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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?

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    $\begingroup$ For a complete beginner, unformalized (but precise) set theory should precede any introduction to ZFC. Indeed, one can do very good work in mathematics without being aware of the existence of ZFC. I would also advise work in other subjects before set theory. some of the subjects will use set-theoretic notation. That's not too hard to absorb while studying something else. $\endgroup$
    – André Nicolas
    Apr 11, 2013 at 1:49
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    $\begingroup$ I'd do some geometry as soon as you feel comfortable with algebra. It's the funnest. And you can prove as much or as little as you like. $\endgroup$
    – Paul Gustafson
    Apr 11, 2013 at 1:55
  • $\begingroup$ This may be helpful. $\endgroup$
    – EuYu
    Apr 11, 2013 at 2:05
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    $\begingroup$ I second @AndréNicolas' suggestion. It will be very hard to self-study set theory without motivation. Perhaps just follow the standard sequence. You can read set theory on the side, but don't worry about too much about forcing yourself to read it if you find your interest waning. $\endgroup$
    – Paul Gustafson
    Apr 11, 2013 at 2:17
  • $\begingroup$ On the whole I agree with @André’s comment above. If you want to have on the side a book that gives a decent but relatively informal introduction to set theory, logic, and some closely related topics, you could do worse than Bob Stoll’s Set Theory and Logic, available in an inexpensive Dover paperback. $\endgroup$
    – Brian M. Scott
    Apr 11, 2013 at 3:46

3 Answers 3

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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.

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  • $\begingroup$ Thanks for the notes. My rationale is that understanding set theory and logic may be a better "bridge" for me as in some ways the philosophical concepts may be easier to understand than jumping right in. When I failed my mathematics courses, I was always confused as to what math really was, or "why" certain things were the way that they were; now understanding math as just a constructive method of dealing with problems has made things easier but i'm very weak on applications. My hypothesis is that understanding set theory and logic might solve these unanswered questions for me. $\endgroup$
    – Arima
    Apr 11, 2013 at 2:17
  • $\begingroup$ @Arima : I have to warn you though that these foundation-type studies often employ the ideas from the fields of set theory and logic. So you'd need those first. And if you're going to study those, then you need some basic skills in arithmetic, algebra etc...That's why I was implying the foundations-type study of math is for more advanced studiers. But far be it from me to discourage you if its the "philosophy-type" stuff that interests you; I just mean to say its not the typical math. $\endgroup$
    – Coffee_Table
    Apr 11, 2013 at 2:24
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It depends on what you want to do with mathematics. If your aim is to build a repertoire of problem-solving tools for applications in science and industry, you will probably get all the set theory and logic you will ever need from the introductory chapter of any good algebra or calculus textbook. If reading and writing detailed proofs (e.g. proving there are an infinite number of prime numbers) is important to you, you may need more than that.

I believe it is possible to get an undergraduate degree in pure mathematics without ever once dealing at any length with foundational issues such as the axioms of set theory (ZFC, etc.). It really is quite a specialized interest. I would stick to application-oriented algebra, calculus and statistics for now. A few years of that and ZFC just might make sense to you. But don't worry if it doesn't.

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  • $\begingroup$ Where I studied for my B.Sc. and M.Sc. you can even get through a Ph.D. without taking a course on axiomatic set theory, or any foundational topic. Of course this much is not very easy, but it's possible. $\endgroup$
    – Asaf Karagila ♦
    Apr 11, 2013 at 15:17
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It sounds like you need an opportunity to evaluate your own strengths and weaknesses as a math learner that meets you where you are today.

If you are in a technical field, and you want to weigh exploring traditional subject matter vs. starting with foundations, I would propose modifying your sequence to percolate down Linear Algebra out from "everything else" to somewhere earlier in the chain, after algebra but perhaps before calculus.

On one hand, the importance of linear algebra in applied math can't be overstated. On the other hand, as a kind of abstract algebra, studying LA is well-adapted to a theorem-proof type of presentation and can provide a first opportunity to see sets and logic in action, with many fewer conceptual difficulties than, say, calculus at a similar level of rigor.

As an introduction to LA for applications, consider Strang's 18.06 online lectures at MIT OpenCourseWare. For an axiomatic approach at an introductory level, there is Axler, Linear Algebra Done Right.

As for math and foundations more broadly, there are excellent textbooks, but don't limit yourself to those. You can learn a lot about math, what it is and what proofs are and the role played by foundations from high-quality popular works; these might be enough to satisfy your curiosity short of college-level coursework (or may whet your appetite for more, and give you a leg up on it too). Here are a few old titles just as seeds for your own search:

  • Eves, Foundations and fundamental concepts of mathematics
  • Newman, The World of Mathematics
  • Boyer, History of the calculus and its conceptual development (archive.org)
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