Sometimes it is hard to be fair to Plato. He is basically a good guy, but his politics are bullshit. That “philosopher-king” notion is so self-serving. Then there is that cave shtick. Most people know that philosophers can be a little klutzy in everyday life. We give them some slack and don’t make a big deal of it. But Plato says the wannabe king has been looking at ultimate reality and absolute truth (and maybe a pretty girl sunbathing?) in bright daylight, so he stumbles in the cave that passes for the real world among ordinary Joes. After his eyes adapt to the dim light, he will govern just fine. No way.
Feeling mellow enough to ignore Plato’s politics, I invited him over to watch a baseball game on TV. He was surprised that the pitcher threw a ball rather than a discus or a javelin, and that nobody was naked. But he is a smart guy and soon understood the duel between the pitcher and the batter. He noticed the (4 balls or 3 strikes) rule for ending an at-bat and said something about the ratio 4:3 in music by The Pythagoreans. Are they a band I don’t know about? He broke into a big grin when a batter sent a long fly ball arcing high above the field. Tho he knows zip about physics, he hangs out with Euclid and knows a parabola when he sees one.
To Plato, the path of the fly ball in the grungy everyday world is an imperfect realization of the timeless perfection of an ideal parabolic form. To me, the description of the path as a parabola is a good approximation that ignores air resistance and wind. Ignoring those things is OK in an introductory physics course. It is not OK in a baseball game.
Using the parabola to describe the fly ball oversimplifies a staggeringly complex everyday world that emerges from a staggeringly weird tarantella of elementary particles. Our use of the parabola is fundamentally a story we tell ourselves. Unlike the story of Pegasus the flying horse, it has been corrected, refined, and integrated with many other stories by scientific processes. The notion of a flying horse is appealing (to people who have not been hit by a bird splat). The parabolic story is ultimately more satisfying, as part of something gloriously predictive and useful (despite not being much help to the outfielder running to catch the fly ball).
Pegasus himself is as limited in time and space as the Pegasus story: an idea created by some people at some time and place, elaborated and spread by other people at other times and places. The Pegasus story will vanish and its starring horse will vanish with it, if we succeed in our current efforts to make the Earth uninhabitable long before we can go elsewhere. Would the parabolic story vanish also? That is a question for another time. The mathematical cast of characters in the parabolic story, on the other hand, is special. Very special.
Three plus two was five
before any mind could know.
Where do numbers live?