THE PHILOSOPHY OF MATHEMATICS: SPECIAL REFERENCE TO GEOMETRY & INFINITESIMAL METHOD

By SOPHIA MARTINEZ

CONTENTS LOGIC; A FIRST IMPRESSION GENERAL PROPOSITIONAL LOGIC SETS; FINITE AND IN?NITE PREDICATE LOGIC ARITHMETIC; G¨ODEL’S INCOMPLETENESS THEOREM MODAL LOGIC PHILOSOPHY OF LANGUAGE INTUITIONISM AND INTUITIONISTIC LOGIC APPLICATIONS PROGRAMMING IN LOGIC: PROLOG RELATIONAL DATABASES SOCIAL CHOICE THEORY 1 10 FALLACIES AND UNFAIR DISCUSSION METHODS 1 11 SOLUTIONS REFERENCES LINGUISTIC CONSIDERATIONS SEMANTICS; TRUTH TABLES VALIDITY SEMANTICS; LOGICAL (VALID) CONSEQUENCE DECIDABILITY SOUND VERSUS PLAUSIBLE ARGUMENTS; ENTHYMEMES SEMANTICS: META-LOGICAL CONSIDERATIONS ABOUT TRUTHFUNCTIONAL CONNECTIVES APPLICATIONS IN ELECTRICAL ENGINEERING AND IN JURISDICTION NORMAL FORM?; LOGIC PROGRAMMING? TRAVELLING SALESMAN PROBLEM (TSP)?; NP-COMPLETENESS? SYNTAX: PROVABILITY AND DEDUCIBILITY SYNTAX: META-LOGICAL RESULTS DEDUCTION THEOREM; INTRODUCTION AND ELIMINATION RULES NATURAL DEDUCTION? TABLEAUX COMPLETENESS OF CLASSICAL PROPOSITIONAL LOGIC 2 10 PARADOXES; HISTORICAL AND PHILOSOPHICAL REMARKS 2 10 1 PARADOXES 2 10 2 HISTORICAL AND PHILOSOPHICAL REMARKS 2 11 SOLUTIONS SETS: ?NITE AND IN?NITE RUSSELL’S PARADOX AXIOMS OF ZERMELO-FRAENKEL FOR SETS HISTORICAL AND PHILOSOPHICAL REMARKS MATHEMATICS AND THEOLOGY ONTOLOGY OF MATHEMATICS ANALYTIC-SYNTHETIC LOGICISM RELATIONS, FUNCTIONS AND ORDERINGS? ORDERED PAIRS AND CARTESIAN PRODUCT RELATIONS EQUIVALENCE RELATIONS FUNCTIONS ORDERINGS STRUCTURES AND ISOMORPHISMS THE HILBERT HOTEL; DENUMERABLE SETS NON-ENUMERABLE SETS SOLUTIONS PREDICATE LANGUAGE QUANTI?ERS, INDIVIDUAL VARIABLES AND CONSTANTS TRANSLATING ENGLISH INTO PREDICATE LOGIC, INTENDED AND NON-INTENDED INTERPRETATION SCOPE, BOUND AND FREE VARIABLES ALPHABET AND FORMULAS SEMANTICS: TARSKI’S TRUTH DE?NITION; LOGICAL (VALID) CONSEQUENCE BASIC RESULTS ABOUT VALIDITY AND LOGICAL CONSEQUENCE QUANTI?ERS AND CONNECTIVES TWO DIFFERENT QUANTI?ERS ABOUT THE AXIOMS AND RULES FOR ?AND ? PREDICATE LOGIC WITH FUNCTION SYMBOLS? PRENEX FORM? SKOLEMIZATION, CLAUSAL FORM? NATURAL DEDUCTION COMPLETENESS, COMPACTNESS AND L¨OWENHEIM-SKOLEM UNDECIDABILITY COMPACTNESS AND L¨OWENHEIM-SKOLEM THEOREMS SECOND-ORDER LOGIC SKOLEM’S PARADOX PREDICATE LOGIC WITH EQUALITY ABOUT THE RELATION OF LOGIC WITH OTHER DISCIPLINES LOGIC AND PHILOSOPHY OF LANGUAGE LOGIC AND PHILOSOPHY OF SCIENCE LOGIC AND ARTI?CIAL INTELLIGENCE; PROLOG ARISTOTLE’S ORGANON ARITHMETIC: G¨ODEL’S INCOMPLETENESS THEOREMS FORMALIZATION OF ELEMENTARY NUMBER THEORY G¨ODEL’S ?RST INCOMPLETENESS THEOREM G¨ODEL-NUMBERING PROVABILITY PREDICATE FOR P G¨ODEL’S SECOND INCOMPLETENESS THEOREM IMPLICATIONS OF G¨ODEL’S INCOMPLETENESS THEOREMS NON-STANDARD MODELS OF PEANO’S ARITHMETIC SECOND-ORDER LOGIC (CONTINUED) MODAL OPERATORS DIFFERENT SYSTEMS OF MODAL LOGIC POSSIBLE WORLD SEMANTICS EPISTEMIC LOGIC MUDDY CHILDREN PUZZLE; REASONING ABOUT KNOWLEDGE TABLEAUX FOR MODAL LOGICS APPLICATIONS OF POSSIBLE WORLD SEMANTICS DIRECT REFERENCE RIGID DESIGNATORS DE DICTO - DE RE DISTINCTION REASONING ABOUT KNOWLEDGE COMMON KNOWLEDGE COMPLETENESS OF MODAL PROPOSITIONAL LOGIC STRICT IMPLICATION COUNTERFACTUALS 6 11 MODAL PREDICATE LOGIC 6 11 1 MODAL PREDICATE LOGIC AND ESSENTIALISM 6 12 THE MODAL LOGIC GL 6 13 SOLUTIONS USE AND MENTION FREGE’S SINN UND BEDEUTUNG (SENSE AND REFERENCE) MANNOURY (1867-1956), SIGNI?CS SPEECH ACTS DE?NITE DESCRIPTIONS BERRY’S AND GRELLING’S PARADOX THE THEORY OF DIRECT REFERENCE ANALYTIC - SYNTHETIC 7 10 LOGICAL POSITIVISM 7 11 PRESUPPOSITIONS 7 12 WITTGENSTEIN ON MEANING 7 13 SYNTAX - SEMANTICS - PRAGNATICS 7 14 CONVERSATIONAL IMPLICATURE 7 15 CONDITIONALS 7 16 LEIBNIZ 7 17 DE DICTO - DE RE 7 18 GRAMMARS 7 19 SOLUTIONS INTUITIONISM VS PLATONISM; BASIC IDEAS LANGUAGE FIRST STEPS IN INTUITIONISTIC REASONING INTUITIONISTIC PROPOSITIONAL LOGIC: SYNTAX TABLEAUX FOR INTUITIONISTIC PROPOSITIONAL LOGIC INTUITIONISTIC PROPOSITIONAL LOGIC: SEMANTICS COMPLETENESS OF INTUITIONISTIC PROPOSITIONAL LOGIC QUANTI?ERS IN INTUITIONISM; INTUITIONISTIC PREDICATE LOGIC DEDUCIBILITY FOR INTUITIONISTIC PREDICATE LOGIC TABLEAUX FOR INTUITIONISTIC PREDICATE LOGIC KRIPKE SEMANTICS FOR INTUITIONISTIC PREDICATE LOGIC SOUNDNESS AND COMPLETENESS SETS IN INTUITIONISM: CONSTRUCTION PROJECTS AND SPREADS THE BROUWER KRIPKE AXIOM 6 10 WEAK AND RELEVANT IMPLICATION; ENTAILMENT? APPLICATIONS: PROLOG; RELATIONAL DATABASES AND SQL; SOCIAL CHOICE THEORY PROGRAMMING IN LOGIC RECURSION DECLARATIVE VERSUS PROCEDURAL PROGRAMMING SYNTAX MATCHING VERSUS UNI?CATION LISTS, ARITHMETIC CUT NEGATION AS FAILURE PITFALLS RELATIONAL DATABASES AND SQL SQL SOCIAL CHOICE THEORY; MAJORITY JUDGMENT INTRODUCTION PLURALITY RULE (PR): MOST VOTES COUNT MAJORITY RULE (MR): PAIRWISE COMPARISON BORDA RULE (BR) OUTCOME DEPENDS ON THE VOTING RULE ARROW’S IMPOSSIBILITY THEOREM DOMINATION MAJORITY JUDGMENT (MJ) PROPERTIES OF MAJORITY JUDGMENT 9 3 10 POINT SUMMING AND APPROVAL VOTING 9 3 11 MAJORITY JUDGMENT WITH MANY VOTERS 9 3 12 PRESIDENTIAL ELECTIONS IN THE USA 9 3 13 PRESIDENTIAL ELECTIONS IN FRANCE 9 3 14 ELECTIONS FOR PARLIAMENT IN THE NETHERLANDS FALLACIES AND UNFAIR DISCUSSION METHODS 10 1 INTRODUCTION 10 2 FALLACIES 10 2 1 CLICH´ES AND KILLERS 10 2 2 IMPROPER OR HASTY GENERALIZATIONS 10 2 3 THINKING SIMPLISTICALLY 10 2 4 APPEAL TO IGNORANCE 10 2 5 SPECULATIVE THINKING 10 2 6 INCREDULITY 10 2 7 THE USE OF TERMS WITH A VAGUE MEANING 10 2 8 THE DANGER OF WORDS WITH MORE THAN ONE MEANING 10 2 12RATIONALIZING 10 2 13AFTER THIS, THEREFORE BECAUSE OF THIS 10 3 UNFAIR DISCUSSION METHODS 10 3 1 PUSHING SOMEONE INTO AN EXTREME CORNER 10 3 2 STRAW MAN ARGUMENT 10 3 3 DIVERSION MANEUVERS 10 3 4 SUGGESTIVE METHODS 10 3 5 EITHER/OR FALLACY 10 3 6 THE TREACHEROUS PARADOX 10 3 7 AD HOMINEM ARGUMENTS 10 3 8 ARGUMENTUM AD BACULUM 10 3 9 SECRECY 10 3 10THE RETIREMENT HOME’S DISCUSSION 10 4 SUMMARY 10 2 9 APRIORISTIC REASONING 10 2 10CIRCULAR REASONING 10 2 11APPLYING DOUBLE STANDARDS