Number sense in animals is the ability of creatures to represent and discriminate quantities of relative sizes by number sense. It has been observed in various species, from fish to primates. Animals are believed to have an approximate number system, the same system for number representation demonstrated by humans, which is more precise for smaller quantities and less so for larger values. An exact representation of numbers higher than 3 has not been attested in wild animals, but can be demonstrated after a period of training in captive animals. In order to distinguish number sense in animals from the symbolic and verbal number system in humans, researchers use the term numerosity, rather than number, to refer to the concept that supports approximate estimation but does not support an exact representation of number quality. Number sense in animals includes the recognition and comparison of number quantities. Some numerical operations, such as addition, have been demonstrated in many species, including rats and great apes. Representing fractions and fraction addition has been observed in chimpanzees. A wide range of species with an approximate number system suggests an early evolutionary origin of this mechanism or multiple convergent evolution events. Like humans, chicks have a left-to-right mental number line.
Early studies
At the beginning of the 20th century, Wilhelm von Osten famously, but prematurely, claimed human-like counting abilities in animals on the example of his horse named Hans. His claim is widely rejected today, as it is attributed to a methodological fallacy, which received the name Clever Hans phenomenon after this case. Von Osten claimed that his horse could perform arithmetic operations presented to the horse in writing or verbally, upon which the horse would knock on the ground with its hoof the number of times that corresponded to the answer. This apparent ability was demonstrated numerous times in the presence of the horse's owner and a wider audience, and was also observed when the owner was absent. However, upon a rigorous investigation by Oskar Pfungst in the first decade of 20th century, Hans' ability was shown to be not arithmetic in nature, but to be the ability to interpret minimal unconscious changes in body language of people when the correct answer was approaching. Today, the arithmetic abilities of Clever Hans are commonly rejected and the case serves as a reminder to the scientific community about the necessity of rigorous control for experimenter expectation in experiments. There were, however, other early and more reliable studies on number sense in animals. A prominent example is the work of Otto Koehler, who conducted a number of studies on number sense in animals between 1920s and 1970s. In one of his studies he showed that a raven named Jacob could reliably distinguish the number 5 across different tasks. This study was remarkable in that Koehler provided a systematic control condition in his experiment, which allowed him to test the number ability of the raven separately from the ability of the raven to encode other features, such as size and location of the objects. However, Koehler's work was largely overlooked in the English-speaking world, due to the limited availability of his publications, which were in German and partially published during World War II. The experimental setup for the study of numerical cognition in animals was further enriched by the work of Francis and Platt and Johnson. In their experiments, the researchers deprived rats of food and then taught them to press a lever a specific number of times to obtain food. The rats learned to press the lever approximately the number of times specified by the researchers. Additionally, the researchers showed that rats' behavior was dependent on the number of required presses, and not for example on the time of pressing, as they varied the experiment to include faster and slower behavior on the rat's part by controlling how hungry the animal was.
Methodology
Examining the representation of numerosity in animals is a challenging task, since it is not possible to use language as a medium. Because of this, carefully designed experimental setups are required to differentiate between numerical abilities and other phenomena, such as the Clever Hans phenomenon, memorization of the single objects or perception of object size and time. Also, these abilities are seen only from the past few decades and not from the time of evolution. One of the ways that numerical ability is thought to be demonstrated is the transfer of the concept of numerosity across modalities. This was for example the case in the experiment of Church and Meck, in which rats learned to "add" the number of light flashes to the number of tones to find out the number of expected lever presses, showing a concept of numerosity independent of visual and auditory modalities. Modern studies in number sense in animals try to control for other possible explanations of animal behavior by establishing control conditions in which the other explanations are tested. For example, when the number sense is investigated on the example of apple pieces, an alternative explanation is tested that assumes that the animal represents the volume of apple rather than a number of apple pieces. To test this alternative, an additional condition is introduced in which the volume of the apple varies and is occasionally smaller in the condition with a greater number of pieces. If the animal prefers a bigger number of pieces also in this condition, the alternative explanation is rejected, and the claim of numerical ability supported.
Approximate number and parallel individuation systems
Numerosity is believed to be represented by two separate systems in animals, similarly to humans. The first system is the approximate number system, an imprecise system used for estimations of quantities. This system is distinguished by distance and magnitude effects, which means that a comparison between numbers is easier and more precise when the distance between them is smaller and when the values of the numbers are smaller. The second system for representation of numerosity is the parallel individuation system which supports the exact representation of numbers from 1 to 4. In addition, humans can represent numbers through symbolic systems, such as language. The distinction between the approximate number system and the parallel individuation system is, however, still disputed, and some experiments record behavior that can be fully explained with the approximate number system, without the need to assume another separate system for smaller numbers. For example, New Zealand robins repeatedly selected larger quantities of cached food with a precision that correlated with the total number of cache pieces. However, there was no significant discontinuity in their performance between small and larger sets, which would be predicted by the parallel individuation system. On the other hand, other experiments only report knowledge of numbers up to 4, supporting the existence of the parallel individuation system and not the approximate number system.
Number sense by species
Fish
An approximate number system has been found in a number of fish species, such as guppies, green swordtails and mosquitofish. For example, preference for a bigger social group in mosquitofish was exploited to test the ability of the fish to discriminate numerosity. The fish successfully discriminated between different amounts up to three, after which they could discriminate groups if the difference between them also increased so that the ratio of the two groups was 1:2. Similarly, guppies discriminated between values up to 4, after which they only detected differences when the ratio between the two quantities was 1:2.
Rats
Rats have demonstrated behavior consistent with an approximate number system in experiments where they had to learn to press a lever a specified number of times to obtain food. While they did learn to press the lever the amount specified by the researchers, between 4 and 16, their behavior was approximate, proportional to the number of lever presses expected from them. This means that for the target number of 4, the rats' responses varied from 3 to 7, and for the target number of 16 the responses varied from 12 to 24, showing a much greater interval. This is compatible with the approximate number system and magnitude and distance effects.
Birds
Birds were one of the first animal species tested on their number sense. A raven named Jacob was able to distinguish the number 5 across different tasks in the experiments by Otto Koehler. Later experiments supported the claim of existence of a number sense in birds, with Alex, a grey parrot, able to label and comprehend labels for sets with up to six elements. Other studies suggest that pigeons can also represent numbers up to 6 after an extensive training.
Dogs
A sense of number has also been found in dogs. For example, dogs were able to perform simple additions of two objects, as revealed by their surprise when the result was incorrect. It is however argued that wolves perform better on quantity discrimination tasks than dogs and that this could be a result of a less demanding natural selection for number sense in dogs.
Monkeys
seem to have an innate understanding of numbers up to 4. This is shown by the study on semi-free-ranging rhesus monkeys in their natural environment in which the monkeys spontaneously discriminated numbers from 1 to 3 but did not demonstrate a numerical ability beyond this number. In another study that included training, rhesus monkeys showed the ability to generalize this knowledge to numerosities up to nine.
Chimpanzees
are believed to have a sense of number, albeit an imprecise one. While they can correctly compare numbers such as 6 and 7 by choosing a tray with more pieces of reward, their performance is more error-free when the difference between the numbers is bigger and when the numbers are smaller; that is when the distance and magnitude effects take place. Chimpanzees also seem to have knowledge of proportions. In an experiment by Woodroof and Premack an adult chimpanzee correctly identified and performed operations on exemplars of numbers and proportions presented on a variety of objects, such as apples, wood disks and water containers. For example, when presented with of an apple and full water container, the chimpanzee chose of a wooden disk, suggesting an addition-like ability. The ability to add fractions has only been demonstrated in primates so far.
Ants
Ants were shown to be able to count up to 20 and add and subtract numbers within 5. In highly social species such as red wood ants scouting individuals can transfer to foragers the information about the number of branches of a special “counting maze” they had to go to in order to obtain syrup. The findings concerning number sense in ants are based on comparisons of duration of information contacts between scouts and foragers which preceded successful trips by the foraging teams. Similar to some archaic human languages, the length of the code of a given number in ants’ communication is proportional to its value. In experiments in which the bait appeared on different branches with different frequencies, the ants used simple additions and subtractions to optimize their messages.
Wild rodents
Striped field mice Apodemus agrarius demonstrated a sense of number consistent with precise relative-quantity judgment: some of these mice exhibit high accuracy in discriminating between quantities that differ only by one. The latter include both small and relatively large quantities of elements.
