Mathematics and the Real World (6 page)

Certain trees, including some types of mangrove, increase in number by a branch taking root in the ground and growing into a new trunk. A year has to pass, however, until a branch of a young mangrove can send out one of its branches from which a new tree will grow. Assume that a young mangrove is planted in the ground. After one year there will still be one mangrove tree, but after two years a branch of the first tree will also be growing, so there will be two mangroves. This is the beginning of the sequence 1, 1, 2. The next year, only the first tree can send out a branch to take root, so in the fourth year there will be three trees. The year after that, the two oldest mangroves will send out a branch each, so there will be a total of 2 + 3 = 5 trees growing, and we already have the sequence 1, 1, 2, 3, 5, and so on. Each year the number of new trunks is equal to the number of older trees (more than a year old), and the sequence describing their number of trees is the Fibonacci sequence. We will not expand the scope of this matter beyond the example quoted, but I will just add that if a number in the sequence is divided by the preceding number, the further along the sequence we go, the closer is the result to the golden proportion discussed above. This is another fact that convinced the ancients that they were observing a divine proportion or ratio. The fact that series whose extensions can be discovered intuitively are reflected in natural phenomena boosted the tendency to develop the ability to identify patterns throughout the generations.

We will summarize the observations in this and the previous section by stating that we can point to, and to some extent corroborate by means of experiments, mathematical abilities that throughout hundreds of thousands
of years of evolution afforded an advantage in the evolutionary struggle for survival. The processes of mutation and selection by which evolution shaped the human race resulted in those abilities being etched into human genes.

5. MATHEMATICS WITH NO EVOLUTIONARY ADVANTAGE

In this section we will examine a number of aspects of mathematics that, apparently, are not carried by our genes because they did not provide an evolutionary advantage during the formation of the human species (other nonnatural aspects of mathematics will be discussed later on). The current discussion is speculative, but further on we will present evidence corroborating the observations made here. We emphasize once again that the lack of an evolutionary advantage we are referring to relates to a period in which the genes determining the human species were developing. That is why mathematics of the type we will discuss here is not natural to intuitive thinking. This does not mean that this aspect of mathematics is not important or useful. Just the opposite. This type of mathematical ability provides a great advantage in the later evolution of human societies, but the time that has elapsed since human societies developed is not long enough for these abilities to have been etched into their genes.

The language of mathematics makes much use of quantifiers, expressions such as “for every,” or “there exists” that appear in mathematical propositions. For example, Pythagoras's famous theorem, which was proved as early as two thousand five hundred years ago, states that
for every
right-angled triangle, the sum of the squares on the two sides equals the square of the hypotenuse. The emphasis is on the quantifier “for every.” Another useful claim states that every positive integer is the product of prime numbers. A recent famous example is Fermat's last theorem. The hypothesis that it was correct was formulated as early as the seventeenth century but was unproven until the proof by mathematician Andrew Wiles of Princeton University, which was not published until 1995. The theorem states that for every four natural numbers (i.e., positive integers)
X
,
Y
,
Z
and
n
, if
n
is greater than or equal to 3, the sum
X
ⁿ
+
Y
ⁿ
cannot equal
Z
ⁿ
. Throughout the thousands of years of development of modern mathematics, the proof that a particular property
always
holds was considered an achievement.

However, is it natural to examine whether a particular property
always
holds? When something occurs repeatedly under certain conditions, does it naturally give rise to the question whether it occurs
every
time those conditions hold? Not so. If experience shows that a tiger is a dangerous predator, the conclusion drawn is that if one meets a tiger one should flee or hide. Losing energy or time in abstract thought about whether that particular tiger always devours its prey, or whether every tiger is a dangerous predator, would not afford an evolutionary advantage.

Another concept often referred to in mathematics is the concept of infinity. The Greeks proved that there is an infinite number of prime numbers. Is the urge to prove this statement a natural one? On observing many elements, is it reasonable to ask whether there is an infinite number of them? Again, I think it is not. Imagine ancient man discovering that a certain region is teeming with tigers. Is it worthwhile for him to consider whether there is an infinite number of them, or would it be preferable for him to get as far away as possible from that area as quickly as possible? The question “Is there an infinite number of tigers?” and even the question “Are there many more tigers than the large and dangerous number that I have already seen?” are academic questions, which will only harm those who devote time and energy to them and hence will impair their chances of surviving in the evolutionary struggle.

Another type of claim developed by mathematics is expressed in the reference to facts that
cannot
exist. A statement such as “If A does not occur, then B will occur” is commonplace among teachers, students, and researchers of mathematics. We will come across many such examples further on. This way of thinking is also not natural. Activity of the human brain is based on association, on the recollection of things that happened. To base oneself on an event that did not take place may be possible and useful, but does not come easily or intuitively. When you enter a room,
you look at what is in it and devote less thought to what is not there. We should repeat that we are not claiming that searching for an infinite number of mathematical elements, or proving that a certain property always holds, or relating to the negation of a possibility is an unworthy, unimportant, or uninteresting activity. What we are claiming is that those activities are not natural and that without a mathematical framework that suggests these possibilities, a reasonable person or an untrained student would not intuitively ask those questions.

Another attribute that is not innate in human nature is the need for rigor and precision. Mathematics is proud that a mathematical proof, provided it does not contain an error, is like an absolute truth. Mathematics therefore developed techniques of rigorous tests intended to lead to that absolute truth. Such an approach cannot have been derived from evolution. Genes do not direct humans to act rigorously to remove any possible doubt. The following anecdote illustrates this convincingly.

A mathematician, a physicist, and a biologist were sitting on a hill in Ireland and looking at the view. Two black sheep wander past them. The biologist says: “Look, the sheep in Ireland are black.” The physicist corrects him: “There are black sheep in Ireland.” “Absolutely not,” says the mathematician, “In Ireland there are sheep that are black at least on one side.”

Is the mathematician's claim, however rigorous and correct it may be, reasonable and useful in daily life? Of course not. In that sense, life is not mathematics. In life, even in ancient times, it is and was worthwhile and desirable to allow a lack of rigor, and even to allow errors, in order to achieve effectiveness. If a tiger's head can be seen above a bush, a man should not insist on being precise and saying that it has not been proven that the specific tiger has legs, but instead he had best distance himself from there as fast as he can.

We have claimed that the use of quantifiers and the interest in negatives or the reference to facts that cannot exist were not absorbed into the human brain during the evolutionary process and are not intuitive. Indirect evidence supporting this claim may be derived from studies that examined how many mathematical operations the human brain can perform consecutively.
Calculations such as addition and subtraction can be performed one after the other almost without limit. A person can be asked to perform a long series of multiplications, additions, division, and so on, and if he manages to remember the order, for instance by discovering a pattern in it, he can internalize the instructions and develop intuition regarding the next operation. This does not apply to quantifiers and negations. “Every dog has a collar that is not green.” That statement uses three concepts of logic:
every
;
has
; and
is not
. Studies have shown that even if someone can remember the order of the operations, the largest number of quantifiers that the brain can absorb is seven. Beyond that, even the most capable person cannot assess the outcome of the operation. It is interesting that the limit to the number of logical operations the human brain can absorb is seven, the same number as the maximum number of elements that animals can identify (see section 2 above). Other indirect evidence is provided by the existence of certain individuals, some of them autistic and some with Asperger's syndrome, who can perform complex arithmetic calculations with amazing speed and accuracy. However no individuals have been found who can similarly perform complex logical operations. The reason is apparently that the ability to perform arithmetic calculations exists in the brain naturally and is strengthened disproportionately in people whose limitations do not allow them to develop other abilities. Logic is not one of those extreme abilities.

Why is it important to identify mathematical abilities that are innate by virtue of evolution and to identify other attributes that are not innate? Humans think intuitively, associatively, and it is possible and easy to develop intuition based on natural abilities. Abilities contained in the genes are easier to develop, nurture, and use. It is harder to do that with abilities that are not natural to the human species. The recognition that there is a distinction between those two types of mathematical operations and understanding the source of that distinction are important to the understanding and utilization of human thought. In the sections that follow, we will see how these differences are significant to the development of mathematics, and in the
last chapter
of the book, we will discuss the implications of recognizing these differences for teaching mathematics.

6. MATHEMATICS IN EARLY CIVILIZATIONS

In this section we will review the mathematics that developed in the Babylonian, Assyrian, and Egyptian kingdoms. We will also look at the mathematics that developed independently and somewhat later in the Chinese dynasties. Although this survey does not cover the mathematics created in those realms exhaustively, it does correctly reflect the type of mathematics that developed. In particular, we will see that its development clearly traces what we have called the evolutionary advantage. These advantages of mathematics not only afforded humans an advantage over other living beings, but they also gave advantages to societies that developed mathematics over others that did not. The societies that ruled were those that developed the most up-to-date mathematics and that used it to establish and expand their power.

Reference to numbers and arithmetic existed prior to the Babylonian and Egyptian civilizations, but there is no direct evidence about where this mathematics existed or its level of development. Based on those remote tribes discovered in the last few centuries whose languages included only the numbers 1, 2, 3, many, we may assume that the mathematics these remote tribes used was minimal. In contrast, in 1960 human bones were discovered in the Belgian Congo that were dated to 20,000 BCE, and on them were signs that archaeologists and anthropologists believe express counting up to and beyond the number twenty. Thus we may conclude that when man lived and developed in small groups, was nomadic, and subsisted mainly by hunting, he used and even developed simple mathematics, which we referred to in previous sections as giving an evolutionary advantage.

The Babylonian kingdom was a mighty one, certainly for its time. Its origins date back to 4700 BCE. Its culture was based on Sumerian culture. Later, the Akkadian civilization became predominant, leading to cultural, economic, and social progress. The Akkadian contribution is attributed mainly to King Hammurabi, who ruled around 1750 BCE and who is famous mainly through the Code of Hammurabi that constituted the first-known comprehensive code of social behavior in the world. Around the
year 1000 BCE, Assyrians started migrating from modern day Iran (Persia) and eventually dominated the Middle East until the Greek conquest under Alexander of Macedon, also known as Alexander the Great, in 330 BCE.

Our knowledge of Babylonian mathematics is based mainly on large numbers of potsherds discovered that served as the main means of written communication throughout the years of the kingdom. A particularly large collection was found on the site of the ancient Sumerian city of Nippur. Much of this was transferred to Yale University, and work on deciphering the writing has not yet been completed. Babylonian writing was in cuneiform, which had signs for numbers. The system used was based on the position of the digit, similar to the current decimal system of writing numbers, but for reasons that are not quite clear, the system of numbers was to the base of 60 (the decimal system was developed in India in about the sixth century CE and was introduced to the West by the Arabs in about the eighth century, but it was fully adopted by Europe only in the sixteenth century). The Babylonians did not have a symbol for the zero of current times. If we were to adopt the Babylonian system, the number 24 would signify twenty-four and also two hundred four. The reader would have to determine from the context what the writer intended to convey. In cases where the writer's intention was not clear, there would be a space to indicate the difference, but this occurred only in the last centuries of the kingdom. Thus, a space between the 2 and the 4 in the above example would indicate that the writer meant two hundred four. (In some potsherds, it was found that in places where we would currently write “0” there was a symbol that in texts served as a space or as a sign of separation. Some interpret this as the first use of the symbol for zero as a number.) Furthermore, the base 60 was not the only one used; sometimes the base 20 or 25 was used. In those cases too the reader had to decide from the context what base the writer was using. As far removed in time as we are from that practice, it seems to us that such usage was strange and must have made it difficult for the reader. We should be aware, however, that we act similarly in nonmathematical writing. Lack of clarity and even ambiguity are very common in both spoken and written language. The reader can generally understand what the writer means from the context. The reason for lack
of clarity is evident. Precise formulations that would leave no room for misunderstanding would require great effort that would in general not be worth the benefit to be gained. Less precision is more efficient and hence is preferable in the evolutionary struggle. The Babylonians considered mathematical expressions as part of their language and did not think that they had to be more exact than the nonmathematical expressions.