The History of Continua: Philosophical and Mathematical Perspectives

Cover
The History of Continua: Philosophical and Mathematical Perspectives
Copyright
Contents
List of Figures
List of Contributors
Introduction
1: Divisibility or Indivisibility: The Notion of Continuity from the Presocratics to Aristotle
	1. Introduction
	2. Parmenides’ Account of Continuity
		2.1 Being Continuous Excludes Any Temporal Differences
		2.2 Being Continuous Implies Being Homogeneous, Full of Being, and No More or Less
		2.3 Conditions That Would Prevent Continuity
	3. Zeno’s Paradoxes: Negative Consequences of Infinite Divisibility
	4. A Mathematical Conception of Continuity
	5. Aristotle’s Account of Continuity
	6. Implications of Aristotle’s Concept of a Continuum
		6.1 A New Understanding of Parts
		6.2 A New Understanding of Infinity
		6.3 A New Twofold Concept of a Limit
	7. Conclusion
	References
2: Contiguity, Continuity, and Continuous Change: Alexander of Aphrodisias on Aristotle
	1. Alexander on Aristotle’s Definition of the Contiguous
	2. Continuity and Unity
	3. Alexander’s Interpretation of the Definition of the Continuous in Physics V.3
	4. Potential Divisibility
	5. Alexander’s Account of Change at One Go in Its Context
	6. Change at One Go and Accidental Divisibility
	References
3: Infinity and Continuity: Thomas Bradwardine and His Contemporaries
	1. Parts I and II of De Continuo
	2. Part III: Bradwardine’s Affirmative Conclusions
	3. Part IV: Arguments against Immediate Indivisibles
	4. Atomism and Its Place in Medieval Philosophy and Theology
	5. Gerard of Odo on Infinity and Continuity
	6. Gerard of Odo and Walter Burley
	7. Infinity and Imagined Cases in Bradwardine’s De Causa Dei
	8. Conclusion
	References
4: Continuous Extension and Indivisibles in Galileo
	1. Galileo on Continuous Motion and Acceleration in the Dialogo
	2. The Composition of the Continuum?
	3. Galileo on the Continuum and Indivisibles in the Discorsi
	4. Why Infinitely Many Unquantifiable Parts?
	5. Galileo and Huygens: Infinity, Indivisibles, and Geometrical Rigour
	References
	Secondary Literature
5: The Indivisibles of the Continuum: Seventeenth-Century Adventures in Infinitesimal Mathematics
	1. Cavalieri and the Method of Indivisibles
	2. Torricelli and the Extension of Indivisibles
	3. Roberval’s Method of Indivisibles
	4. John Wallis and the Arithmetic of Infinities
	5. Conclusion
	References
6: The Continuum, the Infinitely Small, and the Law of Continuity in Leibniz
	1. Leibniz on the Continuum
		1.1 Composition of the Continuum in Pacidius Philalethi, 1676
		1.2 The Continuum as Ideal and a New Definition of Continuity for Magnitudes
	2. Leibniz on the Infinitely Small
		2.1 The Infinitely Small and the Archimedean Principle of Equality in DQA 1676
		2.2 The Archimedean Principle of Equality and the Law of Continuity
	References
	Secondary Literature
7: Continuity and Intuition in Eighteenth-Century Analysis and in Kant
	1. Introduction
	2. A Brief History of Seventeenth- and Eighteenth-Century Analysis
	3. Kant’s Views and Their Place in This History
	4. Toward a Less Whiggish History of Analysis
	5. Foundations of Analysis in Kant’s German Predecessors
	6. Foundations of Analysis in Kant’s Metaphysical Foundations of Natural Science
		6.1 General Remarks
		6.2 Kant on the Infinitely Small and Limits
	7. Changing Notions of Foundations and the Role of Intuition in Eighteenth-Century Analysis
	References
8: Bolzano on Continuity
	1. Introduction
	2. Philosophical Context
		2.1 Methodology
			2.1.1 Early Views: The Contributions and Related Writings
			2.1.2 Later Refinements
		2.2 Metaphysics
	3. Continuous Extension
		3.1 Early Definitions of the Concepts of Line, Surface, and Solid
		3.2 Later Definitions of the Concepts of Extended Objects and Continua
			3.2.1 Sources
			3.2.2 The Concept of Extension in Bolzano’s Later Writings: 1) Versuch einer Erklärung
			3.2.3 The Concept of Extension in Bolzano’s Later Writings: 2) Über Haltung, Richtung usw.
			3.2.4 The Concept of Extension in Bolzano’s Later Writings: 3) The Paradoxes of the Infinite
		3.3 Final Remarks
	4. The Numerical Continuum
	5. Continuous Functions
	6. Conclusion
	7. Appendix: A Note on the Bolzano–Weierstrass Theorem
	References
		Works by Bolzano: (1) Abbreviations
		(2) Other Works
		Works by Other Authors
9: Cantor and Continuity
	1. The Real Numbers
		1.1 Uniqueness of Trigonometric Series
		1.2 Cantor [1872]
		1.3 Heine [1872]
		1.4 Dedekind [1872]
	2. Uncountability and Dimension
		2.1 Uncountability
		2.2 Dimension
	3. Point-Sets and Perfect Sets
		3.1 Point-Sets
		3.2 Perfect Sets
	References
10: Dedekind on Continuity
	1. Introduction
	2. Dedekind’s Definition of Continuity
		2.1 Background
		2.2 Stetigkeit und irrationale Zahlen (1872)
		2.3 Methodological Reflections
			2.3.1 On Definitions and Ideals
		2.4 The Reception of Dedekind’s Account of Continuity
	3. Other Treatments of Continuous Domains
		3.1 Dedekind’s Early Interest in Riemann’s Work
		3.2 Arithmetical Treatment of Riemann Surfaces
			3.2.1 Motivations for Arithmetizing Riemann Surfaces
			3.2.2 Analogy with Number Theory
			3.2.3 The Theory of Ideals of Functions
		3.3 Set-Theoretical Treatments of More General Continuous Domains
	4. Conclusion
	Acknowledgements
	References
11: What Is a Number?: Continua, Magnitudes, Quantities
	Part 1
	Part 2
	Part 3
	Part 4
	Part 5
	Acknowledgements
	References
12: Continuity in Intuitionism
	1. Introduction
	2. Preliminaries
		2.1 Notions and Notations
		2.2 Classes, Sets, Types: A Precautionary Note
		2.3 Continua, Magnitude, Quantity, Number
	3. Foundations of Intuitionistic Analysis
		3.1 Arithmetic Continuity and Brouwer’s Principle for Numbers
		3.2 Brouwer’s Continuity Theorem and Unzerlegbarkeit
		3.3 Analogues of Continuity and Unzerlegbarkeit in Recursive Mathematics and Realizability
		3.4 Uniform Continuity and the Fan Theorem
		3.5 Recursive Mathematics, Realizability, and the Fan Theorem
	4. Brouwer on Intuition and Choice Sequences—also du Bois-Reymond
		4.1 The Intuitive Continuum, Inner Time, and Primordial Intuition
		4.2 Brouwer’s Choice Sequences
		4.3 Du Bois-Reymond on Choice Sequences
		4.4 The Law of the Excluded Third
	5. Weyl’s Intuitionism and Arithmetic Continuity
		5.1 Weyl’s Treachery
		5.2 Weyl on Choice Sequences
		5.3 The Law of the Excluded Third, Again
	6. In Sum
		6.1 Continua
		6.2 Continuity Theorems
		6.3 Logic
	Bibliography
13: The Peircean Continuum
	1. A Model for Peirce’s Continuum
		1.1 Peirce’s Views on Continuity
			1.1.1 Inextensibility
			1.1.2 Reflexivity
			1.1.3 Supermultitudinousness
			1.1.4 Modality
			1.1.5 Genericity
		1.2 The Construction
		1.3 Back to Peirce’s Intuitions
		1.4 Final Remarks
	2. Historical Appendix: Peirce’s Continuum, Unreconstructed
	Acknowledgements
	References
14: Points as Higher-Order Constructs: Whitehead’s Method of Extensive Abstraction
	1. Introduction
	2. Philosophical Motivations
	3. The Basic Idea
	4. Refinements (I)
	5. Refinements (II)
	6. The Final Account
	7. Concluding Remarks
	References
15: The Predicative Conception of the Continuum
	Part I: Origins
		1. Russell
			1.1 The Antinomies
			1.2 The Early Theory of Types: 1903
			1.3 Five Desiderata
			1.4 The Zigzag Theories: 1903–5
			1.5 The Theory of Descriptions: 1905
			1.6 The Substitution Theory: 1906–8
			1.7 The Ramified Theory of Types: 1908
		2. Weyl
	Part II: Analysis
		3. The Starting Point
		4. Reflection
		5. Definability
		6. Merger
		7. Strengthening the Case
			7.1 Definiteness
			7.2 The Need for a New Approach
			7.3 Reflective Closure
			7.4 Unfolding
			7.5 Summary
	Part III: The Continuum
		8. Three Predicative Standpoints
			8.1 The Restricted Predicativist
			8.2 The Ideal Predicativist
			8.3 The Predicativist Sympathist
		9. The Extent of Predicative Mathematics
		10. The Predicative Conception of the Continuum
	References
16: Point-Free Continuum
	Introduction
	1. Mereology for an Inclusion-Based Approach to the Continuum
		1.1 The Basic Notion of Mereology (Leśniewski 1916)
		1.2 Two Pioneering Books (Whitehead 1919 and Whitehead 1920)
		1.3 Two Pioneering Readings of Whitehead’s Books (de Laguna 1920–2, Nicod 1924)
		1.4 An (Apparent) Candidate for a Mereological Approach to Point-Free Geometry: Point-Free Topology (Wallman 1938)
	2. To Go beyond Pure Mereology: The Connection Relation
		2.1 Process and Reality: Whitehead 1929
		2.2 Grzegorczyk’s System (1960)
		2.3 Clarke’s System (1981)
	3. Going beyond Mereotopology: Spheres, Half-Planes, Ovals
		3.1 Regions and Spheres (Tarski 1927)
		3.2 An Approach by Half-Planes (Sniatycki 1968)
		3.3 From Ovals to Half-Planes
		3.4 Continuum and Tessellation (Hellman and Shapiro, 2012)
	4. Metrics and Continuous Logics Approaches
		4.1 Lattices and Diameters (Previale 1966)
		4.2 Frame with a Diameter (Pultr 1984)
		4.3 Distances and Diameters
		4.4 Continuous Logic for a Point-Free Continuum
		4.5 Graded Mereology
		4.6 Naïve Geometry: The Graded Predicates ‘Is Close’ and ‘Is Small’
	5. The Group Isometries for Point-Free Geometry
		5.1 The Group of Displacements in the Physical Geometry of H.J. Schmidt
	6. Conclusion
	References
17: Intuitionistic/Constructive Accounts of the Continuum Today
	1. Brouwer and the Intuitionistic Continuum
	2. The Continuum of Real Numbers in Constructive Mathematics
	3. The Order Relation on the Constructive Reals
	4. Convergence of Sequences and Completeness of the Constructive Reals
	5. Functions on the Constructive Reals
	6. Axiomatizing the Constructive Reals
	7. Smooth Infinitesimal Analysis
	8. Algebraic and Order Structure of ℝin SIA
	9. Comparing SIA and CA
	10. Cohesiveness of the Continuum in SIA
	References
18: Contemporary Infinitesimalist Theories of Continua and Their Late Nineteenth- and Early Twentieth-Century Forerunners
	1. Introduction
	2. The Emergence of Non-Archimedean Systems of Magnitudes
	3. The Archimedean Axiom
	4. Veronese’s Theory of Continua
	5. Hahn’s Non-Archimedean Generalizations of the Archimedean Arithmetic Continuum
	6. The Pantachies of du Bois-Reymond and Hausdorff
	7. Elementary Continua
	8. Nonstandard (Robinsonian) Continua
	9. The Absolute Arithmetic Continuum: Conway’s System of Surreal Numbers
	10. Hjelmslev’s Nilpotent Infinitesimalist Continuum
	11. Infinitesimalist Approaches to Differential Geometry of Smooth Manifolds and Their Underlying Continua
	12. Invertible and Nilpotent Infinitesimals: Afterthoughts
	13. Concluding Remarks
	Acknowledgements
	References
Index