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Pythagoreans was the sobering discovery that their own “numerical religion” was, in fact, pitifully unworkable. The whole numbers 1, 2, 3,…are insufficient even for the construction of mathematics, let alone for a description of the universe. Examine the square in figure 6, in which the length of the side is one unit, and where we denote the length of the diagonal by d. We can easily find the length of the diagonal, using the Pythagorean theorem in any of the two right triangles into which the square is divided. According to the theorem, the squareof the diagonal (the hypotenuse) is equal to the sum of the squares of the two shorter sides: d 2 = 1 2 + 1 2 , or d 2 = 2. Once you know the square of a positive number, you find the number itself by taking the square root (e.g., if x 2 = 9, then the positive x = v 9 = 3). Therefore, d2 = 2 implies d = v2 units. So the ratio of the length of the diagonal to the length of the square’s side is the number v2. Here, however, came the real shock—a discovery that demolished the meticulously constructed Pythagorean discrete-number philosophy. One of the Pythagoreans (possibly Hippasus of Metapontum, who lived in the first half of the fifth century BC) managed to prove that the square root of two cannot be expressed as a ratio of any two whole numbers. In other words, even though we have an infinity of whole numbers to choose from, the search for two of them that give a ratio of v2 is doomed from the start. Numbers that can be expressed as a ratio of two whole numbers (e.g., 3/17; 2/5; 1/10; 6/1) are called rational numbers . The Pythagoreans proved that v2 is not a rational number. In fact, soon after the original discovery it was realized that neither are v3, v17, or the square root of any number that is not a perfect square (such as 16 or 25). The consequences were dramatic—the Pythagoreans showed that to the infinity of rational numbers we are forced to add an infinity of new kinds of numbers—ones that today we call irrational numbers. The importance of this discovery for the subsequent development of mathematical analysis cannot be overemphasized. Among other things, it led to the recognition of the existence of “countable” and “uncountable” infinities in the nineteenth century. The Pythagoreans, however, were so overwhelmed by this philosophical crisis that the philosopher Iamblichus reports that theman who discovered irrational numbers and disclosed their nature to “those unworthy to share in the theory” was “so hated that not only was he banned from [the Pythagoreans’] common association and way of life, but even his tomb was built, as if [their] former comrade was departed from life among mankind.”

    Perhaps even more important than the discovery of irrational numbers was the pioneering Pythagorean insistence on mathematical proof—a procedure based entirely on logical reasoning, by which starting from some postulates, the validity of any mathematical proposition could be unambiguously established. Prior to the Greeks, even mathematicians did not expect anyone to be interested in the least in the mental struggles that had led them to a particular discovery. If a mathematical recipe worked in practice—say for divvying up parcels of land—that was proof enough. The Greeks, on the other hand, wanted to explain why it worked. While the notion of proof may have first been introduced by the philosopher Thales of Miletus (ca. 625–547 BC), the Pythagoreans were the ones who turned this practice into an impeccable tool for ascertaining mathematical truths. The significance of this breakthrough in logic was enormous. Proofs stemming from postulates immediately put mathematics on a much firmer foundation than that of any other discipline discussed by the philosophers of the time. Once a rigorous proof, based on steps in reasoning that left no loopholes, had been presented, the validity of the associated mathematical statement was essentially unassailable.