# The house of symbols and the tree of ideas.

Hey there, friend. New book out today. Best one yet. Well worth buying, oh, 70-80 copies.

Publication day always puts me in a philosophical mood. I’m only going to write so many of these things; each one has to count. And somehow that throws a stark light on my years, my days. I’m only going to live so many of these things. Each one has to count.

So let me tell you about the tree and the house.

When I lived in the U.K., I had an ongoing debate with my math colleagues. Or perhaps not a debate, but a dilemma, a question. *How do pencil-and-paper processes relate to the ideas they encode?* In particular, which should you learn first: the algorithmic *how*, or the conceptual *why*?

I was an ardent “why-first” guy. Algorithms follow from ideas, not vice versa.

I argued from experience. I’d met too many students who knew how to calculate, but not why their process worked, or what the calculations meant. I worried about math devolving into a sequence of meaningless rituals, and I suspected that once you can perform a ritual fluently, it becomes difficult to see it with fresh eyes and find the meaning behind it.

But my colleagues differed, and they too argued from experience. As students, they’d learned procedures first. Only later had those procedures taken on meanings. This learning pathway felt natural to them: you meet a problem-solving algorithm, learn its rhythm, and over time, you discover deeper significance within it.

To insist on the reverse, as I did, was like stopping a reading of Hamlet until every student had mastered every nuance of every speech. You’d never make it past the first page.

I was right, and they were wrong.

I was wrong, and they were right.

Truth is, there is no general answer to the question, no single way to learn math. Teaching is not about following the One True Path. It’s about helping students navigate the myriad, myriad paths (that’s 10^8 paths, if you’re counting).

It’s about the tree and the house.

Let me explain the metaphor.

No matter how you conceive of math–whether as the study of formal systems or as the exploration of deepest reality–there is something beyond our control, something we discover. There is a branching and interwoven splendor of ideas.

This is the tree.

But we struggle to access the tree directly. Formal systems can be accessed only via language and convention; Platonic abstractions, only via shadows on the cave wall. And so, around the tree of ideas, we have built a house of symbols.

The house consists of our notations, conventions, and definitions. In a word: our language. The house is the part of mathematics we invent. It’s the part that allows us to approach the tree.

The symbols “sin” and “cos” and θ? That’s the house.

Periodicity? That’s the tree.

The unit circle definitions of sine and cosine? Well, those are parts of the house, custom-built to access some of the sturdiest and most splendid branches of the tree…

… and this, in a nutshell, is the problem of learning math. To grasp the meaning of the symbols, you need access to a world that’s best accessed through the symbols themselves.

To see the tree, you need to enter the house; and to enter the house, you need to see the tree.

Some students spend a decade clambering around the exterior of the house, wondering what this painful exercise is supposed to teach.

Some students, even after a thousand hours in math class, never catch a glimpse the tree.

I spent half a decade trying to show the tree without the house. There is mathematics that you can access with no fancy algebra, no ornate definitions. I can bring twigs and leaves outside of the house to share. I can show you windows and openings from which you can glimpse the tree, even if you can’t open the door.

For my fourth book, I wanted to try something different. I found myself circling back to the earliest and most primal purpose of my career.

I wanted to bring people inside the symbols, so they could touch the ideas with their own hands.

This book took me two years to write. Along the way, I threw out perhaps two books’ worth of text. It’ll probably take me ten years before I know what to make of the result. Is this the best book I’ve written, or the most banal? The most original or the most obvious? Have I finally made school math intelligible, thereby attaining the holy grail of math popularization? Or is this book like a 19th mission sent to investigate the mysterious doom that befell the first 18 missions?

I have my hopes.

But I know there is no right way to learn math, no one book that will suddenly make a hard subject easy. I’m just grateful to be here in this house of symbols, admiring the tree of ideas, and I hope that each day, a few more people can join us here.