Foundations of Math Textbook Recommendations

(This question could just as well go in Cafe Society, but since it’s a question about technical books I thought it might get a larger audience here. Anyway, feel free to move it if necessary).

So this fall I’m teaching a Mathematical Foundations class (college undergrads). When I taught it before (five years ago), I just used the textbook other professors were using for the class:

Logic, Sets, and Recursion, by Robert L. Causey.

When I took a similar class years ago at Virginia Tech, we used Foundations of Higher Mathematics by Fletcher and Patty. Between the two, I prefer that textbook, but since it’s retailing in the neighborhood of $300 on Amazon, I’m very reluctant to use it for my class this fall.

The reason I prefer the Fletcher/Patty book is that it includes a decent amount of “actual mathematics” (it has an introduction to group theory (definition of a group and modular arithmetic, for example), an introduction to advanced calculus (like metric spaces and epsilon-delta proofs for limits if I remember correctly; I don’t have the book in front of me), some combinatorial stuff, set theoretic stuff like countability/uncountability).

The Causey book, on the other hand, is primarily devoted to propositional calculus and predicate calculus, with a little set theory stuff thrown in the middle (like set operations, formal definition of functions, partial orders, that sort of thing. Primarily just “definitional” stuff, but not doing anything especially interesting with it). Overall, it just seemed a lot drier to me than the other book.

(I know propositional calculus and predicate calculus are important for students learning to write proofs, I would just prefer the textbook didn’t focus on that to the exclusion of “actual” mathematics, if that makes sense. If you think I’m making a big mistake and doing my students a disservice by not wanting to focus on that extensively, feel free to give me your opinion. I’m open to the idea that I’m going about this wrong and it’s possible my mind could be changed).

Anyway, bottom line: I’m looking for a “reasonably priced” Math Foundations textbook (which, at this point, I’m thinking is in the neighborhood of $150, but cheaper is always better) that, while including basic techniques and structures of proofs, also includes a good amount of mathematics like I described above for the Fletcher/Patty book.

So: Suggestions, please?

The same title by Fendel/Resek is substantially cheaper, at least in paperback, and like your preferred text, also goes into some actual mathematics (Cantor-Schroeder-Bernstein is technically set theory, but there’s also discussion of groups, rings, and undergraduate real analysis). Be warned, as hinted at by the Amazon comments, that there’s not much hand-holding for the reader, so constant feedback and monitoring of student progress will demand a great investment of time by the instructor.

Yeah, I’m a bit reluctant to use a book that isn’t more catered to helping the student understand the material. Thanks for the suggestion, though!

I’m not necessarily looking for cheapest (though I don’t want it ridiculously expensive), but clearly written with lots of examples is important to me, too.

Anyone have any other suggestions?

I’ve used three textbooks for teaching a similar class, and there are two that I thinkg are great.

The first is Book of Proof, by Richard Hammack at VCU. It’s a good book, it’s freely available as a PDF, and you can buy a hard copy from Amazon for $14 or so.

This books is also really good. In my opinion it is slightly better than Hammack’s book, but you can’t beat Hammack for value.

Both of those look like they definitely have a lot of potential to be what I’m looking for. Thanks for the suggestions!

I don’t know if suits your purposes, but Introduction to Real Analysis by Bartle and Sherbert I thought was one of the best-written textbooks I’ve ever owned.

edit: reading the OP, this actually probably isn’t what you want

This thread brought back happy memories of my own days as an undergraduate math major, when the class of this sort that I took used a pre-published, draft version of this book: Introduction to Mathematical Structures, by Steve Galovich, who taught the course. (He is, unfortunately, now deceased—he was one of my favorite profs.) I see that there is an updated version of the book, Doing Mathematics: An Introduction to Proofs and Problem-Solving, which is still in print.

In looking this book up online, I ran across one professor’s posted notes on Logic and Proof (PDF) which give references to a few other books that are available freely online, such as
A Gentle Introduction to the Art of Mathematics by Joe Fields
A Introduction to Proofs and the Mathematical Vernacular by M. Day

Perhaps one of these could be usable, either as a primary text or supplementary resource?

As an erstwhile (but hopeless) math student, I think I would have really appreciated having a text like the second book (Chartrand’s Mathematical Proofs: A Transition to Advanced Mathematics) that breezman mentioned (but the price, not so much). In particular, it looks like the type of thing I needed to help me deal with, what was then for me, the totally new territory of fundamental, abstract proofs.

I still remember not even knowing where to start when one of the first questions on our first problem set was to prove that a[sup]2[/sup] was > 0 for all a ∈ R.

Who’s the audience for this class? Is it an early class for potential math majors that introduces the techniques of proof writing, or is it a senior-level elective on the foundations of mathematics? Your description doesn’t make it entirely clear, and those two classes really need different books.

Thanks for all of the suggestions, I should be able to find something to use out of all these.

Ultrafilter, it’s a junior level Math Foundations class for math majors to teach them how to write proofs.

Huh. I’m used to seeing those sorts of classes presented as a baby analysis or baby algebra class. Doing it as logic is different, but not necessarily a bad idea.

I’d be tempted to use the first book and supplement it with my own notes or a Schaum’s outline of whatever. There’s enough good material on the basics of real analysis/group theory/etc. out there on the web that I don’t think that’d be unreasonable. You could also take a look at some of Ken Binmore’s books, which I think have a very good balance of focusing on the basics of proving things while providing some actual mathematical content.

Edit: They’re cheap, too. Amazon has all three books for roughly $150 in total.