^{1}

In the literature on numerical cognition, the presence of the capacity to distinguish between numerosities by attending to the number of items, rather than continuous properties of stimuli that correlate with it, is commonly taken as sufficient indication of numerical abilities in cognitive agents. However, this literature does not take into account that there are non-numerical methods of assessing numerosity, which opens up the possibility that cognitive agents lacking numerical abilities may still be able to represent numerosity. In this paper, I distinguish between numerical and non-numerical methods of assessing numerosity and show that the most common models of the internal mechanisms of the so-called number sense rely on non-numerical methods, despite the claims of their proponents to the contrary. I conclude that, even if it is established that agents attend to numerosity, rather than continuous properties of stimuli correlated with it, an answer to the question of the existence of the number sense is still pending the investigation of a further issue, namely, whether the mechanisms the brain uses to assess numerosity qualify as numerical or non-numerical.

Despite the advances made in the field of numerical cognition in recent decades, the existence of the so-called “number sense” remains controversial. The number sense is usually described as the ability to perceive “numerical information in its nonsymbolic form” (^{1}

E.g.,

One of the most influential arguments against this view claims that participants’ behavior in experiments such as these are best explained by reference to continuous properties of stimuli, such as surface area and element size, rather than the number of discrete items (

One premise of this debate is that, if it is confirmed that participants are undoubtedly attending to the number of items in the stimulus—in one word, numerosity—to identify the largest collection, then they must have some sort of numerical ability—the number sense. That is why supporters of the existence of the number sense put much effort into showing that participants are truly relying on numerosity (e.g.,

This premise, however, is false. As I show in this paper, there are non-numerical methods of evaluating numerosity. I show that numbers provide only one among several other methods of assessing numerosity. Therefore, evidence that participants are undoubtedly relying on numerosity to identify the largest collection does not necessarily imply that they have any kind of numerical ability, since the assessment of the numerosities involved can be achieved by a non-numerical method. I conclude that, even if it is confirmed that participants are undoubtedly attending to the number of items in the stimulus, this cannot be taken as sufficient indication of the presence of a number sense unless it is also shown that they do use a numerical method of assessing numerosity.^{2}

Certainly, the question of the existence of the number sense can still be answered negatively by showing that participants are not relying on numerosity at all, but rather relying on continuous properties that correlate with it, as

The existence of non-numerical methods of assessing numerosity is rarely discussed in the literature. One exception is

This paper is meant to offer a conceptual contribution to the debate about the existence of the number sense. As pointed out by

In this section, I am still not concerned with models of the number sense. The purpose of this section is to establish a conceptual distinction between number and numerosity. Noticing this distinction is the first step to understanding that there are non-numerical methods of assessing numerosity. Only later, in Section 4, this conceptual distinction is applied to analyze models of the number sense.

Numerosity is a technical term proper of the field of numerical cognition. More often than not, cognitive scientists do not present a definition of what they mean by this term. Those who do, however, introduce numerosity as a synonym of cardinality. For example,

In mathematics, the cardinality of a set refers to its “size,” i.e., how many elements it has. Numbers provide one way of determining and expressing the cardinal size of a set, but it is a trivial fact in mathematics that the cardinality of a set can be determined and expressed without even mentioning numbers. The following example illustrates this point. Suppose that someone is organizing a meeting and wants to make sure that there are sufficient chairs for everyone. She can do this in at least two ways. By using numbers, she can count the people, count the chairs, and compare the outcomes; or, without even mentioning numbers, she can just ask people to sit down, each person in a single chair. If no person remains standing and no chair remains empty, she concludes immediately that both sets have the same size. If someone remains standing, then the set of people is larger than the set of chairs; inversely, if any chair remains empty, then the set of chairs is larger than the set of people. No number is involved in this procedure. It can be carried out recruiting only the notion of one-to-one correspondence or equinumerosity, which, despite its name, is defined without invoking numbers. (For a definition of equinumerosity, see

As this example illustrates, cardinality can be assessed without using numbers by establishing a one-to-one correspondence between two collections. Let us call these the ^{3}

In principle, both collections can mutually play the role of

It may be argued that one-to-one correspondence just establishes that the two collections have the same cardinality, but does not reveal which cardinality they share. Supposedly, this could be done only by using numbers. However, even when we use numbers to assess and express cardinality, all we have is a comparison through one-to-one correspondence. Counting consists of establishing a one-to-one correspondence between the target collection and a collection of numbers. In counting, an initial segment of the sequence of natural numbers plays the role of model collection. For example, the collection {1, 2, 3, 4, 5, 6, 7, 8} models the size of the collection of planets orbiting the Sun because they share the same cardinality. But

Set theory is the mathematical field where such questions are answered. What we see in set theory is that there is no privileged entity that could be called

More generally, we can use any set as a “yardstick” (model collection) to evaluate and express cardinalities. For example, I can use the set of pencils on my desk—let us call it P—for this purpose. Because there is an injective and non-surjective mapping from P onto the set of planets orbiting the Sun, I can say that the cardinality of the set of planets is greater than P’s. The existence of a bijective mapping between P and the set of moons orbiting Mars allows me to say that the cardinality of the set of moons orbiting Mars is equal to P’s. P is not a number, but I just used it to express cardinalities in the same way I could have used the number two. In other words, in the second set-theoretic approach mentioned above, I just selected a different, unusual set, P, to be the representative of the class of all pairs. One obvious advantage of using numbers is that almost everyone knows which set size two models, whereas only I know the cardinality P models. However, this does not prevent P from working as a “cardinality ruler.”

At this point, it should be clear that numbers are just one among many possible cardinality rulers. In other words, numbers are just a commonly used measurement scale of cardinality (

True enough, not all model collections are equally good. An important difference between using numbers and using other model collections is that numbers constitute a fully-fledged ratio scale for the measurement of cardinality. My set P of pencils can express only one cardinality with precision, whereas we have numbers to express every cardinality with precision. That number is a ratio scale for measuring cardinality was already noticed back in 1946 by S. S. Stevens, the influential psychophysicist who introduced the theory of measurement which is now standard in the field:

Foremost among the ratio scales is the scale of number itself—cardinal number—the scale we use when we count such things as eggs, pennies, and apples. This scale of the numerosity of aggregates is so basic and so common that it is ordinarily not even mentioned in discussions of measurement (

Stevens is also responsible for introducing the term ‘numerosity’ (

Number is to numerosity as meter is to length and gram is to weight. It is easy to notice the difference between meter and gram—scales—and length and weight—magnitudes—because there are other scales for length and weight, such as feet and pounds. When it comes to cardinality, though, we do not have other scales, which makes the distinction less obvious. However, not having other scales for cardinality does not mean that there are not other, non-numerical ways of measuring it. For example, even if there were not any scale for length other than meter, we could still use, say, a stick to measure the length of a window, for example. By showing the stick, one could say “the window is that long.” This is what happens when we use a non-numerical model collection to determine and express cardinality. Returning to the example above, suppose that the person who was organizing the meeting was able to provide exactly one chair for each person in the room. Then suppose that, after the meeting is finished and everyone is gone, someone comes in and asks: “how many people attended the meeting?” Even if the organizer had not counted the attendees, she could easily answer by pointing to the chairs: “that many.”

The common mistake of conflating number, the scale, and numerosity, the magnitude, is understandable. Because numbers are our only scale for the measurement of cardinality, we usually refer to it indirectly, via its numerical measurement. For example, in the sentence ‘the number of planets in the Solar System is eight,’ the expression ‘the number of planets’ is used in place of ‘the cardinality of the set of planets.’ This is a clear case of metonymy, the figure of speech in which a word is used in place of another associated with it. A similar case of metonymy occurs when we say that “the square footage of the house is 1,200 sq ft.” In this case, the expression ‘the square footage of the house’ is used in place of ‘the area of the house,’ which is the property that is being measured in square feet. Thus, when in a study on numerical cognition an infant is said to be able to perceive “the number of dots on the screen,” this may be charitably interpreted as meaning that the infant is able to perceive the numerosity^{4}

More precisely, we should say that the infant is able to perceive the numerousness of the set of dots on the screen. ‘Numerousness’ was introduced as a technical term by Stevens in the same paper where he introduced the term ‘numerosity’ (

If the distinction drawn in this section between number as a measurement scale and cardinality/numerosity as the magnitude numbers measure is correct, then the argument according to which the number sense is numerical because it allows for the perception of number is not tenable. What is perceived through the so-called number sense is the magnitude, i.e., numerosity. Whether the brain uses a numerical scale for assessing numerosity is a different question. I address this question in Section 4. Before that, though, let us go deeper into the distinction between numerical and non-numerical methods of assessing cardinality.

In this section, I am not yet concerned with models of the number sense. The purpose of this section is to establish a conceptual distinction between numerical and non-numerical methods of assessing numerosity. Only later, in Section 4, this conceptual distinction is applied to analyze models of the number sense.

As we saw above, establishing one-to-one correspondences between target collections and non-numerical model collections is a non-numerical method of assessing numerosity. The nature of the elements belonging to the model collection, however, is not the only difference between numerical and non-numerical methods. There is also an important procedural difference.

The counting procedure is governed by five rules, the so-called Counting Principles, as defined by

Principle | Description |
---|---|

One-to-one Correspondence | Each item of the counted collection must be paired with one and only one number word. |

Stable Order of Counting Words | The order in which number words are used for tagging items must follow a stable order, i.e., the order must be kept constant across different counting events. |

Order Irrelevance of Items | The order in which items are paired with number words is irrelevant, i.e., the order may change across different counting events. |

Abstraction | Counting applies to any collection of all sorts of objects, including sets formed by physical objects or ideas, and heterogeneous sets, formed by a combination of different kinds of objects. |

Cardinality | The number word used for tagging the last item of a collection represents the cardinality of the whole collection. |

The two remaining principles—Stable Order of Counting Words and Cardinality—do not have a tallying version, since they do not apply to tallying. It would be counterproductive to use the items of a model collection in a stable order across different tallying events, since the resulting model collection will be the same, no matter the order. What is more, model collections used for tallying often consist of hardly distinguishable items, such as pebbles, sticks, or beans, which could make a principle such as Stable Order very difficult to follow. For the same reason, the Cardinality principle does not apply to tallying, since the last item of the model collection tallied with the last item of the target collection may be hardly distinguishable from the items used previously in the process. In tallying, it is the whole model collection that represents the cardinality of the target collection.

This brings us to two structural differences between counting and tallying. First, in counting, the model collection consists of easily distinguishable items, which are used in a fixed order; in tallying, by contrast, the model collection may consist of hardly distinguishable, unordered items. Second, in counting, each item of the ordered model collection corresponds to a cardinality—that is, each number (or number word) is associated with a certain set size (a magnitude). In tallying, by contrast, single items are not associated,

This second difference explains why counting is so superior to tallying as a way of denoting and expressing cardinality (although both are equally precise). Since counting uses an ordered model collection—numbers or the sequence of number words—in every counting event where the target collection has the same size, the last number word used is always the same. Thus, that number word becomes associated with that specific cardinality. Furthermore, the model collection used in counting is standard: within a linguistic community, everyone uses the same sequence of words. This facilitates communication, since every numerate person knows this model collection as well as the position each number word occupies in the sequence and, consequently, the set size it represents. Thus, mentioning a number word suffices for referring to a cardinal size. Surely, the use of numbers words is not essential here; body-part counting systems (such as those described in

The fact that each number or number word represents a cardinal size is commonly referred to as the symbolic character of number, often mentioned as an essential property of numerical systems (e.g.,

Symbolic reference is an essential property of numerical systems because the elements of numerical model collections usually cannot refer to the cardinality each one of them represents by iconic or indexical means. For example, the word ‘five’ in English, the thumb in the Kombai body-part counting system (^{5}

As noted by an anonymous reviewer, Zermelo’s definition of the natural numbers “might be interpreted as having some iconic flavor” to our eyes, since the couples of parentheses are somewhat analogous to tallies. However, from

Once a numerical system is established, however, the association between the elements of its model collection and cardinal sizes is no longer totally arbitrary. According to ^{6}

Notice that the choice of a symbol-to-symbol relationship to model a certain object-to-object relationship is arbitrary. For example, the positional relationship that in English means the agent-patient relationship can mean, in other languages, a different relationship between objects or no relationship at all.

The same mode of semantic linking takes place when we use numbers to represent cardinality. The position numbers occupy within the natural-number structure is used to model the relationship of relative size between collections. Through system-dependent linking, the relationship “comes before” (predecessor) between numbers translates into the relationship “has less elements than” between collections, and the relationship “comes after” (successor) between numbers translates into the relationship “has more elements than” between collections (

Wiese points out that numbers are used not only to express cardinality, but also to assign names. This is the case, for example, when we call bus lines by numbers. In such cases, the symbolic association between numbers and the objects they designate is also established by means of system-dependent linking. However, in such cases the relationship exploited within the system of natural numbers is not the one of successor/predecessor, but the relationship of identity. “[W]hen we distinguish different bus lines by different numbers,” ^{7}

This does not mean that properties of numbers other than identity cannot be used to represent other properties of bus lines. For example, as an anonymous reviewer pointed out, proximity between numbers can be used to represent proximity between the areas covered by the buses.

Numbers can be cardinality rulers only when the successor relationship is the one used to establish system-dependent links with collections (I come back to this point below, when discussingUp to this point, we have seen two methods of assessing cardinality and two differences between them. The methods are counting (numerical) and tallying (non-numerical). One difference between them is procedural: counting is governed by five rules (the counting principles), whereas tallying is governed by only three of these. The other difference pertains to semiotics: the numerical method uses symbols for representing cardinal sizes, whereas tallying uses iconic representations. But there is an important similarity between the two: both yield exact representations of cardinal sizes. Each numeral, number or model collection represents exactly and precisely one cardinal size. Are there imprecise methods of representing cardinality? The answer is yes.

In the literature on numerical cognition, the accumulator has been proposed as an imprecise device for the representation of numerosity (

The accumulator owes its plausibility as a model of the number sense from the fact that it captures the observation that the capacity of the number sense to distinguish between two numerosities decreases as the ratio between them approaches 1:1. The imprecision of the number sense makes room for an often-mentioned objection to its numerical character. I address the argument from imprecision and

In this section, we saw three methods of assessing and representing cardinality: counting, tallying, and accumulation. We also saw two essential properties of numbers with regard to their use as cardinality rulers: they are precise—each number represents a single cardinality precisely—and they do so by symbolic reference, relying on the successor relationship to model the “has more/less elements than” relationship between collections. Given these criteria, among the methods considered here only counting (with number words, body parts or any other ordered model collection) qualifies as numerical. Tallying is not numerical because it does not satisfy the second criterion (it is iconic) and accumulation is not numerical because it does not satisfy both criteria (it is imprecise and iconic).

Method | Precise | Form of representation | Procedural rules | Numerical |
---|---|---|---|---|

Tallying | yes | iconic | 1-to-1 correspondence, Order Irrelevance, Abstraction | No |

Counting | yes | symbolic | 1-to-1 correspondence, Stable Order, Order Irrelevance, Abstraction, Cardinality | Yes |

Accumulation | no | iconic | 1-to-some correspondence, Order Irrelevance, Abstraction | No |

There are two ways of viewing the number sense as numerical: (1) the number sense may be said to be numerical because it putatively allows for the perception of numbers; or, (2) it may be said to be numerical because it putatively uses numbers (or representations thereof) to assess and represent numerosities.

According to the first point of view, numbers are perceptual properties of perceptual collections, and the number sense represents number just as perception represents other perceptual properties, such as color and distance. In the final paragraph of Section 2, I mentioned that this view is mistaken, since it is conflating the scale (number) with the magnitude (numerosity). Number is a measurement scale of numerosity, and measurement scales are not the sort of thing that could be perceived. Just as we do not perceive square feet, but area, we do not perceive numbers, but numerosity.^{8}

More precisely, numerousness; see Footnote 2.

And, since perception of numerosity may rely on non-numerical methods, perception of numerosity does not imply,The second point of view does not make this mistake; it admits that the magnitude the number sense represents is numerosity, and it claims that the number sense is numerical because it uses numbers (or representations thereof) to represent numerosity. However, researchers who adopt this point of view often make another mistake. They uncritically assume that every method of assessing and representing numerosity qualifies as a numerical method. In fact, there is no clear understanding in the literature of the existence of non-numerical methods of doing this.^{9}

This is illustrated, for example, by

These points of view are rarely clearly articulated in the literature. Two exceptions are

Measurement processes establish numerical reference by mapping from magnitudes in the world (numerosities, lengths, durations and so on) to the numbers with which we represent those magnitudes [in the brain] (

To suppose otherwise is to confuse what the system is doing (e.g., functioning to track and represent numbers—the computational level description with which we are concerned) for a specific account of how it does this (an algorithmic level description (Marr, 1982)) (

However, as we have seen, to suppose that number is the perceptual property the so-called “number sense” represents is to confuse the measurement scale for the magnitude it measures. Contrary to Clarke and Beck’s opinion, ‘numerosity’ is a well-defined term, and when we clearly distinguish number (the scale) from numerosity (the magnitude), we realize that purely computational models of the number sense should be seen as neutral regarding its numerical character. Even if we assume that it is the function of a cognitive system that determines its conceptual content, the observation that the function of the so-called “number sense” is to represent numerosity only means that its contents are numerosity representations. But, given that there are numerical and non-numerical methods of building numerosity representations, it may well be that the so-called “number sense” builds numerosity representations by non-numerical means. If this is so, it is correct to characterize it as a sense of

Computational models of cognitive systems describe the problem the system aims at solving and provide a mathematical function specifying the relationship between input and output (given stimuli

True enough, certain computational models may provide some clues on the algorithmic level. For example, the model presented by

This is what the algorithmic level is meant to do. Algorithmic models describe the strategies and processes a cognitive system uses to solve a problem (

To date, the prevailing view is that the ability to perceive numerosities is implemented by two different systems: one system for numerosities smaller than four or five, whose fast and accurate perception is called “subitizing,” and one system for larger numerosities, whose fast and only approximate perception is called “estimation” (

According to the most common algorithmic models of subitizing, this ability relies on a domain-general mechanism for individuation of multiple objects in parallel, called the object tracking system (OTS) (

The object tracking system . . . is thought to represent numerical information only in an implicit way: in this system, there is no summary representation of ‘two’; instead, infants form a mental model of two objects by recruiting two attentional indexes or ‘object files’ (

In line with the conclusion drawn in Section 3, that one-to-one correspondence (tallying) is non-numerical,

The most common algorithmic model of estimation is the already mentioned accumulator. At the implementation level, the metaphorical “cupful of water” translates into neural activity: each object in the stimulus produces a variable quantity of neural activity, which is normalized to an approximately constant quantity. These normalized, approximately constant quantities are then summed by “accumulation neurons.” The level of activation in these neurons corresponds to the level of water in the metaphorical beaker: the higher the activation, the higher the numerosity represented (

Unlike the computer, it [the brain] does not rely on a digital code, but on a continuous quantitative internal representation. The brain is not a logical machine, but an analog device. Randy Gallistel has expressed this conclusion with remarkable simplicity: “In effect, the nervous system inverts the representational convention whereby numbers are used to represent linear magnitudes. Instead of using number to represent magnitude, the rat [like the Homo sapiens!] uses magnitude to represent number” (

In this passage, Dehaene and Gallistel make the mistake of confounding number with numerosity. Correcting the final sentence, we could say that “instead of using number to represent magnitude, the rat, like the Homo sapiens, uses magnitude to represent magnitude.” In the accumulator model, one magnitude—level of activation—is used to represent another magnitude — numerosity. This is the analog or iconic aspect of the accumulator, which contrasts with the symbolic mode of representation of numbers. Accordingly, the cognitive system responsible for estimation is often described as the “Analog Number System” (ANS). This is a misnomer, however, since there are no such things as “analog numbers;” numbers can represent numerosity only symbolically.

To be fair, in

“Number neurons” or “numerosity detectors” resemble numbers in one aspect, namely, in that they represent numerosity symbolically. But “number neurons” differ from number in that they are not precise; they fire when presented to neighbor numerosities, whereas each number represents a single numerosity. Since the so-called “number neurons” do not satisfy one of the two criteria for being numerical (precision), the designation “number neuron” is misleading.

Nothing new here. The imprecision of the so-called number sense is often-mentioned by those who deny its numerical character (e.g.,

Precision is not a necessary condition for numerosity representation, but it is a necessary condition for any method of representing numerosity that is to be classified as numerical. Clarke and Beck’s response to the argument from imprecision does not change this requirement. Once we understand the difference between number (scale) and numerosity (magnitude), we understand that what the argument from imprecision shows is that the imprecision of the ANS implies that it does not use a numerical method, otherwise it would be precise. If this is so, “Approximate Number System,” a common alternative reading of the acronym ANS, is also a misnomer, although this acronym may remain unchanged if it is taken to stand for “Approximate Numerosity System.”

Whereas Dehaene and Changeux’s accumulator-based model of the ANS does not qualify as numerical,

The operation of this mechanism [the accumulator] conforms to the principles that define counting processes (

Gallistel and Gelman define numerons as “mental representatives of numerosities,” and take numerons to be symbols (

A remark is in order. If Gallistel and Gelman’s precise accumulator is supplemented with a final step, of the sort found in

Another model of the number sense worth considering here is the one proposed by Laurence and Margolis (

The “small number system” is both precise and symbolic. This system would qualify as numerical, were it not modeling a different, non-numerical relationship. Recall from Section 3 that, when numbers are used as symbols to represent numerosities, the sign-to-sign relationships predecessor/successor are used to model the relationships “has less/more elements than” between collections. The symbols of the “small number system,” however, do not bear predecessor/successor relationships between them, nor do they bear a similar relationship that could be exploited for the same purpose, such as the higher/lower relationship between symbols engraved on a graduated beaker.

Notice that on this minimal account, the symbols for small numerical quantities need not be inherently ordered, and there need not be a procedure that ensures that three is represented as more than two, or two as more than one (unlike conventional counting terms).

Since the symbols used in this system are not ordered, there is no sign-to-sign relationship in this system which could be used to model the relationships “has more/less elements than” between cardinalities. What Laurence and Margolis’s system is modeling is the relationship of identity between numerosities. The system simply assigns a name to each numerosity (up to four), in the same way that we assign numerals as names to bus lines. Different symbols (names) go to different numerosities, and that is all. From this, the agent cannot infer anything about the size of the collections, except that they are equal or different. In terms of the classification of measurement scales used in measurement theory, the scale Laurence and Margolis’s system uses is

In this section, I applied the criteria for being numerical we saw in Sections 2 and 3 to judge whether models of the so-called “number sense” qualify as numerical. I concluded that, among the models analyzed, only a precise accumulator combined with precise numerosity detectors would qualify as truly numerical. A model such as this has never been proposed, though, since it is unable to account for the imprecision of estimation. The more realistic models of the number sense considered here do not qualify as numerical.

I started this paper by showing that number is best conceived of as a measurement scale of cardinality. Cardinality, a.k.a. numerosity, is the magnitude. Consequently, if there exists a number sense, it does not allow for the perception of number, but numerosity, since what we perceive through our senses are magnitudes, and not their measurement scales. Thus, “number sense” is a misnomer, and it is best described as a sense of

Perhaps the expression “number sense” could still be used to describe a sense of numerosity if representations of numerosity were built through numerical means. However, as we have seen, it is possible to build numerosity representations by non-numerical means and, in fact, most of the algorithmic models proposed for the so-called number sense do not display the essential features of numerical methods. Whereas symbolic reference and precision in the representation of the relationship “larger/smaller than” between cardinalities are the hallmarks of numerical methods, the algorithmic models of the so-called number sense use iconic or imprecise methods.

These conclusions have an important consequence for the debate about the existence of the number sense. This debate has focused on establishing whether cognitive agents rely on continuous or discrete properties (numerosity) of stimuli in order to distinguish the size of collections. The question of the existence of the number system can be answered negatively if it turns out that agents rely exclusively on continuous properties. However, if the final conclusion is that agents do rely on numerosity, then, I submit, the question of the existence of the number sense remains open. After all, agents may be using non-numerical methods to assess and represent numerosity. Only conceptual clarity (of the sort put forward here) and the further empirical investigation of the mechanisms the brain uses to assess and represent numerosity can answer this latter question.

Conceptual clarity should not be seen as mere philosophical pedantry. Evidence for the existence of a capacity to perceive numerosity accumulates at the same time that correlation studies about the relationship between this capacity and symbolic arithmetical skills are inconclusive or find only a weak correlation (

Rather than mere pedantry, conceptual analysis can shed light on relevant empirical questions.

This work was partially supported by FAPEMA (Foundation for Research and Scientific and Technological Development of Maranhão) under Grant Number BD-08292/17.

The author has declared that no competing interests exist.

I am grateful to Daniel Schiochett for being a philosophical interlocutor during the elaboration of the ideas presented in this paper.