The Logical Syntax of Greek Mathematics (English and Greek Edition)

Description

The aim of this monograph is to describe Greek mathematics as a literary product, studying its style from a logico-syntactic point of view and setting parallels with logical and grammatical doctrines developed in antiquity. In this way, major philosophical themes such as the expression of mathematical generality and the selection of criteria of validity for arguments can be treated without anachronism. Thus, the book is of interest for both historians of ancient philosophy and specialists in Ancient Greek, in addition to historians of mathematics. This volume is divided into five parts, ordered in decreasing size of the linguistic units involved. The first part describes the three stylistic codes of Greek mathematics; the second expounds in detail the mechanism of "validation"; the third deals with the status of mathematical objects and the problem of mathematical generality; the fourth analyzes the main features of the "deductive machine," i.e. the suprasentential logical system dictated by the traditional division of a mathematical proposition into enunciation, setting-out, construction, and proof; and the fifth deals with the sentential logical system of a mathematical proposition, with special emphasis on quantification, modalities, and connectors. A number of complementary appendices are included as well.  TABLE OF CONTENTS LIMINALIA 1. THE THREE STYLISTIC CODES OF GREEK MATHEMATICS 1.1. THE DEMONSTRATIVE CODE 1.2. THE PROCEDURAL CODE 1.3. THE ALGORITHMIC CODE 1.4. PUNCTUATING GREEK MATHEMATICAL TEXTS 1.5. THE ELEMENTS AND ITS LEXICAL CONTENT 2. VALIDATION AND TEMPLATES 2.1. ARISTOTLE AND GALEN ON LINGUISTIC TEMPLATES 2.2. SUBSENTENTIAL VALIDATION: FORMULAIC TEMPLATES 2.3. SENTENTIAL VALIDATION: SYNTACTIC TEMPLATES 2.4. LARGE-SCALE VALIDATION: ANALYSIS AND SYNTHESIS 2.4.1. Geometric analysis and synthesis 2.4.2. Validating algorithms and procedures by the “givens” 3. THE PROBLEM OF MATHEMATICAL GENERALITY 3.1. THE PRESENTIAL VALUE OF THE VERB “TO BE” IN THE SETTING-OUT 3.2. THE FUNCTION OF THE DENOTATIVE LETTERS 3.3. THE INDEFINITE STRUCTURE 3.4. ONTOLOGICAL COMMITMENT 3.5. OVERSYMMETRIZED DIAGRAMS 4. THE DEDUCTIVE MACHINE 4.1. ENUNCIATION AND CONCLUSION 4.2. SUPPOSITIONS AND “SETTING-OUT” 4.2.1. Determination 4.3. THE ROLE OF CONSTRUCTIONS 4.4. ANAPHORA 4.5. PROOF 4.5.1. The logic of relations 4.5.2. Metamathematical markers: potential and analogical proofs, references to the obvious, optative mood, personal verb forms 4.5.3. Postposed arguments 4.5.4. Instantiated and non-instantiated citations of theorems 4.5.5. Assumptions and coassumptions 5. THE LOGICAL SYNTAX 5.1. QUANTIFICATION; IMPLICIT AND EXPLICIT GENERALITY 5.1.1. Quantifiers 5.1.2. Determiners of arbitrariness 5.1.3. Determiners of indefiniteness 5.1.4. Generalizing qualifiers 5.1.5. The use of the article 5.1.6. Ordinals as variables 5.1.7. The indefinite conditionals of Stoic logic 5.2. MODALS 5.2.1. Reductions to the impossible 5.2.2. Arguments “for a contrapositive” 5.3. SENTENTIAL OPERATORS 5.3.1. Conditional 5.3.2. Paraconditional 5.3.3. Negation 5.3.4. Disjunction 5.3.5. Conjunction 5.3.6. Syllogistic connectors APPENDICES APPENDIX A. PROBLEMS IN THE GREEK MATHEMATICAL CORPUS APPENDIX B. THEOREMS OF THE DATA THAT HAVE A SYNTHETIC COUNTERPART AND EXTANT SOURCES ON GREEK ANALYSIS AND SYNTHESIS APPENDIX C. ONOMASTICON BIBLIOGRAPHY INDICES INDEX NOMINUM INDEX FONTIUM INDEX LOCORUM INDEX RERUM