Do animals know numbers? What are their mathematical abilities?

Number recognition is a basic skill for humans. In the long process of evolution, recognizing numbers and developing mathematics have become the cornerstone of human civilization. However, as a member of the biological world, an important question is, is the ability to recognize numbers unique to humans? Can animals count? In recent years, many studies have found that the ability to use numbers is widely present in the animal world. These observation-based experiments tell us that animals have an innate awareness of comparing sizes and even the abstract 0. But we still know too little about what kind of mathematical or other intellectual abilities animals have.

Written by | Pikachu Bulbasaur

1. "The truly unique function of human language is not that it can convey information about people or lions, but that it can convey information about things that do not exist at all. As far as we know, only Homo sapiens can express things that they have never seen, touched, or heard, and they can speak of them as if they were real." - Yuval Noah Harari, A Brief History of Humankind

2. “The difference in mind between man and the higher animals, great as it is, certainly is one of degree and not of kind,”——The Descent of Man, Charles Darwin

Besides eating, sleeping and having sex, what else can animals do? Talk? Count? Ride a bicycle?

If you were to tell us that animals can count, you might be a little surprised, but there are many examples of animals being able to speak. For example, parrots can imitate people and say simple sentences, such as "Wish you good fortune" and "Your uncle"; when natural enemies appear, meerkats use a specific combination of calls to warn their companions to run away quickly; the ultimate nightmare of "thalassophobia patients" coming from the deep sea - "whale song" is considered to be one of the most complex language systems in the world, so complex that it even has its own grammar. Researchers have analyzed that each whale song starts with the most basic unit (unit), which is composed of a secondary structure short sentence (phrase), and several short sentences are combined into a melody (theme), and multiple melodies are combined into a complete whale song. Compared with the way we speak language, it is similar to words forming sentences, sentences forming paragraphs, and paragraphs forming an entire article [1].

Language, or more conservatively, the ability to communicate, is not unique to humans.

The first sentence of this article is what the author wants to express, that although other animals can also "chat", humans are the only ones with "abstract thinking". In other words, when animals talk, it almost always has to be related to something that actually exists. They may be able to warn their companions after discovering their natural enemies, but they cannot issue a warning when the natural enemies are not there: danger, danger, danger...

In fact, the numbers we use in our daily lives are essentially an abstract concept. Three apples, three elephants, and three cars are completely different things visually, but we can abstract these objects into the same number - 3, which has nothing to do with the characteristics of the objects themselves. So the question is, are we born with the ability to recognize and use numbers? Do animals have this ability to count?

Clever Hans

The story about animal counting begins with a clever horse.

The decades after Darwin's death were called "the eclipse of Darwinism" by later generations. Especially in the early 20th century, although the academic community had widely accepted the theory of "evolution", almost no one agreed with Darwin's theory of "natural selection". In other words, the truth as big as the sun was right above our heads, but everyone turned a blind eye and frantically proposed various other theories that might explain biological evolution. Among them, the more famous ones are "orthogenesis", which says that biological evolution comes from the internal power of organisms, pushing organisms to change in a certain direction; and "mutationism", whose ideas existed long before the publication of "The Origin of Species", believing that evolution occurs in an instant, and a mutation may make a species "soar". Even after scientists knew that genes are genetic material, it is still the main competing theory. Of course, all attempts to challenge "natural selection" ended in failure. However, there was an issue that attracted widespread public attention at the time, and its essence was the study of animal intelligence.

In Germany in the early 1900s, there was a retired mathematics teacher named William Von Osten. Although he had not made much progress in his main job, he was very good at taking care of animals and raised a horse named Hans. Perhaps due to occupational disease, Osten not only fed Hans well, but also taught him various skills.

This teaching is extraordinary. According to records, Osten once asked his horse: "If the 8th day of a month is Tuesday, what day of the week is Friday?" Even a normal person would have to spend time thinking about this question, or even count on his fingers; but this horse was obviously smarter and quickly knocked the ground 11 times with his hoof; after 4 years of teaching, Hans actually learned addition, subtraction, multiplication, division, date calculation, scale recognition and other "magic skills". People who saw the horse's special ability gave it a nickname, "Clever Hans (Kluger Hans)".

Figure 1. "Clever Hans" and his owner | Source: Britannica

The horse must die, either in the manger or on the road, but the reputation of "Clever Hans" spread across the ocean and floated to the headlines of the New York Times. At that time, a large number of people in society, including well-known psychologists, zoologists, horse trainers, etc., went to visit and study this famous horse, but they didn't understand whether it was telepathy, magic, or really hitting the "intelligence ceiling" in the animal world. Among them was Oskar Pfungst, a comparative biologist and psychologist. He observed for a long time and always felt that something was wrong: Maybe the horse was able to give the correct answer to the question because it received some very obscure hints, such as the subtle "snorting" or the specific gestures of the horse owner.

In order to find the answer to the question, he designed a series of rigorous animal behavior experiments, controlling the variables of horse trainers, surrounding environment, horse clothing, and sensory shielding (blindfolding or plugging ears). In a controlled arithmetic operation experiment, von St found that when the questioner knew the answer to the arithmetic question in advance, the horse could answer most of the questions correctly; if the questioner himself did not know, the horse could only answer a few of them correctly. He also tested the horse's memory. First, a person said a number in the horse's ear. After a while, the questioner came over to ask the horse what the number was. As a result, the horse got it wrong 8 out of 10 times. After a series of experiments, von St concluded that no matter what kind of numerical problem, if the questioner knows the answer in advance, the horse can answer it correctly in 90% of cases; if it does not know in advance, it can only answer it correctly in 10% of cases, and in this case, it is most likely guessed.

Fonster felt that there was only one possible explanation for this repeatedly verified conclusion, that is, the horse's counting was not based on its own intelligence, but on some external cues that were not noticed by others, guiding the horse to give the correct answer. To confirm this idea, he blocked the horse's ears and asked the questioner to try to use "ventriloquism" to ask numerical questions, or to recite the question in his mind without speaking it out loud. He was surprised to find that Hans could almost always give the correct answer in these cases. In other words, the horse did not rely on some kind of sound cues, and the cues were not hidden in the process of asking questions.

If this hypothesis is true, von Ster believes that the hint must exist in the process of the horse tapping its hooves to give an answer. He found a way to blindfold the horse and conducted the same test as before. This time he got a magical result: when the horse couldn't see the questioner, it couldn't give the correct answer; when it could see, it could give the correct answer in 90% of cases. This means that during the horse's hooves, there must be some phenomenon on the questioner's body that gave the horse a hint of the correct answer. Finally, he covered the questioner's face and found that in this case, the horse could hardly give the correct answer.

After long and rigorous experiments and meticulous observations, Vonster finally came to a satisfactory conclusion: almost every time the owner asked a question, he would unconsciously bow his head and bend forward slightly, and the horse would get the hint and immediately start tapping its hooves; when the correct answer was tapped, the owner would unconsciously raise his head slightly, and the horse would immediately retract its hooves and stop tapping. In other words, rather than saying that the horse was performing mathematical calculations, it would be better to say that it was performing the magic trick of "mind reading" and was very good at observing the owner (or other people)'s words and expressions.

Figure 3 Results of the digital control experiment designed by Pfungst | Source: Clever Hans, Oskar Pfungst

However, although the truth of the matter was a bit disappointing, the horse was indeed different from others. People's admiration and longing for this "divine horse" did not fade quickly. The horse's owner continued to train and promote it, so that it could react the same way in the dark as in the daytime. Later, two more horses were trained, one specializing in arithmetic and the other in reading. The three-horse team earned a lot of fame. According to legend, Hans ended his legendary life by being drafted into war.

In this story, people thought that the horse's intelligence was high enough to perform abstract numerical calculations, but in the end, it was proved that the horse could not understand the abstract meaning of numbers at all, and could only make some physical reactions based on what it saw with its eyes. Later, the "Clever Hans Effect" or "Observer Expectation Effect" developed into an important concept in psychology, which said that the observer's expectations caused the observation behavior itself to unconsciously manipulate the observation results in some form, resulting in a wrong interpretation. The story of psychologist von ster and horse Hans has also become a landmark event in the history of psychology. On the other hand, people's doubts about "animal counting" have continued to this day, and have indirectly contributed to the extensive research of the scientific community on animal consciousness and intelligence.

Can animals count?

If you randomly ask passers-by on the street, "Do you think animals can count?", most likely everyone will say no, because the ability to intuitively understand numbers is unique to human's advanced intelligence and is one of the characteristics that distinguishes humans from other animals.

In fact, this statement is not entirely accurate. As early as 1988, bioethologist Hank Davis and others published a long review, summarizing the previous decades of research on "animal counting". They pointed out and criticized the confusion in the field of "one word for multiple uses" and "multiple words for one use", which led to confusion and ambiguity in the concept of numerical cognition [2]. They concluded that many animals can actually show primitive "numerical competence", but lack "sense of number".

Figure 4. Analysis of concepts related to digital competence | Source: Screenshot of the review published by Davis et al. in 1988 [2]

For example, at that time, the academic community believed that some animals could distinguish that one object and two objects were different things, but they could not understand which one was more; nor could they distinguish which one was more between a pile of 100 pine cones and a pile of 200 pine cones, and could not make vague estimates. Compared with the human ability to "count", this numerical ability is called "protocounting". Among the numerical abilities, the top one is called "concept of number", which has abstract meaning and can be counted independently of the entity. For example, people can calculate out of thin air "a monkey riding on a tree, a monkey under a tree, a total of two monkeys", but animals do not have this ability. Hank Davis and others believed at the time that the understanding of the size of numbers and the abstract meaning of numbers was one of the characteristics of humans as "high-quality mammals".

After the researchers reached a consensus on basic concepts and summarized the limits of animals' counting abilities, they gradually discovered that animals' use of numbers is quite diverse.

Later studies have shown that industrious bees can count the number of landmarks to estimate the distance they should fly to reach their experimental target (nectar)[3]; lioness can distinguish the number of calls of unfamiliar species in the distance to determine whether to go on a rampage and crush them, or to keep quiet and make a fortune[4]; man-faced spiders (Nephila clavipes) can roughly judge how many "vegetables" they have caught on their web today[5]; sand ants (Cataglyphis) even install WeChat step counters (they have an organ called odometer in their bodies, which bees also have) and navigation maps, so they will not lose their direction and distance in the vast desert, and will always turn back no matter how far they have walked[6]. Similar abilities to use numbers have been found to be widely present in insects, reptiles, amphibians and birds in studies over the past few decades.

However, animals can tell the difference between 1 and 2 and decide their behavior accordingly, but they cannot judge the size of 1 and 2. Is this really the case? What is the limit of animals' understanding and application of numbers?

Innate mathematical ability

In a 2009 study, Rosa Rugani, an animal psychologist at the University of Padua in Italy, and others observed newly hatched chicks in an experiment: when the chicks were given two groups of different numbers of objects that had been "remembered" by them, they always tended to move closer to the group with more objects. In another experiment, they placed two sets of screens in front of the two groups of objects, and then the researchers moved the objects back and forth behind the two screens to change the total number of objects in the two groups; that is, the chicks initially saw the initial number of the two groups of objects, and also saw the number of objects that moved between each other, but could not see the final total number of the two groups of objects.

Surprisingly, the chicks showed a dynamic calculation ability, which allowed them to accurately choose the side with more numbers. Moreover, they did not undergo any training to perform this calculation; they seemed to have this ability innate. Unlike Hans, the "divine horse", this time the experimenters did not give them any possible hints.[7]

In 2011, Sayaka Tsutsumi, a psychologist at Kyoto University, found that long-tailed monkeys have a similar ability. The researchers placed a certain number of bread slices in an opaque box in front of the monkeys, and then took them out in batches until the box was completely empty. Although the monkeys could not see how many slices of bread were left in the box, they would always try to approach the box to get food before the box was completely empty. After the box was completely empty, the monkeys lost interest in it [8].

Figure 5. Chicks that can count | Source: Reference [7]

The common point of these two studies is that the numerical ability of chicks and monkeys is reflected in real objects. Similar animal arithmetic studies based on "real objects" have given researchers the motivation to further explore animals' understanding of the abstract meaning of numbers.

In 2015, Rosa Rugani published another study in which they placed food behind a display showing five white dots and repeatedly trained chicks to identify it. Then the display was removed and two displays showing two dots were placed on the left and right. The researchers found that the chicks almost always went to the left screen to find food. Under the same conditions, if both displays showed eight dots, the chicks would go to the right screen. This result is very interesting. If the three groups of white dots, 2, 5, and 8, are only different in image but not in size in the chicks' minds, there should be no difference between 2 and 8 in the chicks' minds. In this experiment, the number "2" is associated with "left", and the same is true for "8" and "right".

Our understanding of the size of numbers is represented by a straight line from left to right, with larger numbers always to the right of smaller numbers. Although we cannot simply say that chickens also know this pattern, researchers still speculate that chickens may have an innate understanding of the size of numbers, and this understanding may be deeper than we previously thought [9].

Moreover, insects, birds, and primates can match the symbols of numbers with the numbers themselves after a certain degree of training. For example, chimpanzees can touch each number in a pile of randomly placed number symbols in order from small to large.

In fact, human beings’ ability to recognize numbers is also basically derived from acquired learning and experience accumulation. Caleb Everett, an anthropologist at the University of Miami, mentioned in his book Numbers and the Making of Us: Counting and the Course of Human Cultures that there are more than 7,000 languages ​​in the world. In his field research, he found that some rare and ancient language systems lacked expressions for numbers. For example, the Pirahã language of the primitive tribes of Brazil only has words such as “one” or “several”, which caused the Pirahã people to encounter considerable difficulties in dealing with daily activities related to “numbers”, such as not being able to put two groups of the same number of things together one by one. But the Pirahã people are not stupider than other races. If they are placed in a language environment with a mature counting system, they can also learn to count, and everything will be normal. This means that the numerical ability we have, at least most of it, can be regarded as an acquired “tool” [10].

Abstract “0”

This tool acquired by humans includes a very special part, which is the understanding of the most abstract and complex "0". Unlike other numbers that can represent things that exist in reality, 0 represents "nothing" and "non-existence", and has a more complex meaning in mathematics. This abstract meaning even makes Homo sapiens children scratch their heads and can only gradually understand this abstract number through acquired learning. Of course, this is also consistent with the process of humans gradually understanding and giving new meaning to 0 in the long history.

When scientists discovered that animals seemed to have their own opinions about 1, 2, 3, and 4, they wanted to further explore whether they had the ability to appreciate the abstract beauty of "0". In a 2016 monkey study, researchers conducted a simple experiment: 0 to 4 dots appeared randomly on a display screen, and the number of dots was refreshed after an interval of 1 second. Then the monkeys were asked to judge whether the number of dots displayed twice was the same. The study found that when the number of dots displayed twice was similar, such as 3 and 4, the monkeys were more likely to make a misjudgment and think that the number of dots displayed twice was the same, but 1 and 4 were almost never misjudged. Surprisingly, the probability of monkeys misjudging "empty set" with "1" was greater than the probability of misjudging "empty set" with other numbers. This means that in the monkeys' concept, "0" may not be something unrelated to numbers, but something that is closer to "1". In 2018, Scarlett R. Howard from the Royal Melbourne Institute of Technology found that bees have similar abilities [11].

At the same time, when monkeys were shown different numbers, different areas of their prefrontal cortex fired, just like humans.[9] Similarly, a study published in the Journal of Neuroscience in June showed that in an almost identical experiment, small-billed crows (Corvus corone) performed almost identically to previous monkey experiments. The authors concluded that the understanding of the “empty set” evolved independently in at least three completely different groups of animals—mammals, arthropods (insects), and birds—and that specific sets of neurons are at work in this understanding.[12]

Figure 6. The crow that can count | Source: Andreas Nieder[8]

We still know too little

One of the essences of the scientific spirit is curiosity, which is to get to the bottom of things. Can animals count? Can animals think? When your cat or dog is arguing with himself in the mirror, do they know themselves? Do animals have thoughts? These questions seem to be asked by six-year-old children, but it often takes a century for the academic community to get a glimpse of the outline of the answer. Moreover, even today, the way we explore these questions is based on primitive and simple behavioral observations.

On the one hand, we have a strong interest in the behaviors of animals that seem to be related to advanced intelligence, and we are often extremely excited about new discoveries; on the other hand, we are afraid that some animal will be too smart and stage a "Rise of the Planet of the Apes." But the undeniable fact is that too many animals have more diverse behaviors and more complex emotions than we imagine, and many times we who claim to be advanced simply cannot understand them.

Why would the socially anxious Western Scrub Jay (Aphelocoma californica) stand on the treetops, howling to the sky, and spontaneously gather to hold a solemn funeral for a half-hour-long funeral for a deceased companion who they did not know? Why would crows, after escaping from the clutches of the devil by chance, still bear grudges and attack the bird catcher who once put them in a cage a few years later? Why would a lone ant lose vitality and die of depression? What abilities do dolphins, the most powerful brains in the animal world, have that we don't know about? The ultimate question is, are we smart enough to understand how smart animals are?

Figure 7. Are we smart enough?

Perhaps in the end, we will return to the second sentence at the beginning of this article, which Darwin wrote in his 1871 book The Descent of Man: "Although there is a difference in thinking between humans and other higher animals, the difference between the two is only one of size, not black and white."

References

[1] https://www.science.org/doi/10.1126/science.173.3997.585?url_ver=Z39.88-2003&rfr_id=ori:rid:crossref.org&rfr_dat=cr_pub%20%200pubmed

[2] https://www.cambridge.org/core/journals/behavioral-and-brain-sciences/article/abs/protocounting-as-a-last-resort/7425806CA36B689F47D47B686FB220C5

[3] https://www.sciencedirect.com/science/article/abs/pii/0003347295801634?via%3Dihub

[4] https://linkinghub.elsevier.com/retrieve/pii/S0003347284710529

[5] https://link.springer.com/article/10.1007%2Fs10071-014-0801-9

[6] https://www-science-org.pitt.idm.oclc.org/lookup/doi/10.1126/science.1126912

[7] https://royalsocietypublishing.org/doi/10.1098/rspb.2009.0044

[8] https://www.hindawi.com/journals/ijz/2011/806589/

[9] https://www.livescience.com/49633-chicks-count-like-humans.html

[10] https://fivethirtyeight.com/features/why-do-we-count/

[11] https://www.science.org/lookup/doi/10.1126/science.aar4975

[12] https://www.sciencedirect.com/science/article/pii/S0960982216302627#mmc1

[13] https://www-jneurosci-org.pitt.idm.oclc.org/content/41/22/4889

[14] https://www.quantamagazine.org/animals-can-count-and-use-zero-how-far-does-their-number-sense-go-20210809/

[15] https://www.livescience.com/61084-can-animals-count.html

Note: Some animals can be trained to ride a bicycle.