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The Trisectors
Underwood Dudley is well known for his collection of books on mathematical cranks. Here he offers yet another - angle trisectors. It is impossible to trisect angles with straightedge and compass alone, but many people try and think they have succeeded. This book is about angle trisections and the people who attempt them. According to Dudley: 'Hardly any mathematical training is necessary to read this book. There is a little trigonometry here and there, but it may be safely skipped. There are hardly any equations. There are no exercises and there will be no final examination. The worst victim of mathematics anxiety can read this book with profit and dry palms. It is quite suitable to give as a present.'
202 pages, Paperback
First published January 24, 1994
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May 20, 2011
You might agree with me that this statement is true:
"It is impossible for a human to run a mile in thirty seconds."
It's so far beyond the ability of anyone alive that it seems clear it's impossible. But in some philosophical sense, this is nonsense. There is no physical barrier making it absolutely inconceivable that some mutant will some day run a mile in thirty seconds. In that sense, it's not impossible, it's just so hard that we seriously doubt anyone will ever do it.
The word "impossible" means something different in mathematics. In mathematics, we work with very simple systems that have clearly specified rules. These systems are so simple that we can prove properties about them in a rigorous fashion, such that there is no doubt these properties hold. In particular, we can prove that certain things are impossible. Just as in chess, one could easily demonstrate that a bishop that begins on a black square can never end up on a white square during a chess match played according to the rules.
A problem the ancient Greek mathematicians set out to solve was to trisect an angle, using only the tools of compass and straightedge. This problem was unsolved until the 19th century, when it was demonstrated that it is impossible. The proof that this construction is impossible is not particularly difficult as these things go: it requires just some plane geometry, some trigonometry, and a smidge of so-called modern algebra.
Nevertheless, a baffling number of people throughout the 19th and 20th centuries set out to trisect the angle, and in fact believed they had done so. Dudley documents many attempted trisections here in this very interesting book. Amazingly, many of these trisectors are fully aware that there exists a proof that their goal is impossible, but don't seem to understand the concept of mathematical proof, and continue anyway. Many of them seem to hold mathematicians in contempt for "giving up" on the problem. Meanwhile mathematicians see the trisectors about the same way they'd see people who spend their days deep in thought over a chess board, trying to find a sequence of chess moves to get a bishop from a black square to a white square.
The arrogance of many trisectors is staggering. Then again, the whole enterprise can only be born out of arrogance. How else could someone with barely a high school education who can't follow simple proofs believe that every mathematician in the world is wrong about a simple, elementary fact known for over a century?
"It is impossible for a human to run a mile in thirty seconds."
It's so far beyond the ability of anyone alive that it seems clear it's impossible. But in some philosophical sense, this is nonsense. There is no physical barrier making it absolutely inconceivable that some mutant will some day run a mile in thirty seconds. In that sense, it's not impossible, it's just so hard that we seriously doubt anyone will ever do it.
The word "impossible" means something different in mathematics. In mathematics, we work with very simple systems that have clearly specified rules. These systems are so simple that we can prove properties about them in a rigorous fashion, such that there is no doubt these properties hold. In particular, we can prove that certain things are impossible. Just as in chess, one could easily demonstrate that a bishop that begins on a black square can never end up on a white square during a chess match played according to the rules.
A problem the ancient Greek mathematicians set out to solve was to trisect an angle, using only the tools of compass and straightedge. This problem was unsolved until the 19th century, when it was demonstrated that it is impossible. The proof that this construction is impossible is not particularly difficult as these things go: it requires just some plane geometry, some trigonometry, and a smidge of so-called modern algebra.
Nevertheless, a baffling number of people throughout the 19th and 20th centuries set out to trisect the angle, and in fact believed they had done so. Dudley documents many attempted trisections here in this very interesting book. Amazingly, many of these trisectors are fully aware that there exists a proof that their goal is impossible, but don't seem to understand the concept of mathematical proof, and continue anyway. Many of them seem to hold mathematicians in contempt for "giving up" on the problem. Meanwhile mathematicians see the trisectors about the same way they'd see people who spend their days deep in thought over a chess board, trying to find a sequence of chess moves to get a bishop from a black square to a white square.
The arrogance of many trisectors is staggering. Then again, the whole enterprise can only be born out of arrogance. How else could someone with barely a high school education who can't follow simple proofs believe that every mathematician in the world is wrong about a simple, elementary fact known for over a century?
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