What do you think about this proof that was submitted to viXra (something similar to arXiv)? It’s not supposed to be possible…what would be the implications if it were true (in mathematics and physics, in particular)? Do you think the proof is valid?
http://vixra.org/abs/1205.0003
The pdf of the supposed “proof” is on the page.
It’s not possible to prove the parallel postulate from the rest of Euclidean geometry. We know this because we can point to examples that satisfy all the other axioms but not the parallel postulate.
It can’t be proven because there are consistent geometrical systems (e.g., the geometry of the surface of a sphere where the great circles are straight lines) embedded in Euclidean geometry, and satisfying the other axioms of Euclidean geometry. If a triangle on the surface of a sphere must have angles summing to 180 degrees, then three-dimensional Euclidean geometry is inconsistent – indeed, ordinary arithmetic is inconsistent, because you can build 3-D Euclidean space from that – and very little mathematics is left to be consistent.
That’s what I was thinking too.
Me too.
I’m not going to say that everything posted on viXra is nonsense and bullshit, but that’s what I’ll bet on every single time. There’s a reason that these papers aren’t posted on arXiv.
Edit: I should’ve put this on my previous post, but what a lot of people don’t understand is exactly what a mathematician means when they say that something is impossible. It’s not that we don’t know to do it; it’s that we know exactly why it can’t be done. Unfortunately, the explanations are not usually lay-friendly, so it’s a little difficult to impress this on people.
Globally speaking, not all surfaces are planes. But all planes are surfaces.
Locally speaking, all surfaces with a local tangent plane approximation have are homeomorphic to a plane.
A sphere is not a plane globally; it is only a surface that is locally homeomorphic to a plane. Euclid’s fifth postulate, I think, is only meant to apply for surfaces that are planes globally, and not merely locally (i.e. “true” planes).
I am not comfortable with extending Euclid’s fifth postulate to surfaces for which it does not apply. Every conditional statement has its range of validity, and to say that it doesn’t apply outside that range does nothing to either bolster or discredit the conditional statement.
It is quite likely that many people, in my view not entirely without justification, felt that arXiv had a lot of nonsense in it when it first came out too. Peer review and endorsement have their purpose. It’s a question of the signal vs. noise ratio and quality control. If there were a process in which all contributions were “signal” and none of them were “noise,” I would support that process wholeheartedly. But I believe that there is a trend in the opposite direction; namely, that people who choose to submit to viXra, arXiv, or any peer reviewed journal for that matter, consist of self-selected groups.
That’s the key issue IMO. People say peer review is necessary, important, and I agree. I look at peer review as “artificial selection,” and the way viXra does things as “natural selection,” in that the former attempts to solve the problem of quality control with peer review whereas the latter attempts to do so by the principle that those who wish to contribute will “naturally select themselves.”
I just read it, and my conclusion is it’s too good to be true. It probably assumes something equivalent to the parallel postulate/fifth axiom. I didn’t see any blatant errors in logic, but it didn’t even address, let alone disprove, the proof of the impossibility of any proof of Euclid’s fifth postulate.
I agree with this point. It’s not like a proof of impossibility is like “it’s too hard and we’re saying it’s impossible because it’s been 2,000 years and we haven’t managed to find a proof.” I guess it’s like if it’s proven that the number of primes is infinite, it’s proven that it’s impossible to find a largest prime. It’s not because we’ve always found bigger primes than what we thought might have been the largest one that we just gave up and said “there’s no largest one…”
The proof was written up by an undergraduate student from UCLA apparently. And his last name is the same as mine…:dubious:
All that being said, however, I must admit I couldn’t find a logical or factual error in the proof, and I knew this stuff pretty well since middle school (olympiads, competitions, etc.)…
ignore this post
The very first line fails to hold in non-Euclidean geometry… It invokes proposition I.27 of Euclid’s Elements, whose proof is in turn based on proposition I.16, the first in Euclid’s Elements which fails in elliptic geometry.
(I thought, momentarily, that I had perhaps misread it and the first line was simply stating what was about to be proven, but that’s not so: The third paragraph invokes the “Because AB and CD are parallel to one another” provided by the first line)
There must also be some other part which fails even in hyperbolic geometry (which does validate I.27), and it should be fairly easy to spot by just going through line by line with a particular hyperbolic counterexample to the conclusion in mind, but I’m lazy right now.
The initial assumption is impossible for elliptic geometry, so it can’t rule out that non-Euclidean geometry; but it is consistent with hyperbolic geometry, so the result could still have nontrivial content (proving hyperbolic geometry impossible), if it weren’t wrong.
Right, this is the obvious way to localize the error. I’m too busy to look at it closely now, but my first thought is that there’s a confusion between a result holding for “a parallel” and holding for “all parallels,” the distinction only becoming relevant in hyperbolic geometry.
But can it be proven if we (as Euclid and apparently the author) tacitly assume geometry on a plane?
What this line of reasoning will eventually turn into is a definition, with a “true plane” being defined as one for which Euclid’s fifth holds. In which case Euclid’s fifth would be true but uninteresting. One might as well have an axiom that “all true triangles have exactly one right angle”, where “true triangle” is defined as triangles with exactly one right angle.
No, the parallel postulate can’t be proven even if you tacitly assume geometry on a plane. The standard notion of a ‘plane’ is already Euclidean, as Chronos has already noted.
You can have a sort of ‘hyperbolic’ plane, but lines won’t behave on it the way you would expect (which is why the usual assumptions of behavior on a plane are implicitly Euclidean).
I’ve gone through the proof line by line. It’s actually rather cleverly written, but the key flaw comes in the 2nd paragraph.
The crucial conclusion of the 2nd paragraph is only true for geometry on a Euclidean plane. It’s a fairly major assumption, as most of the rest of the conclusions follow from it. If the “world” is hyperbolic, the assumption isn’t necessarily true even on a plane.
He assumes that if two particular line segments were extended indefinitely, they would be equal to each other. While that would certainly be true in a Euclidean plane, it’s not universally true in a general geometric setting relying on just the first 4 postulates. Since that conclusion doesn’t necessarily hold in general, the rest of the proof falls apart.
Other notes I’d include if I were actually reviewing it:
Yes, but that’s not the problem. Euclid, and later geometers, tried to prove various theorems based on a set of axioms. He (and they) found that you need the parallel postulate as an axiom to prove other things that they found to be obviously true (e.g., that the sum of the angles of a triangle is 180 degrees) – or you need some other axiom that is equivalent to it. If your definition of a “plane” includes some equivalent to the parallel postulate, then you haven’t solved the problem.
The problem is not that Euclidean geometry (i.e., geometry with the parallel postulate) is inconsistent: it works fine. The problem is that it’s not the only geometrical system possible: you can define consistent geometrical systems where the parallel postulate is not true. And that includes systems embedded in Euclidean geometry, where the “lines” defined there are well-defined objects that satisfy the other axioms of Euclidean geometry, but are not the “lines” of Euclidean geometry.
So you need to take the parallel postulate (or an equivalent axiom) to define your Euclidean geometry. If you don’t, you might have a system where the angles of a triangle add up to less than 180 degrees, or more than 180 degrees.
Oops. I made an error in my previous analysis.
His actual statement in the 2nd paragraph works just fine.
But his final paragraph has a MAJOR leap of logic that just isn’t true at all.
Here it is:
(Bolding mine)
In absolute geometry (the geometry you can do with just the first 4 postulates), this statement is not supported.
In absolute geometry, the interior angle sum is at most two right angles. The existence of a single example of a triangle with interior angle sum of two right angles does not contradict the possibility of the existence of other triangles with interior angle sum less than 180 degrees.
It’s only if you assume some kind of Parallel postulate (0, exactly 1, > 1 parallels) that the interior angle sum can be used to define a geometry (Elliptic, Euclidean, Hyperbolic, respectively). But, of course, that simply leads to a tautology.
While this error is bad, it doesn’t necessarily prevent the possibility that Hyperbolic geometry is invalid while leaving Euclidean, Elliptic, and absolute geometry alone.
Fortunately, he made an earlier error with a similar generalization that isn’t backed up by the work he already did and is not necessarily true, unless one assumes a Euclidean framework.
The statement he makes is:
His set up is designed to prove this. But what he actually showed was the existence of a specific construction for two such new equal/parallel lines which joined the original parallel lines. The construction relied on a transversal.
The generalization is that such equal/parallel lines can be generated even without the conditions used for the construction (though he clearly didn’t understand he was doing this).
Even though it will work in a Euclidean world, that generalization fails if you don’t have the Parallel Postulate.
Euclid’s fifth postulate was not a postulate for planes at all. It was a postulate for geometry. Coordinates were unknown. He has five (or nine or ten, since there were other notions we would now call axioms) and wanted to know if this one was a consequence of the remaining ones. It isn’t and cannot be. Spherical geometry is one example, but slightly dubious (because one of the postulates that a line can be continued indefinitely and the statement is sufficiently ambiguous that it is not clear that a great circle on a sphere satisfies it), but a hyperbolic space certainly does. Therefore there’s no need to look at some quack paper and find the error. Incidentally, one of the earliest discoveries of non-euclidean geometry was by someone trying to find a contradiction if you assume not PP and finding a perfectly consistent geometry (hyperbolic).
I wouldn’t read angle trisections or cube duplications either because I have read and understood the impossibility proofs.
According to the Wikipedia article on the parallel postulate, it says that both of the following are equivalent statements to Euclid’s fifth postulate:
http://en.wikipedia.org/wiki/Parallel_postulate
With regard to the second point, are you saying that the construction used for that proof is actually a “special case” because of the assumption that a transversal was used? I was thinking that it might be “enough” that the author showed that the transversal does exist rigorously…?
Thanks.