Mathematics for the Adventurous Self-Learner

I'm pretty skeptical about these "best of" lists of books for self-directed mathematics education.

I have my own "best of" list that is very different to this list, although there are a couple of crossovers.

If you are fortunate enough to have access to a university library (or libraries) I would _highly_ recommend inquiring about access to their general collection. I was also fortunate enough to study mathematics to a university-level three-year degree at a research university. So I had an excellent head start.

A HUGE part of my journey of collecting my "perfect library" of mathematics self-tuition and reference books (and course books) was to do my own research on collecting the perfect titles. I started when I was in the early days of my mathematics degree and I used resources like Amazon, Usenet, libraries (already mentioned), and ... that was about it.

Another important question to ask yourself is the following:

"Why am I doing this?"

Life is short and by the time you hit middle age, if you have a family or bills to looks after, are you REALLY going to want to lock yourself away in your study room to learn Lebesgue integration instead of focusing on the rest of your life?

Consider that people fail to emphasise is that mathematics is a social activity much more than many people realize.

Exercise: Find the topics of mathematics that are important to your goals and are missing from the list and find your favorite books or two that cover/s these topics.

Exercise: Consider whether your interest in (self-directed) mathematics is so sincere such that you have a serious application in mind, that you might be better off enroling in a course? Even if it's a night course that last a couple of years, you will meet a LOT of people who can help in ways that are immensely more productive than trying to do this all by yourself.

I recently purchased volume 1 of my favorite calculus and analysis book. It's an incredible masterpiece. The coverage of topics is much broader and more interesting than Aposotol or Spivak. The latter books are both very good but they also have myopic, one-track pedagogical approaches and limited themes in their coverage.

Exercise: Find your own favorite introductory calculus book that is suitable for the motivated student.

Initially I heard about Euler's famous Basel problem. Years later I got to solving it for my self (for curiosity and fun). I guess what intrigued me was to think of trigonometric sine as an infinite polynomial...After I worked it out, I had indeed seen the fire in Euler's own eyes...I could see how excited he was at having discovered something amazing...But this got me into hooked into math history. What I really wanted was how people came about discovering the Taylor's series...the intuition behind it. So that is how I came across John Stillwell's book. I have to warn people it is rather academic. But if, you, as a self-learner, is excited about mathematics, I would suggest Norman J Wildberger's youtube lectures on mathematics history. I find the buildup to calculus quite fascinating. J. Stillwell's book was the recommended reference in those lectures...

There are a lot of charlatans in the Data Science space who lack the necessary mathematical background for their roles. For me it was necessary to get a rigorous understanding of probability theory, and applied probability theory is basically what mathematical statistics is about. My background was CS and software so R, Python, data visualisation and ML operationalization is by and large the easy part of Data Science to me. If you pick up any book like Bishop's or ESL you will be extremely frustrated if your mathematical background is not there. I didn't feel comfortable creating predictive models for production use that I didn't completely understand what was going on "under the hood", the assumptions being made and how they could fail. It's the only ethical thing for any engineer to do.

As someone who dropped out of highscool after 10th grade and never went to university/college one great way I've found for learning mathematics without any foundational basis is trying to learn CG/3D programming.

I always felt like maths was too abstract to keep me engaged, but when the output of your work is immediately observable visually it becomes a lot more engaging. There's just something so much more satisfying being able to "see" the results.

Plus as a self-taught programmer, I find it much easier to learn front-to-back by deciding on a desired outcome and working towards it, rather than progressively building up abstract fundamental skills that can later be combined to achieve a desired outcome (which is essentially the traditional academia path for learning STEM fields)

Just the book for you:

"Who is Fourier: a mathematical adventure" https://www.amazon.com/Who-Fourier-Mathematical-Transnationa...

It is a simply brilliant book that takes you from basic trigonometry, logarithms and so on through calculus and finally fourier series.