Proofs and Refutations is Imre Lakatos' foray into the foundations of mathematics, as well as primer for the aspiring mathematician to learn how to be a good mathematician. It is a dialogue between a teacher and his class, in the socratic style — hardly a class you'd find in today's education system. In fact, if one were to compare Proofs and Refutations to a platonic dialogue, it wouldn't be far from the mark. Interpreting it like this, then, the "theme" of the dialogue is "What is a mathematical proof?"
The dialogue opens with the teacher presenting Leonard Euler's proof of the Polyhedral formula, which reduces the theorem to a statement about graph theory and graphs. Unfortunately, there are some polyhedra which do not adhere to this formula, and the students propose a counterexample. The first half of the dialogue concerns how a mathematician should react to the discovery of a counterexample. Lakatos demonstrates two methods: "monster-barring" and "concept-stretching", and how both of them are undesirable alternatives.
"Monster-barring" works by restricting the domain of the theorem. Goldbach's Conjecture, for example, states that every even number greater than two is the sum of two prime numbers. It's well-known that no sum of prime numbers will add up to two, but it's also considered a "trivial counterexample." Lakatos recognizes that, taken to extremes, monster-barring can over-restrict the theorem and, in the end, weaken it.
"Concept-stretching" involves redefining concepts in the proof so that they apply to a broader domain. The second half of the dialogue revolves around what exactly a polyhedra could be. After quite a bit of refutation and discussion, it occurs to the students that they've made polyhedra such a broad term that it becomes meaningless. Lakatos advocates so-called "proof-generated" terminology, defining objects by their properties instead of with naive intuitions. On the other hand, proof-generated terms can become self-evident. For instance, defining continuous functions as those which have the property that the pullback of an open set is an open set makes the theorem "All continuous functions are such that the pullback of an open set is open" meaningless.
The final chapter of the book reduces the polyhedral formula to a statement about vector algebra, with the assumption that vector algebra is a "perfectly known" language. It confronts some of the problems inherent in translating a model from one language to the next. While I have no problem with the metaplot, I do dispute the use of vector algebra as a "perfectly known" language. It's really a pedantic point and doesn't detract from the book's message.
My last criticism of proofs and refutations is that the book is mostly written in quotes from other mathematicians. Most of these are relevant, but some tread on to arguments from authority. Ignoring these two distractions (the final chapter's dependence on perfectly known languages and the heavy quoting) Proofs and Refutations is an excellent book for people interested in the methodology of mathematicians and issues with the foundations of mathematics. | 
