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; and that therefore, all of non-Euclidean geometry, and anything in the realms of math and science that in any way rely on non-Euclidean stuff, is therefore incorrect and must be discarded. This would include, among numerous other things, all of relativity.
Let’s start with a bit of background. Euclidean geometry is one of the earliest mathematical systems to be axiomatized. What that means is that it consists of a simple set of logical statements called axioms, and given those axioms, anything else within the system can be proven using those axioms and the basic rules of predicate logic.
Euclid had five axioms. The first four are very straightforward, and it’s clear that none of them can be proven using the others:
- Given any two points, there is one line segment that connects them.
- Given any line segment, you can extend it infinitely in either direction.
- Given a point and a radius, there is exactly one circle around that point with that radius.
- All right angles are equal to one another.
The fifth axiom is where things get problematic. It can be expressed in different ways, but ultimately, they all mean the same thing. In modern notation, what it says is: given a line L and a point P, there is exactly one line parallel to L which crosses through P.
(As an aside, there are actually more axioms; Euclid requires some basic arithmetic axioms, but since they’re numerical rather than geometric, they’re generally taken as a given.)
Euclid, and mathematicians for thousands of years after him, thought that there was something fishy about the fifth axiom. That is, it seemed like it shouldn’t be an axiom; it should be a rule provable by the other axioms.
The problem turned out to be one of definitions. If you take the axiomatic definitions of line, point, circle, and parallel from Euclid, they don’t actually mean what everyone thought they meant. That is, the definitions don’t say anything about the surface on which you’re doing geometry. They assume that it’s a flat plane. But that’s not part of the axioms!
You can take the first four axioms, and apply them to the surface of a sphere. If you do that, you get a system which works – only it has the property that parallel lines don’t exist! Or you can apply them to a hyperbolic surface — in which case given a line L and a point P, there are many lines parallel to L that cross through P. Or you can apply it to irregular surfaces, in which case the fifth axiom applies in some places, but not in others.
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.
Remember what I said just a couple of paragraphs ago? The four axioms were built on the assumption that they uniquely described planar geometry. But that’s not the case. They describe geometry, but they can describe not just geometry on a plane, but also geometry on a variety of other surfaces.
What our crackpot friend does is, very simply, assume that we’re talking about planar geometry, and uses the basic properties of geometry on a plane as implicit axioms in his proof. In other words, he assumes that the fifth axiom is true by restricting himself to a plane; then he uses it to prove that the fifth axiom is true on a plane; and from that, he concludes that the fifth axiom is universally true and that only planar geometry exists.
Let’s look at one example – his favorite of the proof, which he named after the village where he was born: the Theorem of Ehmej. About this proof he says:
I hope that the theorem of EHMEJ, my Birth-place, carries the Mathematical Community to reject the Non-Euclidean geometries and to recognize the theorem of EHMEJ as the only true foundation of the geometry. I am ready to defend it before any jury, and to prove that its deductions are infallible. Anyone not convinced must detect a flaw in the theorem of my Birth-place.
Now, I’m not particularly good at geometry. It is, by far, the area of math where I’m weakest. But I’ll take on this challenge from our Soldier of the Truth.
He’s using a more traditional statement of the fifth axiom. It’s usually stated constructively. So, roughly, take any two straight lines. Draw a third line which intersects both of the original two lines. Now, look at the angles at the point of intersection with the two lines. Either both angles will be right angles; or on one side of the intersections, the sum of the two angles will be smaller that the sum of two right angles. In the latter case, if you extend the lines infinitely, they will eventually intersect on the side where the sum of the angles was less than two right angles. (See why I like the modern version of it better?)
Ok. So – this proof of his works using this second form. He says that he’s going to prove that “a straight line cuts all the coplanar lines of different direction”. “Different direction” is just another way of saying the “two right angles” thing above: two lines go in the same direction if the sum of the angles made by any crossing line add up to the sum of two right angles.
So, here’s his proof.
By one given point B, outside of a given straight line (D) in a plane surface, let’s draw any straight line (S) that cuts (D) in A. Take the bundle (F) of all straight lines around A of which each forms a determined angle with (D). The straight line (F1) that superimposes on (D) forms an angle equal to 0⁰, and the straight line (Fn), that superimposes on (S), forms an angle α.
When A and (Fn) translate on (S), the straight line of (F) keep their respective angles with (S) constant, therefore their directions remain fixed, and consequently their angles with (D) remain constant. In sweeping the plane surface, only the positions occupied by (F1) don’t cut (D), while all the positions of the other straight lines of the bundle (F) cut it.
In particular, when A coincide with B, the straight line (D’), occupying the position of (F1), is the only straight line that does not cut (D).
We conclude: In the plane surface, by one given point, passes only one parallel to a given straight line.
It is what it was necessary to demonstrate.
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”.
Yes, in a plane surface, the fifth axiom does hold. But the first four axioms do not specify the plane surface – and in fact, they do hold in non-plane surfaces.
In fact, in real world situations, we do tons of geometry involving non-plane surfaces – that is, we do geometry where the fifth axiom does not hold. For example, when we calculate orbits of satellites, we’re doing geometry over an essentially spherical surface. Even map-making is thoroughly non-euclidean!