About

What I’m trying to do

Mathematical Structures is a planned series of books intended as a foundation for both becoming a mathematician and using mathematics seriously in any field where it matters. My three goals for any reader are coequal: to see the beauty of mathematics, to understand what mathematics actually is, and to become genuinely competent at it.

The current commitment is at least four books. The first, Mathematical Structures I, orients the reader on what it takes to see and do mathematics well, and provides the structural background the rest of the series assumes. Three further volumes follow from that foundation: one on linear algebra and geometric structure, one on calculus and analysis, and one on probability. Each is written with the assumption that the foundation is already in place — and each tries to show what its subject actually looks like when it isn’t compressed to fit a single semester.

Why I’m doing this

The right framings of these subjects already exist, scattered across the literature. What doesn’t exist — at least not as a single coherent body of work — is an attempt to pull them together. Mathematical Structures is that attempt.

The series is category-native: it shows how to learn the major mathematical subjects from a categorical worldview, which makes structures explicit and forces the right questions to surface early. It also takes Clifford and geometric algebra seriously as the right foundation for a wide range of things — calculus, complex analysis, and a lot of the mathematical structure that physics and engineering actually use. The goal is to build all of this up with full context first, so that when the reader encounters a major result, they have the apparatus to see why it’s there and why it looks the way it does.

Most mathematics curricula compress to fit the term. They simplify early courses to reduce friction, then quietly pay for the simplification later when the foundations have to be retrofitted. This series tries not to make that trade.

How I think about math

Mathematics is, among other things, among the most beautiful things a person can encounter. Most people never see it — not because they’re incapable, but because no one teaches them to see. The barrier is real. It is also lowerable.

The analogy I keep coming back to: imagine being deaf, and never having heard music. Then imagine going from that to hearing Bach for the first time. That’s the order-of-magnitude shift available to someone who finally encounters mathematics done properly. A lot of what I’m doing is trying to lower the barrier to that first encounter, and to incentivize people early enough that they want to keep going.

The method underneath all of this is straightforward to state. If you see something from the right angle, and you sequence things in the right order, things become obvious and transparent. Start from a small number of axioms; apply logic to build up. When you encounter something complicated out in the mathematical world, go back to where you started, understand the ground you’re standing on, and work your way up to it. Done well, the complicated thing becomes less complicated — or it remains complicated, but it becomes clear.

So the operating commitment is: always try to give the reader the right perspective. And alongside the mathematics, teach the metamathematics — how to think about it, how to problem-solve, what makes a problem interesting in the first place, what the right strategies are for asking questions and for finding solutions. The texts try to share that perspective explicitly, as the work happens, rather than treating it as something the reader should pick up by osmosis.

Two corollaries.

Use the right tools for the right problems. You don’t use a screwdriver to hammer nails. You also don’t use legacy vector algebra when Clifford and geometric algebra have been available for nearly two centuries and are the right tool for an enormous fraction of the relevant problems. I’d rather the reader learn the right tools from the start than have to unlearn the wrong ones later — the way physicists do when they first encounter spinors and discover that the language they’ve been using doesn’t quite fit.

Pages in service of clarity, not in service of elegance. I’m sympathetic to the rising-sea posture associated with Grothendieck. Where I’m aimed is the polar opposite of the terse, clever, do-the-work-yourself tradition exemplified by Serre or Rudin. Math is hard enough already; I don’t need to make it harder by hiding things, showing half-solutions, or compressing the exposition to the point where the reader has to reconstruct what’s missing. This is also why I refuse to confuse small calculations for what mathematics consists of. Computation is part of the work; it isn’t the work.

Intended audience

The intended reader is anyone who actually wants to learn this material. That includes undergraduates, motivated high school students who are ready for something serious, researchers from adjacent fields (physics, engineering, computer science) who need to learn or relearn parts of the mathematics under the hood, and self-learners outside any formal program.

The books are written to be readable on their own. The expectation is that a sufficiently motivated reader can work through them without a course, a teacher, or assignments. They are not designed as quick references or supplements to a standard sequence — they are designed to be the sequence, structured so that the reader can do the work themselves.

Will this work?

This is an ambitious project, and the realistic odds that I write all four planned books are not high. Life is busy, and the kind of writing this requires is slow.

What’s posted here is what I’m currently trying to do. The honest vision goes wider — I’d like to write further texts that build on this foundation, in physics, engineering, and more advanced mathematics. Whether any of that happens depends on how far the first book gets, and on whether producing it turns out to be sustainable at the scale required.

For now: see if I finish the first book.

Who I am

I’m Kyle Taljan. PhD in mathematics from Case Western Reserve University, where I trained as a probabilist — though probability isn’t where my work has gone since. I currently work as a data scientist.