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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