Mathematics and the Real World (4 page)

We will first address the simplest mathematical operations, that is, counting, addition, and subtraction. One of the basic principles of classical psychology is that babies are born with a brain that evolution has prepared for learning but that is initially void of all information. Babies learn about the world initially via observation and then via a combination of observation and experience. More-abstract learning appears later, with language development. This view was held and taught by no less than Sigmund Freud (1856–1939), the father of modern psychology. What he said related to knowledge in general and to mathematical ability. At first glance it does appear that regarding mathematical elements that description is correct. Only when they are three or four years old do children acquire the ability to count, and later on to add and subtract. At first they only recite what they have heard, one, two, three, and so on, without realizing that they can count. To illustrate, if given three balls, they may count one, two, three, four, five, counting the same ball more than once. Only at a later age do children begin to understand what counting is, and even later do they start performing simple arithmetic operations. A leading researcher
and advocate of this approach was the famous psychologist Jean Piaget (1896–1980), who formulated a complete theory of cognitive development regarding the gradual acquisition of mathematical abilities from childhood to adulthood. (The reader can find elaborations on this issue and other research referred to later in this section in the monographs by Dehaene and Devlin [2000] listed in our sources.) In one of his experiments, Piaget showed children eight flowers, six roses and two chrysanthemums, and asked, “Are there more flowers or more roses?” A significant number of children answered roses. Piaget concluded that children have no intuition of set inclusion; in other words, the children had no understanding that with two sets, one of which includes the other—in our case the set of flowers includes the set of roses—the former is larger. In Piaget's time, it was believed that the relation between sets was the right basis for mathematics (a view that is currently becoming less and less accepted; more on this in the
last chapter
of the book). Accordingly, Piaget concluded that small children have no understanding of the connection between sizes of sets and of one set including another, let alone any ability to count or knowledge of simple arithmetic.

Nevertheless, the fact that the ability to count is not acquired until a child is several years old does not necessarily prove that the characteristic is not innate. In the observations mentioned above, including Piaget's experiments, counting and arithmetic were acquired together with the ability to communicate and to use a given language, generally the mother tongue. It is not surprising that communicating in a given language is not an innate attribute but a learned one. The ability to learn a language is an inborn characteristic, but acquiring the language itself takes several years. Before the child learns a language, his arithmetic abilities do not come into play, as seen in the above experiments. The obstacle is not the child's lack of ability to count but the fact that he has to answer questions that he does not comprehend, or he does not realize what the expected answer is, until he has had some more years of practice. It is easy to devise tests showing that understanding the question plays an important role in interpreting the results.

Children aged three to four years are shown four marbles and, nearby, four buttons; they are asked whether there are more marbles or more
buttons. Most of them will answer that the number of marbles and buttons is the same. The buttons are then spread more widely, that is, spaced farther apart from each other, and again the question is asked, “Now, are there more marbles or more buttons?” Most of the young children will give the same answer, the number is the same. When this exercise is repeated with older children, aged five and six years, many of them answer that there are more buttons.

This does not indicate a drop in their mathematical aptitude. The correct explanation is that older children are not used to being asked the same trivial question more than once. They therefore conclude that the questioner must expect a different answer and assume that the question is about the distance between the items and not their number, and they answer accordingly.

There is clear evidence that very young infants relate to numbers and can even perform simple addition and subtraction. How can such cognitive capabilities be examined in babies only a few months old? Several parameters enable us to see whether a baby is excited or surprised. One is the length of time it looks at something. A baby can look at an object or a situation for a few seconds and then it will divert its gaze to something else. When it looks at something new or surprising—and for babies a few months old, new is also surprising—it holds its gaze on it for a longer period, for a few seconds more. A second parameter is the rate at which the baby sucks, say its pacifier. When it is excited or surprised, it sucks harder and more frequently.

An experiment undertaken by Ranka Bijeljac-Babic and colleagues in Paris (the results of which were published in 1991) showed that even newborn babies have a sense of numbers. They measured the intensity with which babies sucked while hearing meaningless words of three syllables, such as “defantok,” “alovo,” “kamkeman.” At first, when the babies heard the words, they sucked harder until they became accustomed to the sounds,
and then they reverted to normal sucking. Then two-syllable words were spoken, and this led to harder sucking again. This pattern repeated itself. Whenever the number of syllables in a series of words changed, the reaction of the babies changed too. In other words, even at such an early age, babies can recognize that word sounds consist of syllables and react to a change in the number of syllables between one series and another. The syllables in the “words” were chosen randomly to avoid their having any meaning or significance, so that the only explanation for the babies’ reaction was the number of syllables.

A more complex experiment performed in the laboratory of Prentice Starkey of the University of Pennsylvania (the results of which were published in 1980) showed that distinguishing between different numbers is not restricted to one communication channel. Six-month-old babies were shown pairs of pictures with either two or three elements, say two in the picture on the left and three in the picture on the right. Different objects were shown each time, sometimes just geometric shapes, sometimes dots, and so on, and each time the colors were different; this was done to neutralize any possible effect of the content of the pictures. While the pictures were being shown the babies also heard notes or sounds, sometimes two and sometimes three, in random order, in order to cancel any possible effect of any structure in the order in which the notes were heard. When three notes were heard, the babies clearly preferred to look at the picture with the three images, and when they heard two notes, they turned their attention to the picture with two elements. They were exhibiting a counting operation or were at least comparing quantities perceived via two different senses, sight and hearing.

Another sophisticated experiment conducted by Karen Wynn of Yale University (results published in 1992) showed that babies have a natural sense of addition and subtraction. A screen was placed before babies of a few months, and they saw a figure going behind it. The screen was removed, and they saw the figure. Next, one figure went behind the screen, and then another figure followed. The screen was removed, and the babies saw the two figures. This was repeated several times until the babies became accustomed to what was happening. Then an arithmetically incorrect exercise was performed. One figure went behind the screen, and then
a second figure. When the screen was removed, only one figure could be seen. To a highly significant degree, the several-months-old babies showed surprise. They expected two figures, and lo and behold, there was only one! The experiment was repeated with a number of variations to remove the possibility that the infants were simply used to the result of a particular exercise. It was highly significant that results that were arithmetically incorrect gained more of the babies’ attention. Later, a similar experiment was conducted with adult rhesus monkeys. They showed signs of surprise when, for example, a banana was put inside a box, followed by a second one, and when the box was opened, there was only one banana inside.

These experiments were scrupulously and rigorously controlled, and it may be concluded from them that human beings’ arithmetic abilities are genetic. Clearly these operations are performed in undeveloped brains and in no particular language, and there is thus no possibility for the baby to discuss the results with its parents or friends. When the child grows, it will have to learn how to express this mathematical ability in the everyday language it uses to talk to its parents. This learning is a process in itself. But simple arithmetic is innate in babies and is not a by-product of a brain that had been developed for completely different purposes. From this it may be deduced that simple arithmetic afforded an advantage in the evolutionary competition. This is not surprising. For those competing for food, the mathematical ability to distinguish large from small, the many from the few, and even addition and subtraction gives an evolutionary advantage. An individual with this ability will be better suited to a competitive environment than would other members of the same species with lower mathematical abilities.

How is this finding consistent with the finding that some primitive tribes, including some discovered recently in isolated locations, use only the numbers one, two, and three to describe their environment, and any larger quantities are referred to as “many”? If living beings such as birds or rats can differentiate between numbers greater than three, one would expect humans to be able to count better. The answer is simple: language developed much later among human beings in the process of evolution and placed greater emphasis on more important things than the less important. Those primitive tribes apparently are well aware of the difference between sets consisting
of five or six objects, but their language is not rich enough to describe them because they had no need to devote terms to numbers greater than three. This does not contradict the fact that at the intuitive level their arithmetic capability is much higher. As a language develops, so does the ability to express and perform more-extensive arithmetic operations. Language developed relatively late in the general evolutionary process but is itself part of that process. The human brain is distinctive among living beings in its verbal communication abilities. Indirect evidence that arithmetic, counting, and the facility to perform addition and subtraction, for example, are the direct results of evolution and not just by-products of language can be found in recorded cases of people born with a malfunction of the brain that meant they could not count or perform addition or subtraction but whose other verbal capabilities were perfectly normal. Conversely, there are people with defective verbal capabilities who can perform arithmetic operations easily.

It should be noted that similar techniques of research into the evolutionary roots of mathematics can be used to discover abilities and features whose roots are evolutionary and that are unrelated to mathematics. Recently (in 2010) research by Karen Wynn of Yale, mentioned above, and her partner Paul Blum was published, showing that altruism and the aspiration for justice exist in babies a few months old, at an age when it is reasonable to assume they could not have absorbed these characteristics from the environment. This too is not surprising. The preference for a just distribution of resources is an attribute that helps a society survive the evolutionary struggle, and it is reasonable, therefore, that it is inherent at the genetic level.

4. MATHEMATICS THAT YIELDS AN EVOLUTIONARY ADVANTAGE

Mathematics has many aspects. The previous section showed that the ability to perform arithmetic calculations is the result of evolution. In this section we will indicate other branches of mathematical operations that, it may reasonably be assumed, provided an advantage in the evolutionary struggle. We will present evidence that those parts of mathematics were
also incorporated in the genetic heritage. We may refer to this aspect of mathematics as
natural mathematics
. In the next section we will describe mathematical operations that are not natural, as they did not afford any evolutionary advantage in the hundreds of thousands of years during which the human genome was formed.

It is reasonable to assume that the ability to recognize geometrical elements gave an evolutionary advantage. As sources of food and water have typical geometric shapes, being able to recognize those shapes correctly constituted an advantage in the competition for sources of sustenance. But is there any evidence that, as a result of evolution, the recognition of geometric shapes is carried by the genes? We will soon turn our attention to such evidence but will first introduce what is known as the golden cut, or the golden-ratio rectangle.