Understanding the Nature of Numbers: Cardinality vs. Structural Properties, Assignments of Philosophy

Download Understanding the Nature of Numbers: Cardinality vs. Structural Properties and more Assignments Philosophy in PDF only on Docsity! To appear in: E. Davis and P. Davis (Eds.), Mathematics, Substance and Surmise. Berlin: Springer. Beliefs about the Nature of Numbers Lance J. Rips Northwestern University Beliefs about Numbers / 2 Abstract Nearly all psychologists think that cardinality is the basis of number knowledge. When they test infants’ sensitivity to number, they look for evidence that the infants grasp the cardinality of groups of physical objects. And when they test older children’s understanding of the meaning of number words, they look for evidence that the children can, for example, “Give [the experimenter] three pencils” or can “Point to the picture of four balloons.” But when people think about the positive integers, do they single them out by means of the numbers’ cardinality, by means of the ordinal relations that hold among them, or in some other way? This chapter reviews recent research in cognitive psychology that compares people’s judgments about the integers’ cardinal and ordinal properties. It also presents new experimental evidence suggesting that, at least for adults, the integers’ cardinality is less central than their number- theoretic and arithmetic properties. Beliefs about Numbers / 5 Since the numerals are classified in accordance with their ordinal properties, this suggests that the ordinal conception of the natural numbers is more basic than the cardinal one. As Linnebo notes, the appeal to directness in thinking about numbers relies on intuition, but he conjectures that results from cognitive experiments might back the claim for the immediacy of the ordinal conception (which we have been calling the “structural perspective”). This chapter asks whether any psychological evidence favors the structural perspective or the cardinal perspective.2 We can begin by looking at theories and data on how children learn the meaning of the first few positive integers to see whether the evidence supports developmentalists’ emphasis on cardinality. I then turn to recent experiments that have compared adults’ judgments of cardinality to their judgments of order for further insight on whether our understanding of numbers depends more tightly on one or the other of these two types of information. Finally, the chapter describes some new studies that directly probe the properties that people take as central to number knowledge. 2. The Origins of Number Knowledge As an example of the role that the cardinal perspective plays in theories of number knowledge, let’s consider Susan Carey’s influential and detailed proposal about how children learn the meanings of their first few number words [3]. Figure 1 provides a summary of the steps in this process, which Carey calls Quinian bootstrapping. In trying to understand the children’s progress through this learning regime, let’s start by figuring out what the process is supposed to achieve. At two or three years old, kids are able to recite the numerals in order from “one” to some number like “ten” or “twenty,” but they don’t yet know how to produce numerals for arbitrary integers in the way you do. They have just a short, finite list, for example, “one, two, three, four, five, six, seven, eight, nine, ten.” At this age, they don’t understand the cardinal meanings of the words on this list; so if you ask them to give you two balloons from a pile, they can’t do it. Then, over an extended period of time—as long as a year or so—they first work out the meaning for the word “one,” then for the word “two,” then “three,” and sometimes “four,” again in the sense of being able to give you one, two, three, or four objects in response to a command. At that point, something clicks, and suddenly they’re able to give you five things, six things, and so on, up to ten things (or to whatever the last numeral is on their count list). The Quinian bootstrapping theory is supposed to explain this last step, when things finally click: It extends kids’ ability to enumerate objects in response to verbal requests from three or four to ten. Post bootstrap, kids still don’t know many of the important properties of the positive integers, but at least they can count out ten things, more or less correctly. In other words, what they’ve learned is how to count out objects to determine the right cardinality. What are the steps that children are supposed to go through in graduating from their pre-bootstrap state of knowledge of number to their post-bootstrap knowledge? Here’s a quick tour: 2 This issue might be put by asking whether people think of the first few naturals as cardinal or ordinal numbers. However, “cardinal number” and “ordinal number” have special meanings in set theory, and these meanings don’t provide the intended contrast. In their usual development (e.g., [8]), the ordinals and cardinals do not differ in the finite range, with which we will be concerned in this chapter. (Both ordinal and cardinal numbers include transfinite numbers—for which they do differ—and so extend beyond the natural numbers.) Beliefs about Numbers / 6 Step 1: At the beginning of this process, children have two relevant innate mental representations. Representations of type (a) in Figure 1 are representations of individual objects—for example, representations of each of three balloons. Representations of type (b) are representations of sets. I’ll use Step 1 (pre-linguistic representations): Representation a: Representations of individual objects: object1, object2, object3, ... Representation b: Representations of sets: {object1}, {object1, object2,...} Step 2 (initial language learning): Representation d: Singular/plural: “a” (singular) “some” (plural) {object1} {object1, object2,….} Representation c: Count List: <“one,” “two,” “three,” ... “ten”> Step 3 (one-knower stage): Representation e: “one” “two” “three” “ten” “a” “some” {object1} {object1, object2,….} Step 4 (two-knower stage): Representation f: “one” “two” “three” “ten” “a” {object1} {object1,object2} {object1, object2, object3,….} Steps 5 and 6 (three and four-knower stages): similar to the representation in Step 4 Representation i: “one” “two” “three” “four” “ten” “a” {object1} {object1,object2} {object1,object2, object3} {object1, object2, object3, object4,….} Step 7 (pre-bootstrap stage): Step 8 (post bootstrap, Cardinal-Principle-knower stage): Cardinality(n + 1) = Cardinality(n) + 1 Representation j: “one” “two” “three” “four” “ten” ... ... ... ... Figure 1. Steps in the acquisition of the meaning of number words for the first few positive integers, according to Carey’s [3] Quinian bootstrapping theory (figure adapted from [32]). Beliefs about Numbers / 7 objecti to denote the mental representation of a single individual i, {object1, object2} for the representation of a set containing exactly two individuals, and {object1, object2, …} to indicate the representation of a set containing more than one individual but whose total size is unknown. Children at this stage also have representations for approximate cardinality, but they play no role in Carey’s bootstrapping story. We’ll discuss this approximate number system in Section 3 of this chapter. Step 2: Language learning at around age two puts two more representations in play. The representation of type (c) in Figure 1 is the memorized list of number words, in order from “one” to some upper limit, which we’ll fix for concreteness at “ten.” We’ll see that this list of numerals doesn’t become important until quite late in the process, but it’s in place early. The second new representation of type (d) is a mapping between the indefinite determiner “a,” as in “a balloon” (or other singular marker in natural language), and the symbol for a singleton {object1.3 Similarly, there’s a mapping between the word “some” (and other plural markers in language) and the symbol for a set of unknown size, {object1, object2, …}. This is the way the children learn the singular/plural distinction in number-marking languages like English, French, or Russian—the difference between “book” and “books.” Steps 3-6: In the next few stages, children learn the meanings of the words “one” through “three” or “four” by connecting them with mental representations of sets containing the appropriate number of things. First, the children think that the word “one” means what “a” means (i.e., {object1}) and that the rest of the number words mean what “some” means (i.e., {object1, object2, …}). At this stage, they can give you one balloon if you ask them to, but they’re unable to give you two, three, four, or a larger number of balloons. They think all these latter number words mean the same thing, namely, some. But over a period of about a year, they learn to differentiate this second representation. “Two” comes to be connected with {object1, object2}. However, “three,” “four,” and so on, are still associated with an arbitrary set of more than two elements. Then they learn “three” and occasionally “four” in the same way. Step 7: Finally, kids are able to notice that a relation exists between the sequence of numerals in the count list and the sequence they can form from their set-based representations. They understand this connection because their parents and teachers have leaned on them to count small groups of objects. When children count to “two” while pointing to two squirrels in a picture book, their representation of one thing, {object1}, is active, followed by their representation of two things, {object1, object2}. So kids begin to see a relation between the order of the numerals in their count list and the cardinality of the sets that these numerals denote. Step 8: At long last, then, the children can figure out that advancing by one numeral in the count list is coordinated with adding one object to a set. In other words, what they have learned is how counting, by ticking off the objects with the count list, manages to represent a total. They now have a rule that directly gives them the appropriate number of objects for any of the numerals on their count list, and they no longer need to keep track of the set representations for each numeral. They’ve learned that: cardinality(numeral(n)) = cardinality(numeral(n − 1)) + 1, (1) 3 You might wonder about the use of sets in this construction: Do young children have a notion of set that’s comparable to mathematicians’ sets? Attributing to children at this age a concept of set in the full-blooded sense would be fatal to Carey’s claim that Quinian bootstrapping produces new primitive concepts (e.g., the concept FIVE) that can’t in principle be expressed in terms of the child’s pre-bootstrap vocabulary. We know how to express FIVE in terms of sets (see, e.g., [8]). So the notion of set implicit in representations like {object1, object2} is presumably more restrictive than ordinary sets. Beliefs about Numbers / 10 contrast the structural perspective with the cardinal perspective, and the latter perspective is shared by nearly all developmental theories of number learning.5 3. Judgments of Order versus Numerosity Perhaps we can get a better purchase on the cardinal and structural perspectives by looking at the way people make direct judgments about cardinal and structural relations. For example, if people find it easier or more natural to determine the cardinality of sets of five than to determine the relation between five and six (e.g., 5 < 6), then the cardinal perspective may provide a better fit than the structural perspective to people’s apprehension of numbers. But although several studies exist that are relevant to this comparison, some inherent difficulties muddy the implications these studies have for the issue at hand. 3.1. Distance Effects To appreciate the difficulties in untangling these perspectives, consider a well-known and well- replicated finding from the earliest days of cognitive psychology [1; 2; 27]. On each trial in this type of experiment, adult participants see two single-digit numerals (e.g., “3” and “8”), one on each side of a screen, and their task is to press a button on the side of the larger number as quickly as possible. (The larger number is randomly positioned at the right or left across trials; so participants can’t anticipate the correct position.) One finding from this experiment is that the greater the absolute difference between the numbers, the faster participants’ responses. For example, participants are reliably faster to respond that 8 is larger than 3 than that 5 is larger than 3. This may seem surprising, given adults’ familiarity with the small positive integers. The standard explanation for this distance effect is that people automatically map each of the numerals to a mental representation that varies continuously with the size of the number. They then compare the two representations to determine which is larger. In comparing “3” and “5,” for example, they mentally represent 3 as an internal quantity of a particular amount, represent 5 as an internal quantity of a larger amount on the same dimension, and then compare the two quantities to determine their relative size. Imagine, for example, that 3 is represented as some degree of neural activation in a particular brain region, and 5 as a larger degree of activation in an adjacent region. Then people find the correct answer to the problem by comparing these degrees of activation. On this account, the distance effect is due to the fact that people find it easier to compare quantities that are farther apart on the underlying dimension. Just as it’s easier to determine the heavier of a 3 kg and an 8 kg weight than the heavier of a 3 kg and a 5 kg weight by hefting them, it’s easier to determine the larger of 3 and 8 than the larger of 3 and 5. The mental system responsible for this type of comparison goes by a number of aliases (e.g., “mental number line,” “analog magnitude system,” and “number sense”), but these days most researchers call it the “approximate number system.” So will I. Much evidence suggests that the approximate number system is present in human infants and in a variety of non-human animals (see [7] for a review). Experiments along these lines present two groups of dots, tones, or other non-symbolic items to determine the creatures’ sensitivity to differences in the cardinality of these groups. Distance effects appear with these non-symbolic items that echo those found 5 The same is true in education theory, as Sinclair points out in her chapter in this volume. See that chapter for an alternative that is more in line with the structural perspective. Beliefs about Numbers / 11 with adults and numerals: The larger the difference in the cardinality between the two groups of objects, the easier they are to discriminate. So you could reasonably suppose that the approximate number system is an innate device specialized for detecting cardinalities (e.g., the number of edible objects in a region), that this device persists in human adults, and that the symbolic distance effect for numerals, described a moment ago, is the result of the same system. What’s important in the present context is that what appears to be a primitive system for dealing with cardinality—the approximate number system—may underlie adults’ judgments of ordinal relations (e.g., 3 < 8). If so, the cardinal perspective may be more fundamental than the structural one in human cognition. However, these results do not necessarily support the cardinal perspective. Nearly all quantitative physical dimensions—acoustic pressure, luminosity, mass, spatial area, and others—produce distance effects of similar sorts. The perceptual system translates a physical value along these dimensions (e.g., mass) into a perceived value (e.g., felt weight) that can be compared to others of the same type, and comparisons are easier, the greater the absolute value of the difference. The same is true of symbolic stimuli [26]. If participants are asked to decide, for example, which of two animal names (e.g., “horse” or “dog”) denotes the larger-sized animal, times are faster the bigger the difference in the animals’ physical size. But this effect provides no reason to think that we encode animal sizes as cardinalities. So it is not at all clear that distance effects for numerals depend specifically on representing them in terms of cardinality. Instead, the effects may be due to very general psychophysical properties of the perceptual and cognitive systems. 3.2. Judgments of Order To make some headway on the cardinal and structural perspectives, we need a more direct comparison of people’s abilities to judge cardinal and structural properties. Is it easier for people to assess the size of a set associated with a positive integer or to assess the integer’s relation to others? Linnebo’s [21] second argument, mentioned in Section 1, seems to predict a negative answer to this question. Several recent studies have attempted a comparison of this sort, but the implications for our purposes are difficult to interpret because of some inherent features of the procedures. Here’s an example: Lyons and Beilock [22] compared a “cardinal task,” similar to the standard “Which is larger?” method, described in the preceding subsection, to a novel “ordinal task.” Table 1 summarizes the main conditions in the study. In the ordinal task, participants decided as quickly as possible whether triples of single-digit numerals were correctly ordered (in either ascending or descending sequence) or incorrectly ordered. For instance, participants were to answer “yes” to <2, 3, 4> or <4, 3, 2> but “no” to <3, 4, 2>. The elements of the triples could be separated by an absolute difference of one (the “close” triples in Table 1, e.g., <2, 3, 4>) or two (the “far” triples, e.g., <2, 4, 6>). Lyons and Beilock also included a task in which participants made analogous judgments for triples of dots. For example, they decided whether a triple consisting of two dots followed by three dots followed by four dots was correctly ordered. For the cardinal task, participants decided which of two numerals (e.g., 2 or 3) was larger or which of two groups of dots was larger (e.g., a group of two or a group of three dots). As in the ordinal task, the items within a pair could differ by one or by two. Lyons and Beilock found the usual distance effect in the cardinal task, both for numerals and dots. That is, participants were quicker to respond to the far pairs (e.g., 2 vs. 4) than to the close pairs (e.g., 2 vs. 3). For the ordinal task, however, the results were different for numerals than for the dots. Although the dots showed a distance effect, numerals showed the reverse: The close triples were faster than the far triples. (For related findings, see [10; 44].) The investigators conclude from these findings Beliefs about Numbers / 12 that the link between mental representations of numerals and cardinalities is less direct than what one might gather from the typical distance effects. Judgments of order for numerals rely on a process distinct from the one governing judgments of cardinality (presumably, the approximate number system). “At the broadest level, the meaning of 6 may thus be determined by both its relation to other symbolic numbers and the computational context in which it rests. This is in keeping with the view that the meaning of symbolic numbers is fundamentally tied to their relations with other symbolic numbers…” [22, p. 17059]. The emphasis on “relations with other symbolic numbers” goes along with what we have been calling the structural perspective. However, the relations in question raise an issue about the type of structure responsible for the data. Lyons and Beilock plausibly suggest that the reversal they observe for ordinal judgments of numerals—faster times for close triples in Table 1—is the result of familiarity with the list of count words. Because college-student participants have rehearsed the count list (“one, two, three,…”) on many thousands of occasions during their lives, they find it easier to recognize numerals as correctly ordered when they appear in adjacent positions on the list (e.g., <2, 3, 4>) than when they are not in adjacent positions (e.g., <2, 4, 6>). fMRI imaging evidence from the same experiment suggests that “one interpretation of these results is thus that ordinality in symbolic numbers is processed via controlled retrieval of sequential visuomotor associations…” [22, p. 17056]. In the case of ordinal judgments for dots, however, participants can’t directly access the count list, but have to compare successively the Table 1 Summary of Main Conditions from Lyons and Beilock’s Comparison of Ordinal and Cardinal Judgments (Adapted from [22, Figure 1]). Entries Provide an Example in which Participants Should Respond “True” in the Ordinal Task and Push the Right-hand Button in the Cardinal Task Task Type Stimulus Items Ordinal Task (Are the items in ascending or descending order?) Cardinal Task (Which item is larger?) Numerals Close 2 3 4 2 3 Far 2 4 6 2 4 Dots Close Far Beliefs about Numbers / 15 You might argue that the very low frequency for cardinality is due to the fact that each natural number has just one cardinality, whereas it has many properties of other types (e.g., many number- theoretic properties). So perhaps the frequency of mention just reflects the actual number of available tokens per type. But although a natural number has just one cardinality, it nevertheless denotes the size of an infinite number of sets. Participants could have said that three is the number of bears in the fairy tale, the number of stooges in the film comedies, the number of instrumentalists or vocalists in a trio, the number of vertices or angles or sides in a triangle, the number of people that’s a crowd, the number of deities in the trinity, the number of events in a triathlon, the number of rings in a circus, the number of races in the triple crown, the number of children in a set of triplets, the number of outs in a turn at bat, the number of isotopes of carbon, the number of novels in a trilogy, the number of wheels on a tricycle, the number of leaders in a triumvirate, the number of panels in a triptych. But no one mentioned any of these or any other three-membered groups in response to three. Moreover, for the numbers 3, 7193, 29,000,000,091, and (8.3 × 1055) + 76, none of the participants mentioned cardinality even once. Of course, we asked participants to list number “properties,” and perhaps this way of phrasing the question militated against their writing down items related to cardinality. For example, the number of things in a Beliefs about Numbers / 16 triple or of vertices in a triangle might have seemed too extrinsic to qualify as a property of three. But the question at issue in this study is, in fact, what participants believe to be intrinsic to their number concepts. What kind of information about a number is central to people’s beliefs about the number’s nature? If 0 20 40 60 80 100 0 2 4 6 8 10 0 20 40 60 80 100 0 2 4 6 8 10 Fr eq ue nc y of M en tio n (T ok en s) 0 20 40 60 80 100 M ea n Im po rta nc e R at in g (0 -9 S ca le ) 0 2 4 6 8 10 0 20 40 60 80 100 0 2 4 6 8 10 Property Type cardinality magnitude number lin e number system comparison arith metic oper number th eoretic numeral props non-number props other 0 20 40 60 80 100 0 2 4 6 8 10 -849 0 2 3 7193 Frequency Importance Figure 2. Frequency of mention of properties of different types (solid lines and symbols) and mean rated importance of the same properties (dashed lines and open symbols) from Study 1. Points on the importance curves are missing if participants listed no properties of that type. Beliefs about Numbers / 17 participants find “being an integer” and “being divisible by 3” to be number properties but not “being the number of items in a triple,” then this suggests that cardinality may not be what organizes their conception of numbers. Mean importance ratings for the same properties also appear in Figure 2 (dashed lines and open circles). The means for the cardinality and number-line properties are based on only a small number of data points, as the frequency curves show. Omitting these latter categories, a statistical analysis indicates that, across all the numbers, participants rated the number-theory, arithmetic-operations, and number- system properties as more important than the non-numeric properties. They also rated “other” properties lowest in importance, as you would hope. No further reliable differences in importance appeared among the property categories (adjusting for the number of comparisons). However, arithmetic operations are especially important for zero, probably for the reasons mentioned earlier, and number-theory properties (e.g., “are prime”) are important for three. (7193 is also prime, but participants probably didn’t recognize it as such.) These peaks contribute to a statistically reliable difference in the shape of the importance curves in the figure. Keep in mind, though, that the properties contributing to these importance ratings are the ones that the same participants produced in the first part of the experiment. It might be useful to look at an independent measure of the importance of number properties. The study in the next section provides a measure of this sort. 4.2. The Centrality of Number Properties In a second study of number properties, a new group of participants decided whether a given property of an integer “was responsible for” a second property of the same integer. The properties included: Cardinality (phrased as “being able to represent a certain number of objects”) Number system membership (“being an integer”) Arithmetic (“being equal to the immediately preceding integer plus one”) Position in the integer sequence (“being between the immediately preceding and the immediately following integers”), and Numeric symbol (“being represented by a particular written symbol”). On one trial, for example, participants were asked to consider whether “an integer’s ability to represent a certain number of objects is responsible for its being equal to the immediately preceding integer plus one.” The instructions told participants, “By ‘responsible’ we mean that the first property is the basis or reason for the second property.” The participants answered each of the responsibility questions by clicking a “yes” or a “no” button on the screen. In a preliminary version of this experiment, I asked participants about the properties of specific integers from Study 1 (0, 3, 7193, and 29,000,000,091). For example, participants had to decide whether three’s property of representing three objects was responsible for its being equal to 2 + 1. The results from this pilot study, however, suggested that participants were interpreting these questions in a way that depended on the specificity of the properties. Asked whether being an integer was responsible for three’s being able to represent three objects, for example, many participants answered “no,” apparently because Beliefs about Numbers / 20 Responsibility scores for cardinality (“being able to represent a certain number of objects”) were about midway between those for the system property and symbol property. Cardinality does somewhat better here than we might have predicted from its very infrequent mention in the preceding study, where the participants had to produce their own properties for the numbers (see Figure 2, panels 2, 4, and 5). The results from Study 1 could be put down to the obviousness of the connection between an integer and its cardinality. Maybe it literally goes without saying that “three” denotes three entities. But the same is probably true of several of the other properties on our list. Another more likely possibility is that because the cardinality property fixes its referent in a nonarbitrary way, some participants in the present study may regard cardinality as sufficient for the other properties. For example, if a number has cardinality 3, then it is 3, and hence, is the successor of 2, is between 2 and 4, is an integer, and is symbolized by “3.” Other participants, however, may regard cardinality as too “incidental” to an integer to allow it to be responsible for its purely mathematical properties, and this same feeling may explain why cardinality went unmentioned in Study 1. The final section tries to sketch what “incidental” could amount to in the mathematical domain. 5. Conclusions What implications do the two studies of Section 4 have for the cardinal and structural perspectives? On the one hand, Study 1 suggests that cardinality isn’t something that often occurs to people when they are asked to consider the properties of an integer. They don’t immediately take a cardinality perspective under these circumstances. Instead, the properties of the integer that come readily to mind are ones that might be called “number internal,” such as being odd or positive or prime. In the case of special numbers, such as 0, these number-internal properties also include arithmetic or algebraic features, such as being an additive identity. On the other hand, Study 2 shows that some people acknowledge the importance of cardinality in relation to an integer’s other properties. Once we know that an integer has a particular cardinality, say 7193, we can predict all its other properties, for example, that it is the successor of 7192 and the predecessor of 7194. Cardinality can, in this way, single out an integer uniquely, yet people may not typically use it in thinking about the integer as such. Perhaps something of the same is true of numbers in general. In applied math, numbers denote specific times, spatial coordinates, velocities, temperatures, money, IQ, and what have you. But we don’t think of a number as intimately connected with these instances (e.g., a particular position, time, or value). A shift in co-ordinates will assign a different number to the same spatial point. A change in currency will associate a different number with the same value for a good. Numbers provide models for time, space, utility, and other dimensions, as measurement theory makes clear (see, e.g., [19; 40]), but only up to certain well-defined transformations. The numbers themselves have an identity that’s independent of their role in the model. In the same way, you could view the use of integers to denote cardinalities as another application of a number-based model to the size of sets. Proponents of the cardinality perspective still have some cards to play in defending the idea that cardinality is conceptually fundamental. For example, they could take the view that cardinality provides children’s first entry point to the natural numbers, even if children outgrow this perspective as they gain further knowledge of number properties in school. Or they could claim that cardinality is central to adults’ thinking about the naturals, even though adults don’t often mention it when explicitly asked about number properties. But although defenses of this kind may turn out to be right, not much evidence exists to back them. We noticed in Section 2 that it’s hard to see how cardinality can by itself advance children’s knowledge of the naturals. Mere apprehension of numerosity through the approximate number system Beliefs about Numbers / 21 can’t provide the right properties for the naturals (e.g., the perceived difference in numerosity between 14 items and 15 items is smaller than the perceived difference between 4 and 5 in this system). Perhaps the more sophisticated use of cardinality in children’s initial counting creates the link to the naturals, but we found in Section 2 that it’s difficult to make this case without begging the question against the structural perspective (see [31; 37] for more on this theme). Similarly, the results we reviewed in Section 3 show that, although numerosity may influence adults’ judgments about the integers in tasks like numeral comparison, there’s a catch: The mental process most likely responsible such effects is again the approximate number system, which delivers distorted information about cardinality. Maybe true cardinality underlies adult intuitions about the naturals, but what reasons support such an assumption? The structural perspective may be better able to cope with the results. But we should enter a couple of qualifications about the support that these studies lend to the structural view. First, we’ve seen that participants didn’t often mention properties connected to numerals (except for the larger numbers) in Study 1, and they didn’t judge them as especially important in producing a number’s other properties in Study 2. This seems reasonable in view of the arbitrary nature of the symbols. But if “the only perfectly direct and explicit way of specifying a number seems to be by means of some standard numeral in a system of numerals” [21], then you might have expected numerals to play a more important role in the results. Notice, though, that in these experiments no variations occurred in the manner in which the numbers were specified. In Study 1, a numeral was given to participants explicitly (as an English phrase), and they responded based on this numeral. In Study 2, no numerals appeared. So the results are still compatible with the possibility that numerals within a standard system provide the easiest way to denote numbers relative to other possibilities, such as the set of sets containing that number of elements. Second, in Study 1, the frequency of positional properties was fairly low. We classified these properties as arithmetic comparisons in that study (see Figure 2), and they included items such as “is greater than -1” (in the case of 0), “is smaller than four” (in the case of 3), and “is less than 10000” (in the case of 7193). Properties of this sort comprised only 7.3% of tokens across the five numbers in Figure 2 and produced a mean importance rating of 4.5, which is the midpoint of the 0-9 scale. Why did participants fail to produce these positional properties and fail to rate them as important? The reason may be similar to the one that makes cardinality properties unpopular. People may feel that relations like being between 7192 and 7194 in the integer sequence are external to the number 7193 itself. What come to mind more readily for the numbers in Study 1 are properties like being even or positive—properties we classified as “number-theoretic” in the loose sense of the first study. These properties were either the most often mentioned (for -849 and 3) or the second most often mentioned (for 0, √2, and 7193), as Figure 2 indicates. We can unpack properties like these in relational terms—being positive is being greater than 0, and being even is being divisible by 2—but participants may see them (at least at first thought) as something intrinsic to the numbers themselves. My guess is that much the same is true for most other individual concepts (e.g., UNCLE FRED, WALDEN COLLEGE): The properties of these concepts that are easiest to access are ones that we represent as monadic. It may be an important fact about individual concepts that we mentally organize them around central properties of this sort and take other, more peripheral properties to be the products of the central ones. Uncle Fred may have traits like being genteel or pig-headed that we see as non-relational and intrinsic, but that are responsible for many other aspects of his personality and behavior. In planning our interactions with him, or in predicting what he will do at the party or believe about big government or want for his birthday, we consult and extrapolate from these central properties. The suggestion here is that this type of thinking may carry over to individuals in abstract domains, such as numbers. We don’t personify numbers in the way Lewis Carroll did (“‘Look out now, Five! Don't go splashing paint over me Beliefs about Numbers / 22 like that!’ ‘I couldn't help it,’ said Five, in a sulky tone; ‘Seven jogged my elbow.’” [4, chap. 8]). But we might think about five as centrally positive, odd, and prime because these properties are handy in making inferences about this number. This isn’t quite the structural perspective that we get in the philosophy of mathematics, but it has its own structural aspects. Structuralists in philosophy see numbers as atoms with no internal structure, as the quotations from Resnik [30] and Shapiro [43] in Section 1 make clear. Their content is exhausted by the position they have in a relevant number system, as given by an appropriate set of axioms. In one way, this seems right for psychological concepts of numbers, as well. Our mental representations of a natural number, for example, had better conform to the usual Dedekind-Peano axioms, since otherwise it’s difficult to see in what sense they could represent that number [38]. In another way, however, mental representations about a natural number include a richer domain of facts that we use in dealing with typical mathematical tasks, including calculation and proof (see [34] for the distinction between “representations of” and “representations about” objects and categories). What the studies reported here suggest is that information of the latter sort, at least among college students, may organize itself around, not cardinality, but instead properties like primality that may be more helpful in mathematical contexts.