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Right triangle in non euclidean geometry8/24/2023 ![]() From antiquity all the way to the 19th century. Many people tried to “improve” on Euclid in this way. The parallel postulate should be a consequence of the core notions of geometry, not a separate assumption. Many felt that it should not be necessary to assume the parallel postulate as Euclid had done. So, people tried to prove the parallel postulate as a theorem. Very different from the other postulates, which are things like: any line segment can be spun around to make a circle. This particular type of configuration has such-and-such a particular property. It states that two lines will cross if a rather elaborate condition is met. It’s not a primordial intuition like the other postulates. The parallel postulate, by contrast, is not very simple at all. It makes sense that the bedrock axioms of geometry should be the simplest possible things, such as the existence of a line between any two points. The earlier postulates were very straightforward: there’s a line between any two points, stuff like that. Euclid’s postulate had rubbed a lot of people the wrong way even since antiquity. I’m referring to Euclid’s fifth postulate, the parallel postulate. It’s going to be hell to pay for this, as you can imagine. What had been thought to have been proofs were exposed as fallacies. If this armor is compromised, if proofs are fallible, then what’s left of mathematics? It would ruin everything.īut the nightmare came true in the 19th century. That’s what mathematical proofs are supposed to do: eliminate any risk of being wrong. There are thousands of theorems in Greek geometry, and every last one of them is correct. The mathematician has only one nightmare: to claim to have proved something that later turns out to be false. Or subscribe with your favorite app by using the address below
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