[krick]

No, you don't use the proof of Pythagorean's Theorem every day. I guess you never did at all. You use some computational procedure which is developed along with that proof. For a layman it might seem there's no difference, but difference is huge. For Pythagoras himself and other people who used this simple procedure in that time it was more like physical quality of the real world, one of many procedures used to divide land on parcels (hence geometry, by the way). The "mathematical" part of his philosophy was more like religious practice with peculiar ideas, than what anybody calls "math" today.

The problem of "proof" didn't exit back then. What is proof? What looks like correct reasoning to me, that's what it is! And that was like a fine definition for every sane for a very, very long time. Have you read Kant's "Critique of Pure Reason"? It was way later that Pythagoras, Kant was obviously quite smart man and his reasoning was pretty solid for his time. Now it's nonsense. Because Euclidean geometry was considered the only possible back then, and Kant indeed believed it's the only possible. Then some 100 years later it just appeared out of nowhere. That's when a problem of "proof" and "formal reasoning" was actually started to be considered. And only really became a problem after Goëdel. The whole math you use every day by making cell-phones, building atomic reactors and launching rockets to the Moon was mostly invented before that by people who weren't so concerned by proofs, the central object of mathematics today.

Similarly, you use just some formula you was taught at one point, and some semi-intuitive rules by which you can derive other formulas from it and don't really care about proofs. And although you without doubt have heard about Lobachevsky and Riemann and even Klein, and heard of "Erlang" (even if it's just a name of programming language to you) you don't really think much about Pythagorean's Theorem's proof and what it's built on, you use some computational methods you've been taught and what seems like "correct reasoning" to you. Actually, you even reinforced that by your examples, because there's many more ways to calculate a "distance" inside matrices or in a video game (and on plain Earth it's more based on physical properties you care for and would be fine even if only "proved" empirically). Because "distance" is not derived from Pythagorean Theorem, distance is defined and is one of key properties of some geometry. That is, you define geometry by introducing formula of how distance is calculated.