Informal mathematics, also called naïve mathematics, has historically been the predominant form of mathematics at most times and in most cultures, and is the subject of modern ethno-cultural studies of mathematics. The philosopher Imre Lakatos in his Proofs and Refutations aimed to sharpen the formulation of informal mathematics, by reconstructing its role in nineteenth century mathematical debates and concept formation, opposing the predominant assumptions of mathematical formalism. Informality may not discern between statements given by inductive reasoning (as in approximations which are deemed "correct" merely because they are useful), and statements derived by deductive reasoning.
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