In a prior post, “Proving Euclid’s Fifth Postulate,” it was shown that the narrative that Euclid’s Parallel Postulate cannot be proven from neutral geometry is a myth.
The politicized nature of this narrative is evident from a blog post I ran across. It was in a political (recent posts discuss the 2020 U.S. presidential election and COVID-19 herd immunity) blog called “Good Math/Bad Math.” The entry is dated August 13, 2010, and it is entitled “Euclid? Moron!”
The post begins like this:
“A coworker of mine at Google sent me a link this morning to an interesting piece of crackpottery: a guy who calls himself “the Soldier of the Truth” who claims to have proved Euclid’s parallel postulate.”
After giving some background, the blogger continues:
“So, back to our crackpot friend. He claims to have assembled a suite of forty different proofs of Euclid’s fifth postulate. If this were true, it would make this weeks announcement of a proof that P != NP look like chicken feed. It would make Andrew Wiles’ work proving Fermat’s theorem look trivial in comparison.
Unfortunately, it’s a pile of foolish, shallow, amateurish rubbish.“
From the comments, it appears that the “crackpot” referenced in the blog is Rachid Matta MATTA, a Lebanese mathematician. (Matta’s website, “Soldier of the Truth,” does not seem to exist anymore.) Mr. Matta apparently discovered the post a couple years later in 2012 and made some comments. Matta asked for proof he was incorrect and never got a response. The last comment is by a German professor who admits he simply stopped talking to Matta rather than attempt to refute him.
The blogger (after admitting he’s weak at Geometry) dispenses with Matta’s work in the following way:
“So can I do it? Can I find a flaw in his infallible proof, in the only true foundation of the geometry?
Heh. Of course I can. Even someone as bad at geometry as me can find the flaw, right in the very first sentence.
The problem is right there, in the very first line: … in a plane surface. The first four axioms of Euclid don’t give you a plane surface. All of the reasoning in this proof relies on that little assumption: ‘in a plane surface’.“
So what was Matta’s error that deserved so much derision? The blog states it this way: “he assumes that the fifth axiom is true by restricting himself to a plane.” The blog recognizes that the issue concerns definitions. A commenter named John Armstrong stated: “The definitions come before the postulates, and can largely be dispensed with.“
Now that’s interesting. Apparently the math “experts” believe that Euclid’s definitions are irrelevant. But why?
The definitions in Euclid’s Elements are not irrelevant! What other document would be read that way?
It has been pointed out, for example, that in spherical geometry a “triangle” can have three right angles. (Start at the north pole, go due south to the equator, then turn and follow the equator a quarter of the way around, then turn and head back north to the pole.) But is a spherical “triangle” a Euclidean triangle? The answer is No!
Euclid differentiates between “line” and “straight line” in the Definitions 2 and 4. “Rectilineal” is defined to mean that the lines containing angles are straight. (D.9.) A “triangle” is a “trilateral figure,” which in turn is a “rectilineal figure.” Again, “rectilineal figures” must contain straight lines. (D.19-21.)
It is evident from these definitions that Euclid limited his geometry (at least in Book 1) to plane figures. Any three points connected by straight lines will always be in the same plane. If Euclid defined it that way, then it is simply incorrect to say, as the “Good Math/Bad Math” blogger and commenters did, that Euclid’s first four postulates don’t require a plane surface.
One of the few bits of constructive criticism I have received regarding my proof is that the side-side-side (SSS) method of finding congruent triangles doesn’t work. As discussed above, this is because SSS doesn’t work in Non-Euclidean Geometry (such as spherical geometry). Trying to use SSS would be a bit like attempting to construct a bicycle wheel (rim and spokes) using rope.
But of course, Euclid defines triangles to be rectilinear, which means straight lines. So one can use SSS in Euclidean geometry. (In other words, Euclidean Geometry uses spaghetti before it goes in the water.) In fact Euclid’s very first proposition–right after the definitions, postulate, and axioms–constructs an equilateral triangle from the two center points of two equal circles and one intersection of their two circumferences.
The proof the resulting triangle is equilateral derives from the fact that the same radius is used for all three sides of the triangle. In essence, Euclid begins his Elements by proving an equilateral triangle using the SSS method. Those who claim the SSS method is invalid in Euclidean Geometry are simply mistaken.
That being the case, what really is “Good Math/Bad Math’s” problem with Matta? The blog and comments do not uncover any inconsistency in Matta’s reasoning.
Perhaps they took issue with Matta’s conclusion that Non-Euclidean Geometry is invalid. It appears to me that the question whether Euclid’s Fifth Postulate can be proven really has little to do with the relationship between Euclidian and Non-Euclidean Geometry. It would be better to say that Euclidean Geometry is an abstract system that describes points and relationships based upon Cartesian system of straight lines, whereas Non-Euclidean Geometry is a system that describes points and relationships using curved lines.
But if that’s the case, then why all the politics? Why should it be necessary to demean someone and suppress his work over such a trivial distinction? How did the concept of abstract straight lines become controversial?
Perhaps people such as the blogger and commenters mentioned above want crooked lines… Consider this:
John the Baptist said: “Prepare ye the way of the Lord, make his paths straight!” Matt. 3:3.
The Apostle Paul said: “Will you not cease to pervert the straight ways of the Lord?” Acts 13:10.
The crooked lines are in their thinking!