Diagrammatizing Mathematics: Some Remarks on a Revolutionary Aspect of Ancient Greek Mathematics
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Claas Lattmann
Abstract
This paper takes a fresh look at the ancient mathematical diagram from the viewpoint of semiotic model theory. It discusses, first, the ontological nature and epistemic function of diagrams; second, the characteristics of their interrelation and interaction with the textual part of mathematical proofs; third, their epistemic validity; and, fourth, their ‘revolutionary’ nature as per Thomas S. Kuhn’s theory of scientific revolutions. The analysis is based on an understanding of ancient mathematical diagrams as ‘diagrams’ as defined by Charles S. Peirce. Such a ‘diagram’ is a specific sort of iconic sign, and thus ‘model,’ that represents the relational structure of its object by way of its own relational quality. Against this backdrop, this paper shows: (1) that the diagram stands as a holistic model for a general mathematical structure (2) that consists of ‘diagrammatical atoms’ (3) that represent the general entities of mathematical theory; (4) and that this abstract model is being constructed and subsequently analyzed in the textual part of the proof, (5) the purpose of which is, accordingly, to ‘show’ that this model has the specific sought-for, yet unknown mathematical properties. (6) Given this, Greek mathematical proof turns out to be an analysis of static general models, rather than an either implicit or failed (mathematically) inductive argumentation concerned with particular cases.
Abstract
This paper takes a fresh look at the ancient mathematical diagram from the viewpoint of semiotic model theory. It discusses, first, the ontological nature and epistemic function of diagrams; second, the characteristics of their interrelation and interaction with the textual part of mathematical proofs; third, their epistemic validity; and, fourth, their ‘revolutionary’ nature as per Thomas S. Kuhn’s theory of scientific revolutions. The analysis is based on an understanding of ancient mathematical diagrams as ‘diagrams’ as defined by Charles S. Peirce. Such a ‘diagram’ is a specific sort of iconic sign, and thus ‘model,’ that represents the relational structure of its object by way of its own relational quality. Against this backdrop, this paper shows: (1) that the diagram stands as a holistic model for a general mathematical structure (2) that consists of ‘diagrammatical atoms’ (3) that represent the general entities of mathematical theory; (4) and that this abstract model is being constructed and subsequently analyzed in the textual part of the proof, (5) the purpose of which is, accordingly, to ‘show’ that this model has the specific sought-for, yet unknown mathematical properties. (6) Given this, Greek mathematical proof turns out to be an analysis of static general models, rather than an either implicit or failed (mathematically) inductive argumentation concerned with particular cases.
Kapitel in diesem Buch
- 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
Kapitel in diesem Buch
- 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
