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 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. He 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.
To understand how the Egyptians thought about geometry and geometers, you need to understand something very important about the mindset of the ancients: science, religion, and magick had not yet become separate things. Science/religion/magick taught that there is an invisible order behind the visible disorder of the world. The success of geometers in redrawing boundaries after the floods of the Nile was evidence for the power of that invisible order. Geometry, then, was the special (and probably secret) knowledge of a priesthood.
The Pythagoreans were the first known secret society. They seem to have been some kind of graded order, in which each successive initiation gave the seeker access to more of the secret doctrine. Their worldview, as best we can reconstruct it, was based on an elaborate system of geometric and numerical correspondences – a surprising number of which survive to this day in figures of speech like "a square deal". Through these correspondences, a geometric figure or an arithmetic calculation could have an allegorical significance and express philosophical mysteries.
The discovery of irrational numbers was a huge philosophical shock to the Pythagoreans. In some ways this shock prefigures the shock that non-Euclidean geometry had on the world of the Enlightenment.
We are told that Plato put a sign above the door to his academy saying "Let no one enter who is unfamiliar with geometry." Our modern universities might do well to remember this admonition, because the geometry student faces many of the same conceptual difficulties that re-appear more abstractly in Plato's theory of forms. Trying to understand Plato without understanding geometry is like fighting with one hand tied behind your back.
For example: nearly every geometry student at some point wants to prove a theorem by measuring something in the diagram. The teacher must go to some difficulty to explain that the diagram is just a picture of the problem, not the problem itself. Aristotle wrote: “The geometer bases no conclusion on the particular line which he has drawn being that which he has described.” The lines that we can draw in diagrams are not true lines, and the points in diagrams are not true points. They are merely imperfect representations of the true points and lines, which exist merely as ideas, and have never been seen by anyone. And yet, to the geometer, these ideal invisible points and lines are more real and more perfect than anything that can be drawn on paper.
This relationship between the invisible objects of geometry and the visible objects of the real world, including diagrams, is key to understanding the relationship between the Platonic forms and the real world objects that embody them. Your high school geometry class – whether you knew it then or not – was actually an introductory course in Platonic mysticism.
Return to the outline of The Unreasonable Influence of Geometry