Monday, August 28, 2017

233: A Totalitarian Theorem

A couple of weeks ago, on August 15th 2017, we celebrated a rare Pythagorean Theorem Day, since 8 squared + 15 squared equals 17 squared.   This reminded me of an anecdote I read recently in Amir Alexander’s book “Infintesmal”, a history of the controversies over the concept of infinitesimal quantities over several centuries in Italy and England.   Surprisingly, a key figure in this history was Thomas Hobbes, the English political philosopher best known for his treatise “The Leviathan”, which advocated an autocratic form of government controlled by a single ruler.   What’s not as widely known is that Hobbes developed a strong interest in mathematics, directly influencing his philosophical works.  In fact, his philosophical career was jump-started by his unexpected encounter with the Pythagorean Theorem.

These days, nearly every high school student deals with this theorem in geometry class, but such experience was not nearly as universal in Hobbes’ time, the early 1600s.   In the case of Hobbes, he had been thinking about politics for many years, but by the age of 40, had not yet seen the Pythagorean Theorem.   The story of his encounter with this theorem is related by one of his contemporaries, historian John Aubrey.  One day Hobbes had some spare time to browse while visiting a library, and a copy of Euclid’s Elements opened to a page on the Theorem was sitting on a table.   His reaction was “By God, this is impossible!”   Hobbes wondered, could the formula a squared + b squared = c squared really apply to *every* right triangle, even an arbitrary new one he drew right there?     But he read the proof, and the related proofs and definitions leading up to it, and soon was convinced.   He was amazed that such a profound and non-intuitive result could be deduced based on simple axioms and definitions.  From this point forward, he was in love with geometry and Euclid’s methods of proof.   In addition to attempting further mathematical work himself, he used this method as the basis for his philosophical works.

Hobbes’ most famous treatise, the “Leviathan” published in 1651, was then built upon this method of starting with basic definitions and propositions and deriving the consequences.    Most works of philosophy strive for this ideal, though I think the line between valid logic, sophistry, and word games gets very fuzzy  once you leave the realm of pure mathematics.    Back in college, I remember my classmates majoring in philosophy bragging that mathematics, was a mere subset of the vast realm that they studied.    After graduation, many of them applied their broad expertise in logical reasoning to brewing numerous exotic varieties of coffee.    Of course, some works of philosophy are indeed brilliant and convincing, but it is nearly impossible for them to truly exhibit a level of logical rigor comparable with a mathematical proof.

In any case, this attempt at rigorous foundations made the Leviathan very convincing, and it is today regarded as a foundational work of political philosophy.   To Hobbes’ contemporaries, its convincing nature made the work very disturbing when it came to controversial conclusions.   Today most people remember the Leviathan superficially for its advocacy of a strong central ruler, a king or dictator, who must have absolute power.   Thus he is mixed up in people’s minds with the horrific totalitarian regimes that arose in the 20th century.   But we need to keep in mind that he was writing in a very different time, with the opposite problem:  the weakening of the monarchy had led to decades of civil war in England, with multiple factions repeatedly committing mass murder against each other.   A strong central ruler was seen as a much lesser evil than this situation of pre-civilized barbarism into which Hobbes’ country seemed to have sunk.    We also need to keep in mind that the Leviathan introduced many positive concepts of modern Western political philosophy:   individual rights and equality, the idea that whatever is not forbidden by law is implicitly allowed, and the basis of a government in the consent of the governed.     Thus, while his concept of an absolute ruler is not in favor, Hobbes continues to be a philosophical influence on many modern governments.

Hobbes also tried his hand at advancing mathematics, but with much less success than he achieved in the political arena.   He had been disturbed that some classical math problems, such as the squaring of the circle, were still unsolved, and decided that in order to claim completeness of his methods of reasoning (and thus his philosophical system), he needed to solve them.   He then published numerous solutions to the problem of the squaring of the circle, not anticipating that a few hundred years later this problem would be proven definitely unsolvable.   As you may recall from earlier podcasts, this is a consequence of the fact that pi is a transcendental number, and cannot be algebraically derived from unit ratios.   As a result, all his attempts in this area were flawed in one way or another.    The much more talented mathematician John Wallis published a famous series of letters ripping apart Hobbes’ reasoning from many different angles.   It may seem silly that someone like Wallis wasted so much time on this dispute with a lesser mathematician.    But part of the motivation may have been that discrediting Hobbes mathematically would help to discredit him politically, and save politicians of the time from the need to face the powerful challenges of Hobbes’ ideas.

And this has been your math mutation for today.

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