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numerical value to d . The side and the square are "incommensurable"; their ratio a/d cannot be represented by any real numbers or fractions thereof; it is an "irrational" number; it is both odd and even at the same time. * I can easily draw the diagonal of a square, but I cannot express its length in numbers – I cannot count the number of dots it contains. The point-to-point correspondence between arithmetic and geometry has broken down – and with it the universe of numbershapes.
    ____________________
*
The simplest manner of proving this is as follows. Let d be represented by a fraction
    ,
    where m and n are unknown. Let a = 1, then d 2 = 1 2 + 1 2 and

    Then

    If m and n have a common factor, divided it out, then either m or n must be odd. Now m 2 = 2 n 2 , therefore m 2 is even, therefore m is even, therefore n is odd. Suppose m = 2 p . Then 4 p 2 = 2 n 2 , therefore n 2 = 2 p 2 and therefore n is even, contra hyp . Therefore no fraction

    will
measure the diagonal.
    It is said that the Pythagoreans kept the discovery of irrational numbers – they called them arrhētos , unspeakable – a secret, and that Hippasos, the disciple who let the scandal leak out, was put to death. There is also another version, in Proclos: 11
    "It is told that those who first brought out the irrationals from concealment into the open perished in shipwreck, to a man. For the unutterable and the formless must needs be concealed. And those who uncovered and touched this image of life were instantly destroyed and shall remain forever exposed to the play of the eternal waves."
    Yet, Pythagoreanism survived. It had the elastic adaptability of all truly great ideological systems which, when some part is knocked out of them, display the self-regenerating powers of a growing crystal or a living organism. The mathematization of the world by means of atom-like dots proved a premature shortcut; but on a higher turn of the spiral, mathematical equations proved once again the most serviceable symbols for representing the physical aspect of reality. We shall meet with further examples of prophetic intuition supported by the wrong reasons; and we shall find that they are rather the rule than the exception.
    Nobody before the Pythagoreans had thought that mathematical relations held the secret of the universe. Twenty-five centuries later, Europe is still blessed and cursed with their heritage. To non-European civilizations, the idea that numbers are the key to both wisdom and power, seems never to have occurred.
    The second blow was the dissolution of the Brotherhood. We know little of its causes; it probably had something to do with the equalitarian principles and communist practices of the order, the emancipation of women, and its quasi-monotheistic doctrine – the eternal messianic heresy. But persecution remained confined to the Pythagoreans as an organized body – and probably prevented them from degenerating into sectarian orthodoxy. The Master's principal pupils – among them Philolaus and Lysis – who had gone into exile, were soon allowed to return to Southern Italy and to resume teaching. A century later, that teaching became one of the sources of Platonism, and thus entered the mainstream of European thought.
    In the words of a modern scholar: " Pythagoras is the founder of European culture in the Western Mediterranean sphere." 12 Plato and Aristotle, Euclid and Archimedes, are landmarks on the road; but Pythagoras stands at the point of departure, where it is decided which direction the road will take. Before that decision, the future orientation of Greco-European civilization was still undecided: it may have taken the direction of the Chinese, or Indian, or pre-Columbian cultures, all of which were still equally unshaped and undecided at the time of the great sixth-century dawn. I do not mean to say that if Confucius and Pythagoras had exchanged their places of birth, China would have beaten us to the Scientific Revolution, and Europe