No, I think arithmetic and set theory would be safe. It just means there’s something wrong with the first four Euclidean postulates.
But no, I’m not taking this seriously, because if the claim is true it’s more than just proving some old theorem that no one was clever enough to prove before. It means that the foundations of geometry, and hence most of the foundations of mathematical physics, are at risk. It is possible for people to find problems in the foundations that have not been seen before, but I don’t think that it’s happened here.
Hang on, I don’t get this. (I’m not a mathematician.) You’rew saying that the postulate is assumed to be true, everyone belies it to be true, much is based on the assumption that it’s true - but if anyone ever actually *proves *it all other maths will be undermined? How can this be?
Because we’ve proven, using mathematics as we best understand it, that it’s impossible to prove the parallel postulate from the axioms of neutral geometry. Therefore, if someone were to subsequently prove the parallel postulate from the axioms of neutral geometry, the metatheory of geometry would contain as proven theorems a statement (“the parallel postulate cannot be proven”) and also its contradiction (“the parallel postulate can be proven”). That would make the metatheory of geometry inconsistent – and that would shake the foundations of all mathematical thought.
Hyperbolic geometry (specifically, the geometry of hyperbolic 3-manifolds) is my field of research. I can provide at least three complete descriptions of hyperbolic geometry, and demonstrate quite conclusively that hyperbolic geometry satisfies the first four of Euclid’s axioms but not the fifth. And there are plenty of other mathematicians who can do the same. I think I can safely say that this fellow Matta has no more “discredited” hyperbolic geometry than he has “discredited” the colour green.
I am never forget the day I first meet the great Lobachevsky.
In one word he told me secret of success in mathematics:
Plagiarize!
And, just so my post is not completely worthless
All *postulates * are *assumed * to be true. That is what a postulate means. In the case of Euclid’s Fifth Postulate, a lot of energy has gone into proving it from the first Four–because the first four are so simple and neat, and the Fifth is so unwieldly.
However, it has been proved that the Fifth is a true postulate, and so must be assumed. If that proof is false, a lot of other mathematics must also be false.
To put it more strongly (and to directly refute Giles’ assertion that set theory would be safe), we can construct consistent models of Euclidean geometry (and thus non-Euclidean) directly from the axioms of set theory.
Start with the assumption that there exists a consistent model of Zermelo-Fraenkel set theory. This has a natural numbers object (N). From here there’s a very standard construction of integers and then rational numbers. Then define real numbers as Cauchy-equivalent Cauchy sequences of rational numbers (Dedekind cuts have trouble with intuitionistic logic sometimes). Now, R[sup]n[/sup] provides a consistent model of Euclidean geometry as long as we started with a consistent model of set theory. That is, if Euclidean geometry is inconsistent it cannot have a consistent model, and the only way for this construction to fail is to not have a consistent model of set theory. Yes, technically the ZF axioms may be consistent, but there would be no models and thus it’d be a rather useless concept.
Yes, R[sup]n[/sup] with you usual interpretation of coordinate geometry provides a very satisfactory model or Euclidean geometry. So if Euclidean geometry fails, the R (the real numbers) fail, and so does set theory.
It is important to note, however, that while the Fifth is independent, some of the other postulates are not. One can, for instance, extend a straight line segment indefinitely using only the other three absolute postulates.
But this guy is a crank at best, or a con artist at worst. What he has proven is that bears use outhouses, water isn’t wet, Santa Clause is an evil martian bent on world domination, and that I’m the Pope. Seriously. All of those statements (and many, many more) are consequences of a proof of the Fifth Postulate.
I understand that an inconsistent system cannot have a model. I don’t understand what it means to have a consistent system with no models. Is that like a statement being true vacously?
Not quite. More like the statements don’t contradict each other, but there still aren’t any examples. Whether this is even possible is actually a philosophical matter. Some people take the position that any structure (search for my posts using the word “structuralism”) which is consistent has an instantiation (“model”) simply by virtue of being consistent.
Just imagine if the Fifth Postulate had been disproved!!! :eek:
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As we’ve covered, a proof also amounts to a disproof so really it wouldn’t be any worse.
As for effects on physics and stuff, of course physics wouldn’t be altered since it’s what really happens. Our description of it would be shot, though, since a disproof of the fifth postulate leads to an contradiction in set theory and thus brings down all of mathematics.
I can almost see how this could fail in spherical geometry, since things get a bit squirrelly when the triangle takes up over half the space, and maybe the definition of circumscribe is restrictive, but I really can’t see how a triangle in hyperbolic space can’t be circumscribed.
In Euclidean geometry, you find the circumcentre (the centre of the circumscribed circle) by constructing the perpendicular bisectors to each of the three sides of the triangle and seeing where they intersect.
In hyperbolic geometry, you can still construct the perpendicular bisectors to the three sides of the triangle…but those three lines are no longer guaranteed to intersect at all. Instead there are three cases:
(a) The three lines do intersect at a common point. In this case, the point of intersection is the circumcentre of the original triangle.
(b) The three lines are parallel, i.e. they “meet” at a point on the “circle at infinity”. In this case, the original triangle has no circumcircle. Instead, the three vertices of the triangle lie on something called a horocycle, which is a curve of constant curvature (like a circle) but of infinite length (unlike a circle).
(c) The three lines are “ultra-parallel”, i.e. they “meet” at a point outside the “circle at infinity”. In this case, the three lines have a common perpendicular: there is a unique fourth line L which the first three lines meet at right angles. The three vertices of the original triangle lie on a curve C which is at a constant distance from L along its entire length. C is not a line, circle, or horocycle, but it is a curve of constant curvature.
I can almost see how this could fail in spherical geometry, since things get a bit squirrelly when the triangle takes up over half the space, and maybe the definition of circumscribe is restrictive, but I really can’t see how a triangle in hyperbolic space can’t be circumscribed.