Introduction to “The Unreasonable Influence of Geometry”

Part of the article The Unreasonable Influence of Geometry, by Doug Muder

They say that one day Albert Einstein was asked to submit some questions for the Princeton physics qualifying exam. When the chairman looked at the questions Einstein submitted, he complained “These are the same questions you gave me last year.” But Einstein replied, “Oh that’s OK. This year the answers are different.”

That’s what we’re used to in modern subjects. The answers keep changing, and every year there is something new. But tonight I want to talk about a subject where the answers stayed the same for more than 2000 years. It wasn’t just that the essence of the subject stayed the same; its form, its structure, its terms, and even its diagrams didn’t change for thousands of years. The geometry textbook that was written by Euclid in 300 BC was still being used in various translations around the world in the late 19th century.

Euclidean geometry is unique in this respect. Not even the other branches of mathematics can make these claims. Calculus, for example, is less than 400 years old. And although arithmetic may be as old as geometry (or even older), in practical terms a great deal has changed over the years. One thousand years ago calculations were being done in Roman numerals, and only a true master could divide large numbers. But if Euclid himself had walked into my high school geometry class in the 1970s, I believe he would have felt right at home. Many of the same theorems were proved in the same ways, even using the same diagrams. No other area of human thought has held its shape to this extent.

What has changed over the years is what people have thought about geometry: what it means, how we know it, and what it says about the possibilities of human knowledge in general. Throughout western history, philosophers have gone to great lengths to explain how geometric knowledge is possible, and thinkers of all sorts have imitated the form and technique of Euclid in an effort to establish in their own fields the same kind of timeless certainty that Euclid seemed to have established in his. During the period of history known as the Enlightenment, this Euclid-inspired view of reason and its power was bolstered by the prestige of Newton’s accomplishments, and reason remade society on many levels, particularly in the areas of government, religion, and individual rights.

This central pillar of the Enlightenment world came crashing down early in the 19th century with the advent of non-Euclidean geometry. Outside of the mathematical community the importance of this event has never been fully appreciated. But its effects have been sweeping, and are still being felt today. Mathematics spent the next hundred years examining the cracks in its foundations, an effort which led directly to Gödel’s famous Incompleteness Theorem. This in turn led to a complete revisioning of the nature of mathematics and the role of certainty in it.

In the larger society, the door was open to a challenging of reason and common sense in all areas. It was as if Euclidean geometry had been a dam holding back the waters of doubt, protecting people’s faith in the power of human reason. The breaking of that dam started a flood which still has not completely receded.

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