I'd like to become conversant in a wide variety of serious mathematics, but i'm currently one of those students who did very poorly on mathematical subjects in school, never completing even basic algebra. My failure at math has been an ever-present awareness throughout my entire life, and i'd like to go beyond it. I'm in a technical field and my lack of good math foundations haven't crippled me, but i'd like to fill that hole.
I've started to get really serious about (re)learning mathematics, from my somewhat poor arithmetic foundations. I understand that there are essentially two schools of thought: The school of thought that suggests arithmetic->algebra->geometry/calculus (in either order)->analysis->everything else, and the school of thought that suggests an understanding of Set Theory and Logic (being the foundation of mathematics) is important.
I know that i'll eventually learn set theory and logic, but is there some benefit to learning these things before even trying the rest of mathematics? If one were to learn those things, what's the best place for a complete beginner to start? If I were to choose set theory, would ZFC be the only good option for a complete beginner?
Best Answer
A primary goal of "Set Theory and Logic" (I put this in quotations because I get the sense you are referring to a particular school of thought, and not just the pure subjects on their own) is to give foundation and motivation to the structures and systems of numbers that we commonly use. As the most basic example, efforts have been made to define natural numbers in terms of sets. Another basic example is the effort to define Mathematics as an extension of Logic.
While these are highly interesting studies, I would classify these types of studies more under the heading of "meta-mathematics" or foundations of mathematics. In essence, this type of study works backwards from the familiar world of numbers and mathematical areas we know, and attempts to ground these structures in well-defined "fundamental" ideas (sorry I have to be vague here, but this stuff is abstract!).
At any rate, from what I've said, you can get the sense that these types of study are not the typical areas a beginner should engage in, unless that beginner be of a more philosophical disposition; in other words, these areas are of a broader nature, and have a different conceptual "flavor". They seek to unify mathematical structures into more basic structures.
On the other hand, the "typical" mathematician works within established fields of math; that is to say, he uses and manipulates the structures and symbols given to him, in an attempt to discover deeper connections and new relationships. He is not usually concerned with foundations, that is a completely seperate study.
So, after I've said all that, my practical advice is to move along the "Algebra -> Pre-Calculus -> Calculus -> etc." route. That gives one the necessary tools for advanced study, and it familiarizes one (at a nice pace!) with what mathematicians really do. And IMHO, Calculus is absolutely essential in this path, because studying that results in a certain understanding and maturity in math that one will need throughout the rest of his mathematical career (e.g., the notions of limit and derivative in Calculus are really fundamental, and are great examples of mathematical intuition and thought).
Just a side note, I am not implying that the independent fields of Logic and Set Theory, as subjects on their own, are "deeply abstract" in the sense I described above (i.e., relate to the foundations of math); but that being said, I do not think they serve as beginning studies either. I believe they fall under the "etc." in the path I mentioned above.
Hope this helps you figure out how you'd like to proceed! Good Luck.