# The Backstage of Mathematical Thinking: How Our Brains Bring About the Abstract

A brief survey of recent insights from cognitive science

In the recent “ask me anything” streaming, Noam Chomsky pointed to an interesting problem.

Darwin and Wallace were very puzzled and debated the fact that all humans have arithmetical capacity. [That] it’s just a part of our nature to understand that there are infinitely many natural numbers; that when you add them it works this way and not some other way, and so on. This seems to be a part of universal human nature. They were very puzzled by that because it couldn’t possibly have been selected — since it was never used! It’s only been used in the tiny, recent period of human history.

The question is serious. Natural selection is able to select only those mutations which are “put into practice” and whose effects gain their carriers a better fit into the environment. Since any arithmetical capabilities we now possess couldn’t had been used, they could not have been selected by the evolutionary mechanism. So where do they come from?

Chomsky later hypothesized that

one possible answer (which we don’t know to be true) is that our arthmetical knowledge could be an offshoot of language. It turns out that if you take the most elementary principles that yield the linguistic structures and you reduce them to the absolute minimum, a lexicon which contains one element, you get the successor function and something like addition — the rudiments of arithmetic.

The philosophically-minded probably already sense where this is headed: if we can show arithmetic to be just some sort of a mere linguistic structure, we have a serious argument (but maybe not the utterly decisive one) against mathematical Platonism. Why so? Well, Platonists believe that mathematics is independent from human thought, experience and even the existence of the universe. If there was no humans, if there was actually nothing, mathematical theorems would still be true. And since the only criterion for mathematical existence for Platonism is freedom from contradiction, mathematical objects would still exist “in” the non-spatiotemporal realm. But, if on the other hand it was the case that mathematics is in *any* way dependent on the human mode of thinking, for instance on our linguistic faculty, it would mean that mathematics is just *some* tool (if any) for description of reality, and not * the *objective

*language of nature.*

This article is intended to recollect where the research stands right now on that issue and particularly to bring focus to one of the recent promising theories. So, where does our mathematical thinking come from?

But first, what exactly do we mean by “mathematical abilities”? In order to describe how we, humans come to grips with the basics of mathematics, it is likely best to start the journey with other species. Thanks to that, we will be probably able to differentiate between some very basic numerical endowment of, say, birds or apes, and the full-fledged mathematical ability of humans.

## Can animals count?

Scientific journals are nowadays full of studies showing some numerical competences of various species. However, those abilities are by no means what we would call a “mathematical capability”. Let me set some conceptual framework. The cognitive-scientific literature usually distinguisihes at least three separate abilities:

*Estimation*which relies on comparison of the numerosity of two sets in order to find out which one is bigger. It does not require any accurate counting as it operates on rough approximations.*Subitizing*that enables to quickly and accurately state the numerosity of a given set. It is put into practice only with rather small sets. Subitizing is sometimes described as “intuitive determination of numerosity”.*Counting*which, in comparison to subitizing, is much more time-consuming and requiring additional cognitive abilities. Counting allows to determine numerosity with better accuracy than estimation and can be used also on bigger sets (than subitizing).

It turns out that primates have very good estimation skills. In an experiment by D.M. Rumbaugh and others, chimps were presented with two trays. Each of them carried a pile of chocolate cubes. Every chimp could choose one of the trays and eat what was on it. In the 90% of cases the animals chose the tray with a bigger amount of chocolate. The next task was more complicated. Now, each tray carried two piles. One tray had piles substantially differing in size, the other had two piles of the same size, but the overall amount of chocolate was greater than on the first tray. Chimps were not fooled and have once again been choosing trays with more sweets. The experiment has been repeated in different variants and on other apes with similar results. Moreover, the estimation ability seems to be shared by vertebrates.

For instance, it has been shown that pigeons and crows can distinguish the numerosity of compared sets quite well. It turned out also that the number of birds’ correct answers decreased with both higher numerosity of sets and when the compared amounts were close to each other. This shows that the discussed animals really do use their estimative faculty in such situations and none other in conjunction with it. That is, among other reasons, because estimation strongly depends on the ratio of objects in the compared sets.

The ratio which expresses perceptible differences between compared sets, called the **Weber’s fraction**, varies from species to species and, in humans, between ages. It seems that human newborns can tell a 1-element set from a 3-element set (and their multiples, eg. 3 from 9); one year olds can distinguish between 1- and 2-element sets; the fraction for four year olds is 3:4; seven year olds — 4:5; and twenty year olds — 7:8. To give an example: an adult can quickly distinguish between a set with 70 elements and a set with 80 elements, but not between a set of 71 and that of 81 elements. Note that the fraction gets higher with experience (that is, with age).

The ability of estimation must be very useful in various situations, so it isn’t surprising that the natural selection has equipped numerous species with it. Now let’s have a closer look at subitizing.

It is clear that it’s a faculty of different kind than counting, since it operates on smaller numbers only. It can be said that it is a connection between the fast and inexact estimation, and the slow but accurate counting of sorts. The research on subitizing is handled on previously learned numerals (represented verbally in the case of parrots and graphically in the case of primates other than humans).

For instance, the African parrot called Alex was able to immediately determine (in English) the numerosity of sets up to six elements. The parrot managed to state the number of objects in a diversified set (eg. how many blue blocks are there?), as well as to state the joint number of differing objects. In both cases the percentage of correct answers was about 80%. Moreover, the parrot reacted properly when there was no object of certain feature in a set presented to it.

There are various studies about subitizing showing that this ability features other animals than humans and that it’s something different than counting. Now, let’s have a closer look at this most abstract of basic “proto-mathematical” capacities.

The question whether animals are able to count has been the subject of study since 1950s. It has yielded ambiguous results.

S. Boysen and G. Berntson investigated whether chimps use numerical representation. It turned out that they can carry out summation equally well on physical objects and arabic numbers from 1 to 4.

Chimpanzees were presented with a tray and three bowls. There was a number of objects on a tray, and from one to three objects in each bowl. Chimps were conditioned in a way that if they pointed to a bowl containing the number of objects the same as that of objects on the tray, they were rewarded with food. When chimps mastered this task, the object in one of the bowls was replaced with a paper sheet with the arabic “1” written on it. After chimps learned that this sign corresponds to the previous number of objects in that bowl, the same has done with the bowl with two objects. The same was done later with paper sheets with a “3” and a “4”. When the animals learned these correspondences, they were thought to perform numerical operations. Three cases were put into their cages. There were some objects put into the cases. After looking into the cases a chimp was to point to a paper sheet with a numeral on it. It was rewarded if it pointed to a numeral corresponding to the *sum* of objects in all cases. In the second part of the experiment the objects were replaced with paper sheets with numerals. Chimps were awarded if they summed up the numbers and pointed to a sheet with a number which was the result of the summation. They succeeded in 70% of cases.

This would mean that animals can understand the *cardinal* aspect of number, that is, can understand that a certain numerosity corresponds to a certain number. But in order to perform basic operations, they should also be able to understand the *ordinality* of numbers, i.e. that there is a hierarchy of consecutive numbers.

T. Matsuzawa tried to teach a chimp called Ai about the latter aspect of numbers. The cipher “1" was shown on the screen and the chimp was awarded when she touched it. Next, numbers 1 and 2 were shown randomly and Ai was rewarded when she clicked on them in the successive order (firstly on 1, secondly on 2). Other numbers up to 9 were introduced in the analogous fashion. Ai was successful in ordering them in 80% of cases.

However, it is controversial whether what the chimp was doing can be called counting. Indeed, it was “counting” in a very limited scope and one which resulted after a long training initiated by humans. In fact, it rather resembled “Simon says” than regular counting. In their natural habitat, chimpanzees do not and don’t even have occasions to, show the ability to count.

It is unquestionable that animals have some numerical capacities, they are far from what we would mean by even basic human “mathematical ability”. Hence it is at least on that account excluded that there is some anti-Platonic origin of mathematics: animals tend to operate on what they learn from experience and, since they don’t show any abilities that would make for some very rudimentary basis of mathematical thinking, there does not seem to be any empiricistic foundation of maths. But, at that point, we simply cannot know. Now, how does it look in humans? How do we learn the basics of mathematical thinking?

## Is mathematics innate?

It is established that some capabilities related to mathematics are almost inborn when it comes to humans.

As soon as three to five days after birth, children can discern between 2-element and 3-element sets, and between 3-element and 4-element sets. After four months they seem to understand that 1+1 = 2 in terms of objects. Moreover, they probably understand that 2–1 = 1 (note that animals did not show any understanding of this “object-subtraction”). Shortly after, they can do these operations on three objects. Before a year of age, children compare numerical information from different senses (eg. that the number of sounds is the same as the number of perceived objects). So, are we predetermined to become mathematicians?

An interesting insight comes from cultural differences. Here, the subject of study mainly focuses on the usage of numerals in various languages. Although numerals are used in virtually all languages, in many of them they are very limited. It is well known that in the language of the Amazon tribe Pirahã there are just three numerals: “one”, “two” and “many”. Another Amazon tribe, Munduruku, uses numerals up to five. Does this mean that numerical abilities of members of these tribes are confined by the languages they use? Numerous studies indicate that it is *not* the case. Members of these tribes are able to state the numerosity of sets as well as add numbers bigger than those expressed by the numerals of languages used. Nevertheless, the precision of these “extra-linguistic” calculations is severely limited. Also, there are some studies strongly suggesting that subitizing is independent of language. These evidence suggest that mathematics is not fully “the offshoot of language”, as Chomsky put it, but something embedded in a more fundamental mental faculty.

The renowned cognitive neuropsychologist, Brian Butterworth, has been doing research which results point to similar conclusions. In an interview with Brady Haran he described a rather curious case of one of his patients:

The patient that we tested formally and in detail, was a woman who kept the accounts of the hotel that her family ran. But after a stroke, which affected her left hemisphere, she was unable to deal with numbers above four. Any numbers in any way we presented to her. If we gave her a word for a number above four, she wouldn’t know whether it was a real word let alone what it meant.

What was really interesting for us was that everything else seemed to be fine: her language was fine, she could understand what we were saying perfectly well, she could remember what she had for breakfast, she could reason, she undersood symbols (like traffic signs or political party signs) — everything was absolutely fine. It was just numbers that seemed to be selectively affected.

This *could* mean that human’s mathematical capacities are completely independent of language. In fact, Butterworth later in the interview expressed doubt in Chomsky’s idea. The former supports his view with a couple arguments. First, neural systems that are responsible for numerical processes are quite distant from regions of the brain that are connected to speech. Secondly, Butterworth doubts the conclusions from studies of Amazon and Aboriginal tribes that use languages with limited number representation. He believes that the inability to precisely calculate might stem from not understanding the task and not from linguistic limitations. Butterworth has conducted experiments with three groups of children: the English-speaking and the two groups from indigenous Australian tribes, both of which used languages with very few number-words. He made various tests, eg. memorizing the number of tokens, associating the number of token with the number of sounds heard, non-verbal addition and dividing a cake. It turned out that all three groups were similarly successful with their tasks, which would indicate that the language spoken by the children has no effect on the operations they perform.

However, one can have justified doubts whether the tasks Butterworth had the children do were *arithmetical* tasks. Were they not something more practical with a mere *numerical *aspect to it, resembling the studies on animals and newborns? It seems that they might have the latter nature and that they could be interpreted as requiring arithmetical calculations only by a mathematically-educated person. It is not sure whether Butterworth did not impose his conceptual framework on his experiment.

Moreover, Butterworth’s theory does not explain how humans shift from basic proto-arithmetical abilities to the full-fledged mathematical thinking such as theorem proving or manipulation with geometrical structures.

Finally, Butterworth claims that mental representation of numbers is *unrelated* to language. But how exactly is “language” understood here? It seems that Butterworth takes it to be something external to mathematical understanding. In such interpretation the hypothesis that the language forms our mathematical abilities is to be understood as if language was something *prior* to mathematical understanding. His experiments indeed disprove such view. But that is not necessary. The hypothesis could be interpreted in the way that language forms the latter faculty, but develops simultaneously with it. There is no evidence that mathematics as we know it could arise without the conceptual tools language enriches us with. There is a possibility that mathematical capabilities and language are not completely separated mental faculties and that language is not “external” to the mathematical understanding. So maybe we are talking about some kind of a mix?

## The interconnection of language and mathematical understanding

The currently most explanatory theory aiming at elucidating our issue argues in favor of such view. Elisabeth Spelke argues in (2011) that we need something more refined than the basic numerical abilities in order to develop mathematical understanding.

Cognitive scientists and neurobiologists have highlighted two very basic cognitive systems that might lay at the foundation of human’s mathematical abilities. They are *the* *object tracking system *(OTS)* *and *the approximate number system *(ANS). Both of them are prior to language, that is they are developed much earlier than a child develops basic linguistic capacities.

The first system is responsible, as the name says, for precise tracking of objects. The number of objects tracked simultaneously is limited to four. It must be said that OTS did not evolve in the service of mathamatical understanding or even numerical skills. Nevertheless, it is regarded as one of the basic cognitive systems that is necessary to create the exact representations of numbers.

The second system is responsible for the estimation of numerostiy of objects in a given set without counting them. It processes the quantities without precise number representation. The research also suggests that the accuracy of estimations by ANS improves with experience (it pertains to the mentioned Weber’s fraction).

Capabilities of OTS and ANS, interactions between them, together with their roles in forming representations of numbers bigger than four, are the subject of vivid discussion. There is a number of views but I would like to recollect only the most promising one (in my opinion).

Spelke reflects upon the fact that OTS and ANS, however they improve with time and interact with each other are not enough for representations of numbers higher than 9. That is because although the adults can immediately tell seven elements from 8 elements, they can’t discern in this way between a eight-element set and a nine-element set. So, in order to be able to carry out not only operations on numbers much higher than 9 but also the much more complex mathematical operations, we need something more than the said systems. In other words, roughly put, we need some other faculty that gives us access to the representation of, say, 92.

According to Spelke OTS and ANS have their solid role in the foundation of arithmetical understanding, but she believes that two other cognitive systems also play a vital role in the enterprise. These two systems are connected with perception of space. The first one (2D) works on smaller spatial scales. It is responsible for shape recognition. The second one (3D) works on bigger scales and provides recognition of size and direction. In some moment of human’s mental development all four systems, called by Spelke “systems of core knowledge”, begin to cooperate very closely. If she is right then it means that numerical and spatial abilities are closely related. But still, what lets us into the world of abstract arithmetic and geometry? Spelke claims that what enables us to enter these domains is **language .**

Natural language plays a pivotal role in the development of abstract numerical and geometric concepts, and does so by serving as the primary medium for combining information productively across distinct systems of core knowledge.

She believes that the process of mathematical ability is as follows. Children learn to use the basic numerals very early, but they are unable to understand their meaning before they turn three.

Then, they begin to comprehend that the numeral “one” represents the number 1. In the following year this process repreats for other numbers up to 4. Then, the ability to use numbers and numerals develops very rapidly. In that period children learn to solve harder and harder arithmetical problems.

It is the conceptual framework from the natural language that lets the children do the last step in their basic mathematical development:

In particular, it is presumed that they develop a general rule in their minds that connects every numeral with a respective number. Hence, at the age of four, children seem to be aware of the two aspects of number mentioned before: cardinality and ordinality.

Hence, Spelke seems to hypothesize that the inductive nature of **N** comes from the implicit realization of generative aspect of language, i.e. the fact that there are infinite constructible expressions in language. We should note the dissimilarity of this view with the post-Kantian conception of mathematics, where the infinitude of numbers came from the intuition of (infinitely progressing) time and the intuition of space. Spelke’s conception cannot also be reduced to the simple formalism that claims that mathematics is just a language, or that it fully comes from language: there are some basic, in-born pre-linguistic abilities that play a role in enabling us to even talk about mathematics.

## It seems to be a cocktail

There is number of arguments of indispensability of language for the genesis of mathematical understanding. Some of them come also from the anthropological studies and from experiments done with the deaf-mute. For instance, in cultures without (infinitely) many numerals people are able to recognize the equinumerosity of sets bigger than with four elements only in approximation. Similarly, the deaf-mute who use their fingers to represent numbers can recognize equinumerosity only to some approximate extent. It seems that what enables them to do these basic operations is the mentioned pre-linguistic ANS. It seems that *precise* operations on numbers are possible only when we acquire proper conceptual tools, and these are developed with our acquisition of natural language.

Of course, it’s just a theory. And it leaves a lot of questions unanswered, like, for example, the exact role language plays in forming our mathematical understanding. Spelke concluded one of her articles with a series of questions stemming from her ideas:

These suggestions raise many questions, however, concerning the aspects of language that play this role, and the ways in which language interfaces with non-linguistic conceptual representations. Research is also needed to probe whether other symbolic devices can substitute for language and serve to combine core representations productively. Finally, research into other abstract concepts, in domains such as morality, politics or economics, is needed to explore whether core systems and productive combinatorial abilities produce a broad range of abstract concepts or apply more narrowly to the concepts of mathematics. By addressing such questions, research in cognitive science promises to elucidate the mechanisms by which humans go beyond the core knowledge systems that we share with other animals and construct truly abstract knowledge systems that are unique in the living world.

In conclusion, I have to emphasize one more thing: it surely cannot be said that Platonism is shot into the curb even if Spelke was right. Firstly, the scientific research only *indicates* that our mathematical thinking is language-related. It isn’t yet certain. Secondly, one can argue that if mathematical constructions are strongly influenced by linguisitic tools that feature human mode of thinking, and realizing the trivial fact that humans with their brains are themselves parts of the nature, then we can conclude that mathematics *is* an objective part of nature, as the latter plugged it into our minds. Thirdly, and I think that it is the most sensible line of argumentation (though it is maybe too general), we have seen that the basic cognitive numerical systems we have at hand are very imprecise; but natural language, as we know it, is maybe even a less precise thing: there are no strict rules, millions of exceptions and contradictory practices — so how is the child, listening to this drivel, allegedly able to form the basic mathematical categories that lead us to the most exact and certain results? Maybe the picture lacks something more? Maybe we have some “axiom of consistency” installed in our minds, something that would make sense of language and bring it into precise structures? Hopefully, the research will shed some elucidation on this issue in the forthcoming future.

# Reading List

Throughout writing this article, I drew considerably from

Brożek, B., Hohol, M.,(2014) *Umysł matematyczny*, Copernicus Center Press. (Sadly only in Polish)