How do mathematicians know that they're right? How do mathematicians know that they're right? How do they know that there's no flaw in a proof, or know when something has been proved?
Is this a welldefined concept, is it is some kind of intuition that must be developed?
 A: Mathematical proofs are structured upon predicate logic, which is a well formalised area of philosophy. As we are only human, mathematicians make mistakes all the time. The catch is that, not only must they convince themselves something is correct, they have to convince their peers as well. This strict review process ensures that by the time a proof gets published, many people who are experts in the field agree on it's validity.
A: The simple answer is they don't. To verify a proof one has to read through it very carefully and ensure that each logical step is valid. This is not a flawless technique because people make mistakes. This is especially true when one reads his/her own work. Thus the best solution is to have others read your argument, which is why no result is published by a serious paper without first being peer reviewed. However this process isn't flawless either. It's not uncommon for an error to be noticed after a proof has been peer reviewed and even applied by other authors to prove new results.
This question, which is also linked in the comments by FullofDill, has many examples.
A: Proofs are based on definitions, axioms and rules of logic (i.e. rule of inference). A proof is verified by peers as others mentioned. Thus, the proof is reproducible and verifiable. If you think about it, no other science depends on proofs but math can provide certainty. No computer can calculate infinite sums but the mathematicians can. 
