural, some, all), and importantly contains a symbol with content that overlaps considerably with that of the verbal numeral “one” (namely, the singular determiner, “a”), but the singular determiner is not embedded within a system of arithmetical computations. Also, natural language set-based quantification has an upper limit on the sets’ sizes that are quantified with respect to exact cardinal values (singular, dual, trial). Analog magnitude representations include symbols for quantity that are embedded within a system of arithmetical computations, but they represent only approximate cardinal values; there is no representation of exactly 1, and therefore no representation of 1. Analog magnitude representations cannot even resolve the distinction between 10 and 11 (or any two successive integers beyond its discrimination capacity), and so cannot express the successor function. Therefore, none of the CS1s can represent 10, let alone 342,689,455.

The argument appears to be just this: since CP-knowers understand the meanings of count words, they are capable of representing exact cardinalities; yet none of the relevant individual conceptual systems available to CP-knowers – the ANS, the language faculty, and the parallel individuation system – can by itself represent exact cardinalities; thus, representing exact cardinalities constitutes a distributive discontinuity.

Is this claim correct? Though we grant, for the sake of argument, that neither the ANS nor the parallel individuation system have the resources to represent exact cardinalities, we doubt that her claims about language are correct. Specifically, given Carey’s assumptions about what count words denote and what the relevant semantic machinery is like, language alone seemingly provides the resources required to do so.

Here’s why. According to Carey, there is an upper limit of three on the cardinalities “set-based quantification” represents. By set based quantification, Carey intends the same set-based semantics for nouns discussed in Section 3.4.2, illustrated in Figure 3. Thus, Reference CareyCarey (2009a, p. 256) writes:

Linguists such as Genaro Chierchia (1988) and G. Link (1987) show how the quantifiers of all languages, as well as the count/mass distinction, the distinction between nouns in classifier languages and nouns in languages with the count/mass distinction, and much else, can be defined over the semi-lattice depicted in [Figure 3], which creates all of the possible sets from a domain of atoms or individuals. This structure makes explicit the contrast between individuals (the bottom line) and all of the sets composed out of them. One needs explicit symbols with the content [SET] and [INDIVIDUAL], plus distributive and collective computations over those symbols, to capture the meanings of natural language quantifiers, including even the singular/plural distinction. I call this system of representation “set-based quantification.”

Apparently, then, set-based quantification includes the entire semi-lattice structure depicted in Figure 3 – the urelements, the sets of urelements, and the relations on them. Moreover, according to Carey, these resources collectively impose “an upper limit on the sets sizes that are quantified with respect to exact cardinal values.”

Yet this conclusion strikes us as unfounded. Indeed, on Carey’s own assumptions regarding set-based quantification, there is no evident reason why it cannot represent exact cardinalities far greater than three.Footnote ²⁴ On standard analyses of numerical modifiers, including that of Reference Link, Bäuerle, Schwarze and von StechowLink (1983) and Reference Chierchia and RothsteinChierchia (1998a), Reference Chierchia(1998b), cardinality predicates denote collections of precisely the sorts of entities depicted within the relevant semilattice structures. For example, a set-based semantics will naturally provide denotations for cardinality predicates like those depicted in Figure 4.

Figure 4 A set-based semantics for cardinality predicates.

These same denotations are what Carey identifies with cardinal values – collections of equinumerous sets.Footnote ²⁵ Moreover, Carey clearly identifies cardinal values with numbers: “cardinal values of sets are numbers” Reference CareyCarey (2009a, p. 136). Hence, it would appear that set-based quantification alone provides the resources for representing exact cardinal values. Furthermore, it should be evident that there is no clear upper-limit on the cardinal values representable within such a system. Indeed, for any number of urelements, $n$ , the system is capable of representing numbers up to $n$ .

We thus conclude that, by Carey’s own lights, there is little reason to posit a conceptual discontinuity for the case of natural number. More precisely, since a single core system – the language faculty – would alone suffice to represent the relevant sets, it follows that there is no distributive discontinuity in the case of natural number. Moreover, since the absence of a distributive discontinuity implies the absence of a cumulative discontinuity, it also follows that there is no cumulative discontinuity. Thus Carey’s account of natural number concepts fails to secure conceptual discontinuity in either sense.

4.2.1 Cultural Constructionism

Cultural constructionism is the thesis that numbers depend for their existence on quite specific cognitively mediated cultural practices. Reference DehaeneDehaene (2011, p. 260) seemingly adopts this view when writing:

In a similar vein, Reference CareyCarey (2009a, p. 333) writes:

To avoid confusion, Carey and Dehaene’s cultural constructionism is not intended to be a constructivism (or intuitionism) of the sort normally associated with Brouwer or Heyting.Footnote ²⁶ Nor is it a constructivism about psychological processes of the sort associated with Piaget. Rather, they appear to be endorsing a kind of ontological dependency thesis regarding the natural numbers. In a slogan: no natural numbers without human cognition (of the relevant sort). As such, numbers are not discovered by human beings, but rather invented. Specifically, both Dehaene and Carey seemingly suppose that this invention – this bringing into being – depends on the existence of counting procedures.

What should we make of this view? Given its distance from Carey and Dehaene’s primary intellectual concerns, we’re inclined to think of its endorsement is a kind of rhetorical flourish – something not intended to be central to the overall views being defended.Footnote ²⁷ For one thing, the aforementioned quotations occur in the absence of any serious argumentation. In Dehaene’s case, for example, the claim that “the integers are manmade” occurs within the context of claiming that “what occurs in the child’s mind when he suddenly understands that there is a discrete infinity of exact numbers” depends on “a cultural invention.” However, it is clearly one thing for the child’s understanding of natural numbers to rely upon cultural innovations, quite another to claim that numbers are cultural inventions. To suppose otherwise involves something akin to a use-mention confusion – to conflate the origin of numbers with that of mental states representing numbers.

Regardless, there are independent, well-known reasons for rejecting cultural constructionism. Specifically, following familiar arguments from Frege, it seems incapable of accommodating two widely assumed properties of mathematical truths:Footnote ²⁸

• Eternality: Mathematical truths were true before humans existed, and will remain true if humans cease existing. This is why it is permissible to apply mathematics not just to the present and future, but also the distant past.

• Necessity: Mathematical truths are necessary, rather than contingent. Thus, in all conceivable ways the world could be, including those in which no cultural practices exist, 3 + 2 would be equal to 5. This is why it is permissible to apply mathematics to both actual and counterfactual circumstances.

But if numbers only came to exist once humans engaged in counting behavior, then it is hard to see how arithmetic truths could have either property. After all, counting procedures might never have been invented, and mathematical truths presumably held prior to their invention. Altogether, then, cultural constructionism is rather implausible.Footnote ²⁹

4.2.2 Term Formalism

A second familiar metaphysical thesis about numbers, sometimes invoked by NCRs, is term formalism. Crudely, this is the view that natural numbers are symbols we use to talk about or represent them. Thus, (17) is true if ‘two’, which refers to itself, is an even number.

1. (17) Two is an even number.

Sometimes term formalism is explicitly endorsed in NCR. For example, Reference GallistelGallistel (2021, p. 29) recently writes: “Numbers are symbols manipulated in accord with the axioms of arithmetic.” He then proceeds to summarize the earlier view of Reference Gelman and GallistelGelman and Gallistel (1986) – itself a seminal text in NCR – which endorses “the modern formalist view of arithmetic”:

Although Gelman and Galistel’s commitment to term formalism is rather explicit, it can also be found, though perhaps less overtly, in other NCR. For example, the preface to Reference DehaeneDehaene (2011, p. xiii) begins:

Similarly, while speaking of CP-knowers, Reference Davidson, Eng and BarnerDavidson et al. (2012, pp. 3–4) suggest that items comprising a count list are numbers.

The NCR literature is replete with similar comments. However, such pronouncements only make sense if numbers are symbols: if what is etched on credit cards, engraved on coins, or recited are themselves numbers.

Term formalism is well-known within philosophy primarily because of what Reference Weir and ZaltaWeir (2022) colorfully describes as “a demolition job by a great philosopher, Gotlob Frege.” Indeed, Reference FregeFrege’s (1903) polemic against term formalism has long been widely regarded as decisive. Here, we list a couple well-known problems, also due to Frege, as articulated by Reference LinneboLinnebo (2017):Footnote ³⁰

• Identity: According to term formalism, an arithmetic identity statement such as ‘3 + 2 = 5’ (or ‘Three plus two equals five’) is true just in case ‘3 + 2’ is coreferential with ‘5’. However, assuming both terms refer to themselves, this is false. Thus, term formalism is committed to an empirically unsupported ambiguity in the equative copula: it expresses the familiar notion of identity when flanked by nonmathematical terms, e.g. ‘Superman’ and ‘Clark Kent’, and an altogether different notion when featuring mathematical terms.

• Infinity: Initially, the philosophical motivation for adopting term formalism is metaphysical and epistemological. Specifically, how can we know that 3 + 2 = 5 if numbers are abstract [Reference BenacerrafBenacerraf (1973)]? Thus, it is tempting to identify numbers with concretia, in the form of linguistic tokens, e.g. particular inscriptions of, e.g. ‘3 + 2’ and ‘5’. But given the truth of universally quantified statements like ‘Every natural number has a successor’, infinitely many numbers are required. Yet we have no good reason to suppose that infinitely many concrete linguistic tokens exist. Indeed, as Reference FregeFrege (1884, §123) quips: “We have neither an infinite blackboard, nor infinitely much chalk at our disposal.” Hence, the only plausible form of term formalism is one which posits infinitely many expression types, and types are presumably abstract. Hence, the purported epistemic benefit of adopting term formalism – direct perceptual access to mathematical objects – is lost.

Again, such considerations render term formalism rather implausible.

4.2.3 The Frege–Russell Characterization

A third metaphysical thesis, which we encountered in Section 1.1 and again in Section 3.5, is known as the Frege–Russell characterization, because of its association with Reference FregeFrege (1884) and Reference RussellRussell (1919). According to it, natural numbers are collections of equinumerous, or bijective, sets. For example, the number two is just the collection of all two member sets. Again, as Reference Rips, Bloomfield and AsmuthRips et al. (2008) point out, this appears to the be the predominant view among NCRs. For example, as we saw in Section 3.5.2, Reference CareyCarey (2009a, p. 136) adopts this view when identifying numbers with “cardinal values.” Similarly, Reference vanMarle and BanguvanMarle (2018, p. 131) writes: “As described by Frege (Reference FregeFrege, 1884/1980), formally, numbers are a special kind of sets.”

Within NCR, the Frege–Russell characterization is often adopted as a kind of metasemantic thesis. Specifically, NCRs often claim that count words or concepts denote collections of sets, and thus natural numbers. For example, speaking of the denotation of ‘two’, Reference vanMarle and BanguvanMarle (2018, p. 131) writes:

Expanding on this, Reference Bloom and WynnBloom and Wynn (1997, p. 512) write.

The reference here is to Reference FregeFrege’s (1884, §46) well-known contention that “statements of number,” such as (18), are not really about objects, e.g. Elmos, but rather Fregean concepts, such as being two in number.

1. (18) Those are two Elmos.

Quoting Reference FregeFrege (1884, §46) directly:

Speaking in Fregean terms, Frege’s claim is that to assert (18) is to assert that a certain first-level concept – ‘is an Elmo’ – falls under a certain second-level concept – ‘is two (in number)’. Given the Frege–Russell characterization, this second-level concept just is the number two.

In more contemporary semantic parlance, second-level concepts are generalized quantifiers (Section 3.4.2): second-order predicates which take first-order predicates as arguments and return a truth-value. Prototypical examples include the underlined expressions in (19).

1. (19) {Every person/Nothing/Some number} is interesting.

Within Generalized Quantifier Theory [Reference Barwise and CooperBarwise and Cooper (1981)], these take the predicate ‘is interesting’ as argument, and return True just in case the set of interesting things is a superset of the people, or is disjoint with everything, or overlaps with the numbers, respectively. In the previous quote, Bloom and Wynn suggest that ‘two’ in ‘two black cats’ belongs to this same category: it is a second-order predicate true of the first-order predicate ‘is a cat’. Furthermore, and crucially, it denotes a number.

As we saw in Section 4.1, as a metaphysical thesis, the Frege–Russell characterization is only one of several formal characterizations of the naturals. Our primary concern here is with the implications of this thesis for the issue of what number words denote. If the Frege–Russell characterization were correct, then such entities should presumably be the referents of number words used referentially – i.e. numerals. But contrary to what many NCRs suppose, there are good reasons to doubt that such uses of number words denote these entities.

The first objection targets the semantic function of number words. Number words have nonreferential uses, as witnessed by ‘two’ in (18), as well as referential uses, as witnessed by ‘two’ in (20).

1. (20) Two is an even number.

Within contemporary semantic theory, these different occurrences have different semantic types [Reference SnyderSnyder (2021b)]. Moreover, what an expression denotes is determined by its semantic type, so that if different occurrences of the same expression have different semantic types, then they must also have different denotations. It follows that the different occurrences of ‘two’ in (18) and (20) cannot be co-denoting. In particular, if ‘two’ in (18) denotes a collection of sets, then, contrary to fact, ‘two’ in (20) would not refer to a number – not, at any rate, if that number just is that collection of sets.

In more detail, we can formulate the problem based on a principle which is widely assumed within linguistic semantics, namely:

Denotation-to-Type:

For any two occurrences $o_1$ and $o_2$ of an expression $\alpha$ , $o_1$ and $o_2$ have the same denotation only if $o_1$ and $o_2$ have the same semantic type.

1. 1. a. The Elmos on the table are red (in hue).

   2. b. Those are red Elmos (on the table).

   3. c. Red is a primary color.

On standard accounts, the different occurrences of ‘red’ here have different semantic types, and thus different denotations [Reference McNally, de Swart and AloniMcNally et al. (2011)]. Specifically, in (21a), ‘red’ has the type of a predicate, denoting red things; in (21b), ‘red’ has the type of a modifier, denoting subsets of red things, in this case Elmos on the table; and in (21c), ‘red’ has the type of a name, referring to the color red, as an entity.

Of course, as we’ve seen, number words exhibit a similar pattern of uses [Reference Snyder, Samuels and ShapiroSnyder et al. (2022)]. Specifically, while ‘two’ in (20) has the type of a name, referring to a number, as an entity, ‘two’ in (1a) has the type of a predicate, denoting pluralities of two things, and ‘two’ in (18) has the type of a modifier, denoting pluralities which are both two in number and, in this case, Elmos on the table.

1. (1a) The Elmos on the table are two in number.

Now, here’s the argument. The different occurrences of ‘two’ above have different semantic types. Specifically, whereas ‘two’ in (20) has the type of a name, ‘two’ in (18) has the type of a modifier, which according to Bloom and Wynn is that of a generalized quantifier. Thus, by Denotation-to-Type, these different occurrences cannot be co-denoting, contrary to what would be required if numbers were collections of equinumerous sets.

A second problem targets what we call

The Identity Thesis:

Natural numbers are cardinalities.

To see why, consider Reference FregeFrege’s (1884) highly influential analysis, whereby all expressions underlined in (22a-c) are coreferential singular terms, referring to the same number.

1. 1. a. Two is an even number.

   2. b. The number two is even.

   3. c. The number of Elmos (on the table) is two.

Yet despite its considerable influence, there is ample evidence that as a proposal about the semantics of natural language expressions, Frege’s identification of the natural numbers with finite cardinalities is mistaken. Specifically, the evidence suggests that natural languages distinguish the sorts of things referenced by singular terms like those in (22a,b) – numbers – from those apparently referenced by terms like those in (22c) – cardinalities.

Specifically, following Reference MoltmannMoltmann (2013), let’s call phrases of the form ‘the number of Elmos’ the number of-terms, and those like ‘the number two’ explicit number-referring terms. A wealth of semantic contrasts, due mostly to Moltmann, reveal that these are not plausibly coreferential. One kind of data concerns predicates like ‘count’, ‘compare’, and ‘surprise’. As (23a-d) suggest, even if Mary happened to count, compare, or be surprised by two Elmos, it does not follow that she counted, compared, or was surprised by an abstract arithmetic object.

1. 1. a. Mary counted the number {of Elmos/??two}.

   2. b. Mary compared the number {of Elmos/??two} to the number of men.

   3. c. Mary was surprised by the number {of Elmos/??two}.

   4. d. The bridge collapsed because of the number {of cars/??200}.

Yet this is not to be expected if ‘the number of Elmos’ and ‘the number two’ both refer to the number two. We see parallel contrasts in (24a,b).

1. 1. a. The number of Elmos on the table is {two/??the number two}.

   2. b. The number Mary is writing about is {two/the number two}.

On Frege’s analysis, (24a,b) are both identity statements, equating the referents of coreferential singular terms. But since we generally expect coreferential expressions to be acceptably substitutable within identity statements, it is difficult to see how there could be a difference in acceptability in (24a) without there being an analogous difference in (24b). Worse, we see a similar contrast with overt questions [Reference SnyderSnyder (2017)].

1. 1. a. How many Elmos are on the table? {Two/??The number two}.

   2. b. Which number is Mary writing about? {Two/The number two}.

Cardinalities, recall, characteristically answer ‘how many’-questions. Thus, the fact that ‘the number two’ is not acceptable in (25a) suggests that it cannot refer to a cardinality.Footnote ³¹

Based on these and similar contrasts, Reference MoltmannMoltmann (2013) and Reference SnyderSnyder (2017) both conclude that natural language draws an ontological distinction between numbers and cardinalities. For our purposes, however, the crucial point is that the Identity Thesis, when combined with other independently plausible assumptions, implies numerous false predictions. Thus, we have excellent reason for rejecting it.

So far we have argued on semantic grounds that the Frege–Russell characterization is not plausible as an account of what natural numbers are. Still, it might be thought that only a modest refinement of what NCR is about is needed. Specifically, since NCR is principally concerned with our abilities to determine and distinguish cardinalities, one might plausibly suggest that cardinalities, not natural numbers, are the primary denotata of early numerical cognition. Thus, one might propose the Frege–Russell characterization as an adequate account of those entities. However, we conclude this section by sketching reasons for doubting even this position. Specifically, there are reasons to doubt that the denotations of number words and number concepts should be characterized set-theoretically, at all.

Before considering our concerns regarding the aforementioned position, a terminological observation is in order. Within NCR, ‘set’ is used in two importantly different senses. First, there’s an informal sense, roughly synonymous with ‘collection’, ‘plurality’, or ‘group’, often used when discussing various cognitive systems, such as the ANS or the SNS, which are said to track or represents “sets” in this sense. However, NCRs also often use the mathematical sense of ‘set’, applicable to entities described by set theory. As emphasized in Section 3, the latter is often adopted when characterizing NUMBER and various concepts purportedly required to possess it, such as EQUINUMEROSITY or SUCCESSOR. Moreover, it is this latter mathematical sense of ‘set’ that it is used in the Frege–Russell characterization of natural number.

In light of this potential equivocation, it bears emphasizing that while NCRs often suppose that representing cardinality requires possessing mathematical representations, such as SET, ELEMENT, and so on, this is not the only viable option. In fact, cardinality is just as readily represented within mereological frameworks, such as that Reference Link, Bäuerle, Schwarze and von StechowLink (1983), illustrated in Figure 5.

Figure 5 A mereology-based semantics for nouns.

Indeed, most extant analyses of number expressions assume precisely such a framework [Reference SnyderSnyder (2021b)]. Hence, it is entirely possible that understanding the meanings of such expressions depends only on the possession of mereological representations – e.g. ATOM, SUM, and PARTHOOD.Footnote ³²

A second well-known problem concerns perceptual access. The following thesis appears widely assumed within NCR:

Cardinality Perceivability:

Some cardinalities are perceivable.

Set Perceivability:

Some sets are perceivable.

Impurity:

Impure sets whose members are concrete are themselves concrete.Footnote ³³

4.3.1 Synchronic Pluralism

The first kind of pluralism we defend concerns number concepts as possessed by adults having linguistic competence with number words:

Synchronic Pluralism:

Quotidian adult numerical thought is such that for each number word in a natural language, there are multiple corresponding number concepts.

Polysemy:

Different occurrences of the same number word in different syntactic environments express different, related meanings.

1. 1. a. Mars’ moons are two (in number).  (predicative)

   2. b. Those are (Mars’) two moons.   (attributive)

   3. c. Mars has two moons.   (quantificational)

   4. d. Mars’ moons number two (in total).   (verbal complement)

   5. e. The number of Mars’ moons is two.   (specificational)

   6. f. Two is an even number.   (numeral)

   7. g. The number two is even.   (predicative numeral)

   8. h. Michael Jordan is roughly two meters tall.   (measurement)

   9. i. Rafael Nadal is ranked number two in the world.   (ordinal)

The labels here indicate how ‘two’ is being used in the accompanying example, i.e. how it’s functioning semantically. For example, ‘two’ in (26a) is used predicatively, since it functions as a predicate, similar to ‘large’ in (27a). Similarly, ‘two’ in (26a) is used attributively, since it functions as an attributive adjective, similar to ‘large’ in (27a).

1. 1. a. Mars’ moons are large (in size).

   2. a. Those are (Mars’) large moons.

Since an expression’s function is determined by its semantic type in a given syntactic context [Reference Partee, Groenendijk, de Jongh and StokhofPartee (1986)], many of the occurrences of ‘two’ in (26) have different semantic types. Moreover, since the meaning of an expression is also a function of its semantic type, (26) is witness to the fact that ‘two’ is (radically) polysemous [Reference SnyderSnyder (2021b)].

In light of Polysemy, there is a clear route to Synchronic Pluralism. First, it is widely assumed that understanding a natural language expression requires the possession of a corresponding concept, having the same content (Section 2.2). For example, understanding (26a), where ‘two’ is used to predicate a cardinality property, requires possessing a concept having the same content, i.e. one representing a cardinality property. However, on standard assumptions, concepts are at least partially individuated by their contents. Thus, the concept required to understand ‘two’ in (26a) is distinct from the one required to understand ‘two’ in, say, (26f), where ‘two’ is being used to refer. Indeed, the concepts differ not just in their contents, but also their representational functions: one functions predicatively, the other referentially.

Thus, rather than supposing that there is a single concept, TWO, corresponding the word ‘two’, plausibly there are at least two concepts corresponding to the word ‘two’:

• TWO ${}_{\text{num-ref}}$ : A concept referring to the number two, as an entity. Call concepts of this sort numerical concepts.

• TWO ${}_{\text{card-pred}}$ : A concept predicating the property of being two in number of collections. Call concepts of this sort cardinal concepts.

Indeed, though we won’t argue for this here, we think that understanding the various uses of ‘two’ in (26) plausibly requires the possession of even more concepts, including, but not limited to:

• TWO ${}_{\text{card-ref}}$ : A concept referring to the cardinality property of being two in number, as an entity.

• TWO ${}_{\text{ord}}$ : A concept specifying the ordinal position of an object among a linearly ordered class of objects.

In any case, it would appear that the polysemy of number words, along with standard representationalist commitments, jointly entail a commitment to Synchronic Pluralism.

4.3.2 The Complex Transition Thesis

Much of developmental NCR focuses on number cognition up to the age of around $3\frac{1}{2}$ or 4, when children become CP-Knowers. However, development doesn’t end here, of course. In particular, NCR must also explain the subsequent development of arithmetic abilities, from approximately five to eight years, when children are exposed to basic symbolic arithmetic, and learn to manipulate numerical symbols, such as Arabic numerals, to solve arithmetic problems (Section 2.4).

One obvious question concerning this period of development is whether it requires the acquisition of novel concepts, or instead whether those concepts possessed through learning to count suffice. Roughly put, are the concepts CP-knowers possess in virtue of understanding count words – e.g. ‘one’, ‘two’, and ‘three’ – the same as those required to interpret symbolic numerals in basic arithmetic equations – e.g. ‘1 + 2 = 3’?

One potential answer here is

Simple Transition:

Early CP-knowers already possess those number concepts required to understand symbolic numerals.

$3\frac{1}{2}$

To be clear, Simple Transition is “simple” in that it minimizes the stock of concepts required to successfully do elementary arithmetic. Specifically, because the concept required to understand, say, the count word ‘two’ is the same as the one required to understand the symbolic numeral ‘2’, only one collection of number-related concepts is needed to explain the transition from counting to elementary arithmetic. Below, we will entertain a competing hypothesis, which is “complex” in virtue of positing distinct collections of number-related concepts – one for understanding count words, and one for understanding symbolic numerals.

For now, suppose Simple Transition is correct, in which case understanding symbolic numerals only requires possessing count concepts. But what are these concepts, exactly? In light of the empirical evidence, the concepts most plausibly attributed to early counters are those like TWO ${}_{\text{card-pred}}$ , which involve ascribing cardinality properties to collections. To see why, consider the sorts of achievements typically taken as evidence of concept possession. These involve cardinality – e.g. successfully answering ‘how many’-questions, or handing over a specific number of items. Thus, such studies do not provide evidence for the possession of TWO ${}_{\text{num-ref}}$  – a concept referring to the number two. Rather, they more plausibly provide evidence for TWO ${}_{\text{card-pred}}$  – a concept predicating a cardinality property.

Thus, under present assumptions, learning basic arithmetic will only require concepts used for representing cardinality. Indeed, quoting Reference ButterworthButterworth (2005, p. 3):

The suggestion appears to be that the child associates count concepts – those already possessed in virtue of understanding count words – with their symbolic numeral counterparts while associating similar kinds of cardinal meanings with basic arithmetic operations and relations, such as ‘+’ and ‘=’ (Section 2.4). Mastering basic arithmetic is thus not a matter of acquiring new concepts, but rather learning increasingly sophisticated ways of manipulating what the child is already capable of representing – cardinalities.

Simple Transition has much to recommend it. Specifically, in the absence of countervailing evidence, it provides a simple and elegant account of the transition between counting and more sophisticated numerical competences. However, we maintain that there are good reasons to reject it. As Reference FregeFrege (1884) observed long ago, symbolic numerals, as they occur in arithmetic equations like (28), represent numbers as objects.

1. (28) $3+2=5$

Furthermore, these same equations can be equivalently paraphrased in English using nonsymbolic numerals. For example:

1. (29) Three plus two is five.

Yet the thought expressed by (29) most plausibly requires possession of concepts which refer to numbers as objects, e.g. TWO ${}_{num-ref}$ , not concepts predicating cardinality properties to collections (Section 4.2.3). Rather, understanding arithmetic equations most plausibly requires possessing number concepts beyond those plausibly attributed to early CP-Knowers.

Thus, we have reason to instead endorse

Complex Transition:

Understanding symbolic numerals requires number concepts not already possessed by early CP-Knowers.

${}_{num-ref}$

${}_{card-pred}$

4.3.3 Diachronic Pluralism

Our view is that taken altogether, the best available empirical evidence supports

Diachronic Pluralism:

At different stages in development, human beings possess different number concepts required to understand the same number word.

Weak Diachronic Pluralism:

At different developmental stages, humans possess different concepts required to understand different uses of the same number word.

The second, perhaps more radical form of Diachronic Pluralism we accept is

Strong Diachronic Pluralism:

At different developmental stages, humans possess different concepts required to understand the same use of the same number word.

1. (1a) The Elmos on the table are two in number.

Obviously, since ‘two’ serves the same (predicative) semantic function in different utterances of (1a), we are committed to there being distinct numerical concepts deployed in understanding this same use of ‘two’, i.e. different token utterances of the same utterance type.

Why accept this version of Strong Diachronic Pluralism? In our view, that’s because it represents the best solution to an important, albeit largely unappreciated, puzzle resulting from attempting to align our best developmental psychology with our best semantics for number expressions. Specifically, it results from two independently plausible, but jointly inconsistent, theses. The first is supported by research in developmental psychology:

Developmental Priority:

CP-knowers possess cardinal concepts prior to numerical concepts.

The second thesis results from combining independently plausible assumptions about word comprehension with what is, in our view at least, the most plausible available semantics for number words:

Semantic Priority:

One cannot understand cardinal meanings of number words without also possessing corresponding numerical concepts.

Meaning Comprehension:

Understanding the meaning of a word in a given context requires possessing a concept having the same content as expressed by that word in that context.

1. 1. a. Rover is a dog.

   2. b. Mary created a new dish.

Meaning Comprehension appears to be widely assumed within Mainstream developmental psychology. Indeed, given representationalism, it seemingly provides a natural, if not compulsory, explanation for how we can bear the attitude of understanding towards word meanings.

The second assumption required is

Polymorphic Comprehension:

Understanding any meaning of a polymorphic expression requires understanding its lexical meaning.

The final assumption required is

Substantivalism:

The lexical meaning of a number word is that of a numeral, referring to a number.

1. (26f) Two is an even number.

Together, Polymorphic Comprehension and Substantivalism entail that understanding any meaning of ‘two’, including cardinal meanings like the one expressed in (26c), requires understanding the meaning of a numeral.

1. (26c) Mars has two moons.

Thus, given Meaning Comprehension, it follows that one cannot understand ‘two’ in (26c) without also possessing a concept having the same content as expressed by ‘two’ in (26f). In other words, understanding cardinal meanings requires possessing a corresponding numerical concept.

Jointly, Developmental Priority and Semantic Priority generate a puzzle:

The Developmental Puzzle:

How can children understand cardinal meanings of number words, if such understanding requires possessing numerical concepts, and yet children only possess numerical concepts after possessing cardinal concepts?

Proto-Cardinal Concepts:

TWO ${}_{proto-card}$ , applicable to collections in one-to-one correspondence with sequences of natural language count words, such as $⟨$ ‘one’, ‘two’ $⟩$ .

Mature Cardinal Concepts:

TWO ${}_{mature-card}$ , applicable to collections in one-to-one correspondence with the sequence of natural numbers $⟨$ 1, 2 $⟩$ .

${}_{num-ref}$