I am a working programmer who, despite learning basic cs, calculus and linear algebra in high school, was actually trained as a philologist[1]. After receiving my B.A. my path to employment as a 'Software Engineer' involved reading SICP and taking an online web development course about react. I ended up working with clojure for while which was great fun.

It has been 3 years now and I feel that I have become decent at coding, yet my education in math and computer science is sorely lacking. For example, I am working on a semantic search project and I am reading about cross-encoders, and while I understand the problem statement and the solution and could operate the model based on the documentation, I have no idea how it works. Once I understand how to apply the tool, the programs become easy to write and I can resolve errors by directly referencing documentation, either of the language or the library. The last time I wrote python was major version 2 as a 16 year old, but I'm learning python 3 10 years later from the reference manual. This is what I mean by 'decent at coding' but lacking in math/cs background. Put frankly:

- I have gotten to the point where the foundations of the techniques to solve the problems I am working on are beyond my educational level, but I am a decent enough engineer to identify where the technique could be applied and how to implement it.

- I want to learn the math and cs to work on scientific computing. I want to investigate dynamical systems with computers. I think my potential lies not in specializing in a specific natural science (such as geophysics), but rather becoming a skilled implementer.

Is the goal of transitioning to working on scientific or research software as an implementer for someone with little formal education a reasonable one? If it's not, this will still be my hobby. How can I remediate/fill in the gaps of my math education?

- Book list for math: currently reading "A first course in mathematical analysis" by Burkhill for calculus (to refresh and extend high school), and then after that: my college linear algebra book (done wrong), Spivak's "Calculus on Manifolds", and "Differential Equations with Applications and Historical Notes".

- CS/Software Engineering: I gravitate towards functional programming and mathematical logic, so resources like the 'Software Foundations'[2] series are appealing. However my interest in numerical methods and scientific computing tells me that reading a book such as 'Computer Systems: A Programmer's Perspective' and then moving on to an applied numerical analysis book and OpenGL book would be a greater 'return on value' in the short term. There is also the need to continue my professional development as an engineer, and for this I am looking at books like 'Designing Data-Intensive Applications' and 'Software Design for Flexibility'.

How do I balance my interest in foundations with my interest in scientific/numerical computing? How do I balance my math/cs study with my professional development regarding engineering?

TL;DR;

I feel like I have become an 'intermediate' autodidact in the sense that I feel that I have overcome the dunning-kruger effect, I feel like an idiot[3] yet am a working programmer being paid an industry standard wage to work on designing software systems based on mathematics and computer science I do not fully understand, and I am looking for advice on how to actually start mastering the craft with an eye towards both foundations and scientific computing/numerical analysis implementation. I want to continue working part-time and study part-time on my own rather than go to school.

[1] https://www.gutenberg.org/ebooks/18267

[2] https://softwarefoundations.cis.upenn.edu/

[3] https://grugbrain.dev/