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Reductio ad absurdum - The Titfield Thunderbolt
Heisenberg might have stayed here
Reductio ad absurdum
Book Review: The Irrationals - A Story of the Numbers you can't count on, by Julian Havil
In A Brief History of Time, Stephen Hawking says he was advised that every equation in his book would halve the readership. Clearly that memo didn't get to Julian Havil, for this book is full of them; I won't pretend to have followed them all, and I did find it very difficult going, although not without its reward.

Havil begins in ancient Greece and describes that it was well known that the square root of two was not "commensurable" with the natural numbers (i.e. it could not be expressed as a ratio). Havil moves on in time through India and Arabia, then on into Europe with the Renaissance and the Enlightenment, although a surprising amount of the work he refers to is twentieth century.

As well as considering various properties of π and e, there are discussions on the Diophantine approximation of irrational numbers. Curiously the Cantor diagonal is missing by name, although there is some discussion of a method that leads to the same result (the uncountability of real numbers). There are some startling and philosophically troubling insights - such as the somewhat bizarre claim that the Golden Ratio is the most irrational (i.e. least accurately approximable) number, or that one can find the entire past, present, and future encoded in π, if one is prepared to expand it far enough. (It seems to me a similar claim can be made for Fourier transforms, when - in principle - any signal can be represented as a series of sine waves of multiplying frequency extending across all time). No wonder one or two mathematicians have become unhinged along the way.

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