I'm in a similar boat as you. I went up through Calculus in school and hated it. Around 5 years ago I developed a strong interest in learning relevant applied math and have been enjoying it since.
Some things I'd add:
1) Math is fun! If you have the aptitude and disposition to enjoy writing software you'll love working out math problems. They're little nuggets of mental stimulation that you can work on with just some paper, a pencil, and maybe a pocket calculator.
2) You're spot on about an experienced programmer already having an intuitive but non-rigorous understanding of many concepts. It's mostly a matter of learning to read and write comfortably using the notation, which is really similar to learning the syntax and semantics of a big computer language with poor reference material.
3) You really have to have basic math down. This means going and re-learning stuff like applying FOIL to a binomial or dividing by a reciprocal.
4) Calculus and Linear Algebra are the father and mother of applied math. You'll save yourself a ton of grief if you learn them first (and I mean really learn them, maybe you took a calculus class in college but can you apply the Chain rule right now?). I'm learning Linear Algebra currently, which is something I should have done years ago. Part of the problem with self-teaching is getting things out of order.
I agree that getting things in the right order is important, but would argue that the order in which math is usually taken in the US is not the optimal one!
I recently took calc-1,2,3 and linear algebra through my local community college, and then started working my way through a wonderful book on mathematical proofs:(http://www.amazon.com/Mathematical-Proofs-Transition-Advance...), as preparation for working on higher level math. I would now argue that being able to understand and write proofs is a (the?) key mathematical skill to understanding what I would call 'real' (higher) math, and could be learnt by most students following high school algebra. My impression of the calculus series and linear algebra courses was an excessive focus on calculation, the math proof book was way more fun, surprising for a subject that is often thought to be too difficult for first year college students.
For those who are intimidated by the idea of a book on proofs (like I used to be), an example from the third chapter:
Theorem: Let x be an integer. Then x^2 is even if and only if x is even
It seems so simple, and I think would be accessible to anyone who had completed high school algebra but I found that even having done those calculus and linear algebra courses, I had now idea how to go about actually PROVING this! The book however, goes through the thought process step by step, and teaching the skills needed to be able to understand the real math books like Rudin.
I wish there were some sort of open-source prerequisite chain of what order to learn any subjects in. That's the hardest part of self-learning. Elon Musk had a reddit AMA recently where he was asked how he knows so much - he responded that he thought everyone had the capability of learning more than they thought, but that the key was to look at knowledge as a semantic tree. If you learn things in the wrong order, they won't have anything to hang off of.
Number theory. Set theory. Category theory. Combinatorics. Graph theory.
Linear algebra certainly has applications in some of the above. But I don't think that calculus & linear algebra can be fairly described as "father & mother" to these areas. (Am I wrong? I could be missing some connections; I'm not a mathematician.)
Not wrong - the poster above you probably means pure math in the analysis sense - real analysis, topology, functional analysis, algebraic topology. All of which are abstractions/generalizations (zoom out, if you will) from the real life world of 2 and 3 dimensional calculus/LA to N or infinitely many dimensions.
I probably should've thrown in combinatorics, which certainly existed before calculus or linear algebra, and certainly plays a role in applied math.
I would say graph theory is part of combinatorics, and set theory is part of logic.
Category theory was born out of trying to abstract the relationships between different objects in abstract algebra, so is kind of the child of abstract algebra and logic. I think it's fair to say the parents of abstract algebra are combinatorics and linear algebra.
Number theory at an elementary level is combinatorics, but at higher levels branches into analytic number theory (Calculus) and algebraic number theory ((Linear) Algebra).
> 4) Calculus and Linear Algebra are the father and mother of applied math.
> You'll save yourself a ton of grief if you learn them first
Baby Rudin and Axler are used currently by Harvard Math 55 to teach those subjects. Rudin might not be very didactic (I would be happy to hear about alternatives), but Axler is a fantastic choice.
If you liked Axler, you might check out Abbott's Understanding Analysis, also in the Springer UTM series. I think it covers somewhat less than Rudin (e.g. looking at baby Rudin's contents, I'm pretty sure Abbott doesn't touch Lebesgue integration) but it's a pretty great introductory analysis book IMO.
As a math graduate student, I second the choice for Abbott's "Understanding Calculus". It's a wonderful beginning book for analysis. Walter Rudin's "Principle's of Mathematical Analysis" is an amazing book but it's difficult to start with.
For a quick intro to Lebesgue integration you can read the beginning of Rudin's "Real and Complex Analysis" or Halsey Royden's "Real Analysis".
I haven't read Axler's book. I liked Hoffman and Kunze's "Linear Algebra"
It's fun because its incredibly rewarding! The elation of the "a-ha!" moment in math is second to none.
> 4) Calculus and Linear Algebra ...
Though I wouldn't get too caught up in the rigor of analysis or vector spaces right away. If you are self-studying, just spend enough time to feel confident computing and manipulating integrals, differentiation, and matrix math.
Then find a good intro to discrete math textbook covering a wide range of topics: number theory, graph theory, logic, set theory, etc, and learn how to write a "good" proof. This will open up a number of mathematical doors.
This has been my biggest realization as I started learning more math. What before seemed very arbitrary and unrelated becomes much more interesting and exciting once you have a bit of background. Unfortunately, I don't know of any way to get people to see the fun in math until they already know quite a lot of it... this was my experience at least, and seems to be pretty common among people who didn't gravitate towards math immediately.
Some things I'd add:
1) Math is fun! If you have the aptitude and disposition to enjoy writing software you'll love working out math problems. They're little nuggets of mental stimulation that you can work on with just some paper, a pencil, and maybe a pocket calculator.
2) You're spot on about an experienced programmer already having an intuitive but non-rigorous understanding of many concepts. It's mostly a matter of learning to read and write comfortably using the notation, which is really similar to learning the syntax and semantics of a big computer language with poor reference material.
3) You really have to have basic math down. This means going and re-learning stuff like applying FOIL to a binomial or dividing by a reciprocal.
4) Calculus and Linear Algebra are the father and mother of applied math. You'll save yourself a ton of grief if you learn them first (and I mean really learn them, maybe you took a calculus class in college but can you apply the Chain rule right now?). I'm learning Linear Algebra currently, which is something I should have done years ago. Part of the problem with self-teaching is getting things out of order.