Clarke and Beck (C&B) seek to defend the thesis that states of the approximate number system (ANS) represent, or refer to, “not only natural numbers (e.g., 7), but also non-natural rational numbers (e.g., 3.5).” In doing so, however, they rely on some highly controversial, and largely undefended philosophical assumptions.
The assumptions we have in mind arise most clearly in section 2.2, where C&B mention an apparent puzzle for their view: If the ANS often operates perceptually in order to represent numbers, and if numbers are abstract objects, then how could numbers feature in the contents of such perceptions, if we cannot generally perceive abstract objects? In response, they adopt a “standard picture” of perception on which perceptual states both refer to concrete objects and attribute (abstract) properties to them. On this view, “abstract objects enter perceptual content through the attribution of properties, not through reference to objects.” For example, perceiving a red apple involves referring to a concrete apple and ascribing the abstract property of redness to that apple. Similarly, perceiving that there are four apples involves referring to a set of apples and attributing an abstract cardinality property to it. In this way, C&B suggest that states of the ANS refer to numbers “by enabling numbers to enter into contents via [cardinality] property attribution.”
The present view carries two significant, although controversial, implications familiar from a minority view within the philosophy of mathematics known as empiricism (Kim, Reference Kim1981, Maddy, Reference Maddy1990). First, because sets are the objects referenced by certain perceptual states, those sets must be concrete. The likely view, following Kim and Maddy, would be that impure sets – sets whose members are concrete urelements – are themselves concrete. This is controversial, of course, as the predominant view is that sets are uniformly abstract. But perhaps C&B are talking loosely. By “set,” perhaps, they intend some more intuitive notion of collection, which might be elucidated in different ways, for example, mereologically.
The second, more pressing implication is what we call the Identity Thesis:
This view is philosophically controversial, linguistically problematic, and threatens an explanatory regress.
First, IT is philosophically controversial, in part, because it presupposes a cardinal conception of the naturals, whereby natural numbers are essentially the sorts of things we can use when counting. However, this is not the only – or even dominant – characterization available within the philosophy of mathematics. There are also structuralist characterizations (Shapiro, Reference Shapiro1997), ordinal characterizations (Linnebo, Reference Linnebo2018), and geometric characterizations (Tennant, Reference Tennant2009), for instance. Furthermore, even among advocates of the cardinal conception IT is controversial because properties are intensional entities, whereas the naturals on all prominent cardinal characterizations are extensional, identified with (finite) sets or classes (Frege, Reference Frege and Austin1950; Hale and Wright, Reference Hale and Wright2001; Maddy, Reference Maddy1990; Russell, Reference Russell1903).
Second, IT is linguistically problematic because the presumption of its truth appears to yield numerous false semantic predictions. Consider the following semantic contrasts inspired by Moltmann (Reference Moltmann2013) and Snyder (Reference Snyder2017):
(1) (a) The (??rational) number of women at the party is {four/??the number four}.
(b) The (rational) number Mary is writing about is {four/the number four}.
(2) (a) How many women are at the party? {Four/??The number four}.
(b) Which (rational) number is Mary writing about? {Four/The number four/??The number of women}.
(3) (a) The number {of women/??four} is expanding rapidly.
(b) The number {of women/??four} exceeds that of the men.
Plausibly, “the number of women” refers to a cardinality property, whereas “the number four” refers to a number (Snyder, Reference Snyder2017). Thus, pace IT, (1)–(3) strongly suggests that cardinality properties are not numbers.
Furthermore, if this is so, then contra C&B, “numerosities” are plausibly neither “exotic” nor “recherché.” Rather they are, as Butterworth (Reference Butterworth2005) and others appear to assume, cardinality properties – entities for which we already possess mathematically and linguistically well-understood theories (Scontras, Reference Scontras2014; Snyder, Reference Snyder2017).
Finally, even assuming IT, we doubt that, when combined with C&B's presumed account of perception, it resolves the original puzzle of how numbers, qua abstracta, could be referenced by perceptual states. In order to see why, consider what we take to be the two main ways of elaborating the proposal.
The first maintains that to attribute a property just is to reference that property. Thus, numbers enter into perceptual contents in virtue of consisting of two components, a set and a number, both referenced in perception. However, as should be clear from the following examples, this conflates predication and reference:
(4) (a) The apples are four (in number).
(b) Four is an even number.
In (4a), “four” functions as a predicate, whereas in (4b) it functions as a singular term, that is, a referential-type expression. Generally, predicates and singular terms have different semantic types, and thus semantic values. The present suggestion conflates those.
The second option characterizes predication as a relation between objects: “F(a)” is true just in case a instantiates F-ness, where “F-ness” names a property, viewed as an abstract object (Chierchia, Reference Chierchia1985). Thus, numbers enter into perceptual contents in virtue of those contents consisting of three components: a set, the instantiation relation, and a number.
This proposal avoids conflating predication and reference, but only at the cost of an explanatory regress. On the present suggestion, the attribution of cardinality properties in perception requires reference to two objects, related by instantiation: a concrete set and an abstract number. But now the original problem recurs: Unless reference to numbers, qua abstracta, was possible, numbers couldn't feature in perceptual contents.
In the forgoing, we sought to cast doubt on the plausibility and explanatory utility of IT.
In doing so, however, we do not intend to suggest that C&B are idiosyncratic in their adoption of this assumption. On the contrary, we suspect that IT is implicit in much research on number cognition. If this suspicion is correct, then a significant upshot of our discussion is that this and related assumptions merit sustained critical scrutiny.
Clarke and Beck (C&B) seek to defend the thesis that states of the approximate number system (ANS) represent, or refer to, “not only natural numbers (e.g., 7), but also non-natural rational numbers (e.g., 3.5).” In doing so, however, they rely on some highly controversial, and largely undefended philosophical assumptions.
The assumptions we have in mind arise most clearly in section 2.2, where C&B mention an apparent puzzle for their view: If the ANS often operates perceptually in order to represent numbers, and if numbers are abstract objects, then how could numbers feature in the contents of such perceptions, if we cannot generally perceive abstract objects? In response, they adopt a “standard picture” of perception on which perceptual states both refer to concrete objects and attribute (abstract) properties to them. On this view, “abstract objects enter perceptual content through the attribution of properties, not through reference to objects.” For example, perceiving a red apple involves referring to a concrete apple and ascribing the abstract property of redness to that apple. Similarly, perceiving that there are four apples involves referring to a set of apples and attributing an abstract cardinality property to it. In this way, C&B suggest that states of the ANS refer to numbers “by enabling numbers to enter into contents via [cardinality] property attribution.”
The present view carries two significant, although controversial, implications familiar from a minority view within the philosophy of mathematics known as empiricism (Kim, Reference Kim1981, Maddy, Reference Maddy1990). First, because sets are the objects referenced by certain perceptual states, those sets must be concrete. The likely view, following Kim and Maddy, would be that impure sets – sets whose members are concrete urelements – are themselves concrete. This is controversial, of course, as the predominant view is that sets are uniformly abstract. But perhaps C&B are talking loosely. By “set,” perhaps, they intend some more intuitive notion of collection, which might be elucidated in different ways, for example, mereologically.
The second, more pressing implication is what we call the Identity Thesis:
(IT) Natural numbers are identical to cardinality properties.
This view is philosophically controversial, linguistically problematic, and threatens an explanatory regress.
First, IT is philosophically controversial, in part, because it presupposes a cardinal conception of the naturals, whereby natural numbers are essentially the sorts of things we can use when counting. However, this is not the only – or even dominant – characterization available within the philosophy of mathematics. There are also structuralist characterizations (Shapiro, Reference Shapiro1997), ordinal characterizations (Linnebo, Reference Linnebo2018), and geometric characterizations (Tennant, Reference Tennant2009), for instance. Furthermore, even among advocates of the cardinal conception IT is controversial because properties are intensional entities, whereas the naturals on all prominent cardinal characterizations are extensional, identified with (finite) sets or classes (Frege, Reference Frege and Austin1950; Hale and Wright, Reference Hale and Wright2001; Maddy, Reference Maddy1990; Russell, Reference Russell1903).
Second, IT is linguistically problematic because the presumption of its truth appears to yield numerous false semantic predictions. Consider the following semantic contrasts inspired by Moltmann (Reference Moltmann2013) and Snyder (Reference Snyder2017):
(1) (a) The (??rational) number of women at the party is {four/??the number four}.
(b) The (rational) number Mary is writing about is {four/the number four}.
(2) (a) How many women are at the party? {Four/??The number four}.
(b) Which (rational) number is Mary writing about? {Four/The number four/??The number of women}.
(3) (a) The number {of women/??four} is expanding rapidly.
(b) The number {of women/??four} exceeds that of the men.
Plausibly, “the number of women” refers to a cardinality property, whereas “the number four” refers to a number (Snyder, Reference Snyder2017). Thus, pace IT, (1)–(3) strongly suggests that cardinality properties are not numbers.
Furthermore, if this is so, then contra C&B, “numerosities” are plausibly neither “exotic” nor “recherché.” Rather they are, as Butterworth (Reference Butterworth2005) and others appear to assume, cardinality properties – entities for which we already possess mathematically and linguistically well-understood theories (Scontras, Reference Scontras2014; Snyder, Reference Snyder2017).
Finally, even assuming IT, we doubt that, when combined with C&B's presumed account of perception, it resolves the original puzzle of how numbers, qua abstracta, could be referenced by perceptual states. In order to see why, consider what we take to be the two main ways of elaborating the proposal.
The first maintains that to attribute a property just is to reference that property. Thus, numbers enter into perceptual contents in virtue of consisting of two components, a set and a number, both referenced in perception. However, as should be clear from the following examples, this conflates predication and reference:
(4) (a) The apples are four (in number).
(b) Four is an even number.
In (4a), “four” functions as a predicate, whereas in (4b) it functions as a singular term, that is, a referential-type expression. Generally, predicates and singular terms have different semantic types, and thus semantic values. The present suggestion conflates those.
The second option characterizes predication as a relation between objects: “F(a)” is true just in case a instantiates F-ness, where “F-ness” names a property, viewed as an abstract object (Chierchia, Reference Chierchia1985). Thus, numbers enter into perceptual contents in virtue of those contents consisting of three components: a set, the instantiation relation, and a number.
This proposal avoids conflating predication and reference, but only at the cost of an explanatory regress. On the present suggestion, the attribution of cardinality properties in perception requires reference to two objects, related by instantiation: a concrete set and an abstract number. But now the original problem recurs: Unless reference to numbers, qua abstracta, was possible, numbers couldn't feature in perceptual contents.
In the forgoing, we sought to cast doubt on the plausibility and explanatory utility of IT.
In doing so, however, we do not intend to suggest that C&B are idiosyncratic in their adoption of this assumption. On the contrary, we suspect that IT is implicit in much research on number cognition. If this suspicion is correct, then a significant upshot of our discussion is that this and related assumptions merit sustained critical scrutiny.
Financial support
Partial funding of this research was provided by a grant to JO from the Institute of Education Sciences (R305A160295).
Conflict of interest
None.