Proofs and Refutations

<p><i>Proofs and Refutations: The Logic of Mathematical Discovery</i> is a 1976 book by philosopher <a href="/facts/Imre_Lakatos/yA7PxtFW">Imre Lakatos</a> expounding his view of the progress of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>. The book is written as a series of <a href="/facts/Socratic_dialogue/01aDBj6Y">Socratic dialogues</a> involving a group of students who debate the proof of the <a href="/facts/Euler_characteristic/JCjzWUr2">Euler characteristic</a> defined for the <a href="/facts/Polyhedron/UC0p9FTF">polyhedron</a>. A central theme is that <a href="/facts/Definition/5SlGKKDJ">definitions</a> are not carved in stone, but often have to be patched up in the light of later insights, in particular failed <a href="/facts/Mathematical_proof/7ECDrU80">proofs</a>. This gives mathematics a somewhat experimental flavour. At the end of the Introduction, Lakatos explains that his purpose is to challenge <a href="/facts/Formalism_(mathematics)/g0690AVL">formalism in mathematics</a>, and to show that <a href="/facts/Informal_mathematics/fbtJxxg4">informal mathematics</a> grows by a logic of "proofs and refutations".
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Proofs and Refutations