Evidence and Refutation

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Proofs and Refutations (German title: Proofs and Refutations ) is a work by Imre Lakatos (1922–1974) in the philosophy of mathematics . It was published in four parts from 1963 to 1964, and came out as a book in 1976.

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It is a detailed case study of evidence analysis. In an imaginary classroom, a teacher and his "fairly advanced" class try to prove Euler's polyhedron substitution . Lakatos takes up various historical evidence approaches and puts them in the mouths of the students. These in turn are criticized by other students. Some students bring counterexamples that test the boundaries of the concept and the sentence. Various methods are discussed to deal with these examples. These methods are:

Due to a counterexample, the assumption is completely abandoned.
Monster lock
The counterexample is declared a monster. The definitions are subsequently changed (here: the term polyhedron) in order to exclude the monster.
Exception lock
In contrast to the monster lock, the counter-example is not denied the right to exist. Instead, adjust the sentence by listing all exceptions.
Monster customization
The statements of the sentence are reinterpreted until they also apply to the counterexample.
Sentences incorporation
If a proposition shows counterexamples in the proof, then the assumption is added to the assumption that the statement of the proposition is fulfilled.
Evidence and Refutations
Lakatos summarizes the procedure to three heuristic rules:
Rule 1: Develop a naive proof of your guess. Using evidence analysis, break this proof down into clauses. Look for counterexamples for the conjecture itself ("global counterexamples") and for the individual auxiliary clauses ("local counterexamples").
2. Rule: Respond to a global counterexample with surrender or the incorporation of auxiliary sentences. Reject a monster ban. Make sure you haven't overlooked any “hidden” clauses.
3rd rule: Check every local counterexample whether it is not also a global counterexample.

A second case study is discussed in an appendix: The aftermath in the 19th century of Cauchy's “proof” that the limit of a convergent series of continuous functions is always continuous.


Many authors have discussed possible applications in mathematics didactics and in teacher training. The term “monster lock” was also taken up in law.



  • WV Quine : Lakatos, I. [1976]: Proofs and Refutations: The Logic of Mathematical Discovery . In: British Journal for the Philosophy of Science . tape 28 , no. 1 . Oxford University Press, March 1977, ISSN  0007-0882 , pp. 81-82 , doi : 10.1093 / bjps / 1/28/81 .
  • David Corfield : Towards a Philosophy of Real Mathematics . Cambridge University Press, 2003, ISBN 0-521-03525-2 (especially chapter 7).

Individual evidence

  1. Imre Lakatos , edited by John Worrall and Elie Zahar: Evidence and Refutations: The Logic of Mathematical Discoveries (=  philosophy of science, science and philosophy . No. 14 ). Vieweg Verlag , Braunschweig 1979, ISBN 3-528-08392-1 , here pp. 8-43 (English: Proofs and Refutations: The Logic of Mathematical Discovery . London 1976. Translated by Detlef D. Spalt).
  2. ^ Evidence and Refutations , p. 8
  3. ^ Evidence and Refutation , p. 9
  4. ^ Evidence and Refutation , p. 18
  5. ^ Evidence and Refutations , p. 24
  6. ^ Evidence and Refutations , p. 27
  7. ^ Evidence and Refutations , p. 40
  8. ^ A b Albrecht Beutelspacher , Rainer Danckwerts, Gregor Nickel, Susanne Spies, Gabriele Wickel: New thinking in mathematics: impulses for high school teacher training at universities . Vieweg + Teubner, 2011, ISBN 978-3-8348-1648-1 , here p. 77 ( limited preview in Google book search [accessed January 17, 2018]).
  9. David Pimm, Mary Beisiegel, Irene Meglis: Would the Real Lakatos Please Stand Up . In: Interchange . tape 39 , no. 4 , October 2008, ISSN  0826-4805 , p. 469-481 , doi : 10.1007 / s10780-008-9076-x .
  10. Stephan Berendonk: About a gap in Lakatos' "Evidence and Refutation" . In: Explorations on Euler's polyhedron replacement: Genetic, explorative, vivid . Springer Spectrum, 2014, ISBN 978-3-658-04598-2 , chap. 2 , p. 41-84 .
  11. Thomas Schlapp: Theory structures and legal dogmatics: Approaches to a structuralist view of legal theory formation (=  writings on legal theory . No. 138 ). Duncker & Humblot , Berlin 1989, p. 88 ( restricted preview in the Google book search [accessed on August 12, 2016] At the same time dissertation, University of Frankfurt (Main) , 1988).