Ask HN: How or where to begin learning mathematics from first principles?

Now, if he wanted only to get good at the software realm, I'd recommend a more superficial understanding of calculus, saving time to study discrete math and CS (that are mostly the same thing in different languages).

Anyway, I'd also recommend algebra to both specialities, to be learned at the same time as calculus and discrete math. It'll make the learning easier.

I'd really love feedback from you on what you found approachable and what you found unapproachable.

High School

Algebra I

Plane geometry (with emphasis on proofs)

Algebra II

Trigonometry

Solid geometry (if can get a course in it -- terrific for intuition and techniques in 3D)

College

Analytic geometry (conic sections)

Calculus I and II

Linear algebra

Linear algebra II, Halmos, Finite Dimensional Vector Spaces (baby version of Hilbert space theory)

Advanced calculus, e.g., baby Rudin, Principles of Mathematical Analysis -- nice treatment of Fourier series, good for signals in electronic engineering. The first chapters are about continuity, uniform continuity, and compactness which are the main tools used to prove the sufficient conditions for the Riemann integral to exist. At the end Rudin shows that the Riemann integral exists if and only if the function is continuous everywhere but on a set of measure zero. But what Rudin does there at the beginning with metric spaces is more general than he needs for the Riemann integral but is important later in more general treatments in analysis. Rudin does sequences and series because they are standard ways to define and work with some of the important special functions, especially the exponential and sine and cosine further on in the book. The material in the back on exterior algebra is for people interested in differential geometry, especially for relativity theory.

Ordinary differential equations, e.g., Coddington, a beautifully written book, Coddington and Levinson is much more advanced) -- now can do basic AC circuit theory like eating ice cream.

Advanced calculus from one or several more traditional books, e.g., the old MIT favorite Hildebrand, Advanced Calculus for Applications, Fleming, Functions of Several Variables, Buck, Advanced Calculus -- can now look at Maxwell's equations and understand at least the math. And can work with the gradient for steepest descent in the maximum likelihood approach to machine learning.

Maybe take a detour into differential geometry so that can see why Rudin, Fleming, etc. do exterior algebra, and why Halmos does multi-linear algebra, and then will have a start on general relativity.

Royden, Real Analysis. So will learn measure theory, crucial for good work in probability and stochastic processes, and get a start on functional analysis (vector spaces where each point is a function -- good way to see how to use some functions to approximate others). Also will learn about linear operators and, thus, get a solid foundation for linear systems in signal processing and more.

Rudin, Real and Complex Analysis, at least the first, real, half. Here will get a good start on the Fourier transform.

Breiman, Probability -- beautifully written, even fun to read. Measure theory based probability. If that is too big a step up in probability, then take a fast pass through some elementary treatment of probability and statistics and then get back to Breiman for the real stuff. Will finally see what the heck a random variable really is and cover the important cases of convergence and the important classic limit theorems. Will understand conditioning, the Radon-Nikodym theorem (von Neumann's proof is in Rudin, R&CA), conditioning, the Markov assumption, and martingales and the astounding martingale convergence theorem and the martingale inequality, the strongest in mathematics. So will see that with random variables, can look for independence, Markov dependence, and covariance dependence, and these forms of dependence, common in practice, can lead to approximation, estimation, etc.

Now will be able to understand EE treatments of second order stationary stochastic processes, digital filtering, power spectral estimation, etc.

Stochastic processes, e.g., Karatzas and Shreve. Brownian Motion and Stochastic Calculus. Now can get started on mathematical finance.

But there are many side trips available in numerical methods, linear programming, Lagrange multipliers (a surprisingly general technique), integer programming (a way to see the importance of P versus NP), mathematical statistics, partial differential equations, mathematical finance, etc.

For some ice cream, Luenberger, Optimization by Vector Space Methods or how to learn to love the Hahn-Banach theorem and use it to become rich, famous, and popular with girls!