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- In his book *Proofs and Refutations*, Lakatos identifies seven methods by which mathematical discovery and justification can occur. These methods suggest ways in which concept definitions, conjectures and proofs gradually evolve via interaction between mathematicians. Different mathematicians may have different interpretations of a conjecture, examples or counterexamples of it, and beliefs regarding its value or theoremhood. Through discussion, concepts are refined and conjectures and proofs modified. For instance, when a counterexample is found, one might look for general properties which make it fail a conjecture, and then modify the conjecture by excluding that type of counterexample (piecemeal exclusion). Alternatively, one might generalise from the positives and then limit the conjecture to examples of that type (strategic withdrawal). Another reaction might be to deny that the object is a counterexample on the grounds that the conjecture refers to objects of a different type (monster barring). Given a faulty proof, a counterexample may be used to highlight areas of weakness in the proof, and to either modify the proof or the conjecture which it purports to prove (lemma incorporation).