I have a question that I have been wanting to ask for quite some time. I am currently a senior Computer Science student who will be graduating in a few months. I have taken all my required math courses (statistics, calculus 1-3, discrete mathematics, linear algebra, etc.) despite doing fairly well I feel like I lack any real understanding of mathematics. What I want to know is if there are any resources out there to learn math completely from the beginning starting at basic algebra and progressing into advanced topics. Right now I feel like I just know how to follow instructions and plug in formulas and numbers. I would like to eventually have a real understanding of exactly why I am doing what I am doing and how to use mathematics to solve future problems. Thanks for any advice!
Here's a possible ordered list of undergraduate-level books that could help you in your quest, but I have to warn you it will take some time. You're wanting to go all the way from understanding the 'why' to problem-solving and applications. That's quite a broad sweep of mathematics.
- Language, Proof, & Logic, by Barwise and Etchemendy. Gotta have logic to understand advanced math. I prefer the natural deduction system to any other symbolic logic proof system.
- Foundations of Analysis, by Edmund Landau. You can find free electronic copies of this. This book actually proves a lot of the basic properties of the number systems used in advanced mathematics.
- Axiomatic Set Theory, by Patrick Suppes. Set theory and functions undergird all of advanced mathematics.
- Abstract Algebra, by Israel Herstein.
- Linear Algebra, by W. Keith Nicholson.
- Essential Topology, by Martin Crossley.
- Principles of Mathematical Analysis, by Walter Rudin.
- Numerical Analysis, by Burden and Faires.
- How to Solve It, by George Polya.
- How to Solve It: Modern Heuristics, by Michalewicz and Fogel.