QuoteRef: lakaI_1976

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references i-l
philosophy of mathematics
mathematics by proofs and refutations
language and life as a game
philosophy of mind
history of science
minimal manuals and guided exploration
definition by example
philosophy of science
skepticism about knowledge
infinity and infinitesimal
problems with analytic truth
mathematical proof
limitations of formalism
scientific method
mathematics as a formal system
non-constraining system
scientific paradigms and research programs


Lakatos, I., "Proofs and refutations. The logic of mathematical discovery", in Worrall, J., Zahar, E. (ed.), Cambridge, England, Cambridge University Press, 1976 reprinted with corrections 1977, 1979. . Google


in Appendix I shows a similar history of proofs and refutations for the principle of continuity

4 ;;Quote: mathematicians use a situational logic that is neither formal, mechanistic, nor irrational blind guessing
4 ;;Quote: Poincare--study the history of science to learn about the development of minds; like 'embryology recapitulates phylogeny'
5 ;;Quote: informal mathematics does not grow by indubitable theorems but by speculation, proof, and refutation
5 ;;Quote: Lakatos used a Socratic dialogue to reconstruct the history that is referenced in the footnotes
6 ;;Quote: the students guessed that Euler's relation held for all polyhedron, testing didn't falsify; so attempt a proof
8 ;;Quote: Cauchy's proof of Euler's relation used conjectures about planar maps, triangulations, and the effect of removing triangles
9 ;;Quote: proof is a thought-experiment that decomposes a conjecture into lemmas that belong to a body of knowledge
14 ;;Quote: monster-barring: since the theorem was proved, counter-examples can be barred; e.g., a pair of nested cubes is not a polyhedron
24 ;;Quote: a theorem applies to a restricted domain; so counter-examples are exceptions to the conjecture instead of monsters
31 ;;Quote: monster-adjustment: the 'urchin' counterexample was actually a misinterpretation of what are edges, faces, and vertices
32 ;;Quote: Stoic theory of error--false ideas are part of external reality but they can't mature into clear and distinct ideas (science)
34 ;;Quote: lemma-incorporation--modify a conjecture by analyzing why the proof's components failed for a counterexample; i.e., proof and refutations
37 ;;Quote: mathematicians want to improve their conjectures without refuting them; yields a monotonous increase of truth
38 ;;Quote: one should scrutinize a proof to see if any assumptions can be removed; use counterexamples to show the necessary ones
40 ;;Quote: the process of lemma-incorporation yields an infinite regress unless proof is a game or there are trivially true lemmas
41 ;;Quote: lemma-incorporation results in a theorem that is defined by the proof
48 ;;Quote: should prove self-evident conjectures since they may hinge on a dubious lemma whose refutation leads to a global refutation
50 ;;Quote: method of proof and refutation: 1) inspect proof for non-trivial lemmas and seek local and global counterexamples
50 ;;Quote: method of proof and refutations: 2) if a global counterexample, make the hidden lemma explicit and a condition of the conjecture
56 ;;Quote: in Frege's and Russell's systems can give infallible proofs if the axioms and translation are correct; but both are fallible
58 ;;Quote: method of proof and refutations: 4) if a truly local counterexample, try to replace the refuted lemma and increase the theorem's content
62 ;;Quote: SIMPLICIO: heavenly and earthly phenomena must be different; the same proof can not explain the course of planets and projectiles
63 ;;Quote: as well as certainty, want finality in a theorem; e.g., one that covers all polyhedra that satisfy Euler's relation
64 ;;Quote: proofs and refutations--to improve a conjecture may need many proofs; thus expanding the conjecture's domain
64 ;;Quote: Pappian heuristic--derive consequences of a conjecture; if true, rederive the original conjecture, thus yielding a final truth; failed for science
70 ;;Quote: can deductively guess Euler's relation from V=E for polygons; no need for facts and data
73 ;;Quote: naive conjectures are arrived at by trial and error; e.g., used in building a table for Euler's relation
74 ;;Quote: mathematical heuristic is like scientific heuristic but with different conjectures, explanations, and counterexamples
74 ;;Quote: if you have too much respect for facts that refute your conjectures, you may miss conjectures that can be tested with thought-experiments
76 ;;Quote: proofs and refutations: 5) if you have counterexamples, try to find, by deductive guessing, a deeper theorem that handles them
89 ;;Quote: proof-generated concepts replace the original naive concepts completely; very different
107 ;;Quote: in a formalized theory, the tools are completely prescribed by its syntax; in proof-analysis, tools are unconstrained
114 ;;Quote: chain spaces is the proof-generated concept that yields an indubitable formulation of Euler's theorem
122 ;;Quote: if require a Euclidean programme with perfectly known concepts, must translate vague concepts and perhaps lose part of their meaning
123 ;;Quote: a dominant theory is one with perfectly known terms and infallible inferences; incorporating other, non-Euclidean theories enriches their development
124 ;;Quote: a formal proof does not hinge on the meaning of specific terms; the meaning is carried by the non-specific, arithmetic/logical terms
125 ;;Quote: new theories replace the dominant, mathematical theories; e.g., arithmetic to geometry to algebra to arithmetic, probability to measure theory
139 ;;Quote: proofs and refutations--a proof can be respectable without being flawless; Seidel 1847, c.f. Hegel and Popper
152 ;;Quote: should teach mathematics by showing the development of proof-generated concepts from naive conjectures; dispels mysticism of mathematics

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