Number plays a continuous role in humans’ everyday interactions – for many, our early morning routine may involve checking the temperature outside to know whether to grab a jacket on our way out the door, or checking our pockets for correct change to grab a coffee. Likewise, number is also an important function for nonhuman animals in their daily existence – knowing how much food is available, how many offspring one has, or how many predators are approaching are all useful survival skills that may be assessed through the use of numerical discrimination.

Counting in humans is defined as a formal process of enumeration in which each item in an array must be given a unique tag (Piaget 1952). When counting is referred to colloquially in research with nonhuman animals, it is generally referring instead to simple numerical discrimination, as opposed to formal number systems or mathematical ability (the exception to this are studies which explicitly study whether animals are able to discriminate absolute discreet number; however, results from these studies as a whole are inconclusive). Arithmetic is the product of human language and culture, and is generally considered a uniquely human construct. Interestingly, however, two human tribes (the Pirahã and the Mundurukú) have been shown to exist with no formal system or language to represent number (Everett et al. 2005; Sheffler 1978). The Mundurukú have number words up to only five, although each word is not as definite in meaning as number words in English; the Pirahã, on the other hand, do not have words for precise numbers, but rather concepts for small amounts versus larger amounts. Interestingly, in both cases, the lexical limitation is no obstacle to the ability of these individuals to discriminate between larger quantities of items (Gordon 2004; Pica et al. 2004). Likewise, preverbal human infants are able to discriminate quantities, despite lacking linguistic ability to express their choices (Wynn 1992; Xu and Spelke 2000).

Adult humans, nonhuman animals, and preverbal infants are all able to discriminate basic quantities due to an evolutionarily primitive system of numerical discrimination, known as the approximate number system (ANS; Merritt et al. 2012). According to the ANS, number is represented internally on a continuous number line, which allows both humans and nonhuman animals to discriminate approximate magnitudes. Number only takes on a discrete representation in human language when number symbols become mapped onto these approximate magnitudes through learning during childhood language acquisition and subsequent mathematics training. It is from this shared and primitive system that number symbols and the more precise use of number that is typical of adult humans has presumably risen.

A key element of the approximate number system is that it obeys Weber’s law. Weber’s law states that the change in stimulus intensity needed for an organism to detect a change is a constant proportion of the original stimulus intensity, rather than a constant amount. Two effects that are seen as a result of this are the distance effect and the magnitude effect. The distance effect maintains that the greater the distance between two numbers, the easier they will be to discriminate (for example, 9 vs. 1 will be easier to discriminate than 3 vs. 1). The magnitude effect is the common finding that when distance is held constant, larger numbers are harder to discriminate than smaller numbers (for example, 2 vs. 1 is easier to discriminate than 9 vs. 8).

Ratio effects in accordance with Weber’s law have clearly been demonstrated in many species. The most intensive early work in this area was conducted by Otto Köehler, with a variety of bird species (Koehler 1951). Subjects were presented with containers covered with lids showing different numbers of pieces of grain glued to the lid. The birds were rewarded for pecking at the lid with fewer pieces of grain, and punished for choosing an incorrect lid (punishment in this case ranged from being shooed away from the container by the experimenter, to having the incorrect lid thrown up into the air by a device known as the “frightening springboard” upon selection). Birds learned to choose the lid with fewer items; however, there are several alternate ways in which they could have solved this problem. If comparing, for example, a lid with 2 pieces of grain to a lid with 5 pieces of grain, the 2 pieces of grain will take up less space than the 5 pieces of grain, making it possible for the birds to solve this problem based on a judgment of surface area, rather than number. With enough trials, the birds could also simply learn to recognize the lid with two items as the correct response, without actually making a judgment based on relative numerosity. While it is not known if Köehler took any measures to eliminate these potential confounds, current studies of number in animals typically control for these issues by manipulating the size of stimuli such that it cannot be used as a reliable source of information. Animals are also typically trained with one set of ratios, and then tested with a new set of ratios. If an animal has simply memorized correct responses, then their performance should fall to chance when given new ratios. If, on the other hand, they understand numerosity they should apply the rule “choose the smaller number of items,” making it possible for the animal to choose correctly when given a ratio they have never seen before (it is worth noting that Köehler taught animals to choose the smaller number of items, whereas most studies typically teach animals to choose the larger number of items).

Number has also been studied through the use of the operant conditioning chamber. Roberts (2010) showed pigeons three different combinations of red and green light flashes, using ratios of 2 versus 1, 3 versus 2, and 4 versus 2. The use of these particular ratios was important because the distance between numbers is 1 for 2 versus 1 and 3 versus 2, but increases to 2 for 4 versus 2. The ratio between numbers, on the other hand, was equal for 2 versus 1 and 4 versus 2, but smaller for 3 versus 2. If distance was controlling the performance of the pigeons, then it should have been found that 4 versus 2 was most easily discriminated, while 2 versus 1 and 3 versus 2 were harder but equally discriminable. Instead, it was found that 2 versus 1 and 4 versus 2 were equally discriminable, while 3 versus 2 was significantly more difficult to discriminate. This finding suggests that ratio, not distance, was controlling the performance of the birds.

Number takes three forms – cardinal, ordinal, and nominal numbers. Cardinality refers to the elements of a set, and essentially asks the question “how many.” Ordinality refers to the rank of an item (e.g., third place). Nominality concerns assigned numbers (e.g., zip codes in the United States) used for labeling purposes only, and is therefore a uniquely human practice. Studies of numerosity in nonhuman animals are thus concerned exclusively with cardinal and ordinal number. When animals choose between two discrete numbers (e.g., 2 vs. 4), this constitutes a study of cardinality. Ordinality, however, has also been assessed in nonhuman animals. In a landmark study, Brannon and Terrace (1998) trained two monkeys to respond to displays containing 1–4 items in an ascending numerical order. As a control for non-numerical cues, exemplars were varied with respect to size, shape, and color, such that the correct response could only reliably be obtained through the use of number. The monkeys were later tested, without reward, on their ability to order stimulus pairs composed of the novel displays of 5–9 items. Both monkeys were found to correctly order the novel items. In another study, Cantlon and Brannon (2006) trained monkeys to order pairs of numerical stimuli with the values of 1–9. Once the monkeys learned to order these values, they were introduced to pairs of novel displays of 10, 15, 20, and 30 items. Once again, monkeys were able to spontaneously order the novel values, suggesting that there is no known upper limit on the numerical capacity of these animals.

A number of different procedures have been used in studying number in nonhuman animals. Often, these paradigms have been borrowed from developmental psychology studies of human infants, as in both cases procedures need to be nonverbal. In one study (West and Young 2002), a version of the preferential looking technique that had previously been used with infants (Wynn 1992) and monkeys (Flombaum et al. 2005) was adapted for use with dogs. In this task, dogs watched as food items were placed, one at a time, behind a screen and out of their view. The screen was then lifted so that the dog could see the resulting number of items. In one condition, dogs saw a simple, correct calculation (1 + 1 = 2). In two other conditions, however, dogs either saw an unexpected outcome in which fewer objects were present when the screen was lifted than should be expected (1 + 1 = 1), or saw an outcome in which more objects resulted than should be expected (1 + 1 = 3). When an expected outcome was used, dogs spent as much time looking at the outcome as they did looking at the initial presentation of items, but when a result was unexpected, dogs spent significantly longer looking at the resulting amount of food. This suggests that, like infants and monkeys, dogs may have been anticipating the outcome of the calculations, which would require them to employ some form of numerical system.

While the majority of numerical experiments with nonhuman animals take place in a laboratory setting (where arguments could be made that numerical ability is the result of thousands of trials in an artificial scenario), animals have also been shown to be sensitive to number in the wild. In one study using playback recordings of lioness roars, defending adult female lions were found to be more likely to approach the location of the recordings if they heard the roar of a single female lion, than if they heard the roars of three female lions (McComb et al. 1994). Avoiding the costs of engaging in territory fights with larger groups of assailants presents one of many selective advantages presented by the use of numerical information in the wild.

There is substantial support for the existence of the ANS, meaning that numbers are represented internally in an indiscrete fashion on a number line. The mathematical formulation of this number line, however, has been debated. Some researchers (Whalen et al. 1999) have proposed that this number line uses scalar variability. In this case, the numbers on the number line are equally spread, with variability that increases across numbers, making discrimination more difficult. Proponents of the logarithmic model (Nieder and Dehaene 2009), on the other hand, maintain that variability is fixed, while numbers themselves are numerically compressed as magnitude increases across the number line. Predictions made by both of the models, however, are quite similar.

While the ANS remains the most dominant explanation for numerical processing, arguments have also been made for a secondary number system, often referred to as the object file system (Brannon 2006; Carey 1998; Feigenson et al. 2004). While the analogue magnitude system is generally an accepted account of numerical representation, the object file system is more controversial. The object file system (sometimes referred to as “subitizing”) deals only with small numbers, specifically numbers 1–4. These numbers are thought to be mapped discretely in a one-to-one representation, making them instantly accessible. For example, if subjects are shown four dots on a screen, they do not need to systematically count the dots because they will immediately recognize that there are four dots.

In support of the object file system, Hauser et al. (2000) found that wild, untrained monkeys were able to discriminate and successfully choose a larger number of food items over a smaller number of food items. Over 200 semi-free-ranging rhesus monkeys watched as experimenters placed pieces of apple into each of two containers. The experimenters then walked away so that the monkeys could approach the containers. When the containers contained ratios of 1 versus 2, 2 versus 3, 3 versus 4, or 3 versus 5 slices of apple, the monkeys chose the container with the greater quantity of food. Interestingly, however, when the ratios exceeded the number four (4 vs. 5, 4 vs. 6, 4 vs. 8, or 3 vs. 8) the monkeys were unable to reliably choose the container with the most food. Hauser et al. suggested that the breakdown in performance of the monkeys when the number exceeded four items is evidence that the monkeys were using a spontaneous number system (i.e., the object file system) as opposed to an analogue magnitude system in order to solve the problem. In subsequent laboratory studies, (Beran 2001; Beran and Beran 2004), however, experiments analogous to those of Hauser et al. (2000) were conducted, in which chimps watched as an experimenter sequentially dropped pieces of food (M&Ms. or pieces of fruit), one-by-one, into each of two bowls. The chimp was then allowed to choose one of the two bowls and consume its contents. In this laboratory task, chimps discriminated magnitude well beyond 4 items, and up to 10 items.

The most persuasive argument against the use of an object file system is that even with numbers 1–4, both humans and nonhuman animals have been shown to demonstrate the ratio-effects that are the signature of the ANS (Beran and Rumbaugh 2001). If the one-to-one representation suggested by the object file system were in fact being used, then these ratio effects should not be seen for the numbers 1–4. As a result, some researchers question the evolutionary need for this secondary system. An alternate explanation for the extremely rapid processing of numbers 1–4 would be that the ANS is simply very quick in discriminating such small quantities, because of their lesser magnitude.

Rapid advances in the development of functional magnetic resonance imaging (fMRI) technology has provided further insight into the understanding of numerical cognition, implicating the intraparietal sulcus (IPS) as the primary brain structure involved in numerical processing (Nieder 2005). Although it was originally thought that the IPS might contain a specified number module, evidence now suggests that the IPS serves a “patchwork” of different functions (Ansari 2008). The prefrontal cortex (PFC) is also involved in numerical processing; however, response latencies are faster in the IPS than in the PFC, suggesting that the IPS extracts numerical information and subsequently sends the information to the PFC for processing (Nieder and Dehaene 2009).

Single cell recordings have also proven to be a great asset in understanding numerical processing. In a delayed matching to sample task (Nieder and Miller 2004), monkeys were shown an array of 1–5 items and subsequently had to decide if a sample array matched this number. Single-cell recordings showed that the monkeys had number-selective neurons, which fired preferentially to particular numerical values and thus appeared to be “tuned” for specific numbers. These tuning curves are imprecise, such that a neuron with a “preference” for 6 may also fire for 5 and 7. The resulting tuning curves help explain the distance and magnitude effects associated with the analogue magnitude system. When two numbers are farther apart, their respective tuning curves will overlap less, creating less “noise” around the number and making them easier to discriminate. Tuning curves, however, get wider and less precise as numerical magnitude increases. Thus, while 1 versus 2 will have narrower tuning curves that are more precise, 8 versus 9 will have broader tuning curves that overlap more with surrounding numbers, making discrimination more difficult.

The question of numerical processing in animals can be traced back as far as the misadventures of Clever Hans (Pfungst 1911). While it was proven that Hans was not able to perform any type of arithmetic, we know today through the comparative study of number that many nonhuman species are sensitive to numerosity, and can make decisions based on this information. Advances in technology through the use of both fMRI and single cell recordings have also thoroughly increased our understanding of numerical processing, allowing researchers not only to pinpoint regions of the brain associated with number, but also to identify number selective neurons, which elegantly explain behavioral data in terms of Weber’s law and the ANS. To date, there is little information as to what the upper limit of numbers that can be discriminated may be in nonhuman animals. Additionally, further comparative investigation may be useful in understanding species-related differences in numerical discrimination, and how they may be related to memory systems. For example, although both chimps and monkeys have been successful in numerical tasks with sequential presentation of items (Beran 2001; Beran and Beran 2004), there is evidence that monkeys more easily discriminate numerosity in tasks that use simultaneous presentation of items (Nieder et al. 2006). Furthermore, in a study with dogs (Macpherson and Roberts 2013), subjects were found to be incapable of discriminating number in several variations of a sequential task, yet discriminated number easily when stimuli were presented simultaneously. This apparent difficulty in discriminating sequentially presented items may reflect a limitation in working memory, as subjects in these experiments must mentally keep track of how many items they have seen presented to them. Further studies are needed to more fully understand any cognitive or physiological differences in processing number across species.