Now that the night had stretched on and I was scrolling through my blog, I noticed a friend had left a private comment on the previous post, saying:
But aren’t all mathematical axioms unprovable? Then why say the parallel postulate can’t be proven?
Reductio ad absurdum, and other things
That comment struck me as a nice opportunity—especially for those interested in the history of science—to dig into something beautiful. So I figured I’d write a few clarifying notes here.
The parallel postulate… In ancient times, it was the final riddle—one that must have consumed Greek mathematics long before Euclid. — Hans Freudenthal
Around 300 BCE, Euclid gathered the mathematical heritage of those before him, added a few things of his own, and the result became one of the greatest masterpieces of mathematical history: The Elements.
From just five axioms, Euclid derived around 465 theorems—none of which were, strictly speaking, obvious. The significance of his work was in this: to build so much from so little. His method was solid and reliable, so much so that even great philosophers like Descartes and Kant borrowed from it.
For centuries, all of Euclid’s axioms were universally accepted—except one. The one that was shaky from the very beginning: the parallel postulate.
Why? Aren’t all axioms on equal footing?
Here are the first four postulates of Euclidean geometry, according to Wikipedia:
- A straight line segment can be drawn joining any two points.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are equal to one another.
And the fifth, the infamous parallel postulate, can be summarized like this:
- Two lines are parallel if they do not intersect—i.e., if no point lies on both.
So what makes this fifth axiom so different? Why did it keep mathematicians awake for centuries?
Maybe it feels obvious to us because we’ve been raised on Euclidean assumptions since grade school. But if you strip away those assumptions, the fifth postulate doesn’t feel nearly as intuitive as the first four.
The first two come from experience with rulers. The third, we’ve witnessed with compasses. The fourth might be more abstract, but even that one we can verify with a protractor.
But the fifth? We can’t test it experimentally. We can only draw line segments—never actual infinite lines. So what can we do? We try to define the postulate using other, more intuitive axioms. And for centuries, mathematicians tried to prove the fifth postulate using the first four—or at least something “more self-evident.” But every attempt hit a wall. Eventually, it would become clear that the argument had—somehow—sneaked in the parallel postulate again, indirectly.
(Feel free to search for these efforts—they’re wonderfully creative but always end up going in circles, landing right back on the postulate they aimed to eliminate.)
It seems even Euclid himself was aware of this issue—he avoided invoking the fifth postulate until Proposition 29 of The Elements.
In the 19th century, something revolutionary happened. The parallel postulate was reimagined—and in doing so, a new, astonishing world was discovered. A world in which the sum of a triangle’s angles could differ, rectangles didn’t exist, and parallel lines could converge or diverge. (There’s also an exciting story about how Gauss, Bolyai, and Lobachevsky all independently discovered hyperbolic geometry—if I’m ever in the mood, I’ll tell that story too, complete with philosophical digressions.)
This shift was so fundamental that it pushed science forward in a massive way. You probably already know: Einstein’s theory of relativity is built on the foundations of non-Euclidean geometry.
We learned that the world isn’t how we thought it was. That things are, in fact, different. Even though Euclidean geometry still describes part of the world, it’s not the whole story.
But what’s most interesting to me here is a certain strategy—a truly elegant one. The strategy of changing perspective, and it traces back to Euclid himself. This strategy is about thinking beyond all assumptions. When the usual methods fail to solve a problem, it asks us to throw away all the rules we’ve taken for granted, and to look at the problem with fresh eyes. Only then might the path forward appear.
The 19th century’s strange and brilliant progress owes a lot to this approach. Europe’s thousand-year night ended when a collective change of perspective occurred—about nature, religion, philosophy, and science. The prerequisite of this strategy, of course, is courage. The courage to abandon what we’ve accepted and re-examine everything. And if our beliefs really are true—well, we’ll come back to them.
To close, I’d like to quote one of the lights of the Enlightenment. In response to the question What is Enlightenment?, Kant wrote:
Enlightenment is mankind’s exit from its self-imposed immaturity. Immaturity is the inability to use one’s own understanding without guidance from another. This immaturity is self-imposed if its cause lies not in lack of understanding but in lack of resolve and courage to use it. The motto of Enlightenment is this: Dare to know!
And regarding the comment from that unseen friend: yes, axioms are accepted without proof. So, perhaps instead of writing “it can never be proven,” I should have said: “it can never be trusted to be proven.”