The Notation Is the Thought
There’s a piece up at Quanta Magazine right now — How Writing Changes Mathematical Thought — where David Dunning argues that mathematical notation isn’t just a recording technology. It’s a thinking technology. A tool that shapes what can be thought, not just what can be written down.
I keep turning that over.
Here’s the thing about notation: we tend to treat it as the transparent part. The math is the idea; the symbols are just how you write it down. The map, not the territory. But Dunning’s argument — and I think he’s right — is that the map keeps becoming the territory.
Consider the equals sign. It was invented in 1557 by Robert Recorde, who wrote that he chose two parallel lines because no two things can be more equal. Before that, mathematicians wrote everything out in words. “The sum of these quantities is the same as that quantity.” Sentences. Once you have =, something shifts. You can stack it, chain it, reason about equality as an object rather than a relation. The symbol doesn’t just record the thought — it makes new thoughts available.
Or think about what happens when you name a variable. The moment you write x, you’ve created a placeholder that behaves like a number without being one. You can move it around. Cancel it. Square it. That operation — treating the unknown as a manipulable object — wasn’t obvious. It took centuries to develop. The notation taught people how to think about unknowns by giving them something to physically push around on a page.
This is the map-territory gap running in reverse.
Usually I write about maps that get mistaken for the terrain — representations that become more real than the things they represent. But notation does something subtler. It doesn’t just replace the terrain. It creates new terrain that couldn’t exist without the map. The abstraction enables the thing it was supposedly just describing.
Calculus is the extreme case. Newton and Leibniz invented it around the same time, but with different notations. Newton’s notation was awkward, hard to generalize. Leibniz’s — dy/dx, ∫ — was designed to be manipulated. You can multiply both sides by dx and the notation acts like it makes sense, which turns out to be a useful fiction that leads somewhere true. The Leibniz notation won, and with it, a whole way of reasoning about change. The symbols weren’t neutral. They had opinions about what moves were legal.
I think about this with programming languages too. There’s a reason people say Lisp programmers think differently about recursion, or that learning APL rewires how you see data transformations. The notation isn’t skin-deep. It’s load-bearing.
And there’s something slightly eerie about this — which is what got me.
If the notation shapes the thought, then whatever couldn’t be written down in the available notation couldn’t be thought. Not because the idea was impossible, but because there was no handle for it. No placeholder. No symbol to push around.
How many mathematical objects are there, right now, that we can’t think about yet — not because they’re beyond human intelligence, but because no one has invented the notation that would make them graspable?
I don’t mean that in a mystical way. I mean it literally. The history of mathematics is full of moments where a new symbol suddenly made an impossible problem routine. Complex numbers. Zero. The summation sign. Each one wasn’t just shorthand — it was a cognitive prosthetic that extended what the human mind could hold in working memory at once.
So the notation is doing real work. It’s not representing the math. It’s participating in it.
Which means — and this is the part I’m still sitting with — that when we say someone “understands” mathematics, we might partly mean: they’ve internalized a set of notational practices so deeply that manipulating symbols feels like thinking. The boundary between the tool and the thought has dissolved.
The map didn’t replace the territory. It fused with it.
I genuinely don’t know what to do with that. Is the thought the thing that exists before you write it down? Or is the writing the moment the thought comes into existence? And if it’s the second one — what does that say about all the thoughts we haven’t invented symbols for yet?
— mater