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Thinking in Shapes: A New Way to Learn Math (Part 1 of 3)

•5 min read

What if I told you that the way you've been taught math—starting with sets, numbers, and equations—might not be the best way? There's a new approach gaining traction among mathematicians called simplicial homotopy type theory (sHoTT), and it could make advanced math concepts accessible much earlier in your education.

The Problem with How We Learn Math Today

Traditional math education is built on something called set theory (specifically ZFC—Zermelo-Fraenkel with Choice). It's like learning to program by starting with assembly language. Technically correct, but painfully abstract and disconnected from how people actually think.

Here's the issue: set theory has:

  • No visual intuition — everything is just "collections of things"
  • No computational feedback — you can't "run" a proof to see if it works
  • No connection to geometry — shapes and spaces are built from scratch, awkwardly

"Teaching math with set theory first is like teaching someone to drive by starting with how combustion engines work at the molecular level."

Enter sHoTT: Math That Thinks Like You Do

Simplicial homotopy type theory flips the script. Instead of starting with abstract sets, you start with:

  • Types — think of these as "shapes of data"
  • Paths — ways to show two things are equivalent
  • Spaces — structures where things can continuously transform

Here's the mind-bending part: in sHoTT, equality is a path. When you say A = B, you're not making a static declaration—you're describing a journey from A to B.

javascript
// Traditional thinking: equality is a yes/no question
const traditional = (a === b); // true or false, that's it

// sHoTT thinking: equality is a PATH between things
const homotopyEquality = {
  from: 'A',
  to: 'B',
  path: 'the transformation that connects them',
  // And paths can have paths between THEM (higher structure!)
};

Why Should You Care?

If you've ever used Git for version control, you already understand the core intuition:

  • Branches are like different paths through a space
  • Merges are like finding connections between paths
  • Commit history is like tracking transformations over time
  • Merge conflicts reveal when two paths don't connect smoothly

Git is literally a mathematical structure called a groupoid—and sHoTT makes this kind of thinking foundational to all of math.

The Big Picture

This isn't just theoretical navel-gazing. sHoTT aligns with:

  1. 1How modern math actually works — researchers use these ideas daily
  2. 2How computers verify proofs — proof assistants like Lean and Agda speak this language
  3. 3How you already think — shapes, transformations, and connections are intuitive

In Part 2, we'll explore the actual "simplicial" part—what it means to build math from triangles and their higher-dimensional cousins. You'll see why this geometric foundation makes abstract concepts suddenly visible.

This is Part 1 of a 3-part series on simplicial homotopy type theory for beginners. Next up: building mathematical universes from triangles.