In a world of flux and change, Euclidean geometry maintained its structure for more than 2,000 years. Over time, this apparent changelessness made Euclidean geometry the poster child of Reason. It showed that timeless certainty was possible, and it tempted thinkers in all areas of thought to imitate its forms. With the success of Newton and the Enlightenment political program, Reason gained the prestige to remake the world in its own image, which was very much the image of Euclid.
Then non-Euclidean geometry was discovered, and we are still sifting through the wreckage of the Enlightenment world view almost 200 years later.
This essay is not about geometry per se, but about what geometry has meant to the larger culture. It has meant different things at different times. And because it has so often symbolized something larger than itself, changes in geometry have had an unreasonable influence on fields far removed from the study of lines and planes.
The geometry that we all know from high school is Euclidean geometry, which goes back to the book Elements of Geometry written by Euclid in about 300 BC, just after the time of Alexander the Great. But Euclid did not invent his geometry, he collected and systematized the geometry that was already known in his time. Plato, who lived a couple of generations before Euclid, clearly knew a great deal of geometry. He probably learned it from the Pythagoreans, a secret society of mystic mathematicians who were active in Greece at that time. Think about that phrase for a minute: a secret society of mystic mathematicians. Already that should tell you that the world worked very differently in those days.
The Pythagoreans trace their origin back to Pythagoras, a semi-legendary figure who lived sometime before 500 BC – later than Homer, but earlier than Socrates. The Pythagorean theorem is named after him. Pythagoras is supposed to have learned his geometry during his travels in Egypt. This may or may not be true. Compare, for example, the story of Atlantis. We know the story from Plato. Plato is supposed to have heard it from a friend, who heard it from his grandfather. The grandfather is supposed to have heard it from Solon (another semi-legendary figure), who learned it during his travels in – guess where? – Egypt. In short, the Greeks thought of Egypt as the source of all things wise and mysterious – somewhat like our image of Tibet.
There's undoubtedly a lot of myth in the Greeks' ideas about the history of geometry, but the line of descent that goes roughly from Egypt to the Pythagoreans to Plato's Academy to Euclid may well be true.
If Euclid didn't invent geometry, what was so important about him? Euclid perfected the Greek notion of a system of knowledge. Egyptian geometry was just a grab-bag of tricks for getting answers. Euclidean geometry has structure. Euclid's proofs are the first known examples of what we might call "public argument": They aren't intended to convince this person or that person, making use of whatever assumptions that the audience is willing to grant. Euclid's assumptions are intended to be unassailable by anyone, whether they are present or not, and his proofs are intended to be universally convincing.
With Euclid, reason becomes coercive: All reasonable people must come to the same conclusions, whether they want to or not. This idea changed the world.
From Plato to Kant, philosophers viewed Euclidean geometry as a special kind of knowledge – perfectly certain, independent of experience, yet somehow useful in the world. Geometry, then, proved the existence of an incredibly powerful kind of knowledge, and invited philosophers to seek more of it.
If you want your ideas to last thousands of years, why not model them after a system of thought that already has? If you want your private convictions to become public truths, which all reasonable people will be forced to accept, why not model them after the only such public truths you know? In one field after another, thinkers imitated the form and style of Euclid's Elements. The most famous imitation is Jefferson's Declaration of Independence, which begins by listing the truths that the author holds to be self-evident.
Given a line and a point not on the line, how many lines parallel to the first line can be drawn through the point? It seemed clear that the answer was one. But was it self-evident enough to deserve to be a postulate? Generations of mathematicians tried to prove this statement indirectly – by assuming that the answer was not one and searching for a contradiction. Along the way they deduced all kinds of strange and unlikely statements, but those statements stubbornly refused to contradict each other.
And then Gauss began to suspect that something else was happening: He wasn't finding a contradiction in this system because there wasn't one. It was a new geometry – one that contradicted common sense, but not logic.
You might wonder why anyone other than a mathematician should care how many geometries there are. Like the first small leak in a huge dam, it doesn't seem to amount to much. But non-Euclidean geometry marked the beginning of the end of the Enlightenment worldview. If you could doubt geometry, you could doubt anything – and people have.
So what about that geometric intuition of ours? If we no longer think of it the way Spinoza did, as our participation in the mind of God, how do we account for it? Why does it seem so unjustifiably solid to us?
The contemporary view borrows a little from several philosophers. Like Kant, we now believe that we picture space as Euclidean and 3-dimensional because our brains are set up to do it that way. And yet this structure is not divorced from the world outside our minds. Like Hume, we now believe that our concept of space is based on experience – but not our experience. Like Plato, we now say that our geometric intuition is based on the experience of previous lives – but not our previous lives.
Instead, we now say that evolution makes us the beneficiaries of the life-or-death experiences of countless generations of humans and prehuman ancestors. The development of geometric intuition gave our ancestors a survival advantage, and that's why we have it. Earth, it turns out, is the kind of place where a Euclidean imagination comes in handy. And that’s about the strongest claim we can make on the subject.