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Much of the work of mathematical antirealists has been destructive rather than constructive -- the goal has been to show that the epistemology of abstract objects (the sorts of things that mathematical objects are generally taken to be -- non-spatiotemporal things) is untenable.

The attempts to be a nominalist about math have been unsatisfying to many. One famous nominalist strategy was to claim that science doesn't actually need math in order to work -- that math is just a helpful but dispensable tool. Thus, math objects don't need to be quantified over in scientific theories; thus math objects don't exist. Hartry Field's famous attempt at this has generally been deemed a failure (though it's a fascinating project). It did open the floodgates to fictionalism in mathematics: the idea that mathematical objects are fictional, but play an indispensable role in the story of mathematics, in much the same way that Holden Caulfield is a fictional character in The Catcher in the Rye. The idea, basically, is that we can get truth and falsity out of mathematical claims by saying that, e.g., "2+2=4" is true in the story of math, just like "Holden Caulfield is depressed" is true in The Catcher in the Rye.

One of the great defenses of fictionalism is Mary Leng's recent book, Mathematics and Reality. Check out the SEP page for a better explanation and some great references.

It might be helpful to read an introductory text first. My first philosophy of math course used Stewart Shapiro's [Thinking about mathematics] (http://www.amazon.com/Thinking-about-Mathematics-The-Philosophy/dp/0192893068/ref=sr_1_1?ie=UTF8&qid=1341687670&sr=8-1&keywords=stewart+shapiro) as a supplementary text. I didn't use it too much, but it is pretty good and quite approachable from what I recall. Shapiro is a very well regarded contemporary philosopher of mathematics.

You could also start with the [SEP article] (http://plato.stanford.edu/entries/philosophy-mathematics/). This will give you an overview of the area, its history, and the various sub-disciplines. That can help you narrow down what in particularly you are interested in which will make it easier for you to find appropriate books.

I've only taken one module in philosophy of mathematics (also the only actual philosophy class I've taken) but Shapiro has a good book we used as a go-to text. Link below bc I don't know how to format on mobile. As far as prerequisite knowledge, you shouldn't need much beyond set/model theory and some mathematical logic, and even that isn't necessary depending on how far your studies are.

https://www.amazon.co.uk/Thinking-About-Mathematics-Philosophy/dp/0192893068

Gives a good overview of various topics in PoM, mainly questions of either:
• Ontology - Do mathematical objects exist? If so, in what sense?
• Epistemology - How do we have mathematical knowledge? How does it apply to the real world?

Aside from the book mentioned above, just do a quick Google and see what you can find in your library catalogue! Ayer, Kant, and Quine are some prominent authors.

Hope that helps some :)

To quote what is probably the best math book ever written, Peter Rozsa's Playing with Infinity http://www.amazon.com/Playing-Infinity-R%C3%B3zsa-P%C3%A9ter/dp/0486232654

> if a mathematician has proved something about points and lines, he communicates his findings to his fellows as follows: ‘I do not know what kind of pictures you have of geometrical figures. My idea is that through any two points whatever I can draw one straight line. Does this agree with your idea?’ If the answer is in the affirmative, then he can proceed thus: ‘I have proved something and during the proof I did not make use of any other property of points and straight lines apart from the ones about which we are already agreed. You can now think about your points and lines; you will still understand what I have to say.’

And

> Mathematics does not pretend to enunciate absolute truths. Mathematical theorems are always put in the more humble form: ‘If, . . . then . . .’ ‘If we can use only ruler and compass, then the circle cannot be squared. If by points and lines we mean figures with such and such properties, then the following things are true of them.’

My take on this: mathematics is a totally artificial construct and it is not even a fixed construct. There are axioms we agree on, there are rules of reasoning we agree on but Godel has proven that for every (usable) axiom system we can create two new ones (one by adding the unprovable statement and another by adding the negated statement).

Polity Press just released a new introductory text on paradoxes that I recommend thoroughly. In my opinion it is better than the Sainsbury: Roy T. Cook - Paradoxes.

Introduction to Meta-Mathematics is a great book for this (considered by some as the bible of meta-mathematics).

http://www.amazon.com/Introduction-Metamathematics-Stephen-Cole-Kleene/dp/0923891579/

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.

Following this up I would recommend "Empiricism, Semantics, and Ontology" by Carnap, "Critique of Pure Reason" by Kant, "Philosophy of Mathematics" compiled by Putnam and Benacerraf, "Philosophy of Mathematics: Structure and Ontology" by Shapiro, "Mathematics in Kant’s Critical Philosophy: Reflections on Mathematical Practice" by Shabel, and "On the Infinite" by Hilbert.

Also I would recommend looking into the lesser works of Shapiro, Shabel, and Yablo

Edit: I forgot to mention that Aristotle and Badiou also have writings on mathematics.

Before the Princeton Companion to Mathematics, there were:

What Is Mathematics? by Courant and Robbins

Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrent'ev

Concepts of Modern Mathematics by Ian Stewart

It's actually part of a series, a lot of others philosophers and logicians have done it too
http://www.amazon.com/Philosophy-Mathematics-Questions-Vincent-Hendricks/dp/8799101351
you can find most of the others on google

http://www.amazon.com/Computability-Logic-George-S-Boolos/dp/0521701465

Everyone here thinks you're crazy (and maybe they aren't wrong), but indeed these kinds of issues have been thought about and discussed for millenia, by the Babylonians, Mayans, Greeks, Medieval Europeans, Indians, Arabs, Renaissance Europeans, and more. You might like to read Zero: The Biography of a Dangerous Idea (and the references contained in it) for more.

First math book I read for pleasure was Zero: The Biography of a Dangerous Idea. Its focus is more on the history side of things, which come to think of it makes it weird that I liked it since I normally am not interested in history.

But it's pretty amazing to imagine living in a society where zero was not was an accepted concept--in fact, it hadn't just not been thought of, it was actively denounced by the Church.

It's been maybe a decade since I read it, but I still remember the BS proof they used back then.

God cannot do evil.
There is nothing God cannot do.
Therefore,"nothing" is evil.

And with that, you were disallowed from using the concept of 0. Which makes a lot of math really difficult.