Unlocking AI with Math: Update
The highest activity a human being can attain is learning for understanding, because to understand is to be free. — Spinoza
As promised in my last email, I want to briefly cover some of the math topics I’ve been exploring over the past few months and share why I’m diving into math at this stage.
Being in my forties, I’m not approaching this to become a professional mathematician. My goal is more pragmatic: I want to see how things work under the hood, especially in AI. There’s a lot of hype and noise out there, and understanding the math helps cut through it. It will let me assess claims more critically and grasp what’s really going on in research papers. Plus, AI is a big part of robotics, so studying math tackles both fields at once.
Why focus on “pure” math? These days, many books mix math with Python code and real-world applications, and that’s where I started as well. But after a while, I felt I needed to step back and understand the core ideas on a deeper level. I also wanted a more intuitive, yet rigorous, introduction to these concepts.
That’s where the trio of books by Seth Braver came in:
The Dark Art of Linear Algebra: An Intuitive Geometric Approach
Full Frontal Calculus: An Infinitesimal Approach
Precalculus Made Difficult
I was lucky to find these books because they were exactly what I needed at this point. Your experience may vary, of course, but if you’re self-studying and want an accessible introduction, I highly recommend them.
I started with linear algebra, then added calculus. Since calculus depends heavily on precalculus, especially trigonometry, I dove into precalc as well.
Here’s what I’ve covered so far by reading those three books:
Linear Algebra
Vector addition and scalar multiplication
Standard basis vectors
Vector length
Dot product
Vector basis and subspace
Parametric representations of subspaces and affine spaces
Linear maps, matrices, and matrix algebra
Gaussian elimination for solving linear systems, matrix inversion, image and kernel
Calculus
Basic ideas of differential calculus (infinitesimals, rates of change, the derivative, derivatives of polynomials)
Product, quotient, and chain rules
Concavity
Precalculus
Various topics to support calculus, like slope formulas and trigonometry
Seth Braver says these books are meant to be read slowly and carefully. I’ve definitely taken that to heart, maybe too much. I’m doing most of the exercises that have answers in the book. Understanding math, like programming, requires getting your hands dirty. It’s a “contact sport,” and working through exercises is the only way to truly understand it.
Recently, I discovered that the author has video lectures to accompany the books. I’m now revisiting the topics through his YouTube channel, which I also highly recommend after you’ve spent some time with the text and exercises.
Some key ideas that have stuck with me: in linear algebra, matrix-vector multiplication can be understood as mapping one vector to another via a linear transformation. In calculus, the idea that on an infinitesimal scale, curves become straight lines is both paradoxical and illuminating.
So far, it’s been a slow climb, but each step forward is a step closer to the freedom of understanding Spinoza spoke of.
Dig deeper. Until next time!
— Ruslan