Ebook: The Growth of Mathematical Knowledge

Mathematics has stood as a bridge between the Humanities and the Sciences since the days of classical antiquity. For Plato, mathematics was evidence of Being in the midst of Becoming, garden variety evidence apparent even to small children and the unphilosophical, and therefore of the highest educational significance. In the great central similes of The Republic it is the touchstone ofintelligibility for discourse, and in the Timaeus it provides in an oddly literal sense the framework of nature, insuring the intelligibility ofthe material world. For Descartes, mathematical ideas had a clarity and distinctness akin to the idea of God, as the fifth of the Meditations makes especially clear. Cartesian mathematicals are constructions as well as objects envisioned by the soul; in the Principles, the work ofthe physicist who provides a quantified account ofthe machines of nature hovers between description and constitution. For Kant, mathematics reveals the possibility of universal and necessary knowledge that is neither the logical unpacking ofconcepts nor the record of perceptual experience. In the Critique ofPure Reason, mathematics is one of the transcendental instruments the human mind uses to apprehend nature, and by apprehending to construct it under the universal and necessary lawsofNewtonian mechanics.

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This book draws its inspiration from Hilbert, Wittgenstein, Cavaill?s and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate or question it.
The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how progress may be understood.
Readership: Students and scholars concerned with the history and philosophy of mathematics and the formal sciences.

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This book draws its inspiration from Hilbert, Wittgenstein, Cavaill?s and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate or question it.
The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how progress may be understood.
Readership: Students and scholars concerned with the history and philosophy of mathematics and the formal sciences.
Content:
Front Matter....Pages i-xli
Knowledge of Functions in the Growth of Mathematical Knowledge....Pages 1-15
Huygens and the Pendulum: From Device to Mathematical Relation....Pages 17-39
An Empiricist Philosophy of Mathematics and Its Implications for the History of Mathematics....Pages 41-57
The Mathematization of Chance in the Middle of the 17th Century....Pages 59-75
Mathematical Empiricism and the Mathematization of Chance: Comment on Gillies and Schneider....Pages 77-80
The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge....Pages 81-91
Hamilton-Jacobi Methods and Weierstrassian Field Theory in the Calculus of Variations: A Study in the Interaction of Mathematics and Physics....Pages 93-101
On Mathematical Explanation....Pages 103-119
Mathematics and the Reelaboration of Truths....Pages 121-132
Penrose and Platonism....Pages 133-141
On the Mathematics of Spilt Milk....Pages 143-152
The Growth of Mathematical Knowledge: An Open World View....Pages 153-176
Controversies about Numbers and Functions....Pages 177-198
Epistemology, Ontology and the Continuum....Pages 199-219
Tacit Knowledge and Mathematical Progress....Pages 221-230
The Quadrature of Parabolic Segments 1635 – 1658: A Response to Herbert Breger....Pages 231-256
Mathematical Progress: Ariadne’s Thread....Pages 257-268
Voir-Dire in the Case of Mathematical Progress....Pages 269-280
The Nature of Progress in Mathematics: The Significance of Analogy....Pages 281-293
Analogy and the Growth of Mathematical Knowledge....Pages 295-314
Evolution of the Modes of Systematization of Mathematical Knowledge....Pages 315-329
Geometry: the First Universal Language of Mathematics....Pages 331-340
Mathematical Progress....Pages 341-352
Some Remarks on Mathematical Progress from a Structuralist’s Perspective....Pages 353-362
Scientific Progress and Changes in Hierarchies of Scientific Disciplines....Pages 363-376
On the Progress of Mathematics....Pages 377-386
Attractors of Mathematical Progress — the Complex Dynamics of Mathematical Research....Pages 387-406
On Some Determinants of Mathematical Progress....Pages 407-416
Back Matter....Pages 417-428

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This book draws its inspiration from Hilbert, Wittgenstein, Cavaill?s and Lakatos and is designed to reconfigure contemporary philosophy of mathematics by making the growth of knowledge rather than its foundations central to the study of mathematical rationality, and by analyzing the notion of growth in historical as well as logical terms. Not a mere compendium of opinions, it is organised in dialogical forms, with each philosophical thesis answered by one or more historical case studies designed to support, complicate or question it.
The first part of the book examines the role of scientific theory and empirical fact in the growth of mathematical knowledge. The second examines the role of abstraction, analysis and axiomatization. The third raises the question of whether the growth of mathematical knowledge constitutes progress, and how progress may be understood.
Readership: Students and scholars concerned with the history and philosophy of mathematics and the formal sciences.
Content:
Front Matter....Pages i-xli
Knowledge of Functions in the Growth of Mathematical Knowledge....Pages 1-15
Huygens and the Pendulum: From Device to Mathematical Relation....Pages 17-39
An Empiricist Philosophy of Mathematics and Its Implications for the History of Mathematics....Pages 41-57
The Mathematization of Chance in the Middle of the 17th Century....Pages 59-75
Mathematical Empiricism and the Mathematization of Chance: Comment on Gillies and Schneider....Pages 77-80
The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge....Pages 81-91
Hamilton-Jacobi Methods and Weierstrassian Field Theory in the Calculus of Variations: A Study in the Interaction of Mathematics and Physics....Pages 93-101
On Mathematical Explanation....Pages 103-119
Mathematics and the Reelaboration of Truths....Pages 121-132
Penrose and Platonism....Pages 133-141
On the Mathematics of Spilt Milk....Pages 143-152
The Growth of Mathematical Knowledge: An Open World View....Pages 153-176
Controversies about Numbers and Functions....Pages 177-198
Epistemology, Ontology and the Continuum....Pages 199-219
Tacit Knowledge and Mathematical Progress....Pages 221-230
The Quadrature of Parabolic Segments 1635 – 1658: A Response to Herbert Breger....Pages 231-256
Mathematical Progress: Ariadne’s Thread....Pages 257-268
Voir-Dire in the Case of Mathematical Progress....Pages 269-280
The Nature of Progress in Mathematics: The Significance of Analogy....Pages 281-293
Analogy and the Growth of Mathematical Knowledge....Pages 295-314
Evolution of the Modes of Systematization of Mathematical Knowledge....Pages 315-329
Geometry: the First Universal Language of Mathematics....Pages 331-340
Mathematical Progress....Pages 341-352
Some Remarks on Mathematical Progress from a Structuralist’s Perspective....Pages 353-362
Scientific Progress and Changes in Hierarchies of Scientific Disciplines....Pages 363-376
On the Progress of Mathematics....Pages 377-386
Attractors of Mathematical Progress — the Complex Dynamics of Mathematical Research....Pages 387-406
On Some Determinants of Mathematical Progress....Pages 407-416
Back Matter....Pages 417-428
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