Irrational / Everything’s relative

One popular (though maybe apocryphal) story from the history of mathematics involves the discovery of irrational numbers by Pythagoras and his disciples. The following quote is from the book Fermat’s Last Theorem by Simon Singh:

One story claims that a young student by the name of Hippasus was idly toying with the number \sqrt{2}, attempting to find the equivalent fraction. Eventually he came to realize that no such fraction existed, i.e. that \sqrt{2} is an irrational number. Hippasus must have been overjoyed by his discovery, but his master was not. Pythagoras had defined the universe in terms of rational numbers, and the existence of irrational numbers brought his ideal into question. The consequence of Hippasus’ insight should have been a period of discussion and contemplation during which Pythagoras ought to have come to terms with this new source of numbers. However, Pythagoras was unwilling to accept that he was wrong, but at the same time he was unable to destroy Hippasus’ argument by the power of logic. To his eternal shame he sentenced Hippasus to death by drowning.

When I was a boy, the story was told that Pythagoras could not accept irrational (i.e.., cannot be written as the ratio of two integers) numbers because their existence would mean that we live in an irrational (i.e., insane, crazy) world, and so he had the unfortunate discoverer silenced.

When I present this story to my own students, they’re usually incredulous about the story, doubting that someone so smart could act so stupidly (or irrationally). Then I’ll tell them a much more recent story, from less than 100 years ago, about how a scientific principle morphed into a statement of ethics. Einstein’s theories of special relativity and general relativity were developed in the early 1900s; his theory of general relativity explained precession in the orbit of Mercury and predicted the deflection of starlight by the Sun’s gravity, which were both unexplained by Newtonian mechanics.

Writing to a popular audience, Einstein summarized his theory as follows:

The ‘Principle of Relativity’ in its widest sense is contained in the statement: The totality of physical phenomena is of such a character that it gives no basis for the introduction to the concept of “absolute motion”; or, shorter but less precise: There is no absolute motion.

The following sentences from Paul Johnson’s Modern Times summarize the popular reaction to Einstein’s work:

But for most people, to whom Newtonian physics, with their straight lines and right angles, were perfectly comprehensible, relativity never became more than a vague source of unease. It was grasped that absolute time and absolute length had been dethroned; that motion was curvilinear… At the beginning of the 1920s the belief began to circulate, for the first time at a popular level, that there were no longer any absolutes: of time and space, of good and evil, of knowledge, above all of value. Mistakenly, but perhaps inevitably, relativity became confused with relativism.

Indeed, the modern catchphrase “everything’s relative” was spawned shortly after the discovery of special and general relativity, a moral principle that Einstein himself abhorred.

So, after telling the story about Pythagoras and \sqrt{2}, I’ll use this story to hold up a mirror to ourselves, demonstrating that the passage of time has not made us immune from translating mathematical or scientific principles into statements of ethics.

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1 Comment

  1. The problem with “people” is that they want absolutes and they see science as producing these. The absolutes are treated as “the truth”, whereas science is about explanation and prediction. It turned out in Galileo’s time that seeing the sun as the centre of the solar system yielded a simpler explanation for a lot of stuff, but “truth” has nothing to do with it.

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