The Geometry of Thinking

I have a friend who’s an artist, and he’s always giving me a hard time. He holds up a flower and says, “Look how beautiful this is,” and I agree. But then he says, “You, as a scientist, you take this all apart and it becomes a dull thing.” And I think he’s kind of nutty.

First of all, the beauty he sees is available to me and to everyone else, I believe. But I see more. I see the cells inside, the complicated actions of the molecules, the way the colors evolved to attract insects. It’s a fascinating system! The science doesn’t subtract; it adds. It adds the mystery of how it works.

And when you try to understand how things work—really understand them—you find out pretty quickly that words are slippery. They’re not good enough. If I tell you “the electron interacts with the photon,” you might picture a billiard ball hitting a marble. But that’s wrong. It’s not a collision; it’s a dance. It’s a probability. It’s a mess!

So, when I was trying to figure out Quantum Electrodynamics (QED), the math was getting out of hand. Pages and pages of integrals. You couldn’t see what was happening. You were lost in the algebra. So I started drawing pictures. Not to make it pretty, but to keep track of the score.

The Diagram: Just Keeping Score

People look at my diagrams now—these little arrows and squiggles—and they think, “Ah, that’s what an electron looks like!” No! It’s not a photograph. It’s a map of what could happen.

Here’s the trick: you draw a line. That’s a particle, say, an electron, moving through time. Then you draw a wavy line. That’s a photon, a piece of light. Where they meet—bloop!—that’s a vertex. That’s an interaction.

In ScienceClic’s explanation, they put it perfectly. The diagram isn’t the territory; it’s the calculation tool. Every line is a piece of the equation. Every vertex is a multiplication. You draw the picture, and you know exactly what math to do.

If you have an electron and a positron (that’s an anti-electron, going backwards in time—don’t panic, it’s just a direction!), and they smash together, they might annihilate and turn into a photon. Then that photon might turn back into a new pair. You draw the lines: arrow in, arrow out, squiggle in the middle.

And here’s the really wild part: nature doesn’t just pick one picture. It picks all of them. Quantum theory tells us that every possible way the particles could interact—the simple ways, the complicated ways where they do loop-de-loops—they all happen at once! We have to add them all up.

But the diagram lets us see the structure. It turns a nightmare of calculus into a geometry problem. You can see the “virtual particles” in the middle—those are the lines that start and end inside the diagram. We call them “mathematical intermediaries.” They’re like the money changing hands in a bank transfer; you don’t see the cash, you just see the balance change. They are the ghosts in the machine that make the force happen.

The Network: A Diagram for Thoughts

Now, look at what the computer people are doing today. They’re building these “Neural Networks” to make machines think. And if you look at a computational graph, what do you see?

It’s the same picture!

They draw circles (neurons) and connect them with lines (weights). Information flows from the left, travels along the lines, gets mixed up at the nodes, and comes out the right. It’s a computational graph. It’s a Feynman diagram for logic.

In physics, the lines carry energy and momentum. In these neural networks, the lines carry information and importance. A thick line means “this matters a lot.” A thin line means “ignore this.”

When the machine tries to recognize a cat, it takes the pixels—that’s the input particles—and sends them through this massive tangle of interactions. Each layer of the network is like a new set of vertices. The information interacts, combines, splits, and recombines.

And how does it learn? That’s the beautiful part. It uses something called Backpropagation. The mathematicians explain it, but think of it like this:

Imagine you run the diagram forward. You put in a picture of a cat, and the machine says “Dog.” Wrong! The machine looks at the answer and says, “Who messed up?”

It sends a signal backwards through the graph. It walks back along the lines, visiting every node, and says, “You! You shouted too loud. Quiet down.” “You! You didn’t say anything. Speak up!”

This is exactly what the “Chain Rule” is talking about. The “Chain Rule” sounds fancy, but it’s just tracing the blame back to the source. Because the whole thing is connected—because it’s a graph—you can calculate exactly how much each little line contributed to the mistake.

The Insight: The Geometry of Thinking

So why does this matter? Why am I excited about lines and squiggles?

Because we are finding out that thinking has a geometry.

We used to think of thoughts as these ghostly, floaty things. But recent thinking hits on something profound: thought isn’t the source of the river. It’s a “middleman.” It’s a process. It’s what happens between the input (the world) and the output (your action).

Just like the virtual particles in my diagrams are the “middlemen” of the electromagnetic force, your thoughts are the “middlemen” of consciousness. They are the hidden layers in the neural network of your brain.

When you have a “gut feeling,” that’s a forward pass through a graph you can’t see. When you learn from a mistake, that’s backpropagation rewiring your edges.

We are building a map of the invisible. In physics, we mapped the invisible forces that hold the universe together by drawing diagrams. In AI, we are mapping the invisible forces that hold intelligence together by drawing graphs.

Reflection

It turns out nature is pretty economical. It uses the same trick twice.

To make the universe work, it uses a web of interactions—particles exchanging information to decide where to move. To make a mind work, it uses a web of interactions—neurons exchanging signals to decide what to think.

If you can draw it, you can understand it. The math is just the bookkeeping. The reality is the connection. So, don’t be afraid of the pictures. The pictures are where the truth is hiding.