(Part 2) Best combinatorics books according to redditors

A good one for combinatorics is A Course in Combinatorics by J.H. van Lint & R.M. Wilson.

I don't know if category theory is your thing but maybe you'd be interested in categorical logic?
http://www.amazon.com/Introduction-Higher-Order-Categorical-Cambridge-Mathematics/dp/0521356539

that book goes a lot into various ways to study logics inside categories and indeed categories as certain kinds of deduction systems. It also has a nice introduction to toposes which are kinds of categories which have a kind of inner logic (usually multivalued and always intuitionistic). It links nicely to some newer branches of math too.

If you're unfamiliar with categories they are abstractions of collections of mathematical structures and the functions between them(eg. sets and functions, or vector spaces and linear functions).

Depends on your background. Mac Lane is the standard text and he is a phenomenal author in general, but it builds off knowledge of concepts such as modules, tensor products and homotopy (I still don't have a sufficient background in AT to be honest though). For a more modest background, I would recommend the book "Sets for Mathematics" by Lawvere and Rosebrugh. The book is entirely on category theory, the title is because there is a focus on the category of sets. The first chapter or so is deceptively simple, it gets very difficult as it goes on, but still doesn't require much specific background.

I'll also note that I first got into the subject through a whim purchase in a local Borders of a cheap dover book Topoi by Robert Goldblatt when I was very into mathematical logic. It's 500 pages and requires pretty much no background (I'd know what a topological space is, but I can't think of anything else). It gets very challenging though, and I never got more than 250 pages in before getting overwhelmed, but the first hundred pages really sparked my interest in category theory. Functors (and especially adjoint functors) are postponed much later than you will see in many other sources though. You can find a link to an online version free from the author's webpage too.

Since you're studying comp sci, combinatorics might be a really good topic--most of the difficulty students have studying algorithms comes from not having a strong enough background in combinatorics. It's also something that you can delve right into. I don't really have any book recommendations, but my undergrad class used Roberts and Tesman. I don't remember enough about this book to say much about it.

Alternatively, logic is super cool! A book on computability or mathematical logic would work. Herbert Enderton has written a book on each. I find his writing a little too informal, but they're good for getting a grip on the basic notions.

Ah, the "least read great book of modern science."

Laugh if you will, but when I was in high school I nearly blew a substantial chunk of my savings on the (rare and monstrously overpriced) Principia Mathematica. Fortunately, I found and bought the abridged edition, which cured me of the temptation forever.

You should check out the books published in the New Mathematical Library series.

Here are a few I think might be really awesome:

Geometric Inequalities by Kazarinoff

Invitation to Number Theory by Ore

Numbers: Rational and Irrational by Niven

Mathematics of Choice: Or, How to Count Without Counting by Niven

Episodes from the Early History of Mathematics by Aaboe

Episodes in Nineteenth and Twentieth Century Euclidean Geometry by Honsberger

I wish I knew about these books when I was in high school.

I just finished studying Gödel's incompleteness theorems. The tutor made us work from this book and it was not great. Full of errors and sometimes left proofs that were less than obvious as exercises. However, I picked up this book from the library and found it to be reasonably good (just be sure to look up the errata). The first half of it is quite basic if you have some previous experience with logic or mathematical logic. If you want something less formal on Gödel then read Nagel and Newman's book. One of the best informal books about formal logic I have ever read in terms of clarity of exposition.

I don't want to recommend any other books, as all the others I have used I have not used in their entirety. I would say that most logic books have mistakes or bad exposition. I think it occurs due to the people writing the books being so fluent in the topic that some proofs or points seem obvious to them but may not be to the reader. I recommend this website. There are a lot of good links there. Personally, I think the best way to learn logic is to have someone teach it to you since you will always need to be asking questions or having someone check your workings. And, like maths, just practice exercises and proofs until you can do them without thinking.

I know it may not be exactly what you were looking for but I hope it helps all the same. Good luck.

I like this one. You basically get to develop set theory yourself as a sequence of problems.

I guess it's not really introductory... hmm.

> The distinction is that in math, all foundational meta-theories are require to get the right answers on simple object-level questions like "What's 1 + 1?". If your mathematical metatheory answers, "-3.7" rather than "2", then it is not "different", it is simply wrong. We can thus say that Foundations of Mathematics is always done with a realist view.

The natural numbers are an interesting example, which goes back to ADefiniteDescription's point about a privileged model. The basic axioms of arithmetic are categorical, i.e. have only one model, up to isomorphism. Not all theories have this property, though.

If it could be shown that some moral theory similarly has only one correct interpretation - that all alternative interpretations end up being isomorphic - then that could support a kind of realism, at least in the context of that theory. A lot would depend on the nature and scope of the theory in question, and its interpretation.

So perhaps Parfit's position would be better captured by saying that he believes there are unique true answers to moral questions, as there are for questions in categorical mathematical theories.

> What's a good textbook for that field, anyway?

The books I studied are quite outdated now, but a classic modern text is Model Theory by Chang & Keisler. That might be more comprehensive than you're looking for. You could try Model Theory: An Introduction - its first chapter is quite a concise basic intro. There's also A Shorter Model Theory.

After this coming semester, all or almost all of the math courses that you take will be heavily proof-based. Analysis, topology, and algebra all involve a lot of reading and writing proofs. So I would recommend using this time to become acquainted with mathematical proof techniques. Hammack's Book of Proof is freely available online. I haven't read it myself but I've heard good things about it. Eccles's An Introduction to Mathematical Reasoning is another good choice and it is the intro-to-proof book that I learned from.

Whatever books or resources you choose, just remember that it is very important to do the exercises.