Reddit reviews Philosophies of Mathematics

For Frege, complete generality was a key part of logic so his logical system was designed to be completely general. For that reason his system reduced math not to a theory of sets but a theory of concepts and their extensions (concepts are what is denoted by a phrase like "is blue" or "was born before Frege" while the extension of a concept is the collection of all objects for which a given concept is true). Russell's paradox demonstrated that Frege's system was inconsistent which made it unsuitable as a foundation for mathematics.

In response, mathematicians came up with an alternative foundation for mathematics called Zermelo Fraenkel set theory (or ZF). To the best of our knowledge ZF is consistent but it seems to lack the full generality that is a hallmark of logic. Here are two reasons:

• ZF makes statements about the existence of objects without compelling justification by the "rules of thought" e.g. the axiom of infinity.
  
• ZF is a theory of sets and not a theory of concepts like Frege's original system was. In Frege's system each concept, regardless of whether it was a concept of math or something else, had an extension. But in ZF there are concepts that have no corresponding set. This was done in an effort to avoid contradictory objects like "the set of all sets that do not contain themselves" that was constructed by Russell but had the effect of also prohibiting the construction of other seemingly innocuous sets. For example, there is no set of all sets though there is a class of all sets. These seemingly arbitrary restrictions on what sets did and did not exist have compelling mathematical justification but lack the generality requisite to be considered logical.
  
  
  If you're interested in reading about the foundations of math and already know the basics of logic and naive set theory I recommend "Philosophies of Mathematics".
  
  
  
  Edit: your --> you're
  
  Edit 2: Fixed error caught by /u/fractal_shark

Frege's Foundations of Arithmetic is a classic "primary text" that advocates a specific point of view (that arithmetic can be reduced to logic in some sense).

These are a couple of contemporary introductory books that provide decent surveys of some major views:

http://www.amazon.com/Philosophies-Mathematics-Alexander-George/dp/0631195440

http://www.amazon.com/Thinking-about-Mathematics-The-Philosophy/dp/0192893068/ref=pd_bxgy_b_img_z/181-3737012-4965247

EDIT: If I had to choose, I would pick the Velleman/Alexander book.

I took a course on Philosophy of Mathematics with the authors of this book. It contains an entire section on intuitionism, which is well-written and would serve as a nice place to start.

http://www.amazon.com/Philosophies-Mathematics-Alexander-George/dp/0631195440

http://www.amazon.com/Philosophies-Mathematics-Alexander-George/dp/0631195440/ref=sr_1_3?ie=UTF8&qid=1412983653&sr=8-3&keywords=dummett+mathematics

This book has a good presentation of the most popular philosophical conceptions of mathematics, but might not be exactly what you had in mind. It's fairly technical and require a little bit of logic and set theory. If you'd like something a bit lighter I can try to think of something.

I think it would be good one.