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How the brain senses numbers

2025-01-19 Update From: SLTechnology News&Howtos shulou NAV: SLTechnology News&Howtos > IT Information >

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This article comes from the official account of Wechat: back to Park (ID:fanpu2019), author: Xu Zilong

A philosophical problem gradually evolved into a scientific problem. There's still a long way to go.

Write an article | Xu Zilong

What's the number? How do we perceive numbers?

This seems to be a very simple and naive question. But in fact, in the philosophy of mathematics, this is the most basic question, which has been unanswered. As early as in ancient Greece, some sages raised this question.

This paper attempts to see how human beings try to find the answer to this question from the perspectives of biological evolution, psychology, neuroscience and philosophy.

Smart "BMW" Hans can recognize numbers and do calculations. Is this a unique human ability, or is it an evolutionary family heirloom inherited from our animal ancestors through genes?

Is it true that only humans can perceive and count? Are animals all right?

Listen to a story about BMW Hans a hundred years ago.

Figure 1. Mathematics teacher Hans and his horse "Hans"? source: wikipedia1904 year, German mathematics teacher William von Austen (Wilhelm von Osten) introduced a horse he trained to the audience in Berlin, named Hans (Hans), known as "Smart Hans" (figure 1). When Feng Austen wrote a formula on the blackboard, such as "2x 3", Hans would accurately tap five times with his hoof and then stop. And Hans can also solve more complex arithmetic problems (see figure 2). At that time, some people suspected that this was a circus trick, and thirteen famous experts and scholars (including philosophers and psychologists) formed a "Hans committee" to investigate the matter, but they did not find any scams in the end.

Figure 2. Wikipedia: later, German psychologist Oscar Foster (Oskar Pfungst,1874-1932) surveyed Hans again. Fonster found that the horse could give the right answer even if it was not asked by Feng Austen himself, which ruled out the possibility of the horse owner cheating. It is important, however, that Hans will get the right answer only if the questioner knows what the answer is and the horse can see the questioner. Hans could not answer the question if the questioner did not know the result or was out of the horse's line of sight. As a result, Foster concluded that Hans didn't really do arithmetic, but used the questioner's unintentional physical cues to answer the questions. When Hans clicks the hoof, it will observe slight changes in the posture, facial expression or breathing pattern of the questioner and onlookers, stopping the hoof at the right time. Based on this, he proposed the so-called "Hans effect" (figure 3).

Figure 3. Smart and unsmart Hans. Left: Hans can identify the number when there is a large audience; right: Hans cannot complete the task without the audience and the questioner out of the horse's line of sight. Source: wikipedia shows that Hansma is not that smart. He doesn't really do mathematical calculations, but reacts by observing the movements and expressions of the people around him. From this point of view, we can think that animals are spiritual, but we cannot prove the concept that animals have numbers.

Psychologist Foster wrote in the article [1]:

"counting is essential in our daily life. But what was the use of the perception of numbers to our ancestors before they became Homo sapiens? Do animals use numbers in the first place? According to the principle that biological evolution needs to adapt to the environment, it is clear that only when the perception of numbers is beneficial (or at least harmless) to the individual will this ability be maintained for generations in the population, sometimes for millions of years in large animal taxa. "

From the perspective of biological evolution, human digital perception animals must adopt certain strategies to ensure their own survival until the individual is mature and can reproduce in order to survive and reproduce. For some species, they also need to take care of their offspring to ensure that they survive long enough. For an individual, this means first of all finding food and avoiding becoming food, choosing the right path in a messy environment, and helping each other with friends in daily affairs.

Logarithmic perception can help animals achieve these goals. Studies have shown that logarithmic perception can enhance animals' ability to find food, grab prey, avoid prey, navigate through habitats and survive social interactions.

1. Navigational animals often use counting landmarks to find a suitable route. For example, bees rely on landmarks to measure the distance between food and the hive. In an experiment to study the behavior of bees [2], the researchers placed four tents and placed feeders with sugar water between the third and fourth tents. Bees identify tents as landmarks and use them as navigation grounds to find food. If you change the number of tents and the distance between tents, it will affect the bees' judgment of distance. It is not clear whether bees record the absolute distance directly or measure the distance by counting landmarks (in this case, tents). But the number of landmarks is still an important factor.

two。 Hunting ordinary spiders are solitary animals, while spider-eating spiders are social animals, and some individuals gather together for a relatively long time. This is true of a spider-eating spider in Kenya called Portia africana (hereinafter referred to as Portia). The small spiders they hunt like to build tent-shaped silk nests on boulders, tree trunks and the walls of buildings. Portia usually uses quantitative cues when hunting. Take a typical scenario: two Portia settle next to the nest of one prey, and when one Portia catches the prey, the other Portia joins in and eats together. How does Portia decide which prey spider's nest to settle near? It is based on the number of companions who have settled there. They prefer to hunt in pairs rather than hunt alone, or with two or three more companions, because the more members of a hunting team, the more likely some members are not to cooperate. As a result, large groups are often worse at catching prey than small groups.

It seems that the truth that "two monks carry water and three monks have no water to drink" is understood not only by people but also by spiders.

3. Animals that avoid predation and lack the ability to defend themselves often seek refuge in large groups of social partners. By joining a large group, each individual is less likely to be prey. Therefore, for many fish, joining schools is the main anti-predation strategy. The larger the school of fish, the better for the fish. When a single fish or several fish come into an unfamiliar and potentially dangerous environment, other fish of the same species are often added. If there are two schools of fish, they usually join larger schools, which means they can distinguish between larger schools and smaller schools. Therefore, in the case of life and death, the ability to compare the number of the same species may be crucial.

4. Social territorial defense if individuals cannot defend resources alone, groups and group size are very important. Many animals live in social groups and work together to resist invaders. Defending territory usually means that there may be a fatal conflict with hostile groups, so animals need to be able to assess the number of their own and enemy groups-an ability that is clearly of adaptive value-which is the basis for deciding whether to attack or retreat. The assessment of group size is obviously a perception of the number of individuals in the group.

"there are many people and great strength." Animals also understand this truth.

The concept of number and its cognitive psychological mechanism in human daily life, numbers and numbers are everywhere. When we were babbling, our parents began to teach us to recognize numbers, to count our fingers and to read numbers. So, what exactly is the concept of number? What is the expression of number in the human mind?

To answer these questions, we need to first understand what types of numbers are, what they mean, and what are the differences between different types.

The concept of the first kind of number is the cardinality (Cardinal number). The cardinality represents the quantity, that is, the concept of "how much". The basic function of cardinality is counting [4].

The cardinality in daily life is all too common. For example, a fruit basket contains five mangoes, and the account still has 12.34 yuan. The "5" and "12.34" here indicate how many mangoes and how much money there are, respectively, so they are cardinal figures.

In mathematical set theory, the number of elements in a set is called the cardinality of the set.

The concept of the second kind of number that is often encountered is the ordinal number (Ordinal number). The ordinal number represents the sequential relationship of entities, or before and after the ranking. For example, playing the game of PUBG, I was lucky enough to eat a "chicken" and won the first place. The 1 here is obviously not the number of the number, but the ranking, so it is the ordinal.

The concept of the third kind of number is the so-called label. The label indicates neither the number nor the ranking order, but simply uses the number as a mark to distinguish between different objects or objects. Some of the most common examples are ID card numbers, bank account numbers and QQ numbers.

Several types of the concept of number actually derive from people's abstraction of different empirical properties.

The following figure fully represents the conceptual structure of the above three numbers.

Figure 4. We often use three types of number concepts to represent three corresponding empirical properties [1] so, how does human psychology express and process information about numbers?

There is a very important concept in cognitive psychology, namely Representation. Representation means the psychological model and diagram of the external real world in the human brain, or the abstraction of the real object by the human cognitive system.

To take a simple example, there is a water cup on the table at home, and we know it is a water cup. When we go to work, we see a water cup on someone else's desk. Although it is different from the one at home, we also know that it is a water cup. Or on the store shelves, we see all kinds of water cups, although they are different from the one at home, but we fully know that they are water cups, not as chairs, flowers or cats. When we close our eyes, we can see nothing, but our minds can also show the appearance of a water cup. In essence, we have the concept of a water cup in our hearts, which is abstracted from real concrete objects.

There are two basic types of psychological representation of logarithm: symbolic representation and non-symbolic representation. Both cardinality and ordinal numbers can be expressed in these two ways. On the other hand, the number of label types is only represented by symbols [1].

Symbolic representation refers to the concept of using symbols (such as Arabic numerals) to represent numbers [5]. Non-symbolic representation refers to the intuitive representation of numbers through related graphics (such as an array of points) involving the actual number of elements without symbols.

The representation of non-symbolic number is a perceptual representation of the size of the set, and it is the corresponding relationship between the number of elements actually perceived and the number represented. A typical representation is to use several elements scattered in space, such as points in a lattice, to represent the corresponding number. At this point, people can perceive the number of points in parallel in an instant.

Whether from the point of view of animal evolution or human development, the representation of non-symbolic number is a primitive representation method. Animals have non-symbolic representations, as do humans in infancy. If a person has never learned the symbols of numbers, then he can only use non-symbolic representations [1].

The two numerical representations are shown in figure 5. Figure 5 (A) represents two numbers by the number of dots: 8 and 12, while figure 5 (B) represents the quantity directly through the symbol-Arabic numerals.

Figure 5. Symbolic and non-symbolic representations of numbers [6]. (a) non-symbolic representation; (B) symbolic representation that the cognitive development of logarithm in infants and young children has been studied a long time ago. The researchers believe that [7], the logarithmic representation of infants has gone through three stages of development. The first stage, the infant will remember the phonetic pattern of the number; the second stage, remember the writing pattern of the number, and related to the pronunciation; the third stage is called the symbolic representation stage, the infant will form the internal representation of the number.

Other studies have shown that in early childhood, the symbolic representation of numbers is the main factor affecting the level of mathematics in young children. With the increase of age, the influence of symbolic and non-symbolic skills on the level of mathematics shows a downward trend [6].

There is also an interesting question: is there an automated process for people's perception of numerical values?

The study found the following conclusion: if the number is presented as a graph, when the value is less than 4 (that is, 1, 2, 3) (figure 6), people can feel the number of patterns almost instantly, a process called subitizing.

Figure 6. When the unsigned number is less than 4, you can instantly estimate [6], but if the value is greater than or equal to 4 (figure 7), it can not be estimated instantly. You can only count one by one. The process of counting is called counting. When counting, we may read the voice, or we may just recite it in our minds. In short, at this point, it is necessary to find out how many graphics there are through language symbols.

Figure 7. When the number of non-symbolic representation is greater than or equal to 4, it cannot be estimated instantly, but can only be counted [6].

Figure 8. The effect of non-symbolic numerical size on perceptual reaction time and error rate [6] the results (figure 8) show the effect of non-symbolic numerical size on human perceptual numbers. The horizontal axis of the graph represents the numerical value, the vertical axis of the left represents the reaction, and the vertical axis of the right represents the error rate. From the figure, we can clearly find that with the increase of the numerical value, the reaction time increases and the error rate increases. When the value is 1, 2, 3, the change of both is very small, when the value is near 4, the increase of both is very obvious. This may have something to do with the breadth of attention.

Neural mechanism of the concept of number the physiological basis of cognitive activity is the nervous system. So, what kind of neural mechanism is the human perception of the concept of logarithm and the psychological representation of logarithm produced?

In order to answer this question, cognitive neuroscience researchers use a variety of non-invasive techniques to measure the activity of neurons in the process of mathematics. Now one of the most commonly used methods is functional magnetic resonance imaging (fMRI). The technique can detect activation of the cerebral cortex so that researchers can observe which parts of the brain are more active when performing a cognitive task. Obviously, one or more active brain regions may be closely related to the cognitive activities currently taking place.

In 2003, a French research team used fMRI to scan the brain activity of subjects while perceiving numbers [8]. The scan images (shown in figure 8) show that neurons are more active in three brain regions, namely, the horizontal part of the bilateral parietal sulcus (bilateral horizontal segment of intraparietal sulcus, red area), the posterior superior parietal lobule (bilateral posterior superior parietal lobe, blue area), and the left angular gyrus (left angular gyrus, green area). In other words, all these regions participate in the cognitive / psychological process of logarithmic perception.

Figure 9. Active brain regions in the number perception experiment [8] from the picture, we can also see that there are a little more activated areas in the left brain than in the right brain. This suggests that the left brain may provide more neural resources than the right brain when perceiving the concept of number.

An Italian research team studied logarithmic brain regions in a different way in 2004 [9]. They used transcranial magnetic stimuli (TMS) technology: experimental equipment can generate a magnetic field that passes through the skull, interfering with specific brain areas, inhibiting neuronal activity and causing "virtual damage". This process is also non-invasive (figure 10).

The experimenter asked the subjects to do a simple number sensing task while interfering with the inferior parietal lobule (IPL) region of the lower parietal lobe (equivalent to the red area in figure 9). The results showed that the accuracy of the task decreased and the reaction time increased. Once the equipment was removed, the accuracy of the subjects in solving the problem recovered rapidly, and the reaction speed returned to normal. As a contrast, the experimenter also focused the device on the brain areas that were not related to number perception, and sure enough, interference in these brain areas did not affect the subjects' ability to complete the number perception task.

Figure 10. Transcranial magnetic stimulation device and operation | Picture Source: network further, scientists have also studied the activation of brain regions when we do addition, subtraction, multiplication and division. This can help us understand whether there is a difference in the neural activity on which the cognitive process depends when doing four different operations [10].

Figure 11. The active state of the brain area under the four operations. From top to bottom is addition, subtraction, multiplication and division. [10] figure 11 shows the experimental results. Color indicates the areas of the brain that are activated (red and yellow) and suppressed (turquoise) when doing calculations. As can be seen from the figure, when doing multiplication and division, there are far more brain areas activated than addition and subtraction, which means that multiplication and division requires more cognitive resources-in other words, multiplication and division is more difficult. This result is completely in line with our common sense.

At the same time, we can also observe that the left parietal sulcus (Left IPS) region is activated under each operation. Therefore, it can be considered that this region is involved in every operation and is a very important brain region related to computing.

In addition to fMRI and TMS, there are many non-invasive techniques to study brain activity, such as electroencephalogram (EEG), magnetoencephalogram (EMG), near infrared and so on. There are also a large number of neuroscience literatures using these techniques to study the perception of numbers and mathematical ability. Many questions have not been determined and unified answers, and many conclusions are temporary. The above research examples are only ten thousand, to answer "how the brain perceives numbers", we need to do more exploration.

To sum up, is the "number" that we come into contact with every day in our daily life already existing and discovered, or is it the product of human invention and creation?

As Mario Livio, author of "is God a mathematician?", this problem has long plagued mathematicians.

Figure 14. The Last Mathematical problem (people's Post and Telecommunications Publishing House, September 2019) insists on realism that numbers exist independent of human thought, and we just found them. The other supports anti-realism, believing that numbers are not independent of our perception, but that we invented them.

Perhaps this issue will continue to be debated for quite a long time to come. However, with the development of cognitive psychology and cognitive neuroscience, people are getting closer and closer to the mystery behind the cognition of numbers.

reference

[1] Nieder, A brain for numbers: the biology of the number instinct. MIT press, 2019.

[2] L. Chittka and K. Geiger, "Can honey bees count landmarks?," Animal Behaviour, vol. 49, no. 1, pp. 159-164,1995, doi: https://doi.org/10.1016/0003-3472(95)80163-4.

[3] X. J. Nelson and R. R. Jackson, "The role of numerical competence in a specialized predatory strategy of an araneophagic spider," Animal cognition, vol. 15, pp. 699-710, 2012.

[4] https://en.wikipedia.org/wiki/Cardinal_number

[5] https://dictionary.apa.org/symbolic-representation

[6] Y. Li, M. Zhang, Y. Chen, Z. Deng, X. Zhu, and S. Yan, "Children's Non-symbolic and Symbolic Numerical Representations and Their Associations With Mathematical Ability," Frontiers in Psychology, vol. 9, 2018.

[7] E. Bialystok, "Symbolic representation of letters and numbers," Cognitive Development, vol. 7, Art. No. 3, 1992.

S. Dehaene, M. Piazza, P. Pinel, and L. Cohen, "Three Parietal Circuits For Number Processing," Cognitive Neuropsychology, vol. 20, no. 3-6, pp. 487-506, 2003, doi: 02643290244000239.

[9] M. Sandrini, P. Rossini, and C. Miniussi, "The differential involvement of inferior parietal lobule in number comparison: a rTMS study," Neuropsychologia, vol. 42, no. 14, pp. 1902-1909, 2004, doi: https://doi.org/10.1016/j.neuropsychologia.2004.05.005.

[10] M. Rosenberg-Lee, T. Chang, C. B. Young, S. Wu, and V. Menon, "Functional dissociations between four basic arithmetic operations in the human posterior parietal cortex: A cytoarchitectonic mapping study," Neuropsychologia, vol. 49, no. 9, pp. 2592-2608, 2011, doi: https://doi.org/10.1016/j.neuropsychologia.2011.04.035.

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