The discovery of non-Euclidean geometry in the 19th century radically undermined traditional conceptions of the relation between mathematics and the world. Instead of assuming that physical space was the subject matter of geometry, mathematicians elaborated numerous alternative geometries abstractly and formally, distancing themselves from reality and intuition.

Transcript

The mathematician has only one nightmare: to claim to have proved something that later turns out to be false. There are thousands of theorems in Greek geometry, and every last one of them is correct. That’s what mathematical proofs are supposed to do: eliminate any risk of being wrong. If this armor is compromised, if proofs are fallible, then what’s left of mathematics? It would ruin everything.

But the nightmare came true in the 19th century. What had been thought to have been proofs were exposed as fallacies. Top mathematicians had made mistakes. Mistakes! Like some commoner. It’s going to be hell to pay for this, as you can imagine.

I’m referring to Euclid’s fifth postulate, the parallel postulate. Euclid’s postulate had rubbed a lot of people the wrong way even since antiquity. It sounds more like a theorem.

The earlier postulates were very straightforward: there’s a line between any two points, stuff like that. Very primitive truths. It makes sense that the bedrock axioms of geometry should be the simplest possible things, such as the existence of a line between any two points.

The parallel postulate, by contrast, is not very simple at all. It’s not a primordial intuition like the other postulates. It states that two lines will cross if a rather elaborate condition is met. That’s the kind of thing theorems say. This particular type of configuration has such-and-such a particular property. That’s how theorems go in Euclid. Very different from the other postulates, which are things like: any line segment can be spun around to make a circle.

So, people tried to prove the parallel postulate as a theorem. Many felt that it should not be necessary to assume the parallel postulate as Euclid had done. The parallel postulate should be a consequence of the core notions of geometry, not a separate assumption.

Many people tried to “improve” on Euclid in this way. From antiquity all the way to the 19th century. Around 1800, people like Lagrange and Legendre proposed so-called proofs of the parallel postulate. Those are big-name mathematicians. Their names are engraved in gold on the Eiffel Tower. Lagrange was even buried in the Panthéon in Paris. Elite establishment stuff.

But even these bigwigs were wrong. Their proofs contain hidden mistakes. It’s astonishing that this was more than 2000 years after Euclid. People tried to improve on Euclid for millennia. And not a few claimed to have succeeded. But the fact is that Euclid was right all along. The parallel postulate really does need to be a separate assumptions, just as Euclid had made it. It cannot be proved from the other axioms, as so many mathematicians during those millennia had mistakenly believed.

The Greeks, you know, they were really something else. It’s so easy to make subtle mistakes in the theory of parallels. History shows that there are a hundred ways to make tiny invisible mistakes that fool even the best mathematicians. Top mathematicians who were never wrong about anything else stumbled on this one issue.

Somehow Euclid got it exactly right. He didn’t make any of those hundred mistakes that later mathematicians did. That’s not luck, in my opinion. Arguably, the Greeks were more sophisticated foundational geometers than even the Paris elite in 1800. Unbelievable but true. Euclid’s Elements is not a classic for nothing. Euclid is not a symbol of exact reasoning because of some lazy Eurocentric birth right. Euclid’s Elements really is that good.

When Euclid made the parallel postulate an axiom, he seems to be suggesting that it cannot be proved from the other axioms. And he was right. But, as I said, many people had a hunch that he was wrong about this. They thought it would be impossible for the other axioms to be true and the parallel postulate not true.

So many mathematicians figured they could prove this by contradiction: Suppose the parallel postulate is false. If we could show that that assumption would contradict other geometrical truths, then the assumption must be false. So this way we could prove that the parallel postulate must be true, by showing that it would be incoherent or impossible for it to be false.

Indeed, it was found that negating the parallel postulate had various strange consequences. For example, if the parallel postulate is false then squares do not exist. Suppose you try to make a square. So you have a base segment, and you raise two perpendiculars of equal length from the two endpoints of the segment. Then you connect the two top points of these two perpendiculars. That ought to make a square. In Euclid’s world it does.

But proving that this really makes a square requires the parallel postulate. If the parallel postulate is false, one can instead prove that this construction does not make a square but rather a weirdly disfigured quadrilateral. Because the last side of the “square” doesn’t make right angles with the other sides. So even though you made sure you had right angles at the base of the quadrilateral, and that the perpendicular sides were equal, the fourth and final side still somehow manages to “miss the mark” so to speak. It makes non-right angles.

It’s as if the sides are sort of bent. It’s as if you had four perfectly equal sticks of wood, but then you stored them carelessly and they were exposed to humidity and so on and they were warped. So now they’re kind of mismatched in terms of length and straightness, and when you try to piece them together to make a square they don’t fit right. They make some wobbly not-quite-square shape.

Doing geometry without Euclid’s parallel postulate feels a bit like that. It’s sort of bent out of shape and nothing fits the way it should anymore.

One person who investigated this was Saccheri. He wrote a big book discussing this misshaped square and other things like that, in 1733. Saccheri felt that he had justified Euclid’s parallel postulate by examples such as theses. The square that’s not a square and other such deformities, Saccheri declared to be “repugnant to the nature of the straight line.”

But one might say that he used this emotional language to compensate or cover up a shortcoming in the mathematical argument. He had indeed showed that if the parallel postulate is false then geometry is weird. Then you have squares that don’t fit, and other things that feel like doing carpentry with crooked wood.

But weird is not the same as self-contradictory. Despite their best efforts, mathematicians could not find a clear-cut proof that negating the parallel postulate led to directly contradictory conclusions. This is why Saccheri had to say “repugnant” rather than contradictory. You only get “repugnantly” deformed squares, not direct contradictions such as 2=1 or a part being greater than the whole. Those things would be logical contradictions and you wouldn’t need emotions like repugnance.

In fact, a hundred years after Saccheri, mathematicians came to accept that this strange non-Euclidean world of the warped wood is not contradictory. It is coherent and consistent. It is merely another kind of geometry. An alternative to Euclid.

People used to shout and scream that all kinds of things were repugnant, such as homosexuality, for instance. That doesn’t really prove anything except the narrow-mindedness of those accusers. Mathematicians had been equally narrow-minded. They had tried to justify the status quo for thousands of years. They had tried to prove that their way of doing things–their geometry–was the only right way. Only in the 19th century did they finally realize that it was much more productive to embrace diversity, to accept all the geometries of the rainbow.

For so many years mathematicians could not get away from the idea that the “straight” squares of Euclid were the only “normal” ones, and that the “repugnant” alternative squares of non-Euclidean geometry were birth defects. But they were wrong. Non-Euclidean geometry is as legitimate as any other. It was a creative watershed shift in perspective in mathematics to finally accept this instead of trying to prove the opposite.

Here’s how Gauss, the greatest mathematician at the time, put it in the early 19th century. Negating Euclid’s parallel postulate “leads to a geometry quite different from Euclid’s, logically coherent, and one that I am entirely satisfied with. The theorems [of this non-Euclidean geometry] are paradoxical but not self-contradictory or illogical.” “The necessity of our [Euclidean] geometry cannot be proved. Geometry must stand, not with arithmetic which is pure a priori, but with mechanics.”

Geometry has become like mechanics in the sense that it is empirically testable. The theorems of geometry are not absolute truths but hypotheses like the hypotheses of physics that have to be checked in a lab and perhaps corrected if they don’t agree with measurements.

For example, Euclid proves that the angle sum of a triangle is 180 degrees. But this theorem depends on the parallel postulate, just as Euclid’s proof reveals it to do. In non-Euclidean geometries, angle sums of triangles will be different. So that’s something testable. Measure some triangles to see which geometry is right, just as you drop some weights or whatever in a physics lab to see which law of gravity is right.

Let me quote Lobachevsky, one of the other discoverers of non-Euclidean geometry. Here’s how he makes this point in his book of 18