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pres_280611_lakatos.odt Heuristics in mathematical discovery 1/21Imre Lakatos: Proofs and Refutations "One of the things were sure to discover is that the psychological processes dont parallel the logical processes." "[...] the steps of justification are not the steps of discovery." (Larry E. Travis: The value of introspection to the designer of mechanical problem solvers)Topic: • a historic account of how mathematical proofs are being found • in contrast to formalist accounts of metamathematics / philosophy of mathematics
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pres_280611_lakatos.odt Heuristics in mathematical discovery 2/21Imre LakatosHungarian mathematician & philosopherBorn Lipschitz, later Molnártranslated Polya (how to solve it) into hungarianFlew to GB after WW21960 at London School of Economics, where he met Popper and other philosophers of scienceWas friends with P. FeyerabendPhilosophy of science similar to KuhnDissertation in CambridgePost-mortem published: “Proofs and Refutations”
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pres_280611_lakatos.odt Heuristics in mathematical discovery 3/21Metamathematicsformalized proofs (Eucledian methodology)→ Mathematical notions are defined by a set of axioms, and theorems are derived by a fixed set of inference rules . Russel & Whiteheads Principia Mathematica (1910-1913) demonstrated the feasibility of such an approach. → automated theorem provers possibleOne consequence of the formalist foundation of mathematics is that strong proponentsdo only consider theorems derived by deductive proofs as mathematically meaningful. → therefore, “Gödelian theorems” would be mathematically not meaningful! (Über formal unentscheidbare Sätze der Principia Mathematica ... 1931)
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pres_280611_lakatos.odt Heuristics in mathematical discovery 4/21MetamathematicsDoes this form of proof reflect their discovery? • Lakatos: No mathematician would embark into enumerating theorems. • Automated proofs are syntactic, while human reasoning is semantic (but what is “semantic”?) • Examples and counter-examples play an important role in the development of mathematical concepts • Errors in faulty proofs may remain undetected (even over generations)Lakatos e.g. complains that one is never being told how funding axioms arose. Mathbooks dont show how proofs are found, and they hinder to develop creative, explorativethinking.→ Lakatos claim: Informal mathematics grows by a logic of proofs and refutations.→ mathematics in an experimental style
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pres_280611_lakatos.odt Heuristics in mathematical discovery 5/21Lakatos, I. (1976).Proofs and Refutations: The Logic of Mathematical Discovery 1. A Problem and a Conjecture 2. A Proof 3. Criticism of the Proof by Counterexamples which are Local but not Global 4. Criticism of the Conjecture by Global Counterexamples
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pres_280611_lakatos.odt Heuristics in mathematical discovery 6/21Eulers formula for polyhedra(1752) V+E–F=2Where does the initial conjecture come from? Polya: “You have to guess a mathematical theorem before you prove it” → Lakatos starts where Polya ends.
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pres_280611_lakatos.odt Heuristics in mathematical discovery 7/21Cauchys proof(1813) • Step 1: remove a face and stretch it to 2-D • Step 2: triangulate the planar graph • Step 3: remove triangles ▪ a) one edge & face ▪ b) one vertex, two edges, one face → decomposes the original conjecture into three lemmas
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pres_280611_lakatos.odt Heuristics in mathematical discovery 8/21socratic dialogueCriticism of the proof by a local counterexample • A proposal from the class: call it a theorem, not a conjecture. • But the teacher prefers to call it a thought experiment / quasi-experiment.→ The students doubt proof steps 1, 2, 3Gamma: A counterexample to step 3: Remove a triangle from the middle. Only one face is subtracted !Teacher: Right. Lemma 3 has to be defined more precisely. But this doesnt mean that the initial conjecture does not hold. If possible, proof step 3 could be modified.
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pres_280611_lakatos.odt Heuristics in mathematical discovery 9/21socratic dialogueCriticism of the proof by a local counterexample→ "local counterexample": one that refutes a lemma, and therefore the proof, but not the main conjecture.Teacher: Remove only external boundaries !But Gamma finds a way to remove them in an order which changes V-E+F=2 .Teacher: Ok, so Ill restrict removal order such that V-E+F=2 holds. I claim that this is possible in general!Gamma: But how to find that order? Can you proof that there is a general way?
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pres_280611_lakatos.odt Heuristics in mathematical discovery 10/21socratic dialogueCriticism of the proof by a local counterexample→ what is the sense of such a proof? Arent we worse of because we now have manyconjectures about which we are uncertain?Teacher: No, we have embedded our conjecture into knowledge about graphs, triangulation, and so forth and therefore have more “working points” for our proof. "decomposition opens new vistas for testing [...] deploys the conjecture on a wider front so that our criticism has more targets [...] we now have at least three opportunities for counterexamples instead of one"
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pres_280611_lakatos.odt Heuristics in mathematical discovery 11/21Criticism of the proof by a global counterexample
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pres_280611_lakatos.odt Heuristics in mathematical discovery 12/21Criticism of the proof by a global counterexampleCounterexample 1: A cube within a cube(Lhuilier 1812-13)V-E+F=2 does not hold ?→ (a) Rejection of the conjecture. The method of surrender
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pres_280611_lakatos.odt Heuristics in mathematical discovery 13/21Criticism of the proof by a global counterexampleIdea: Define the initial concepts such that they exclude the counterexamplesDef. 1: A polyhedron is a solid bounded by polygonal surfacesDef. 2: ... is a surface, on which a point can continuously move (topological). The nested-cube is in fact two cubes!→ (b) Rejection of the counterexample. The method of monster-barring
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pres_280611_lakatos.odt Heuristics in mathematical discovery 14/21Criticism of the proof by a global counterexampleMore counterexamples: (connected tetrahedrons; picture frame; urchin)→ V – E + F = 2 does not hold!
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pres_280611_lakatos.odt Heuristics in mathematical discovery 15/21Criticism of the proof by a global counterexampleMore counterexamples: (connected tetrahedrons; picture frame; urchin)Def 3: a) exactly two polygons adjunct at an edge; b) every crossing an edge from on polygonn to the other does not cross a vertexDef 4: for every point on the surface, there is a plane which dissects the polyhedron such that the dissection is one single polygon... - convex polyhedrons!The method of monster-barring creates ad-hoc adjustments to every new counter- example. But it cannot guarantee that there wont be any more counterexamples!
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pres_280611_lakatos.odt Heuristics in mathematical discovery 16/21Criticism of the proof by a global counterexampleAnother idea: exceptions“All polyhedra that have no cavities, tunnels, or multiple structure” → Exception-barring methods: piecemeal exclusions“All convex polyhedra are Eulerian.” → Strategic withdrawal or playing for safety→ (c) Improving the conjecture by exception-barring methods. Piecemeal exclusions. Strategic withdrawal or playing for safety
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pres_280611_lakatos.odt Heuristics in mathematical discovery 17/21Criticism of the proof by a global counterexampleIdea: Monsters are misinterpreted. Re-interpret them correctly, and the conjecture andproof will hold. (Example: urchin)→ (d) The method of monster-adjustment
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pres_280611_lakatos.odt Heuristics in mathematical discovery 18/21Criticism of the proof by a global counterexampleIdea: Establish a unity between proofs and counterexamples.→ Lemma 1 does not holdSolution: Adjust the lemma on the basis of the counterexample.Teacher: The Euler conjecture holds good for simple polyhedra, i.e. for those which, after having had a face removed, can be stretched onto a plane.→ (e) Improving the conjecture by the method of lemma-incorporation. Proof-generated theorem versus naive conjecturePease et al.: “This method evolves into proofs and refutations, which is used to findcounterexamples by considering how areas of the proof may be violated.”
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pres_280611_lakatos.odt Heuristics in mathematical discovery 19/21Methods to cope with counterexamples (a) Rejection of the conjecture. The method of surrender (b) Rejection of the counterexample. The method of monster-barring (c) Improving the conjecture by exception-barring methods. Piecemeal exclusions. Strategic withdrawal or playing for safety (d) The method of monster-adjustment (e) Improving the conjecture by the method of lemma-incorporation. Proof-generated theorem versus naive conjecture
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pres_280611_lakatos.odt Heuristics in mathematical discovery 20/21“simple pattern of mathematical discovery”: stages wrap-up (1) Primitive conjecture. (2) Proof (a rough thought experiment or argument, decomposing the primitive conjecture into subconjectures of lemmas). (3) Global counterexamples (counterexamples to the primitive conjecture) emerge. (4) Proof re-examined: The guilty lemma to which the global counterexample is a local counterexample is spotted. This guilty lemma may have been previously hidden or may have been misidentified. Now it is made explicit, and built into the primitive conjecture as a condition. The theorem - the improved conjecture - supersedes the primitive conjecture with the new proof-generated concept as its paramount new feature.Additional stages frequently occur: (5) Proofs of other theorems are examined to see if the newly found lemma or the new proof-generated concept occurs in them: this concept may be found lying at cross-roads of different proofs, and thus emerge as of basic importance. (6) The hitherto accepted consequences of the original and now refuted conjecture are checked. (7) Counterexamples are turned into new examples - new fields of inquiry open up.
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pres_280611_lakatos.odt Heuristics in mathematical discovery 21/21“simple pattern of mathematical discovery”: holds for science, tooLakatos generalises to science:"Now while Popper showed that those who claim that induction is the logic of scientificdiscovery are wrong, these essays intend to show that those who claim that deduction isthe logic of mathematical discovery are wrong. While Popper criticised inductivist stylethese essays try to criticise deductive style.” (p. 143)"Inductivist style reflects the pretence that the scientist starts his investigation with anempty mind whereas in fact he starts with a mind full of ideas.” (p. 143) • in discovery (e.g. of a conjecture and a proof) humans start with a problem, to which they try to find a solution • “problem” and “solution” are more psychological notionsTo conclude, Lakatos gives several examples of mathematical proofs and their “heuristichistory”.
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