I am very interested in finding out about the foundations of math. When I feel foolish enough to try, I visit Wikipedia and end up in a truly endless rabbit hole. There are dozens of terms that I don't know: group, abelian group, field, structure, ring, algebraic structure, commutative ring, to name a very few. My background is very very shallow. I'm only now about to start taking my upper division classes for my bachelor's degree (so far I've only taken 2000 level classes such as calculus, linear algebra, discrete math, etc.) Could someone give me a very brief overview of how we start at the foundations and arrive to the algebra I know and love? I am looking for an explanation that starts from the very basics. Let's say I am an alien from a different universe. What is the first axiom you tell me, and how do you go from that to everyday math?

  • $\begingroup$ Foundations of math is philosophy, not math. If your goal is not to learn philosophy but just to see how math starts "at the foundations," sort of like the nineteenth century Europeans' search for the sources of the Nile or the search for a college course with zero prerequisites, the following may be helpful: math.stackexchange.com/questions/140681/… $\endgroup$ – ForgotALot Jul 21 '16 at 1:40
  • $\begingroup$ At least historically, it's the other way around, and bi-directional: You start at the algebra you know and love, and go in two directions: Down, to the foundations (probably only after you encounter something scary, like infinitesimals), and up, towards abstracting out the concept of field, ring, group, etc. $\endgroup$ – pjs36 Jul 21 '16 at 1:54
  • $\begingroup$ @ForgotALot Well if a super intelligent alien came from another universe and knew nothing of our universe (except our language) where you start? $\endgroup$ – Ovi Jul 21 '16 at 1:58
  • $\begingroup$ @Ovi If they were super intelligent, by which I mean absurdely intelligent, then it would probably be enough to write down axioms and definitions of a given field and leave the rest as homework over the weekend. $\endgroup$ – Stefan Mesken Jul 21 '16 at 2:10
  • $\begingroup$ Like, historical foundations or logical foundations? Historical starts in geometry, moves into algebra, then calculus, then things probably jump to set theory and everything we know today. Logical can probably start in a couple different places, such as first order logic, ZFC axiomatic set theory, Peano arithmetic, or even Category Theory, and then moves into topology and group theory, which then moves to metric spaces and ring theory respectively, and then they fan out into a bunch of theories. There's also a tremendous amount of overlap. $\endgroup$ – JasonM Jul 21 '16 at 2:12

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