One of the abilities that sets humans apart from other species is our outstanding capacity for mathematical knowledge. Our capacity to manipulate abstract quantities and perform complex calculations not only underlies many of our insights into the nature of the universe, but allows us to design technologies to explore it.

As with other domains in which humans seem to be unique among the species, the question has arisen as to the origins of our mathematical abilities. Researchers have addressed this issue through two broad classes of comparative approach. Developmental studies demonstrate mathematical abilities in human infants from an early age. Studies with other species provide a means of deducing the evolutionary course of mathematical knowledge. Studies with both prelinguistic human infants and non-human species also provide a means of examining numerical representation in the absence of human language. Several models for the representation of numbers have been suggested in order to explain how mathematical knowledge can be encoded without the aid of language.

In this introductory chapter, I will start by defining what we mean by "mathematical knowledge" and then discuss the logic of the comparative approach to the study of mathematical abilities. I will present the issue of whether numerical reasoning is a modularized domain of knowledge, discuss the relationship of language to this module, and review current models of nonlinguistic numerical representation. I will then review the empirical literature that pertains to the above issues in both human infants and other species. I will aim to show the relevance of these experiments to the origins of mathematical knowledge as well as which models best account for the results. In concluding this introductory chapter, I will outline the general approach of the experiments I have conducted with regard to the spontaneous numerical abilities of rhesus macaques, which I will present in the following chapters.

It is important to distinguish "mathematical" abilities from other kinds of computational abilities. All species that possess a neural system are capable of some computation. For the chordates, nearly all perception and response is conferred by fundamentally similar neural systems and their computational activity. The range of abilities exhibited across the animal kingdom, from the simple stimulus detection and gill retraction of the sea slug, aplysia, to the exotic echolocation and navigation of bats, can be accounted for almost entirely by the computations of neural systems.

Humans are capable of a wide range of computational abilities that are similar to other species, such as the abilities to gauge the distance of a perceived object, to determine the location of an acoustic source, and to discriminate the duration of time between events. We utilize these skills on a daily basis. However, these abilities do not set us apart from other animals. It is our "higher-level" abilities, such as the capacity for language and abstract thought, that seem to set us apart. One such ability that we tend to think of as uniquely human is our capacity for mathematical knowledge.

I will rely on Kitcher’s description of the concept of mathematics. In his words, mathematics, in its most basic form of arithmetic, "describes those structural features of the world in virtue of which we are able to segregate and recombine objects: the operations of segregation and recombination bring about the manifestation of underlying dispositional traits"(p.108). Thus, a child could learn that when he places 1 block and another block in a box, the box contains more than 1 block, revealing the rule "1 + 1 > 1." Note that this is the most basic of Kitcher’s descriptions of mathematics, which he admits does not "exhaust our mathematical performances"(p.112).

At this point, I must address one criticism of Kitcher’s account of the nature of mathematical knowledge. Wynn argues, on the basis of the numerical abilities she has revealed in infants (which I will review later) that Kitcher’s account of numerical knowledge cannot be true because it makes the claim that mathematical knowledge is gleaned solely from experience. True, Kitcher does state that children come to "accept basic truths of arithmetic by engaging in activities of collecting and segregating"(p.107). However, his argument is more sophisticated than Wynn seems to imply, given that he takes into account cultural transmission of mathematical knowledge: "We can view the activities of contemporary children as indicating the ways in which our ancestors … began the mathematical tradition"(p.108).

Kitcher’s philosophy of numerical knowledge can be reconciled with Wynn’s evidence of apparently inborn knowledge of mathematics by one crucial realization: mathematical knowledge can be genetically as well as culturally transmitted. Given this fact, one may interpret Kitcher’s "ancestors" as our evolutionary ancestors, and whatever knowledge they transmit to us is in the form of adaptive neural circuitry passed down by the process of natural selection. Wynn , in fact, supports the thesis that a mechanism of numerical representation has been passed to us through natural selection. A crucial question is: at what point did mathematical knowledge begin evolving as part of the human repertoire of cognitive abilities?

The empirical literature has used two general forms of comparative approach to investigate the origins of numerical knowledge. One line of experiments has studied the development of this cognitive ability in humans across the lifespan . These kinds of studies can reveal what cognitive abilities seem to be present from birth as well as how they are shaped and refined during development.

Another set of studies has addressed the evolutionary origins of mathematical knowledge by examining this ability in other species. The logic of cross-species cognitive comparisons is identical to that of the comparative approach used by biologists to determine the evolutionary history of the physical traits of species . By analyzing the similarities and differences among living species, one can deduce when and in which species a trait originated. Cognitive studies based on the comparative phylogenetic method require the same caution as studies of physical traits. Similarities between species may appear for at least two entirely different reasons. Homologies, such as the similarity between the bone structure of a human hand and a bat’s wing, stem from a shared evolutionary ancestor. Homoplasies, such as the similarity between the wings of birds and bats, arise from similar selection pressures.

These two different possible reasons for observing similarities between species are relevant to the study of mathematical knowledge in that we must be careful not to assume that animals that show evidence of numerical abilities necessarily share this trait because of common heritage, whether with each other or with humans. Such a conclusion must be the result of careful comparison of a variety of closely and distantly related species in order to determine the most parsimonious explanation of the evolution of this trait.

One difficulty of comparative cognitive research, as opposed to comparative studies of physical traits, is the design of methods that can be applied across species. In determining the degree of similarity in the above example between human and bat bone structures, biologists can count the number of bones and joints or come up with some such measure of the degree of physical similarity. However, cognitive researchers must measure less easily quantified variables such as "ability." The most obvious way to test ability is to measure an individual’s performance on a task designed to require that ability. If the ultimate purpose of such a task is to compare abilities across species, it would make sense to devise tasks that can be applied across different species, as Macphail has argued. However, until recent years, most research has failed to use analogous methods between humans and other species. Cognitive experimental paradigms with animals tend to rely heavily on training procedures, which are rarely used with humans. Similarly, methods applied to human subjects rely heavily on the use of language, which precludes their application to animals.

In recent years, certain researchers have begun to adapt methods that have been used with human infants to the study of animal cognition. These methods, such as Looking Time and Active Search , allow for more direct comparisons between humans and other species. Besides providing comparative data, the examination of numerical abilities in prelinguistic infants and non-human animals also allows us to determine the independence of these abilities from language, an issue discussed in the next section.

Does it make sense to consider numerical knowledge independently of other reasoning abilities? This question raises the general issue of modularity, which concerns the architecture of the human mind. According to one widely held view , the mind is made up of a general set of abilities that are applied to any given task. On the other hand, it has been increasingly argued , that the mind is composed of independently functioning "modules" that are specialized for particular domains. Modules are thought to process information quickly and automatically, using a fixed neural architecture.

Evidence for the modular view of the human mind includes the observation of aphasias and agnosias, which are characteristic and specific breakdowns in cognitive function, usually due to localized brain damage. For example, aphasias in the domain of language are particularly striking because they can occur with virtually no effect on a person’s other cognitive functions . In a similar way, Dehaene reviews striking evidence of "acalculias," specific deficits in numerical abilities due to localized brain damage. With this and other evidence from a wide range of studies, Dehaene dedicates his entire book, The Number Sense, to explaining how mathematics seems to be a modularized domain of knowledge. One source of evidence which he discusses, and to which this thesis pertains, is research that demonstrates common mathematical abilities across other species, suggesting a "number module" that has biological foundations and can be traced through phylogeny.

Another major issue that concerns the cognitive domain of numerical abilities is the extent to which the number module interacts with language abilities. Dehaene et al. present recent neuroimaging evidence in humans that implicates language areas of the cerebral cortex in the execution of exact arithmetic. Hauser and Carey point out that language encodes numerical information in three different ways: explicitly ("three oranges"), grammatically (" three oranges"), and in terms of labeling objects as entities to be counted in the first place ("three oranges"). Given these close relationships between processing of numbers and language in humans, this raises the question of what kind of numerical abilities can exist without language.

Three different major models have been suggested for how numerical knowledge could be represented in a nonlinguistic mind. The Numeron List Model suggests that a nonlinguistic mind may have at its disposal an innate, mentally represented list of arbitrary symbols, or numerons, that have an established order (for instance, !,@,#,$ …). When this mind "counts" objects, it simply places these objects in one-to-one correspondence with the numeron list. Once the last object in a set is "counted," the mind can determine the number of objects by determining the ordinal position of the numeron in the list. For example, an infant might see 3 oranges. It looks at one, and thinks "!," looks at the second and thinks "@," and then looks at the third and thinks "#" (based on the sample numeron list above). Since the last symbol is "#," and this is the third numeron in the list, this represents the number that we would refer to as "3."

Meck and Church have suggested another nonlinguistic representational system that would make use of a timing mechanism to represent numbers. This Accumulator Model can be most easily understood by the idea of using a mental "measuring cup" to gauge quantities. Imagine that some part of the mind serves as a continuous source of input, analogous to a kitchen faucet left running. One could represent numbers of events or objects by holding the measuring cup under the faucet for a fixed amount of time for each thing to be counted. If each counted object filled the measuring cup by 1 ounce, for example, then 3 oranges would be represented by 3 ounces of mental "water."

Dehaene states that the Accumulator should exhibit certain limitations because of the inherent variability of natural systems. Specifically, the mental measuring cup will give progressively less accurate measurements as it is filled up because the "faucet," being a natural timing mechanism, delivers slightly different bursts of "water" for each entity that is counted. This account predicts magnitude and distance effects for discriminating numbers, since the cumulative inaccuracy of the Accumulator will render numbers higher in magnitude and closer in distance more difficult to discriminate.

The Object File Model suggests that nonlinguistic minds might represent numbers by constructing a simplified mental image of perceived objects. In this scheme, the mind would use basic object knowledge to open an "object file" for each thing perceived to be an object. This model would make use of no integer symbols (as in the Numeron Model) or analogue quantities (as in the Accumulator Model) for representing number, but would rather represent the counted objects themselves as mental representations of objects, or files. The Object File Model, however, is inherently limited in the numerical quantities that it can represent because of the limit to parallel individuation of objects. In this model, objects are "individuated in parallel" in the sense that they are maintained simultaneously in short-term memory. The numerical limit to parallel individuation, and thus the Object File Model, is at about the number 4, as has been shown with adult human subjects .

These models of nonlinguistic numerical representation are important because they suggest what kind of numerical abilities could have evolved in the absence of language. Depending on their validity, these models can illuminate the limitations of nonlinguistic mathematical abilities in animals and human infants. There is current debate as to which of these models of numerical representation is most consistent with empirical data from nonlinguistic subjects. I review some of the empirical evidence for and against these various models below.

In reviewing the research on mathematical abilities, I am excluding studies with adult humans and focusing on the evidence from prelinguistic human infants and non-human animals. The reason for this focus is primarily to avoid the confound of language, since I wish to review evidence for numerical abilities alone. Experiments with adult humans almost invariably require the use of language, a skill not possessed by non-human subjects. This reduces the comparative value of such studies. By this same logic, I will only summarize the animal studies that have depended on extensive training of subjects. These studies are, in general, not comparable to the human infant research, since human infants are not subjected to such paradigms. I will begin by summarizing and offering current interpretations of the operant animal literature.

Wynn points out that Meck and Church conceived of the Accumulator Model because of the similarities they saw between animals’ counting and timing abilities. These similarities, noted across various operant studies, include: 1) the fact that animals transfer learned numerical and duration discriminations to novel stimuli to a similar extent; 2) trained animals that are administered with methamphetamine and then allowed to perform in the same tasks will overestimate both duration and number to the same extent; and 3) measurements of duration and of quantity seem to be interchangeable, for example, a rat that has been trained to respond to 5 one-second tones will respond in a similar way to a 5-second tone. These three points suggest that animals may use a similar mechanism for both duration and number representation, which would be consistent with the Accumulator Model.

Wynn also notes that animals show a greater precision for discriminating smaller quantities, but can discriminate larger quantities if they are sufficiently far apart. Pigeons, for example, can be trained to peck a given number of times for a reward . Although they can discriminate 4 pecks from 5 pecks consistently, they fail at tasks that require discrimination of the same distance between numbers of greater magnitude, such as 49 and 50. However, pigeons can succeed at higher magnitudes if the distance between numbers is increased, such as in the discrimination of 40 and 50 pecks. These magnitude and distance effects are consistent with the Accumulator Model, as discussed above.

One should not, however, place too much weight on the above evidence from human infants, in light of recent evidence that they may not be attending to the number of objects in such tasks. Clearfield and Mix have demonstrated that human infants of 6- to 8-months of age discriminate small sets of visual stimuli on the basis of their contours — or combined perimeter length — but not the number of stimuli. This evidence calls the data from infant habituation studies into question because these studies had been designed to test numerical discriminations, but may, in fact, have obtained results based on the infants attending to the perimeters of visual figures. This confounding factor was not previously considered because, in most situations — such as when objects are of equal size — number of objects correlates with combined perimeter length.

There are a handful of infant habituation experiments that escape the above criticism, because they do not rely specifically on visual figures as stimuli. These studies show that infants can also discriminate numbers of non-object stimuli, such as sounds in a sequence and numbers of actions of a puppet . These are more convincing evidence of numerical discrimination, since they cannot be confounded by a factor such as contour length. These data cannot be accounted for by an Object File mechanism, which can only represent numbers of objects, but are consistent with the Accumulator Model, which can represent any entity that can be measured with a timing mechanism.

The learning of numerical symbols in non-human species provides similar evidence against the Numeron List Model. Hauser and Carey cite studies that have taught non-human primates symbols and relations of the numbers 0 through 9, which have typically involved chimpanzees. However, squirrel monkeys, a new world primate even more distantly related to humans, have also been taught the ordinal relations between paired numerals up to 9 . One experimenter has even taught a parrot to verbally name numbers from 1 to 6 .

The above-mentioned studies required an overwhelming amount of training to teach animals symbols for number, as many as 95 hours of continuous training in Matsuzawa’s and 500,000 trials in Thomas’s chimpanzee studies. Given that these experiments were training the chimps to use a Numeral List system, the amount and detailed nature of the training makes it unlikely that the chimps naturally represent quantities using a Numeron List. The Numeron List, although widely expressed in human languages, is most likely a human cultural construction, since it is inconsistent with the evidence from human children and non-human animals.

At this point, it is necessary to reiterate that the above operant and symbol-learning data from non-human animals have all relied on extensive training of subjects. Such paradigms have come under criticism because of the large amounts of training involved and the artificial demands and environment of the tasks. Some critics have gone so far as to suggest that animals do not spontaneously encode numerical information at all, since they require so much training in order to succeed at numerical tasks. In light of such criticism, the experiments in this thesis will address not whether animals can be trained to encode such information, but whether they do spontaneously encode it. As the title of my thesis implies, I also intend to focus on the capacity of non-human primates to represent mathematical information, as defined earlier in this chapter. Whereas the above studies have addressed numerical abilities, they have not dealt with operations, such as addition and subtraction, that embody what is meant by "mathematical knowledge." I will now review several studies that approach the issue of mathematical knowledge through paradigms that depend less directly on training.

Some studies have made use of the learned symbolic numerical abilities of non-human primates to test more spontaneous abilities. Boysen describes one such study that has been performed with the particularly gifted chimpanzee Sheba. Sheba had been trained to match Arabic numerals (for this experiment, 1 through 4) with numbers of presented objects. Boysen used this acquired skill to test Sheba’s ability to add with the following scheme. Sheba was released into a room in which oranges were hidden in two places. She looked in one location, presumably noting the quantity of oranges there, then looked in the second location, and again saw how many oranges were there. After thus surveying the two separate quantities of oranges, Sheba would consistently point to the card out of a set of four Arabic numerals (1-4) that symbolized the sum of the oranges in the two locations. A subsequent study replaced the oranges with Arabic numeral cards (1-4). In this case, Sheba would go to two locations, presumably noting the numerals, and would then consistently point to a card that portrayed the sum of the two digits. These results are particularly convincing because Sheba consistently arrived at the correct sum of two symbolic numerals. However, one might still argue that Sheba learned to represent numbers during the course of her symbol training, but might not have spontaneously been able to add numbers had she not been exposed to this artificial experience.

Brannon and Terrace present evidence of spontaneous application of learned mathematical concepts in rhesus macaques. In this case, rhesus were trained to respond to visual arrays of 1 through 4 stimuli by touching them in ascending order. After this initial training, the monkeys were able to correctly order not only novel exemplars of the numbers 1-4, but were able to correctly order exemplars of the numbers 5-9, on which they had not been trained. Note that their ability to succeed with numbers greater than 4 is inconsistent with the Object File Model. The monkeys did, however, exhibit magnitude and distance effects, behaving consistently with the Accumulator Model. This experiment demonstrated the ability of rhesus to transfer the trained mathematical concept of ordinal relations to numbers that were not involved in the training paradigm. Nevertheless, a critic could still argue that this performance was not, strictly speaking, spontaneous, since the rhesus required training in order to participate in the experiment.

Rumbaugh, Savage-Rumbaugh, and Hegel present a study that bears on mathematical knowledge and utilized no previously trained abilities. Chimpanzees were presented with two trays of candies, with each tray holding two separate wells that contained candy. The task consisted of simply presenting the chimpanzees with various quantities of candies in the wells and allowing them to pick whichever tray they wanted. The subjects consistently chose the tray with the greatest total number of candies in both wells. The task controlled for the possibility that the chimpanzees were using simpler rules, such as choosing the tray with the greatest number of candies in either well, or avoiding the tray with the smallest number of candies in either well.

The experimenters interpreted this behavior to show that the chimpanzees were using addition skills to mentally add the quantities in each well of each tray and thus pick the tray with more candies. However, I argue that these results should not necessarily be interpreted as addition skills, since all the food was presented simultaneously within view of the subjects. The chimpanzees could have used basic counting across both wells on a tray, or even just low-level perceptual assessments of quantity (e.g., total perceived surface area of candy on a tray). The significant point of this criticism is that when quantities are presented all at once to a subject, the subject can make use of lower-level mechanisms of quantity assessment. Experiments that present objects sequentially and then hide them out of sight can provide more convincing evidence of arithmetic skills.

More convincing data for spontaneous mathematical abilities come from the Looking Time paradigm, which has been used extensively with human infants and requires no prior training of subjects. This method works like a magic trick, performing "expected" and "unexpected" events for subjects, and then measuring the amount of time that they spend scrutinizing the outcomes. A difference in this measurement between expected and unexpected events is thought to indicate a violation of subjects’ expectations. A series of experiments by Wynn made use of the Looking Time paradigm to determine whether infants will expect that operations such as "1 + 1" and "2 — 1" result in the respective quantities 2 and 1. In the first case, Wynn showed infants a Mickey Mouse doll on a stage, introduced a screen, and presented a second Mickey Mouse doll, which she placed behind the screen. Her four and a half-month old subjects then witnessed the screen drop to reveal either the expected outcome of 2 Mickey Mouse dolls or the unexpected outcome of 1 or 3 Mickey Mouse dolls. The infants looked significantly longer at the unexpected outcomes than at the expected one, suggesting that infants have some expectations as to the outcome of these basic operations of addition and subtraction.

With regard to the above studies, Koechlin, Dehaene, and Mehler have addressed the possibility that infants could merely be predicting the spatial location of specific objects, becoming surprised when objects are either missing or appear in new locations. They controlled for this possibility by placing a large rotating plate on the stage area of a looking-time apparatus. Infants still looked longer at the unexpected events of "1 + 1 = 1" and "2 — 1 = 2," although objects could no longer be attributed to specific locations, since they were in constant motion.

Recently, the Looking Time paradigm has been performed with non-human primates , using methods almost identical to Wynn’s . Free-ranging rhesus macaques were shown sequentially 2 eggplants (novel and interesting objects to the monkeys) being placed behind a screen onto an empty stage and witnessed the unexpected result of only 1 eggplant appearing when the screen was removed. The rhesus looked much longer at this "magic trick" of "0 + 1 + 1 = 1" than at other trials portraying "0 + 1 = 1" and "0 + 1 + 1 = 2." Another experiment in Hauser et al. compared looking times of two subtraction events using the same methodology. The rhesus looked longer at the impossible outcome of "2 — 1 = 2" than at the expected outcome of "2 — 1 = 1." Hauser and Carey went on to demonstrate similar Looking Time effects with cotton-top tamarins, a primate species far more evolutionarily distant from humans than rhesus. The tamarins looked longer at analogous unexpected versus expected addition events.

Hauser and Carey, having demonstrated the same effect with rhesus and tamarins that Wynn did with infants, disagree with her conclusion that these results mean that the subjects expect "1 + 1" to equal precisely 2. They point out the alternative explanation that both infants and monkeys could merely be assessing quantities of "stuff" rather than numbers of objects. To rule out this hypothesis, they performed an additional study with tamarins with the operations "1 + 1 = 2" versus "1 + 1 = double-size 1." In this experiment, the "double-size 1" was a single object twice the size of the "1" objects. Thus, both presentations were consistent in terms of amount of stuff, but the unexpected condition violated numerical principles. The tamarins looked longer at the unexpected result of the "double-size 1" showing that they were not merely attending to the amount of "stuff," but actually tracking the number of objects. Recently, the same result has been demonstrated with rhesus macaques (Hauser, unpublished data).

Hauser et al. have recently demonstrated spontaneous mathematical abilities in free-ranging rhesus macaques using another paradigm that requires no training of subjects. Individual subjects were presented with empty opaque boxes, to which pieces of apple were added, one at a time. The monkeys were allowed to approach the boxes, with the measurement of interest being which box they chose first. The rhesus chose the box with the greater number of apple slices when the comparisons were 1 vs. 2, 2 vs. 3, 3 vs. 4 and 3 vs. 5. They chose randomly when the boxes were filled with the larger numbers of 4 vs. 5, 4 vs. 6, 4 vs. 8 and 3 vs. 8 apple pieces.

These data seem to reveal an inability for the rhesus to discriminate numbers larger than 3, a limit that is consistent with the use of an Object File representational system, but not consistent with the Numeron List Model. The performance of the rhesus was also inconsistent with the Accumulator Model, since the rhesus failed to discriminate larger numbers even when they were comparable in relative magnitude to smaller discriminations. Specifically, the rhesus chose randomly in discriminations of 4 versus 6 food objects, but chose the larger quantity in a 2 versus 3 task. This is not consistent with an analogue numerical representation system, such as the Accumulator, since the ratio of 3:2 is equivalent to 6:4. This was the first systematic study of spontaneous numerical representation in animals. The failure of the rhesus to discriminate numbers greater than 3 stands in contrast to the results of studies that have required extensive training. Whereas these studies have supported the Accumulator Model, this recent evidence from a task requiring no training favors an Object File mechanism.

This chapter was designed to provide a general background for the experiments to come, but has not provided a complete account. Because my experiments address several different specific issues regarding the validity of the different models of numerical representation, I have saved certain studies to introduce the experiments that pertain to them. Specifically, I have explicitly left out some of the major evidence in favor of the Object File Model, which my experiments will further support. As the debate now stands, the Numeron List may model the representation of numbers in human languages, but is not consistent with the data from human infants and non-human animals. The Accumulator Model is consistent with much of the evidence that I have presented, but relies heavily on data from paradigms that have required extensive training of subjects. The Object File Model has emerged as a possible candidate, based on a paradigm that has tested numerical discrimination in a non-human primate without any training.

The experiments that I present in the following chapters will similarly use no training of subjects, and will test spontaneous mathematical abilities in rhesus macaques. The general aims of these experiments are to: 1) provide further comparative data relevant to the evolutionary origins of mathematical knowledge; 2) demonstrate novel experimental designs that require no training of subjects and could be applied across species; 3) further demonstrate the range of mathematical abilities that can exist in the absence of language; and 4) provide further data to discriminate the various models of nonlinguistic numerical representation.