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In Pólya's view, ''understanding'' involves restating unfamiliar definitions in your own words, resorting to geometrical figures, and questioning what we know and do not know already; ''analysis'', which Pólya takes from [[Pappus of Alexandria|Pappus]],{{sfnp|Pólya|1957|p=142}} involves free and heuristic construction of plausible arguments, [[working backward from the goal]], and devising a plan for constructing the proof; ''synthesis'' is the strict [[Euclid]]ean exposition of step-by-step details{{sfnp|Pólya|1957|p=144}} of the proof; ''review'' involves reconsidering and re-examining the result and the path taken to it.
{{anchor|proofsAndRefutations}}Building on Pólya's work, [[Imre Lakatos]] argued that mathematicians actually use contradiction, criticism, and revision as principles for improving their work.<ref>{{harvp|Lakatos|1976}} documents the development, by generations of mathematicians, of [[Euler's formula for polyhedra]].</ref>{{efn-lg|name= stillwell'sReviewOfGray'sBioOfPoincaré}} In like manner to science, where truth is sought, but certainty is not found, in ''[[Proofs and Refutations]]'', what Lakatos tried to establish was that no theorem of [[informal mathematics]] is final or perfect. This means that, in non-axiomatic mathematics, we should not think that a theorem is ultimately true, only that no [[counterexample]] has yet been found. Once a counterexample, i.e. an entity contradicting/not explained by the theorem is found, we adjust the theorem, possibly extending the ___domain of its validity. This is a continuous way our knowledge accumulates, through the logic and process of proofs and refutations. (However, if axioms are given for a branch of mathematics, this creates a logical system —Wittgenstein 1921 ''Tractatus Logico-Philosophicus'' 5.13; Lakatos claimed that proofs from such a system were [[Tautology (logic)|tautological]], i.e. [[logical truth|internally logically true]], by [[string rewriting system|rewriting forms]], as shown by Poincaré, who demonstrated the technique of transforming tautologically true forms (viz. the [[Euler characteristic]]) into or out of forms from [[homology (mathematics)|homology]],<ref name= eulerPoincaré >H.S.M. Coxeter (1973) ''Regular Polytopes'' {{ISBN| 9780486614809}}, Chapter IX "Poincaré's proof of Euler's formula"</ref> or more abstractly, from [[homological algebra]].<ref>{{cite web| url = https://faculty.math.illinois.edu/K-theory/0245/survey.pdf| author = [[Charles A. Weibel]] |date=
Lakatos proposed an account of mathematical knowledge based on Polya's idea of [[heuristic]]s. In ''Proofs and Refutations'', Lakatos gave several basic rules for finding proofs and counterexamples to conjectures. He thought that mathematical '[[thought experiment]]s' are a valid way to discover mathematical conjectures and proofs.{{sfnp|Lakatos|1976|p=55}}
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