Ordering of the Numerosities 1 to 9 by Monkeys

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Science  23 Oct 1998:
Vol. 282, Issue 5389, pp. 746-749
DOI: 10.1126/science.282.5389.746


A fundamental question in cognitive science is whether animals can represent numerosity (a property of a stimulus that is defined by the number of discriminable elements it contains) and use numerical representations computationally. Here, it was shown that rhesus monkeys represent the numerosity of visual stimuli and detect their ordinal disparity. Two monkeys were first trained to respond to exemplars of the numerosities 1 to 4 in an ascending numerical order (1 → 2 → 3 → 4). As a control for non-numerical cues, exemplars were varied with respect to size, shape, and color. The monkeys were later tested, without reward, on their ability to order stimulus pairs composed of the novel numerosities 5 to 9. Both monkeys responded in an ascending order to the novel numerosities. These results show that rhesus monkeys represent the numerosities 1 to 9 on an ordinal scale.

Many animal taxa can discriminate stimuli differing in numerosity (1). The importance of this capacity has evoked considerable controversy. Some have argued that animals have a natural ability to discriminate numerosity (2, 3); others maintain that animals attend to numerosity as a “last resort,” that is, only if all other bases for discrimination are eliminated (for example, the shape, color, brightness, size, frequency, or duration of a stimulus) (4).

To defend either position, it is necessary to show that an animal's behavior is controlled by numerosity rather than by one or more non-numerical features of a test stimulus, such as density, surface area, or duration. This is best accomplished by analyzing the first-trial accuracy of responses to exemplars of the numerosity in question (5). Here, we show that monkeys can discriminate exemplars of the numerosities 1 to 4 when non-numerical cues are controlled.

Another important question about the numerical ability of animals is whether they represent ordinal relations among numerosities or, instead, represent each numerosity as a nominal category (6). To evaluate knowledge of numerical ordinal relations, we tested monkeys who learned to discriminate the numerosities 1 to 4 on their ability to order pairs of the novel numerosities 5 to 9.

The subjects were two rhesus monkeys, Rosencrantz and Macduff. They were first trained to order exemplars of the numerosities 1 to 4 (7). Four exemplars, one of each numerosity, were displayed simultaneously on a touch-sensitive video monitor. The configuration of the exemplars was varied randomly between trials (8). The subjects' task was to respond to each exemplar in an ascending numerical order. Subjects had to learn the required sequence by trial and error by remembering the consequences of their responses to each stimulus. Any error ended the trial, correct responses produced brief auditory and visual feedback, and food reinforcement was given only after a correct response to the last stimulus. The same stimulus set was presented on each trial for at least 60 consecutive trials. During the initial phase of training, subjects were trained on 35 different stimulus sets of exemplars of the numerosities 1 to 4. Examples are shown in Fig. 1A.

Figure 1

(A) Exemplars of the seven different types of stimulus sets: equal size (elements were of the same size and shape); equal surface area (cumulative area of elements was equal); random size (element size varied randomly across stimuli); clip art (identical nongeometric elements selected from clip art software); clip art mixed (clip art elements of variable shape); random size and shape (elements within a stimulus were varied randomly in size and shape); and random size, shape, and color (same as random size and shape, but with background and foreground colors varied between stimuli). All types were used with equal frequency in both four-item training and four-item testing. (B) Examples of stimulus sets used in the pairwise numerosity test. The smaller numerosity had a larger cumulative surface area than the larger numerosity on 50% of all trials, and elements within each stimulus were identical in size, shape, and color.

The percentage of trials on which subjects responded to each numerosity in the correct order was well above the chance level of accuracy for each of the 35 training sets. As shown in Fig. 2A, performance also increased with each new set. This increase could reflect either or both of the following factors: (i) Subjects learned the order in which to respond to each stimulus more rapidly, and (ii) subjects learned to use the relative numerosity of each stimulus to predict the required response order for each new stimulus set. The first explanation is plausible because subjects were trained for at least 60 trials on each stimulus set. Repeated exposure to each set provided ample opportunity to associate some non-numerical feature of each stimulus (for example, the configuration of the elements) with its ordinal position (9).

Figure 2

(A) Percentage of correctly completed trials during the first session for each of 35 training stimulus sets in blocks of five sessions. Performance was above chance on the training sets [Rosencrantz, t(34) = 11.9,P < 0.001; Macduff, t(34) = 8.8,P < 0.001] and improved across blocks (Rosencrantz,r 2 = 0.425, P < 0.01; Macduff, r 2 = 0.63, P < 0.01). (B) Percentage of correctly completed trials on the 150 test sets. Performance exceeded chance levels [Rosencrantz,t(4) = 12.7, P < 0.001; Macduff,t(4) = 12.8, P < 0.001]. There was no decrement in performance from the last five training blocks to the five transfer sessions [Rosencrantz, t(4) = –0.69,P > 0.5; Macduff, t(4) = –1.0,P > 0.36]. The percentage of correctly completed trials varied across stimulus types (equal size, 60%; equal surface area, 57%; random size, 36%; clip art, 42%; clip art mixed, 42%; random size and shape, 32%; random size, shape, and color, 24%.).

The opportunity to learn the correct order in which to respond to a new set of stimuli was eliminated during test sessions in which 150 new stimulus sets were presented only once (30 sets per session for five consecutive sessions) (10). Figure 2B shows the percentage of correctly completed trials on the 150 test sets. Numerosity was the only basis for ordering items on the test sets. Accuracy substantially exceeded the level predicted by chance and did not differ from subjects' accuracy during the last five blocks of the 35 training sets (in which subjects could have used non-numerical features of the stimuli to learn the correct order) (11). Rosencrantz's and Macduff's performance on the test sets shows that they learned to discriminate numerosity during training even when a non-numerical strategy would have sufficed. It should also be clear that their performance cannot be attributed to a “last resort” strategy.

In addition to providing unequivocal first-trial evidence that monkeys can discriminate the numerosities 1 to 4 categorically, Rosencrantz's and Macduff's ability to order new exemplars of numerosity suggests that they learned an ordinal rule. An alternative explanation of their performance on the 150 test sets is that they discriminated each numerosity as a nominal category (12) and that they then applied an arbitrary rule to order four unrelated categories (13). To rule out this alternative explanation, we evaluated our subjects' ability to respond correctly to stimulus pairs of novel numerosities in an ascending numerical order (14).

Both monkeys were tested on each of the 36 numerosity pairs that could be generated from the numerosities 1 to 9 (Fig. 3) (15). The numerosities 1 to 4 were familiar by virtue of the subjects' previous training; the numerosities 5 to 9 were novel. Subjects were reinforced for responding in an ascending order on trials on which the six familiar-familiar pairs were presented (Fig. 3, red symbols), but no reinforcement was provided for the familiar-novel or novel-novel pairs (Fig. 3, black symbols). The restriction of reinforcement to familiar-familiar pairs prevented subjects from learning the ordinal relations among the novel numerosities. To control for non-numerical cues, we used new exemplars of each numerosity on each trial. The size of the elements within each stimulus was also varied to eliminate size or surface area as a non-numerical cue (Fig. 1B).

Figure 3

The 36 pairs of the numerosities 1 to 9 used in the pairwise test. These are segregated into three types that were defined with respect to the subjects' prior experience with the constituent numerosities: familiar-familiar (FF), familiar-novel (FN), or novel-novel (NN).

Rosencrantz's and Macduff's performance on familiar-familiar, familiar-novel, and novel-novel pairs is shown in Table 1. Both subjects responded in an ascending order on each type of numerical pair (16). Their use of an ascending rule on the 26 pairs that contained a familiar numerosity can, to some extent, be attributed to prior training on sequences of the numerosities 1 to 4. However, experience with familiar numerosities cannot explain Rosencrantz's and Macduff's ability to respond to novel-novel pairs in an ascending order. Nor can transitive inference explain this ability. Although nonhuman primates are capable of transitive inference (17), the absence of any overlap between the familiar-familiar and novel-novel pairs precludes the possibility that subjects could logically deduce the order of novel-novel pairs (for example, if A > B and B > C, then A > C). To respond to novel-novel pairs, subjects must be proficient in detecting ordinal disparities among novel numerosities and must be able to apply the ascending numerosity rule— learned previously with respect to the numerosities 1 to 4—to the numerosities 5 to 9 (18).

Table 1

Percent correct for the three types of numerical pairs. Both subjects' accuracy exceeded the chance-level accuracy on familiar-familiar pairs [Rosencrantz, t(19) = 70.0,P < 0.0001; Macduff, t(19) = 40.9, P< 0.0001], familiar-novel pairs [Rosencrantz, t(19) = 25.7, P < 0.0001; Macduff, t(19) = 32.3,P < 0.0001], and novel-novel pairs [Rosencrantz,t(19) = 9.1, P < 0.0001; Macduff,t(19) = 7.3, P < 0.0001].

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Further evidence that monkeys represent the ordinal relations among the numerosities 1 to 9 was obtained by analyzing accuracy as a function of the numerical distance between the two test stimuli (Fig. 4). The positive relation between accuracy and numerical distance is similar to that obtained from experiments with human subjects (20). This relation has been interpreted as evidence that numerosities are represented in an analog manner.

Figure 4

Effect of numerical distance on accuracy in the pairwise test (16). A numerical distance of 1 includes all pairs of adjacent numerosities (1 versus 2, 2 versus 3, and so forth), whereas a numerical distance of 8 includes only the pair 1 versus 9. The dashed lines represent the best-fit linear models. The linear fits were significant for both monkeys [(Rosencrantz,r 2 = 0.84, P < 0.05; Macduff, r 2 = 0.51, P < 0.05)].

Our results demonstrate that rhesus monkeys can spontaneously represent the numerosity of novel visual stimuli and that they can extrapolate an ordinal rule to novel numerosities. The process or processes that a monkey uses to detect the direction of ordinal disparities remain to be determined. Our subjects could have used a counting algorithm (19) to judge the relative magnitude of numerosities (20). Alternatively, they could have used a one-to-one correspondence matching algorithm whereby the elements of each stimulus were compared (21). Extensions of the nonverbal numerical tasks used in our experiment should provide a basis for assessing the extent to which a monkey's performance satisfies an operational definition of counting and may also clarify the nature of basic numerical abilities in animals and preverbal human infants.


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