The Cult of Pythagoras

Copyright

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Contents

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pp. vii-xii

Acknowledgments

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pp. xiii-xvi

I warmly thank the following friends and colleagues for kindly sharing thoughtful comments and helpful suggestions in various aspects of this project: Eric Almaraz, Ronald Anderson, Casey Baker, Nicole Banacka, Frank Benn, Susan Boettcher, Jochen Büttner, Richard Chantlos, Alexis R. Conn, Robert Crease, Chandler Davis, ...

Introduction

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pp. xvii-xxiv

The international bestseller The Secret claims 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 ...

1. Triangle Sacrifi ce to the Gods

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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. ...

2. An Irrational Murder at Sea

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pp. 16-28

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 ...

3. Ugly Old Socrates on Eternal Truth

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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. ...

4. The Death of Archimedes

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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 purifi ed geometry, although Archimedes later mixed it with practical things. ...

5. Gauss, Galois, and the Golden Ratio

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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. ...

6. From Nothing to Infinity

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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, ...

7. Euler’s Imaginary Mistakes

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pp. 101-112

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 fi rst explain the forgotten arguments of a famous mathematician, Leonhard Euler. ...

8. The Four of Pythagoras

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pp. 113-133

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 the end of this sentence corresponds to a single complex number. ...

9. The War over the Infinitely Small

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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, ...

10. Impossible Triangles

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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 confi rm 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, ...

11. Inventing Mathematics?

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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, ...

12. The Cult of Pythagoras

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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, ...

Notes

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pp. 217-256

Illustration Sources and Credits

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pp. 257-258

Index

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pp. 259-264