My mathematics journey so far

While my interest in mathematics formed years ago, my commitment to learning the craft is fairly recent. Even so, I thought it’d be worthwhile to spell out how my relationship with mathematics has unfolded over time.

2014

Way back in 2014, I built an embarrassing math site. I was bad at math and at building software—websites included. But I had enthusiasm.

Here’s a screenshot of a totally unhinged post I made about our decimal number system.

Somewhat sadly, this enthusiasm would need to be tucked away for a number of years. I was twenty-two and needed to find a way to earn a living. I chose to double down on software.

2014 - 2023

Over the next decade, I focused on building up my skills as a software engineer. It was an arduous journey. There were meaningful highs—like when I got my first “real” engineering job and stared at the offer in disbelief—and lows that made me want to give up on software engineering altogether.

However, I stuck with it. And I’m glad I did. Learning to build software taught me that growing a skill is like growing a tree. It takes a long time, and no amount of rushing will cause the tree to grow faster.

It also gave me an intellectual foundation. I became comfortable dealing with abstractions. I learned to build systems. And it turns out the same principles that make software systems work also appear in mathematics. For example, in software, you want your abstractions to be loosely coupled. The same is true in mathematics. It might even be more important in mathematics. No one expects their software to run for hundreds of years. Mathematics, on the other hand, must stand the test of time.

2023

In 2023, I decided that mathematics belonged in my life. I started my journey by reviewing James Stewart’s Algebra and Trigonometry. The book contains content that you would find in a pre-calculus course.

The main point of digging through Stewart’s text was to re-acquaint myself with things I had forgotten. But I also took the opportunity to go a little deeper, particularly with trigonometry. In doing so, I learned a valuable lesson: going deeper doesn’t mean you’ll find more beauty. If you peel the cover off of trigonometry, the subject is quite ugly. Prior to calculus, our methods for computing trigonometric ratios were inelegant. It takes work to make a subject beautiful.

2023 was also the year I connected with my good friend and mentor, Matt. He has been my Virgil, helping guide me through the mathematical inferno.

2024

I think of proof writing as being the gateway to modern mathematics. It’s our best tool for demonstrating that what we believe to be true actually is. Proof writing is an expression of logic’s lofty promise. Logic is a truth preservation machine; it promises that if you start with valid premises and apply its principles, it will always give you truth in return.

In late 2024, I started reading through Daniel Velleman’s How to Prove it. Velleman’s book was my launchpad into proof writing. Velleman takes what is to most people an arcane subject and unpacks it in such a way that proof writing begins to feel like structured, logical poetry. I’m not sure that there exists an entry point to mathematics that would have been better for me than Velleman’s text.

In spite of my high praise for Velleman’s book, I haven’t finished it. But that’s a compliment in disguise. Getting just partway through the book equipped me well enough to go and start exploring other subjects.

End of 2025 - Today

In Unknown Quantity: A Real and Imaginary History of Algebra, John Derbyshire states the following.

The adjective “algebraic,” when used by mathematicians, can usually be translated as “concerned with polynomials.”

When I first read that statement, I was taken aback by its profundity. I believe most people would say that algebra is about manipulating symbols or balancing equations. Derbyshire’s statement was very much a mathematician’s statement, rich in perspective.

In a similar vein, I decided I wanted to be able to say—in my own words and as a mathematician would—what topology is about. I started reading Bert Mendelson’s Introduction to Topology.

Progress has felt slow—painstakingly slow. I hit a point in Mendelson’s text at which it became clear that my proof writing wasn’t up to snuff. I made all kinds of different errors. I would frequently make what software engineers call type errors. In practice, that means I’d make assumptions about mathematical objects that weren’t true, and it would shipwreck my proofs. I would also sometimes botch notation in strange ways, or invoke theorems incorrectly. Perhaps unsurprisingly, most of what I got wrong were things that a computer would catch for you.

Velleman taught me footwork and how to hold my hands by telling me, but Mendelson taught me footwork and how to hold my hands by hitting me in the face repeatedly. Nobody ever said the journey would be easy! Fortunately, the path to improvement is tried and true: practice and fast feedback.

Anyway, I set out to learn topology from Mendelson, but I have really been learning proof writing. I believe this suggests that as long as you’re moving in the right direction, the path you take to get there matters much less. The information you need will find you.

I’ll drop an update once I can talk more about topology.