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Computerbewijzen in De Wiskundige Praktijk

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Computerbewijzen in De Wiskundige Praktijk HOGER INSTITUUT VOOR WIJSBEGEERTE KATHOLIEKE UNIVERSITEIT LEUVEN COMPUTERBEWIJZEN IN DE WISKUNDIGE PRAKTIJK Promotor: prof. dr. Leon Horsten verhandeling aangeboden tot het verkrijgen van de graad van Licentiaat in de Wijsbegeerte door: Koen Vervloesem Leuven, 2007 HOGER INSTITUUT VOOR WIJSBEGEERTE KATHOLIEKE UNIVERSITEIT LEUVEN COMPUTERBEWIJZEN IN DE WISKUNDIGE PRAKTIJK Promotor: prof. dr. Leon Horsten verhandeling aangeboden tot het verkrijgen van de graad van Licentiaat in de Wijsbegeerte door: Koen Vervloesem Leuven, 2007 Dankwoord Deze eindverhandeling is niet alleen het resultaat van mijn eigen inspanningen, maar ook van heel wat hulp die ik gekregen heb. Ik wil dan ook graag enkele mensen bedanken die mij tijdens het schrijven van mijn eindverhandeling hebben gesteund. Ik bedank in het bijzonder mijn promotor Leon Horsten, die me voor het kiezen van dit onderwerp geïn- spireerd heeft door me de boeken New Directions in the Philosophy of Mathematics en Philosophy of Mathematics: an Anthology in mijn handen te drukken. Ik bedank hem ook voor de vruchtbare discussies en de waardevolle commentaar op mijn ideeën en kladver- sies. Hij spoorde me ook aan om mijn idee voor Hoofdstuk 9, Wiskundige concepten in computerbewijzen uit te werken voor een presentatie op de Perspectives on Mathematical Practices 2007 conferentie. Dank ook aan de organisatoren van de Perspectives on Mathematical Practices 2007 conferentie die mijn voorstel voor een presentatie aanvaardden. Dank ook aan verschil- lende deelnemers van de conferentie, waarmee ik interessante discussies had over mijn presentatie. Mijn bijzondere dank gaat uit naar Paolo Mancosu, die me aanraadde om mijn bevindingen meer te situeren binnen de bestaande filosofische literatuur van mathe- matical explanation en die me daarover een uitgebreide bibliografie bezorgde. De moge- lijkheid om een hoofdstuk uit mijn eindverhandeling te presenteren op deze conferentie heeft me verplicht om het duidelijker uit te werken en dat is de kwaliteit ervan zeker ten goede gekomen. Dank ook aan Stijn Janssens, Marijke Van Bogaert, Lorenz Demey en Jo Allemeersch voor onze boeiende en gezellige leesgroep. Tot slot wil ik mijn vriendin Liesbeth bedan- ken die me een jaar lang heel wat avonden heeft zien doorbrengen achter de computer, al zoekend, lezend en typend voor deze eindverhandeling, en ook op andere momenten vaak met mijn gedachten verkerend in een platonische wereld. Ik bedank haar ook voor het minutieus nalezen van mijn verhandeling en de hulp bij het opstellen van de bijlagen. i Inhoudsopgave 1. Inleiding ................................................................................................................ 1 1.1. Wat is een bewijs? ..................................................................................... 2 1.2. De status van bepaalde bewijsmethodes .................................................... 5 1.3. Wat is een computerbewijs? ...................................................................... 6 1.4. De mogelijkheid van computerbewijzen na Gödel en Turing ................... 6 1.5. Berekeningen versus redeneringen. ........................................................... 8 1.6. Computerbewijzen als een nieuwe ontwikkeling in wiskunde .................. 9 1.7. Methodologie ........................................................................................... 10 1.8. Het computerbewijs van de stelling van Pappus ...................................... 12 1.9. Inhoud ...................................................................................................... 16 I. Computerbewijzen en hun receptie ..................................................................... 17 2. De vierkleurenstelling ................................................................................. 18 2.1. Het probleem en de eerste pogingen tot bewijs ............................... 18 2.2. De basisideeën van het bewijs ......................................................... 21 2.3. Het bewijs van Appel en Haken ....................................................... 23 2.4. Het bewijs van Allaire ..................................................................... 24 2.5. Het bewijs van Robertson ................................................................ 25 2.6. Het formeel bewijs van Gonthier ..................................................... 26 2.7. Het intrinsieke belang van de vierkleurenstelling ............................ 27 2.8. De complexiteit van de vierkleurenstelling ..................................... 28 2.9. De receptie van de bewijzen bij filosofen en wiskundigen .............. 28 2.9.1. Een computerbewijs is geen bewijs ...................................... 29 2.9.2. Computerbewijzen veranderen het concept van bewijs ........ 30 2.9.3. Een computerbewijs is een bewijs ........................................ 33 2.9.4. Een computerbewijs is geen bevredigend bewijs ................. 38 2.9.5. A priori kennis door computerbewijzen ............................... 39 2.9.6. Locale versus globale inspecteerbaarheid ............................. 41 3. Er bestaat geen eindig projectief vlak van orde 10 ..................................... 43 3.1. Het probleem en zijn geschiedenis .................................................. 43 3.2. Het computerbewijs ......................................................................... 48 3.3. De receptie van het bewijs ............................................................... 50 4. Het Robbinsprobleem ................................................................................. 52 4.1. Het probleem en zijn geschiedenis .................................................. 52 4.2. Het computerbewijs van McCune .................................................... 53 4.3. Logisch redeneren ............................................................................ 55 4.4. De receptie van het bewijs ............................................................... 56 4.4.1. Vertalingen van het bewijs .................................................... 57 4.4.2. De waarde van het bewijs ..................................................... 59 ii 5. Het Keplervermoeden ................................................................................. 61 5.1. Het probleem en zijn geschiedenis .................................................. 61 5.2. Het computerbewijs van Hales ........................................................ 64 5.3. Naar een formeel bewijs van het Keplervermoeden ........................ 66 5.4. De receptie van het bewijs ............................................................... 68 6. Probabilistische computerbewijzen ............................................................ 72 6.1. Probabilistische priembewijzen ....................................................... 72 6.1.1. De bewijzen .......................................................................... 72 6.1.2. De receptie van de bewijzen ................................................. 73 6.2. DNA-berekeningen .......................................................................... 78 6.2.1. Het probleem en zijn DNA-bewijs ....................................... 78 6.2.2. De receptie van de methode .................................................. 81 7. Andere computerbewijzen .......................................................................... 83 7.1. Symbolische berekeningen .............................................................. 83 7.2. Combinatorische oplossingen .......................................................... 84 7.3. Numerieke benaderingen van ongelijkheden ................................... 87 7.4. Axiomatische bewijsproblemen ....................................................... 90 II. Inzicht in computerbewijzen .............................................................................. 93 8. Een classificatie van bewijstechnieken in computerbewijzen .................... 97 8.1. Symbolische berekeningen .............................................................. 98 8.2. Combinatorische oplossingen .......................................................... 98 8.3. Numerieke benaderingen van ongelijkheden ................................... 99 8.4. Logische bewijstechnieken ............................................................ 100 8.5. Probabilistische bewijstechnieken ................................................. 101 8.6. Conclusie ........................................................................................ 101 9. Wiskundige concepten in computerbewijzen ........................................... 104 9.1. Inleiding ......................................................................................... 104 9.2. Bewijzen en concepten .................................................................. 105 9.2.1. Cassini's identiteit bij Fibonaccigetallen ............................. 105 9.2.1.1. Bewijs door inductie ................................................ 106 9.2.1.2. Bewijs door Binets formule ..................................... 106 9.2.1.3. Bewijs door determinant .......................................... 107 9.2.1.4. De concepten in de bewijzen ................................... 107 9.2.2. Fermatpriemgetallen ........................................................... 108 9.2.2.1. Bewijs door berekening ........................................... 109 9.2.2.2. Bewijs door modulaire congruenties ....................... 109 9.2.2.3. Bewijs door eigenschappen van delers .................... 109 9.2.2.4. De concepten
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