Scientific method

==Relationship with mathematics==
Science is the process of gathering, comparing, and evaluating proposed models against [[observable]]s. {{anchor|aModel}}A model can be a simulation, mathematical or chemical formula, or set of proposed steps. Science is like mathematics in that researchers in both disciplines try to distinguish what is ''known'' from what is ''unknown'' at each stage of discovery. Models, in both science and mathematics, need to be internally consistent and also ought to be ''[[falsifiable]]'' (capable of disproof). In mathematics, a statement need not yet be proved; at such a stage, that statement would be called a [[conjecture]].<ref>{{harvp|Pólya|1957|p=131}} in the section on 'Modern [[heuristic]]':
"When we are working intensively, we feel keenly the progress of our work; we are elated when our progress is rapid, we are depressed when it is slow."</ref>

Mathematical work and scientific work can inspire each other.<ref name= ilSaggiatore >
"Philosophy [i.e., physics] is written in this grand book – I mean the universe – which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometrical figures, without which it is humanly impossible to understand a single word of it; without these, one is wandering around in a dark labyrinth." – Galileo Galilei, ''Il Saggiatore'' (''[[The Assayer]]'', 1623), as translated by [[Stillman Drake]] (1957), ''Discoveries and Opinions of Galileo'' pp. 237–238,
as quoted by {{harvp|di Francia|1981|p=10}}.
</ref> For example, the technical concept of [[time]] arose in [[science]], and timelessness was a hallmark of a mathematical topic. But today, the [[Poincaré conjecture]] has been proved using time as a mathematical concept in which objects can flow (see [[Ricci flow]]).<ref>Huai-Dong Cao and Xi-Ping Zhu [https://arxiv.org/pdf/math/0612069.pdf (3 Dec 2006) Hamilton-Perelman’s Proof of the Poincaré Conjecture and the Geometrization Conjecture] 
*revised from H.D.Cao and X.P.Zhu ''Asian J. Math.'', '''10'''(2) (2006), 165–492.</ref>

Nevertheless, the connection between mathematics and reality (and so science to the extent it describes reality) remains obscure. [[Eugene Wigner]]'s paper, "[[The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]", is a very well-known account of the issue from a Nobel Prize-winning physicist. In fact, some observers (including some well-known mathematicians such as [[Gregory Chaitin]], and others such as [[Where Mathematics Comes From|Lakoff and Núñez]]) have suggested that mathematics is the result of practitioner bias and human limitation (including cultural ones), somewhat like the post-modernist view of science.<ref name= WMCF >George Lakoff and Rafael E. Núñez (2000) [[Where Mathematics Comes From]] </ref>

[[George Pólya]]'s work on [[problem solving]],<ref name= findIt >"If you can't solve a problem, then there is an easier problem you can solve: find it." —{{harvp|Pólya|1957|p=114}}</ref> the construction of mathematical [[Mathematical proof|proofs]], and [[heuristic]]<ref>
George Pólya (1954), ''Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics''.
</ref><ref>
George Pólya (1954), ''Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Reasoning''.
</ref> show that the mathematical method and the scientific method differ in detail, while nevertheless resembling each other in using iterative or recursive steps.

{| class="wikitable"
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!scope="col"|[[How to Solve It|Mathematical method]]
!scope="col"|[[#Elements of the scientific method|Scientific method]]
|-
!scope="row"|1
| [[Understanding]]
| [[#Characterizations|Characterization from experience and observation]]
|-
!scope="row"|2
| [[Analysis]]
| [[#Hypothesis development|Hypothesis: a proposed explanation]]
|-
!scope="row"|3
| [[wikt:synthesis|Synthesis]]
| [[#Predictions from the hypothesis|Deduction: prediction from the hypothesis]]
|-
!scope="row"|4
| [[Review]]/[[Generalization|Extend]]
| [[#Experiments|Test and experiment]]
|}
{{anchor|polyaFirstUnderstand}}
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| title = Charles A. Weibel (ca. 1995) 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}}

[[Carl Friedrich Gauss|Gauss]], when asked how he came about his [[theorem]]s, once replied "durch planmässiges Tattonieren" (through [[Constructivism (mathematics)|systematic palpable experimentation]]).{{sfnp|Mackay|1991|p=100}}
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