Scientific method: Difference between revisions

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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| titleauthor = [[Charles A. Weibel]] (|date=ca. 1995)| title = History of Homological Algebra| access-date = 2021-08-28 | archive-date = 2021-09-06 | archive-url = https://web.archive.org/web/20210906014123/https://faculty.math.illinois.edu/K-theory/0245/survey.pdf| url-status = live}}</ref><ref>Henri Poincaré, Sur l’[[Analysis Situs (paper)|analysis situs]], ''Comptes rendusde l’Academie des Sciences'' '''115''' (1892), 633–636. as cited by {{harvp| Lakatos| 1976 |p=162}}</ref>{{efn-lg|name= stillwell'sReviewOfGray'sBioOfPoincaré|Stillwell's review (p. 381) of Poincaré's efforts on the [[Euler characteristic]] notes that it took ''five'' iterations for Poincaré to arrive at the ''[[homology sphere#Poincaré homology sphere|Poincaré homology sphere]]''.<ref name= stillwell>John Stillwell, reviewer (Apr 2014). ''Notices of the AMS.'' '''61''' (4), pp. 378–383, on Jeremy Gray's (2013) ''Henri Poincaré: A Scientific Biography'' ([http://www.ams.org/notices/201404/rnoti-p378.pdf PDF] {{Webarchive|url=https://web.archive.org/web/20210704205514/http://www.ams.org/notices/201404/rnoti-p378.pdf |date=2021-07-04 }}).</ref>}}

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}}