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the properties of this sequence. It reflects many aspects of development and growth in nature, as well as mathematical properties that are interesting in themselves. We describe one use of the sequence here.
    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 thousandsof 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