Rikai Funo · Track 4 · middle

Ode to Homotopy Type Theory

A foundation for all of mathematics built on the idea that equality is not a proposition but a path — a continuous deformation from one thing to another. Type theory meets homotopy theory; proofs become programs; objects become spaces. The Univalent Foundations program, championed by Vladimir Voevodsky, is rewriting what it means for mathematics to be true. An ode to the path between equals — and the space that path traces through.

Lyrics

The hum of the server rack in the winter dark.
Princeton, asleep outside the window.

Before you, it was just a line.
A equals B. A switch thrown.
A fact, brittle and clean.
Until the day you found the crack in your own paper.
A proof that held for years,
and then it didn't.
A foundation of sand.
And you swore, never again.

So you came to the machine.
To the terminal glow of Coq.
Teaching it what a point was.
Teaching it what a space was.
Building a world from the axioms up,
where every step had to be true.
Not just believed.
Checked.

You said equality is not a proposition.
You said equality is a path.
A continuous line from A to B.
And the proof is the walking.
The space between them is the reason.
You gave us a universe where things are not just the same,
they are connected.

And the Univalent Axiom...
...that if two things are the same shape,
they are the same thing.
Identity is equivalence.
Isomorphism is equality.
You built the library, UniMath,
with hands that knew the old ways were broken.
You drew a new map.

You said equality is not a proposition.
You said equality is a path.
A continuous line from A to B.
And the proof is the walking.
The space between them is the reason.
You gave us a universe where things are not just the same,
they are connected.

1966 to 2017.
Not enough time.
You found the error, and you started walking.
You showed us the path between.
The homotopy.
The type that holds the reason why.
The space traced by the proof itself.

The server hums on.
The cursor blinks on an empty line.
The path is still there.
Waiting.