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