<P> 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, 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 Euclidean exposition of step - by - step details of the proof; review involves reconsidering and re-examining the result and the path taken to it . </P> <P> Gauss, when asked how he came about his theorems, once replied "durch planmässiges Tattonieren" (through systematic palpable experimentation). </P> <P> Imre Lakatos argued that mathematicians actually use contradiction, criticism and revision as principles for improving their work . In like manner to science, where truth is sought, but certainty is not found, in Proofs and refutations (1976), what Lakatos tried to establish was that no theorem of informal mathematics is final or perfect . This means that 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 . (If axioms are given for a branch of mathematics, however, Lakatos claimed that proofs from those axioms were tautological, i.e. logically true, by rewriting them, as did Poincaré (Proofs and Refutations, 1976).) </P> <P> Lakatos proposed an account of mathematical knowledge based on Polya's idea of heuristics . In Proofs and Refutations, Lakatos gave several basic rules for finding proofs and counterexamples to conjectures . He thought that mathematical' thought experiments' are a valid way to discover mathematical conjectures and proofs . </P>

What is the planned study and methods used by scientists to gain information