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I have just finished my second year of a pure mathematics degree and am intrigued by philosophy, and would like to spend my summer learning more about philosophy to accompany my mathematical interest.
I have no prior knowledge or study of philosophy, so I am very much looking for resources for beginners.
The question of where to start in philosophy is controversial, and has many different answers already on this site. However, in specifically bridging between mathematics and philosophy, I'd recommend a combination of Plato --specifically the Meno --and advanced symbolic logic.
Plato extensively uses mathematics as a metaphor for his philosophical concepts and was in turn an influence on the development of mathematics. Advanced symbolic logic was born out of an attempt (most famously by Russell and Whitehead) to unite mathematics and philosophy, and has strong implications for both fields.
You might also enjoy the work of Descartes, a towering giant in both mathematics and philosophy. Pascal is likewise noted in both fields.
Since you have an interest in philosophy and ability in mathematics, I would suggest picking up some logic, set theory, etc. (as suggested elsewhere) -- and you might consider looking into the philosophy of language and other allied areas, typically classed as "formal philosophy", where the "formal" indicates occasional, if not constant recourse to mathematics or formal methods in the context of philosophical research. Because you have an inclination towards mathematics it makes sense for you to lean towards formal subjects. (One interesting catch-all source is More Precisely:The Math you Need to do Philosophy by Steinhart. This would help you see what non-logical, formal methods are of relevance to philosophy in a digestible way. You could read it in the summer. Actually it's a really straightforward, and I would say easy book, especially for someone with training in maths.)
Yet, no one comes to philosophy without an interest in something. There has to be a draw. And this draw might not be formal. (This very same thing has happened before, say, in Gian Carlo Rota, a great mathematician yet one who had no interest in analytic philosophy; a strange, yet similar story can be told about Kurt Godel, who did not understand analytic philosophy.)
For instance, you may have read some Sartre in high school and that really appealed to you. So I would encourage you to be specific about what is drawing you to philosophy, and follow that up (if you can discern what that is -- if you have any specific sources, those of us on the stack can help you discern what you appear to be interested in). It's a little more organic that way. Stanford Encyclopedia of Philosophy is really good for this. It can help you both with formal and non-formal philosophy, it's really a great resource. And the bibliographies of those articles are well-compiled and provide you an idea for where you might wish to go next.
But I will still encourage you to look at formal philosophy, because I think it may appeal to you. The trouble, of course, is that if you're reading Sartre on the one hand and Principia Mathematica on the other you might wonder what the common ground is. But if you do philosophy at all you're gonna have to wrestle with that someday anyway and that common ground is for you to find.
Learning about philosophy in the context of the field's historical development and as it relates to math can be a fun approach. To this end, Gödel, Escher, Bach and Logicomix have been fun, stimulating reads. If you're looking for something more comprehensive, consider Bertrand Russell's A History of Western Philosophy (can't post a third link as a new user...).
