Mathematics and the Real World (3 page)

The rock cormorant in the picture on the left cannot fly. It lives on cliffs close to the seashore that are exposed to strong winds. Its abilities to find twigs and to build a proper nest are vital to the survival of the species in these tough conditions. In its courtship, the male cormorant demonstrates to his potential mate his ability to gather twigs to build the nest they would share. The courted female will respond to the male's advances only if he can prove he has that capability. The middle picture is of the frigate bird. In its courting display, the male inflates his gular pouch until it becomes an enormous red balloon. He does this to show his intended mate the strength of his lungs and his ability to fly long distances to scoop fish out of the water. The picture on the right, the blue-footed booby, demonstrates entirely different qualities in its courtship display. The male of this species incubates the eggs and protects them by covering them with his large blue feet. He therefore tries to woo his potential mate by flaunting the size and shape of its feet, thus proving his ability to protect the eggs they will produce together against enemies and unsettled or rough weather.

These examples illustrate that the characteristics and behavioral patterns we can discern today indicate the characteristics that were of evolutionary importance, and those show us how each species survived the evolutionary struggle.

The essential characteristics that helped any particular population to win the battle for survival during the formation of the species are etched into its genes, and we may identify them as innate attributes. Cheetahs’ speed, eagles’ “eagle eyes” and cats’ tree-climbing ability are all innate traits. A cheetah cub is born with the ability and basic instinct to run fast. It will need help from its parents to learn what it needs to fear, how to hunt, and even how to run more effectively, but the basic features of speed and hunting are carried in its genes. Similarly, the genes of a cat enable it to
learn how to catch mice, and the innate characteristics of an eagle include its keen vision and ability to identify potential prey from a great height. Learning merely refines and improves the innate attributes. The attributes of each species enable us to learn about the conditions in which the species developed; similarly, knowing the conditions in which the species developed enables us to learn about the characteristics that evolved.

It is reasonable to assume that just as physical attributes of animal species are innate features etched into their genes, the same will apply to at least some mental attributes. Mental and social skills also play a role in the battle for survival, so that in these too, the selection process strengthens the features that help the species to overcome its rivals. Specifically, in the reproduction process mental attributes can also be changed and improved by mutation. In the following sections we will examine the mathematical capabilities of the human species from an evolutionary aspect. We will ask whether the abilities to understand and to use mathematics are the results of evolutionary development, or whether they may be by-products of a brain that developed to cope with other needs.

2. MATHEMATICAL ABILITY IN THE ANIMAL WORLD

If mathematical ability made a contribution in the evolutionary struggle that brought the human race to the position it currently occupies among the species, it may be assumed that other living beings would possess a certain degree of mathematical ability. But what does mathematical ability mean? Mathematics encompasses a broad range of topics and conceptual methods. The question to ask, therefore, is which of those mathematical features provide an evolutionary advantage? And the follow-up question is how can we identify these mathematical abilities in animals?

The most basic mathematical ability is counting. It is followed by the understanding of the concept of a number as an abstract object and the ability to perform simple arithmetic operations, such as addition and subtraction. We will start by discussing the existence of these simple elements
in adult animals. A mother cat moves her kittens from place to place and generally does not forget a kitten or two, and when she has finished moving them, she does not usually go back again to check whether she has moved all of them. She may remember them individually, but it seems reasonable to state that the mother cat has a sense of quantity. The instinct of quantitative estimation clearly provides an evolutionary advantage, so we should not be surprised that adult animals possess that ability. But does that ability extend to the ability to count and to the possibility of performing arithmetical manipulations?

Before presenting several convincing examples showing that some animal species do have mathematical ability, a warning is in order. The results of experiments in general, and of animal experiments in particular, should be interpreted with great caution. A well-known illustration of this is the case of “Clever Hans.” (More details and references concerning this story, and concerning the research mentioned later in this section, can be found in the monographs by Dehaene and by Devlin [2000] listed in the sources.) Toward the end of the nineteenth century a horse known as Clever Hans was exhibited on tour in Germany with its trainer, Wilhelm von Osten. The horse showed remarkable ability in adding, subtracting, finding the squares of numbers, simple division, and so on, all with a very high degree of success. The horse was wrong occasionally, but such errors occurred infrequently. The method by which the horse showed its abilities was that when an exercise was read out or written on a board, it would tap its hoof the number of times corresponding with the right answer. It was suspected that the act was simply a clever deception by which the trainer somehow or other managed to give the right answer to the horse. An official committee was appointed, headed by a psychologist named Carl Stumpf and whose members included the director of the Berlin Zoo. The committee checked, among other things, whether the horse could solve the problems if the trainer was not present, and it found that even then the horse could still give the correct answers. The conclusion was that some animals have a fairly advanced mathematical ability. Subsequently, more detailed examinations in 1907 by another psychologist, named Oscar Pfungst, showed that the horse did not know mathematics. The trainer
was indeed reliable and honest, but the horse had learned to distinguish involuntary changes in his facial expressions and in the facial expressions of the audience when the trainer was not present. The horse understood from those facial expressions when it had reached the correct number of taps of its hoof. The presence of the trainer or an audience was essential. Pfungst found that if the trainer looked tense at a wrong answer, the horse answered according to the expression and not the correct answer. The research methods developed by Pfungst as a result of this case are now recognized as a breakthrough in psychological research.

Scientific experiments that were more soundly based have proven that some animals do indeed possess mathematical ability. The German zoologist Otto Koehler (1889–1974) proved as early as in the 1930s that some species of birds can identify a collection with a given number of elements. It is apparently not difficult to train a pigeon to choose every third seed when faced with a row of seeds. A squirrel can be trained so that when faced with boxes containing different quantities of nuts, it will choose the box with exactly five nuts. There is a limit to the numerical-identification ability of these animals. Koehler himself found that even the most capable animals could not identify collections with more than seven elements. The number appears in the literature also as a bound to the number of information units that a human brain can process. We will meet the number seven again later on in similar contexts. Still, these experiments demonstrate the mathematical ability to estimate quantity but do not yet prove an ability to count or to grasp the abstract concept of a number.

Adult crows are known to be able to count, within certain limitations. Food is placed near a building. The crow learns very quickly that it is dangerous to attempt to approach the food while someone is in the building. It cannot see into the building to check if anyone is inside or not, but it can see when someone enters or leaves it. The popular literature (without scientific checks, it must be said) reports situations in which several people enter the building one after the other. As long as they remain in the building, the crow keeps away. The people in the building then leave, one by one. With surprising accuracy the crow knows when all those it saw enter the building have left, and only then does it approach the food. Clearly there
is a limit to crows’ ability to be exact, just as there is a limit to humans’ ability to keep track on exact large numbers. Crows managed to count up to five or six in this manner, with a high degree of accuracy. The ability to identify a collection with a given number of elements demonstrated by crows in this example and by other species is consistent with an evolutionary advantage.

The ability to count is clearly an advantage in the battle for survival, but its origin in the avian world is unclear. After all, how often in the evolution of crows did they encounter a situation in which they had to count the number of dangerous animals entering and leaving a building? Specifically, it is unclear whether this apparent counting is in fact counting in the mathematical sense. In other words, does the crow have the ability, whether conscious or not, to comprehend the number of the people entering the building, or does it simply remember who went in and who came out?

Monkeys were found to have a greater mathematical ability to count and compare. The following experiments were carried out by Guy Woodruff and David Premack of the University of Pennsylvania (their paper was published in 1981). A chimpanzee was shown a full glass and a half-full glass, and it was taught to choose the half-full glass every time. The same chimpanzee was then offered the choice of a whole apple or half an apple, and it chose the half apple. In other words, it generalized the mathematical principle from the glass to the apple. In a similar fashion, the chimpanzee was taught to demonstrate simple mathematical abilities, such as recognizing that the combination of half an apple and a quarter of an apple is three-quarters of an apple. In another experiment, two trays were placed before a chimpanzee. The first tray had two piles of pieces of chocolate, one pile with three pieces, and the other with four. The second tray had a pile of five pieces of chocolate and then a separate, single piece. In most cases, the chimpanzee chose the tray with the larger total number of pieces. This does not yet constitute proof that the chimpanzee understood the abstract concept of numbers or the addition of numbers, but it is evidence of mathematical abilities. This is not surprising, as such abilities constitute an evolutionary advantage.

Another experiment with animals proves that the concept of numbers
in the abstract does exist to some degree among some, even among less-developed animals. The experiments were conducted by Russell Church and Warren Meck of Brown University (the research was published in 1984). It is not difficult to train rats so that when they hear two beeps, one after the other, they are given enough tasty food to satisfy them. Similarly, when they see two flashes of light, they can also safely eat the food. They were taught, however, that when they hear four beeps or see four light flashes, it is dangerous to eat the food, as they get an electric shock. The aural or visual signals, that is, the beeps or flashes, are received and processed in the brain via two different senses, hearing and sight. The rats reached a high level of reacting correctly, approaching the food if they heard two beeps or saw two flashes, and avoided doing so if they heard four beeps or saw four flashes. When the rats had been trained sufficiently, they heard two beeps that were immediately followed by two light flashes. How do you think they reacted? Did the rats consider the signals as a double invitation to eat the food, or did they interpret them as a four-signal warning to refrain? If they reacted according to the latter, it may be assumed that they recognized the number four as an independent concept, even though the signals received were of two different types. The answer: the rats clearly identified the number four and did not approach the food when they received four signals, even when they were received via different senses.

This experiment with rats still does not indicate arithmetic ability in these animals, nor does it give a definite proof that such abstract counting is an innate attribute, that is, a characteristic carried in their genes, as it may be the result of training made possible by the development of the brain for other purposes. It seems reasonable, however, that this ability is innate, mainly because of the evolutionary advantage given by the abilities to count and to recognize the concept of numbers. To be convinced beyond all doubt that a particular ability is innate, it should be identified in the animal when it is still very young. Such experiments with cubs and other animal young are obviously very difficult to perform. With human cubs, that is, babies, such experiments can be performed.

3. MATHEMATICAL ABILITY IN HUMANS

Before we present the evidence that mathematical ability is inborn in human beings, that is, embedded in their genes, we need to make two comments about the nature of the discussion. First, our use of the term
genes
from here on is a conceptual one and does not relate to any specific gene or set of genes. We will leave the identification of the genes responsible for mathematical ability to our biologist colleagues. For us, establishing the fact that it is innate is sufficient. Second, in the examples of animals and in the discussion in this section, we do not relate to the ability of any individuals. We do not ask whether the success in mathematics of a specific student is determined by his genes alone or whether it is due to environmental conditions or to his having good or less-good mathematics teachers. The discussion is concerned with the mathematical capabilities of the human race and the connection between that capability and the process of evolution, a process that has continued for millions of years, in the course of which the abilities under discussion were formed.