September 18, 2019 at 12:55 pm
Jeff Moden wrote:hjp wrote:You can never prove something true!
Now there's an interesting hypothesis... I wonder how one would prove that. 😀
Along the same lines as "a car can't fly". Of course it can. You only have to drive it over a cliff-top. The point is that it is not the supposed use. Neither is a "proof of truth" or my statement (untested hypothesis 😉 ) that one can't exist - but surely we enter fast the domain of physics and philosophy, and I am not trained there 😉 There are many paradoxes and dilemmas in science history, and this surely is one.
OK. Then in that same vein, would you not agree that, when applied to the length of the sides of a triangle, that the Pythagorean formula only works for Right Triangles?
--Jeff Moden
September 18, 2019 at 3:12 pm
hjp wrote:Jeff Moden wrote:hjp wrote:You can never prove something true!
Now there's an interesting hypothesis... I wonder how one would prove that. 😀
Along the same lines as "a car can't fly". Of course it can. You only have to drive it over a cliff-top. The point is that it is not the supposed use. Neither is a "proof of truth" or my statement (untested hypothesis 😉 ) that one can't exist - but surely we enter fast the domain of physics and philosophy, and I am not trained there 😉 There are many paradoxes and dilemmas in science history, and this surely is one.
OK. Then in that same vein, would you not agree that, when applied to the length of the sides of a triangle, that the Pythagorean formula only works for Right Triangles?
I think that one has to worry about what the "givens" are. Personally, as a mathematician who studied mathematical logic and semantics I tend to be very careful to apply even something as simple as Pythagoras' theorem only when I know that all the assumptions that are needed to make it valid are actually applicable to the cases I am concerned with. So I need more than a restriction to "Right Triangles" to allow the Pythagorean formula to be used.
The restriction to right Triangles is not adequate, the Pythagorean formula doesn't work for all Right Triangles. The formula works for right triangles in a Flat (i.e. uncurved) two dimensional surface. In a non-flat surface it is possible to have, for example, a right triangle in which all three sides have the same length - the surface of a sphere is one curved surface that allows such triangles easily. As well as the square on each side being only half the sum of the squares on the other two sides the sum of the angles of some such triangles (including the rather obvious ones on in a spherical surface where each side of the triangle has length equal to one quarter of a great circle in that surface) is one and a half times the sum of the angles of a flat Pythagorean triangle. It's also easy to produce a right triangle in a spherical surface where the square of the hypotenuse is less that a billionth of the sum of the squares on the other two sides.
Tom
September 18, 2019 at 4:10 pm
hjp wrote:You can never prove something true!
Now there's an interesting hypothesis... I wonder how one would prove that. 😀
I do get the spirit of that statement, though.
It is indeed an interesting hypothesis. If one could prove it, it would be false. Provided the proof was valid. Or maybe even if the proof wsn't valid. Or maybe it could be true, even if we could prove it - it depends on what our rules of proof are.
In mathematics, there are a lot of arguments over what the rules of proof are:-
Back in the late 19th century, there was a rather brilliant mathematician (Georg Cantor) who in effect invented set theory, and pissed off rather a lot of his fellow mathematicians who hated the idea of different levels of infinity. He used the interesting technique of proving that certan things were false - for example, that it was false to say that there were as many integers as the were real numbers; that is actually rather easy to prove true - but a lot of the mathematicians of the day hated the concept of different sorts of infinity, and assumed (based on their emotional reactions) that he had got it wrong. What he didn't do was use the excluded middle rule of classical logic to prove that things existed even if there was no imaginable way of constructing those things. But some (not many) mathematicians claimed he did. This started something of rift in the mathematical community. A couple of decades later David Hilbert (another brilliant mathematician) began a series of "proofs" of the existence of various things for which we had no imaginable way of constructing them, and that had a massive impact on mathematics. Many mathematicians believed that such a proof was just nonsense.
Most mathematicians now believe that there are two valid forms of mathemtics - constructive, where proof of existence by the rule of the excluded middle is absolutely unacceptable, and classical, where such proofs are acceptable. But when mathematicians are split into two groups with absolutely different rules about logical proof, how is one to prove that something is (or is not) true?
Actually, we are not split into two contrary factions. Most mathematicians now (or 52 years ago, when I was still a mathematician) see both versions of logic as useful, are very happy when both versions deliver the same result, and are interested in finding constructive proofs for things that currently have only classical proofs and in finding errors in classical proofs where constructive maths has found no proofs.
What we do know, as mathematicians, is that for a lot of stuff we don't know whether it is true or not. But we don't believe that you can never prove that anything is true.
Tom
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