What Euclid Did, from “The Unreasonable Influence of Geometry”

From the article “The Unreasonable Influence of Geometry” by Doug Muder

If Euclid did not invent his geometry, but just recorded knowledge that had been around for centuries before him, what did he do that was so important that his text was preserved and used for thousands of years?

Euclidean geometry was history’s first example of a system of knowledge. Egyptian and Babylonian geometry texts are little more than lists of facts and formulas. But Euclid had not just put together a set of facts, he had assembled those facts into a structure. All the terms were defined, all the assumptions listed, and all the other statements were derived rigorously from those assumptions. By comparison all other fields of human thought were little more than grab-bags of ideas and tricks.

The assumptions – which are called postulates – are examples of what became known as self-evident truth. In other words, once these propositions were understood, they could not be denied. For a truth to be self-evident meant that even imagining its contradiction involved the mind in absurdities. Euclid’s postulates include statements like “Any two points can be connected by a line” and “Any circle divides the plane into an inside and an outside.” These statements seemed to be fundamentally different from a truth of experience, like “Snow is white.” You can easily imagine green snow or purple snow, even if you have never seen it. But how could a circle not divide a plane into inside and outside?

Euclid also did something more subtle, something that changed the course of human thought for all time. Prior to Euclid, arguments were private affairs. If I wanted to use reason to convince you of something, I would begin with premises that you would accept, and progress logically from there. The particular premises that you would grant might be very different from those of another person, and the argument that would result from our conversation would be unique to us. (Plato’s dialogues are examples of such arguments. Changing any one of the characters would change the argument.) But Euclid’s assumptions are not intended to be granted by you or me or any other particular person; they are intended to be universally acceptable. Likewise, the steps of Euclid’s proofs are not intended to convince any particular person; they are intended to be beyond anyone’s reproach. Euclid’s structure establishes geometry as a public truth, not a private conviction.

Another way to say the same thing is that with Euclid, reason becomes coercive. If your assumptions are unquestionable, and your logic is rigorous, then all reasonable people are forced to agree with your conclusions, even if they would rather not.

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