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I just did this Google search and the first hit was along the lines of what I was looking for, which is a set of statements, each one building on the ones before it that start from nothing and go on to describe math.

Another way to phrase my question is this: what is a good way to get into Mathematics as a hobby, coming from an undergraduate engineering education ending ~20 years ago?

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  • $\begingroup$ This really depends on how much you remember. I assume as an engineer playing around on this site that you have some familiarity with mathematics. So, what do you think is the most reasonable starting point? $\endgroup$ – 123 Jan 21 '15 at 0:28
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    $\begingroup$ Read books. Lots of books. Good books! Some suggestions here and here. Edit. This is an answer to the question in your second paragraph, which I'm not entirely sure is the same as the one in your first. $\endgroup$ – David Jan 21 '15 at 0:31
  • $\begingroup$ Since you're interested in an axiomatic approach, one good starting point is Spivak's book Calculus, which builds calculus from the axioms for the real numbers. Another starting point could be an introductory number theory book. $\endgroup$ – littleO Jan 21 '15 at 0:34
  • $\begingroup$ Do not confuse axiomatic set theory (what you find in the link you posted) with an introduction to mathematics. What you're looking at in that link is the product of 19th century attempts to answer philosophical questions about whether or not mathematics could be unified under a single, very precise logical framework. It's all very interesting, but only if you already basically understand mathematics. What it boils down to is: set theory and the foundations of math are a topic within math, they're not math itself. $\endgroup$ – Jack M Jan 21 '15 at 0:39
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It really depends on how far back you want to start. For example, you could start from axiomatic set theory, which of late is interesting to me, but which will probably turn you off from boredom pretty quickly.

An excellent way to get into math as a hobby is to read any of a number of good books on elementary number theory, and progress from there to modern algebra.

But something I think you may find more fun is to start with the book "Concrete Mathematics" by Graham, Knuth, and Patashkin.

Here, I am talking about things you would find interesting, rather than foundational topics which I am afraid might not be so interesting.

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What you're talking about here is what mathematicians refer to as the foundations of math. You should start with books on mathematical logic and on set theory. Enderton's "A Mathematical Introduction to Logic" and "The Elements of Set Theory" are a good place to start.

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