Why Did the Greeks Develop Proportion Theory? A Conjecture
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Henry Mendell
Abstract
Greek mathematicians in the late fifth and fourth centuries BC developed number theory based in whole numbers and proportion theory. Their neighbors, Egyptians and Babylonians, did not. We cannot know why, but one significant difference is that the Greeks had no representation system for expressing and calculating with fractions, although they had no difficulty speaking of them as multiple parts. Greek mathematicians would have worked with the acrophonic system, which lacked a system of fractions, and even if they had used Ionic (alphabetic) numerals, there is no evidence of the Ionic fractional system before the end of the fourth century BC. On the other hand, ratios are conceptually a part of Greek thinking, while employment of sub-units and super-units was foundational to arithmetic calculations and puzzles. My suggestion is Greeks worked with what they had, so that these form the foundation for the theory of numbers that we find in Elements VII-IX.20. The evidence for this picture is, admittedly, weak. I consider evidence from Plato’s Republic VIII for a playing with number puzzles, the Archimedes’ Cattle Problem for a class of false position puzzles, and the Anthologia Palatina for Euclid’s Elements as providing a foundation for such puzzles, and Euclid’s Sectio Canonis and Ptolemy’s Harmonica for working from fractions to sub-units.
Abstract
Greek mathematicians in the late fifth and fourth centuries BC developed number theory based in whole numbers and proportion theory. Their neighbors, Egyptians and Babylonians, did not. We cannot know why, but one significant difference is that the Greeks had no representation system for expressing and calculating with fractions, although they had no difficulty speaking of them as multiple parts. Greek mathematicians would have worked with the acrophonic system, which lacked a system of fractions, and even if they had used Ionic (alphabetic) numerals, there is no evidence of the Ionic fractional system before the end of the fourth century BC. On the other hand, ratios are conceptually a part of Greek thinking, while employment of sub-units and super-units was foundational to arithmetic calculations and puzzles. My suggestion is Greeks worked with what they had, so that these form the foundation for the theory of numbers that we find in Elements VII-IX.20. The evidence for this picture is, admittedly, weak. I consider evidence from Plato’s Republic VIII for a playing with number puzzles, the Archimedes’ Cattle Problem for a class of false position puzzles, and the Anthologia Palatina for Euclid’s Elements as providing a foundation for such puzzles, and Euclid’s Sectio Canonis and Ptolemy’s Harmonica for working from fractions to sub-units.
Chapters in this book
- Frontmatter i
- Preface v
- Contents vii
- Notes on Contributors ix
- Introduction: Revolutions in Greek Mathematics 1
- Counter-Revolutions in Mathematics 17
- Diophantus and Premodern Algebra: New Light on an Old Image 35
- Geometer, in a Landscape: Embodied Mathematics in Hero’s Dioptra 67
- How Much Does a Theorem Cost? 89
- Diagrammatizing Mathematics: Some Remarks on a Revolutionary Aspect of Ancient Greek Mathematics 107
- Composition and Removal of Ratios in Geometric and Logistic Texts from the Hellenistic to the Byzantine Period 131
- Why Did the Greeks Develop Proportion Theory? A Conjecture 189
- Recursive Knowledge Procedures Informing the Design of the Parthenon : One Instance of Continuity between Greek and Near Eastern Mathematical Practices 235
- Diophantus, al-Karajī, and Quadratic Equations 271
- Substantiae sunt sicut numeri: Aristotle on the Structure of Numbers 295
- The Axiomatization of Mathematics and Plato’s Conception of Knowledge in the Meno and the Republic 319
- The Anthyphairetic Revolutions of the Platonic Ideas 335
- Name index 383
- General index 387
Chapters in this book
- Frontmatter i
- Preface v
- Contents vii
- Notes on Contributors ix
- Introduction: Revolutions in Greek Mathematics 1
- Counter-Revolutions in Mathematics 17
- Diophantus and Premodern Algebra: New Light on an Old Image 35
- Geometer, in a Landscape: Embodied Mathematics in Hero’s Dioptra 67
- How Much Does a Theorem Cost? 89
- Diagrammatizing Mathematics: Some Remarks on a Revolutionary Aspect of Ancient Greek Mathematics 107
- Composition and Removal of Ratios in Geometric and Logistic Texts from the Hellenistic to the Byzantine Period 131
- Why Did the Greeks Develop Proportion Theory? A Conjecture 189
- Recursive Knowledge Procedures Informing the Design of the Parthenon : One Instance of Continuity between Greek and Near Eastern Mathematical Practices 235
- Diophantus, al-Karajī, and Quadratic Equations 271
- Substantiae sunt sicut numeri: Aristotle on the Structure of Numbers 295
- The Axiomatization of Mathematics and Plato’s Conception of Knowledge in the Meno and the Republic 319
- The Anthyphairetic Revolutions of the Platonic Ideas 335
- Name index 383
- General index 387
