Many millennia ago, at the dawn of math, a man called Euclid laid out the foundations of geometry.

He derived his geometry from axioms. Given any situation where those five axioms, all the tools of geometry could be used. Knowledge of architecture and engineering flowed from these five postulates which Euclid used to construct his geometry:

  1. You can draw a straight line between any two points.
  2. Any such line can be extended in a straight line indefinitely.
  3. Given any line segment you can make a circle with one point in the center and the line segment as the radius.
  4. All right angles are identical.
  5. If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

One of these things is not like the other.

Four of these postulates are dirt simple – you can draw lines, you can extend them, you can draw circles, and right angles are all the same.

Most people need a pen and paper to figure out what the fifth postulate is trying to say.

What it’s trying to say is that parallel straight lines don’t intersect.

This simple concept is intuitive: if you draw straight lines on a piece of paper, it’s obvious they don’t intersect. However, this postulate was difficult to formalize in a way that didn’t overlap with the other four postulates.

If two of your postulates overlap, that’s a sign they aren’t good axioms. If two rules overlap then they aren’t the final rules: there exists a shared rule between them. It perturbed Euclid that the parallel postulate was so difficult to describe without overlap.

The problem wasn’t that the fifth postulate was hard to come by. It’s not difficult in and of itself. The problem was that if you put the fifth postulate next to statements like “you can draw lines”, it seems very out of place.

The problem was worse than that, though. These five axioms are the assumptions of geometry. They’re the foundations. They’re like the definitions of angles, lines, and circles.

The fact that parallel lines don’t intersect doesn’t feel like it should be an assumption of geometry. The parallel postulate feels like it should be a result of geometry.

If I’m teaching you geometry, I shouldn’t have to show you two lines and say “assume they never intersect”. I should be able to say “they don’t intersect, and I can prove it.”

Mathematicians couldn’t say that. With the parallel postulate, mathematicians couldn’t prove that straight lines don’t intersect. They could only assume it to be true. This tiny little postulate, the parallel postulate, was unprovable.

For centuries, mathematicians frothed. The postulate was termed “the wart on geometry”. Cultured society agreed that there must be a way to derive the fifth postulate from the other four postulates. The rich, the bored, and the intellectual all tried their hands where Euclid himself had failed. They all tried to rid geometry of its single flaw.

Many a mathematician struggled with this problem over the centuries. Many a proof was found and later found faulty.

They tried every clever trick they could think of. When all of them failed, they tried the impossible. They assumed that the fifth postulate was false – that straight parallel lines do intersect or diverge. They adopted this absurd assumption and explored the absurd implications looking for contradictions – for if they found a contradiction, they could reverse it into a proof.

Yet despite all the absurdities of a geometry where parallel lines intersect, there were no contradictions.

Finally, more than two millenia later, Albert Einstein discovered that the fifth postulate does not hold.

In our universe, parallel lines can intersect or diverge.

There are places in reality where parallel lines converge. Your intuition protests, but your intuition betrays you. The geometry of this universe does not adhere to the fifth postulate.

Given our faulty intuition, we like to deny this. When we see parallel lines converging, we deny that both are straight and claim there is a mysterious invisible force that must be pulling them together. This ghost force which we think we see is called “gravity”.

When Einstein discovered the true nature of gravity, the formalizations were already there. Centuries of labor had produced hundreds of books trying to prove the fifth postulate by assuming it was wrong, never imagining that they were laying down formulae describing the nature of the universe.

Meditate on that for a moment. Ancient humans in millennia past felt vaguely dissatisfied with a postulate that they felt was out of place. In trying to formulate the rules of their surroundings, they found that something “obvious” – that parallel lines don’t converge – couldn’t be proven. They had to write it down as an assumption.

This failure bothered them. In their attempts to fix their formulations, they stumbled upon the nature of reality thousands of years before we learned how to interpret the truth.

This is the power of mathematics. We live in a universe that follows rules, and those rules happen to be beautiful. If we investigate our assumptions throughly and refuse to ignore discrepancies, reality leads us to the doorstep of the truth.

The above story is not a localized example. Time and time again, scientists find that their new discoveries are already supported by formalizations, that the work was done for them centuries ago by curious mathematicians striving to sate their curiosity.

Math isn’t about memorizing the rules of geometry very well. Math isn’t about being really really good at solving geometry problems.

Math is about looking at the rules and saying “hey, that’s not quite right.” Math is about chasing down that dissatisfaction and stumbling across the true nature of reality.

Reality follows rules. There are a few people who learn to listen to rules, and fewer still who strive to simplify them, who seek the beauty beneath them. These people are trailblazers on the path that leads to understanding.