The Cult of Pythagoras

The Cult of Pythagoras: Math and Myths

ALBERTO A . MARTÍNEZ
Copyright Date: 2012
Pages: 288
https://www.jstor.org/stable/j.ctt7zw974
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    The Cult of Pythagoras
    Book Description:

    In this follow-up to his popularScience Secrets,Alberto A. Martínez discusses various popular myths from the history of mathematics: that Pythagoras proved the hypotenuse theorem, that Archimedes figured out how to test the purity of a gold crown while he was in a bathtub, that the Golden Ratio is in nature and ancient architecture, that the young Galois created group theory the night before the pistol duel that killed him, and more. Some stories are partly true, others are entirely false, but all show the power of invention in history. Pythagoras emerges as a symbol of the urge to conjecture and "fill in the gaps" of history. He has been credited with fundamental discoveries in mathematics and the sciences, yet there is nearly no evidence that he really contributed anything to such fields at all. This book asks: how does history change when we subtract the many small exaggerations and interpolations that writers have added for over two thousand years?The Cult of Pythagorasis also about invention in a positive sense. Most people view mathematical breakthroughs as "discoveries" rather than invention or creativity, believing that mathematics describes a realm of eternal ideas. But mathematicians have disagreed about what is possible and impossible, about what counts as a proof, and even about the results of certain operations. Was there ever invention in the history of concepts such as zero, negative numbers, imaginary numbers, quaternions, infinity, and infinitesimals?Martínez inspects a wealth of primary sources, in several languages, over a span of many centuries. By exploring disagreements and ambiguities in the history of the elements of mathematics,The Cult of Pythagorasdispels myths that obscure the actual origins of mathematical concepts. Martínez argues that an accurate history that analyzes myths reveals neglected aspects of mathematics that can encourage creativity in students and mathematicians.

    eISBN: 978-0-8229-7853-4
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. LIST OF MYTHS AND APPARENT MYTHS
    (pp. ix-xii)
  4. ACKNOWLEDGMENTS
    (pp. xiii-xvi)
  5. INTRODUCTION
    (pp. xvii-xxiv)

    The international bestsellerThe Secretclaims that Pythagoras knew the secret to happiness, the powerful law of attraction: that you can get what you want by thinking about it. Less recently, in one of the most popular science books ever, Carl Sagan noted that on the island of Samos local tradition says that their native son Pythagoras was “the first person in the history of the world to deduce that the Earth is a sphere.” Earlier, the mystic architect John Wood, having studied megalithic ruins such as Stonehenge, concluded that they were “a Model of the Pythagorean World,” that such...

  6. 1 TRIANGLE SACRIFICE TO THE GODS
    (pp. 1-15)

    Legends say that in ancient times a secretive cult of vegetarians was led by a man who had a strange birthmark on his thigh and who taught that we should not eat beans. He believed that when a person dies, the soul can be reborn in another body, even as an animal. So he said that we should not eat animals because they might be our dead relatives or friends. And he said that he had been born five times, even before the Trojan War. And when he died his fifth death, his followers later said that he was reborn...

  7. 2 AN IRRATIONAL MURDER AT SEA
    (pp. 16-28)

    Long after Pythagoras died, some of his admirers were fascinated by numbers. This tradition eventually generated a captivating murder story.

    If not Pythagoras himself, at least some of his admirers seemed to be interested in mathematics. Yet the earliest evidence is not complimentary. It suggests that some Pythagoreans focused on numbers not too thoughtfully. Plato criticized the Pythagoreans for analyzing numerically the harmonies of plucked strings, rather than analyzing relations among numbers themselves.¹ Aristotle repeatedly criticized “the so-called Pythagoreans” for believing that things, material things, are made of numbers.² And we too might think: “Things are made ofnumbers? Nonsense!”...

  8. 3 UGLY OLD SOCRATES ON ETERNAL TRUTH
    (pp. 29-42)

    There is no good evidence that Pythagoras linked mathematics and religion, but apparently someone else did. Socrates lived in Athens in the fifth century BCE. According to ancient accounts, he was very ugly, with bulging eyes and a flat, upturned nose with wide-open nostrils. Allegedly he became a soldier and fought bravely in some battles. In old age he was bald, poor, constantly barefoot, and spent much time discussing philosophy in the city streets, questioning men who presumed to be wise, criticizing their ideas, annoying them by showing the contrary. Although we do not have any works by Socrates, some...

  9. 4 THE DEATH OF ARCHIMEDES
    (pp. 43-58)

    The idea that geometry is timeless led people to think that there can be no change in mathematics. There can be discovery, they thought, but not invention. It also encouraged the idea that change and moving things are foreign to pure mathematics. Euclid seemed to have purified geometry, although Archimedes later mixed it with practical things.

    Euclid’s books on geometry, theElements,began to circulate roughly around 250 BCE. The earliest extant fragments and copies of these books specify no author. The earliest historical trace of a Euclid who worked on geometry is a brief critical mention in a work...

  10. 5 GAUSS, GALOIS, AND THE GOLDEN RATIO
    (pp. 59-81)

    We tend to fit history into the forms of traditional stories about heroes, victims, and martyrs, struggle, success, and injustice. Hence we read: “From of old it has been the custom, and not in our time only, for vice to make war on virtue. Thus Pythagoras, with three hundred others, was burnt to death.”¹ But really, we do not know just how Pythagoras died. Certain stories resonate: the martyr who was punished for speaking truth to power, the unassuming guy who did good against all odds, the wonder boy who cleverly solved the daunting problem, the unappreciated worker whose brilliant...

  11. 6 FROM NOTHING TO INFINITY
    (pp. 82-100)

    Mathematicians have the distinction of agreeing about results more often than members of most other professions. I think that mathematicians usually agree with one another more than the members of any science, any political party, and even any religion. But nevertheless, we can consider instances in which mathematicians have disagreed about various things, even the result of an operation. We would expect that performing basic operations on basic numbers such as 0, 1, 2, 3, would not yield disagreements. And it is well known that you cannot divide a number by zero. Math teachers write, for example, 24 ÷ 0...

  12. 7 EULER’S IMAGINARY MISTAKES
    (pp. 101-112)

    Like zero, negative numbers have sometimes been used to derive apparent contradictions. For example, consider the following:

    $\frac{{{\rm{ - 1}}}}{{\rm{1}}}{\rm{ = }}\frac{{\rm{1}}}{{{\rm{ - 1}}}}$

    $\sqrt {\frac{{{\rm{ - 1}}}}{{\rm{1}}}} {\rm{ = }}\sqrt {\frac{{\rm{1}}}{{{\rm{ - 1}}}}} $

    $\frac{{\sqrt { - 1} }}{{\sqrt 1 }} = \frac{{\sqrt 1 }}{{\sqrt { - 1} }}$

    $\sqrt { - 1} \sqrt { - 1} = \sqrt 1 \sqrt 1 $

    -1=1

    These steps seem to prove the impossible equation, that 1 is equal to its opposite. We expect that something in the sequence of operations must be a mistake. What is it? I will give an original solution to this apparent paradox, and to do so, I’ll first explain the forgotten arguments of a famous mathematician, Leonhard Euler.

    The paradox above involves operations with square roots of negative numbers, the so-called imaginary numbers. While nowadays mathematicians value these numbers...

  13. 8 THE FOUR OF PYTHAGORAS
    (pp. 113-133)

    It is well known that any complex number, such as 4 + 2i,can be represented by a point or a line in a plane. The real and imaginary parts of this number correspond toxandycoordinates on the so-called complex plane, as illustrated in the figure.

    Some teachers love this: it seems to clearly give meaning to complex numbers by connecting numbers and geometry: every single number, real or complex, corresponds uniquely to a single point in a plane. If we take this sheet of paper, this page, as representing the complex plane, then the period at...

  14. 9 THE WAR OVER THE INFINITELY SMALL
    (pp. 134-155)

    Scientists used to say that matter is made of indivisible units, atoms. But some thought that matter is divisible into fragments much smaller. In 1896, physicist Emil Wiechert commented: “We might have to forever abandon the idea that by going toward the Small we shall eventually reach the ultimate foundations of the universe, and I believe we can do so comfortably. This universe is indeed ‘infinite’ in all directions, not only outward in its Greatness, but also down, into the Smallness within.”¹ Soon, radioactivity seemed to show “the almost infinite divisibility of matter.”²

    While physicists divided nature into subatomic particles,...

  15. 10 IMPOSSIBLE TRIANGLES
    (pp. 156-180)

    When Albert Einstein was a solitary boy, less than twelve years old, his uncle told him about the Pythagorean theorem. The boy struggled to confirm it until he devised a way to prove it to himself.¹ By reflex, one might be tempted to construe this anecdote as early evidence that Einstein was a genius, but no—he didn’t see it that way, and there were already very many proofs of the hypotenuse theorem, made by ordinary people, young and old. What matters is that, as the young Einstein realized, certain geometric propositions seem compellingly true.

    For example: “The sum of...

  16. 11 INVENTING MATHEMATICS?
    (pp. 181-200)

    Through the so-called Platonist outlook, many people construed mathematics in religious ways. They assumed that its principles were eternal truths discovered by special men, geniuses, and they accepted that these truths were valid everywhere and could never change. The laws of geometry and numbers seemed like the laws of God, and therefore mathematics was valued as a preparation to discipline the mind for studies of metaphysics and theology. In the 1730s, Bishop Berkeley complained that some people accepted strange mathematical propositions on the basis of faith instead of reason. Some students learned rules on the basis of authority, “because the...

  17. 12 THE CULT OF PYTHAGORAS
    (pp. 201-216)

    Mythology deals with gods and heroes, tales that are passed down especially in popular oral traditions. We began with Pythagoras, in a time when religion and science mixed. I don’t know if he really contributed anything to mathematics, but he became portrayed in the form of classic myths: a wise demigod who started a Golden Age, a hero who solved problems and knew the secret of immortality. In time, stories about him increasingly included science and mathematics, and teachers learned to ignore his older tales about gods, sacrifices, and magical powers.

    But still, other elements from the mythical tradition continued...

  18. NOTES
    (pp. 217-256)
  19. ILLUSTRATION SOURCES AND CREDITS
    (pp. 257-258)
  20. INDEX
    (pp. 259-264)