* The Art of Probability by Hamming. An opinionated, slightly quirky text on probability. Unlike the text used in my university course its explanations were clear and rigourous without being pedantic. The exercises were both interesting and enlightening. The only book in this list that taught skills I've actually used in the real world.
* Calculus by Spivak. This was used in my intro calculus course in university. It's very much a bottom-up, first-principles construction of calculus. Very proof-based, so you have to be into that. Tons of exercises, including some that sneakily introduce pretty advanced concepts not explicitly covered in the main text. This book, along with the course, rearranged by brain. Not sure how useful it would be for self-study though.
* Measurement by Lockhart. I haven't read the whole thing, but have enjoyed working through some of the exercises. A good book for really grokking geometric proofs and understanding "mathematical beauty", rather than just cranking through algebraic proofs step by step.
* Naive Set Theory by Halmos. Somewhat spare, but a nice, concise introduction to axiomatic set theory. Brings you from nothing up to the Continuum Hypothesis. I read this somewhere around my first year in university and it was another brain-rearranger.
These are good recommendations, but I think beginners tend to burn out due to the lack of a structured program and/or exercise solutions if they are trying to study on their own. The simplest structured program I can think of that satisfies both is:
* Basic Mathematics by Lang. Covers basic algebra and geometry at high school level.
Then one of these two, depending on your interests, or both:
* Vector Calculus, Linear Algebra and Differential Forms by Hubbard and Hubbard. Takes you through linear algebra, single-variable calculus and multiple variable calculus. Analysis is discussed in an appendix. All proofs have a constructive bias, so it's very algorithmic and natural for a CS-minded student. Solutions are in a separate volume.
Meh, Hubbard and Hubbard is good, but it certainly does not take you through single-variable calculus. For example, the following topics are assumed and not treated in any detail:
- power series and analytic functions
- the various properties of exponentials, logarithms, trigonometric functions, etc.
- L'Hopital's rule
- Integration by parts
You should definitely work through something like Spivak or Tao in addition to Hubbard.
I don't think this is a big concern. Hubbard discusses L'Hôpital and integration by parts, but it doesn't place a great deal of emphasis on them because the way topics are presented is a bit unusual.
If this is a concern, you can always use another text. I don't think this is a problem. Some freshman courses jump straight into Hubbard.
I disagree. L'Hôpital's rule is not discussed, it is simply presented without proof. And I think it's very important to carefully construct logarithms, exponentials and trigonometric functions, e.g. through power series - else the properties of these functions are just arbitrary axioms. Don't Hubbard and Hubbard claim themselves that the reader should be familiar with power series (somewhere in the chapter on Taylor series)?
Don't get me wrong, I really like the book, but I don't believe it replaces a book on single-variable calculus. I think the authors would agree.
Hubbard & Hubbard, which I referred to in my previous comment, discusses the basics of probability up to the Central Limit Theorem and a bit of Monte Carlo methods. Probability and statistics are really broad. What areas are you interested in?
If you are trying to do inference on large datasets, I think Andrew Gelman's books are superb and will teach you invaluable skills using multi-level Stan models, i.e. generative models and Bayesian inference.
If you want to study basic probability theory and stochastic processes, I really like Elementary Probability Theory by Chung. His more advanced book is really famous, but has a measure-theoretic approach and that's not very practical unless you are interested in developing theories a bit further.
Taleb is really fond of Probability, Random Variables and Stochastic Processes by Papoulis. But I find the typesetting in later editions to be really disorganized and confusing. Take a look. It's a good alternative to the first Chung book.
There is a large number of html format books on bookdown.org, mostly related to Data Science and R, lots of Bayesian too.
There are some more theoretical math books there, one of which I found to be very well written: Theory of Distributions by Peter K. Dunn
* Calculus by Spivak. This was used in my intro calculus course in university. It's very much a bottom-up, first-principles construction of calculus. Very proof-based, so you have to be into that. Tons of exercises, including some that sneakily introduce pretty advanced concepts not explicitly covered in the main text. This book, along with the course, rearranged by brain. Not sure how useful it would be for self-study though.
* Measurement by Lockhart. I haven't read the whole thing, but have enjoyed working through some of the exercises. A good book for really grokking geometric proofs and understanding "mathematical beauty", rather than just cranking through algebraic proofs step by step.
* Naive Set Theory by Halmos. Somewhat spare, but a nice, concise introduction to axiomatic set theory. Brings you from nothing up to the Continuum Hypothesis. I read this somewhere around my first year in university and it was another brain-rearranger.