http://www.dailystar.com.lb/article.asp?article_ID=5486&categ_ID=1&edition_id=1
The above was also mentioned in Philosophy Now magazine.
This would be… huge… right? Like, as in not only math but what we know of Physics would have to be completely rethought, right?
Does anyone know where we can go to find the purported proof, or at least a summary of it?
Surely this can’t be right… can it?
-FrL-
The more I think about it, the more I think this just can’t be right.
Of course, I’m no mathematician.
But we are able to describe the geometry of, for example, spherical surfaces, right? And doing so is just a matter of replacing Euclid’s fifth with a suitable alternative, right? So if you were to disprove Euclid’s fifth, you would have to say it is impossible to develop a geometry for spherical surfaces, right? And wouldn’t that be absurd? (Or would it not?) Or at least, contrary to, I guess, empirical evidence, or something? (Since we actually do accurately describe the geometry of spherical surfaces, if I undertand correctly.)
-FrL-
These are the five axioms Euclid used in his geometric work. They can’t be proven in that they are assumed to be true for the purposes of the whole mathematical field of Euclidean geometry. There’s no way to prove an axiom from within the math based on that axiom.
The fifth one does, indeed, stick out like a sore thumb: It’s way too complex, and it seems like something you should be able to prove from the other four. (Euclid even limited himself to axia 1-4 when he worked on his absolute geometry.) But a lot of people have tried and failed to do just that. If this guy has done it, yes, it’s huge. It would probably alter a lot of mathematics. It would also be a huge ego-trip and lead to a hell of a lot of book sales. (Can you smell the skepticism?)
But wait, that’s not all. People have developed whole self-consistent geometries by negating the fifth postulate. If the fifth has been proven, where does that leave non-Euclidean geometry? It would still be valid, since it leads to self-consistent math, but it might be seen as a bit odd.
I don’t think Lobachevsky is in any danger.
You are correct. Though technically it’s three dimensional elliptical geometry not spherical geometry you’re citing. There is also hyperbolic geometry. In Euclidean geometry exactly one line can be drawn through a point parallel to a line no tincluding the point (one form of the parallel postulate). In ellipitcal geometry there is no line through a given point that doesn’t intersect a given line. In hyperbolic geometry, more than one line can be drawn.
Beltrami (in the 1800s) proved that these systems were as logically consistent as Euclidean geometry. I don’t see how these new proofs could eb valid unless Beltrami’s proofs are wrong.
The proof is false. I haven’t read it, but I can assure you that it’s false: there are at least two separate proofs, both well over a century old, that there can be no proof of Euclid’s parallel postulate.
Several mathematicians, including Eugenio Beltrami, Felix Klein, and Henri Poincaré, have produced projective models from hyperbolic space to Euclidean space that prove the equiconsistency of Euclidean and hyperbolic geometry. Since the only difference between Euclidean and hyperbolic geometry is whether or not the parallel postulate holds (it is true in Euclidean geometry, but false in hyperbolic geometry), this means that the truth of the parallel postulate is independent of the other geometric axioms – and thus can never be proved.
There is a small but steady trickle of people claiming to have new proofs of the parallel postulate; they join the elite company of thousands who have gone before, including such luminaries as Ptolemy, Naser Eddin al-Tusi, John Wallis, Girolamo Saccheri (who concluded that the parallel postulate must be true, for to embrace its negation led to results that were “repugnant to the nature of the straight line”), Alexis Claude Clairaut, Adrien Marie Legendre, Johann Heinrich Lambert, Farkas Bolyai (whose son, János, was the first to develop a systematic theory of hyperbolic geometry), and Charles Dodgson (also known as Lewis Carroll).
The difference between the aforementioned luminaries (excepting Dodgson) and Mr. Matta is that they had no way of knowing of the Beltrami-Klein or Poincaré models that prove the equiconsistency of hyperbolic geometry and the independence of the parallel postulate in Euclidean geometry. Apparently none of Mr. Matta’s hundreds of references included any text on geometry authored after 1900 (by which time everyone who mattered had come to be convinced of the correctness of the various proofs of the independence of the parallel postulate).
For an excellent treatise on the history and development of non-Euclidean geometry, see Euclidean and Non-Euclidean Geometries: Development and History, by Marvin Jay Greenberg (Freeman, 1993).
If non-euclidean geometries are provably self-consistent, doesn’t this make a proof of the fifth postulate impossible? Since if the fifth postulate were provable from the first four, then any postulate incompatible with it would provably inconsistent with the first four? And this in turn would mean non-euclidean geometries could not be self consistent?
-FrL-
Sorry, that last post was meant in response to Derleth, not those who posted after him.
-FrL-
A side note: the Daily Star article called Gauss a “French” mathematician. Anyone with any knowledge of the history of mathematics at all knows that Gauss was German. Does the Daily Star lack a research department?
I started out with a couple brief comments, but what I want to say depends a lot on whether its known that you can’t prove the negation of the fifth axiom from the first four. Has any research been done on that?
Never mind, I didn’t read KellyM’s post closely enough. Since the fifth postulate is known to be independent of the first four, there is no proof of it. It’s just like someone claiming to have squared the circle with a straightedge and compass.
There are several warning signs that this proof is bogus. Why would only a single newspaper have an article about this? If it were true, it would be publicized all over the place. Why would someone come up with 10 proofs? If there was even one correct proof, that would be sufficient. Why is this proof being announced without being peer-reviewed? The proof has apparently been submitted to math journals but hasn’t yet been accepted. It’s suspicious that this discoverer is an engineer and not a mathematician. This sounds like someone who’s vaguely mathematically talented but unfamiliar with how a mathematical proof works. It’s suspicious that this is a Lebanese newspaper praising a Lebanese man on work that’s not accepted anywhere else. It’s suspicious that this man is claiming that his proof has all sorts of philosophical implications. In general, mathematicians never claim that their theorems have any philosophical importance.
Wendell, I think you’re right.
I first saw this mentioned in the magazine Philosophy Now, and then looked it up on the internet. The Daily Star article was the only source I could find.
I am really suprised Philosophy Now even mentioned this, now that I look at the situation. Oh well. Some of their articles are a little off, too, at times, so there it is.
-FrL-
The following is taken from an article in the Math in the Media, a publication of the American Mathematical Society:
Basically, it seems that Matta’s proof is “The parallel postulate is true because God requires it.” Philosophically, this may have some appeal to some, but mathematically this is quite lacking (unless, of course, you go to Bob Jones University).
It can’t be proven from the other four and the other definitions and such. This itself has been proven. It’s also known that it doesn’t “obtain” (to use the philosophers’ term) since gravity makes the spacetime geometry noneuclidean.
Well, since Euclidean geometry is also consistent. It has models. They’re Euclidean spaces…
Frankly, I don’t see how it can philosophically appeal to anyone unless they want a particularly timid, inelegant God.
Given that you can model non-Euclidean geometries within Euclidean geometry (using different definnitions of straight lines, angles, etc.), if you can prove the parallel postulate, then you can also disprove it as well. That means that if the person has done what he has claimed to do, then he has proven that the first four postulates of Euclidean geometry lead to an inconsistency.
If that is true, then there is a big research project ahead to salvage what can be salvaged of geometry (Euclidean and non-Euclidean) to make what remains consistent. One would have to ask, what in the first fouir postulates needs to be changed or dropped to fix things?
(There was a similar project to salvage set theory early in the 20th century, when the Russell paradix was discovered. There, the solution was to limit what you ould say about sets of sets.)
However, given that geometry is such a good model of the physical world, this inconsistency has a pretty fundamental effect on all of mathematical physics too. How does this effect the Newtonian and the Einsteinian world view?
From the article :
Improvable? Anyone else have an image of Euclid’s teacher grading his paper with “C+ could do better.”
You aren’t taking this seriously, are you?
Look, you can build models of any geometry with an altered version of the fifth from a Euclidean one. You can build a model of Euclidean from foundations. Drop by Dunham Labs (in beautiful New Haven, CT) and I’ll whip it out in about an hour or two on the board.
What this means is that you can build a model for any geometry with the first four postulates and any variant of the fifth directly form set theory and algebra (using intuitionistic logic, even). A proof of the fifth postulate from the first four would, as you say, disprove itself as well. This would mean that no models are possible. This would mean that there’s something wrong with the foundations, and all of mathematics would be worthless.