There are two dolls on a table, which are covered by a curtain. An adult puts her hand behind the curtain and removes one of the dolls. The curtain is then raised and behind it there are ... still two dolls. A watching child's gaze rests on the dolls for more than the usual time, because the situation is surprising. There should only have been one doll left on the table. The number of dolls seems surprising for even one or two-day old children.
Karen Wynn, Professor of Psychology and Cognitive Science at Yale University, was the first to perform this doll test. She published the results in Nature in 1992. The results suggest that some numerical concepts are innate to humans. On the other hand, the same effect has been observed in many animals, ranging from chimpanzees to goldfish.
What about the entire number line? Do we have an innate understanding that numbers are arranged from left to right? We at least know that a number line is more than a mere mental image. The so-called SNARC effect (Spatio Numeric Association of Response Codes) refers to the fact that we subconsciously group large numbers to the right of the body and small numbers to the left.
Mathematical concepts are the key to the philosophy of abstract thinking.
However, this does not guarantee that the number line concept is innate to humans. The Chilean American cognitive scientist Rafael Nuñez has argued that the SNARC effect is based on learned, culturally related issues. Alongside linguist George Lakoff, Nuñez has also theorised that mathematical concepts are essentially a kind of metaphor. Nuñez and Lakoff presented their theory in a book published in 2002, "Where Mathematics Comes From? How the Embodied Mind Brings Mathematics into Being”.
Why, in general, are researchers interested in the nature of mathematical concepts in the human brain and mind?
Mathematical concepts provide a key to the philosophy of abstract thinking. By the early twentieth century, mathematicians and philosophers were already worried that concepts, such as those of the number line and functions, were overly based on intuition. What if the assumption of an infinite number of points is contradictory? Do infinities exist only in the minds of mathematicians, or are they part of the real world?
Such questions led to the formalisation of mathematics at the beginning of the twentieth century. This grounds the indisputable nature of mathematical concepts in precise rules of symbolic manipulation. Such rules establish which strings are accepted (axioms) and how new strings (theorems) can be deducted from them. The set theory is an example of an attempt to ground mathematics in a simple list of axioms and rules.
However, the set theory does not tell us where these concepts (such as real numbers, functions or the number line) originate. Nor does it tell us why we want to discuss them in the first place. Cognitive science provides a way of approaching mathematical concepts from the other side, as it were. It begins by posing questions about where ideas come from and the essence of abstract thinking. This approach also brings us to another eternal question: what is it that makes us human.
What are the minimal features of the cognitive system or brain that make mathematics or abstract thinking possible?
Chimpanzees too are surprised if the wrong number of dolls is revealed behind the curtain. On the other hand, only humans seem capable of understanding the number line and the application of rules to equations.
In the spirit of the set theory, a cognitive scientist might ask: ‘What minimal features of the cognitive system or brain make mathematics or abstract thinking possible?'
In 2002, the cognitive scientists Hauser, Chomsky and Fitch suggested that our ability to understand
recursion, i.e., the application of rules over and over again, is the primary factor behind our linguistic and mathematical abilities, which distinguishes us from other animals. Human imagination, thought and planning (i.e., not just language) also seem to be based on recursion.
New light is being shed on these issues in the context of artificial intelligence. What would it mean for a computer to be capable of abstract thought? We are unlikely to be able to build such a computer without understanding the nature of abstract in humans.
Karen Wynn's experiments, which I mentioned at the beginning of this article, show that in the case of infants small numbers – such as two and three – are independent of the sensory channel. If two consecutive sounds are played back to a baby, but three points are displayed, the baby will notice the numerical difference. The concept of number is therefore innate, regardless of the senses and reference frameworks used. The philosopher David Chalmers has argued that our conscious understanding of an issue comprises the ability to use it within different reference frameworks. In the previous example, this would mean applying the concept of number via both sight and hearing.
This last idea returns us to the traditional, age-old question of what consciousness is.
The above questions are bound together by semantics, which explores how the meaning of concepts arises. Abstract mathematical concepts provide an excellent way of studying this. If we can learn how the understanding of mathematical concepts occurs in the brain, we will be a step closer to understanding the essence of consciousness and the nature of meaning itself. Only then can we ask how we can ever get a computer to understand anything.
It is at precisely these interfaces that artificial intelligence is seeking its current form. A computer can output the string “1+1=2”, but does not have a broader understanding of the idea. But if we can induce artificial intelligence to understand that two sounds are somehow the same as two seconds and two points, we may have come a step closer to genuinely understanding numbers – and the mathematical essence of our brains.