Duality in 19th and 20th Century Mathematical Thinking

Preface
Contents
Chapter 1 Introduction
	1.1 Historical issues
	1.2 Epistemological aspects
	1.3 Duality in geometry: an overview
		1.3.1 The theory of polyhedra
		1.3.2 Spherical geometry and trigonometry
		1.3.3 Projective geometry
	1.4 The contents of this book
		1.4.1 Geometry
		1.4.2 Logic, Boolean algebra, Lattice theory
		1.4.3 Topology and topological groups
		1.4.4 Finite-dimensional linear algebra: a missing link?
		1.4.5 Functional analysis and related fields
		1.4.6 Applied mathematics
		1.4.7 Category theory
	1.5 Some answers, some open questions
	1.6 Acknowledgements

Part I The 19th century heritage
	Chapter 2 On “contour apparent”, “courbe de contact” and ramification curves: Duality between a principle and a tool
		2.1 A mathematical detour: what is a ramification curve?
		2.2 1826–7: Étienne Bobillier and the implicit introduction of ramification curves
		2.3 1827–8: The explicit appearance of duality as a formal principle: Gergonne’s rewriting of Bobillier’s results
		2.4 1847–9, 1862, and an epilogue in 1937: Duality as a tool and the ramification curve of the cubic surface
		2.5 Conclusion: The different interweaving of duality and ramification curve
		Acknowledgement
	Chapter 3 De Morgan’s De Morgan’s Laws
		3.1 Introduction
		3.2 The Many Classes of Negation
			3.2.1 The Analysis of Logical (Pr)oppositions
			3.2.2 Putting Syllogistics in Order
			3.2.3 From syllogistic logic to the logic of names. . .
			3.2.4 . . . and back
			3.2.5 The Triple Root of Logical Negation
		3.3 Conjunction and Disjunction
			3.3.1 Complex Propositions
			3.3.2 Dual Behavior of Propositional Disjunction and Conjunction
			3.3.3 Propositions as Classes
		3.4 De Morgan’s De Morgan’s Laws
		3.5 Conclusion
	Chapter 4 The emergence of duality in 19th-century algebra of logic
		4.1 Introduction
		4.2 Boole lays the groundwork for the algebra of logic
			4.2.1 Boole’s application of symbolical algebra to logic
			4.2.2 The law of duality and the dual interpretation
			4.2.3 Boole’s system is not a Boolean algebra
			4.2.4 De Morgan’s laws
		4.3 Jevons and MacColl pave the way to duality
			4.3.1 Jevons’ reinterpretation of logical addition
			4.3.2 MacColl’s algebra of propositions
		4.4 Peirce and Grassmann see the duality
			4.4.1 Peirce’s early work on logic
			4.4.2 Grassmann’s theory of forms
		4.5 Schröder brings out the duality with double columns
		4.6 Some concerns with regard to duality
			4.6.1 Venn sides with Boole
			4.6.2 Peirce’s later work on the algebra of logic
			4.6.3 Ladd-Franklin’s criticism
			4.6.4 Russell on duality
		4.7 Boolean algebras
			4.7.1 Whitehead’s universal algebra
			4.7.2 Huntington on the principle of duality
			4.7.3 Sheffer on ‘Boolean algebras’
		4.8 Duality in logic
			4.8.1 Duality finds its way into Keynes’ Logic
			4.8.2 Duality in some 20th-century and contemporary logic textbooks
		4.9 Conclusion
	Chapter 5 Duality as a guiding light in the genesis of Dedekind’s Dualgruppen
		5.1 Introduction
			5.1.1 The initial impetus
			5.1.2 Some possible external influences regarding duality
			5.1.3 Dedekind’s drafts and the genesis of Dualgruppen
			5.1.4 Dedekind’s main results on Dualgruppen
		5.2 Earliest considerations on duality in module theory
			5.2.1 “Theory of modules”
			5.2.2 “Dualism in the laws of modules of numbers”
		5.3 How duality emerges in Dedekind’s mathematical research
			5.3.1 A short aside on groups
			5.3.2 Notations
			5.3.3 Layouts and research devices
			5.3.4 The Modulgesetz, the source of dualism?
		5.4 Generalisation of the research
			5.4.1 More generality in module theory
			5.4.2 Parallelism between modules and (Abelian) groups
			5.4.3 Reading Schröder and the “logical” theory
			5.4.4 “Generalisation of a part of module theory”
			5.4.5 The case of the first draft of (Dedekind, 1897b)
		5.5 The last steps
			5.5.1 “Some propositions on Modul-Gruppen”
			5.5.2 “On the dualism in module theory”
			5.5.3 “More general (logical) theory”
			5.5.4 Dualgruppen, finally
		5.6 Conclusion
		Acknowledgments
	Chapter 6 From Grassmann complements to Hodge-duality
		Introduction
		6.1 Grassmann’s complement of “extensive quantities”
			6.1.1 Grassmann’s Ausdehnungslehre (1862)
			6.1.2 Dual complement (Ergänzung)
			6.1.3 Vector operations in R3 in a Grassmannian perspective
		6.2 Dual complements in physics of the early 20th century
			6.2.1 Background: Maxwellian electrodynamics in the 19th century
			6.2.2 Einstein’s relativistic electrodynamics, Minkowski’s “six-vectors” and their duals
			6.2.3 Dual complements in the general theory of relativity
			6.2.4 Cartan
		6.3 The birth of Hodge duality
			6.3.1 Background: Riemann’s theory of Abelian integrals
			6.3.2 Hodge’s first steps towards generalizing Riemann’s theory of Abelian integrals
			6.3.3 The ∗-operation, the Hodge theorem and algebraic surfaces
			6.3.4 Hodge’s definition of harmonic forms, Hodge duality, and the Maxwell equation
			6.3.5 Short remarks on the further development of Hodge’s theory
		6.4 Finally an outlook on Hodge duality after 1950
			6.4.1 Hodge theory and Hodge duality become sheaf cohomological
			6.4.2 From Hodge duality in physics via Yang-Mills theory back to mathematics

Part II From topology to groups
	Chapter 7 Duality theorems in topology
		7.1 The early history of Poincaré duality
			7.1.1 “Analysis situs”, 1895
			7.1.2 Poincaré’s sources: missing links
			7.1.3 Heegaard’s counterexample
			7.1.4 The first complément: second proof by using a dual decomposition
			7.1.5 The second complément
		7.2 Poincaré’s theorem stimulating further developments
			7.2.1 The Princeton school
			7.2.2 Vietoris 1928
		7.3 Alexander’s 1922 theorem
			7.3.1 The theorem
			7.3.2 The methods used for the proof
			7.3.3 Further developments
		7.4 Mannoury’s duality theorem
			7.4.1 The structure of Lois cyclomatiques
			7.4.2 Influences on Mannoury
			7.4.3 Criticism of the methods
			7.4.4 Evaluation of Mannoury’s accomplishments
	Chapter 8 The historical development of Pontrjagin duality
		8.1 Point of departure
			8.1.1 Lev Pontrjagin: biographical data
			8.1.2 The role of Alexandroff
			8.1.3 Pontrjagin’s 1927 paper on Alexander’s duality theorem
			8.1.4 The contributions of Felix Frankl
		8.2 Pontrjagin 1931 on topological duality theorems
			8.2.1 Influences and echoes
			8.2.2 An algebraic tool: group pairings
			8.2.3 Duality theorems in the new formulation
			8.2.4 From complexes to arbitrary closed sets
			8.2.5 An open question: what about torsion?
		8.3 The “final question” and the first chapter of “T.T.G.”
			8.3.1 Pontrjagin’s 1934 papers in the Annals of mathematics
			8.3.2 Topological groups as coefficient groups in homology
			8.3.3 The contents of chapter I of T.T.G.
			8.3.4 Proof of the ICM theorem
			8.3.5 Comparison of the results of 1931 and 1934
		8.4 Group characters and character groups up to Pontrjagin
			8.4.1 Dedekind, Weber, Frobenius
			8.4.2 Characters in Weyl’s work
			8.4.3 Haar’s 1931 paper
			8.4.4 The Paley-Wiener Announcement
		8.5 T.T.G.: duality of commutative groups
			8.5.1 The theory of Peter-Weyl
			8.5.2 The introduction of the Haar measure
			8.5.3 Two Comptes rendus notes by Pontrjagin
			8.5.4 The proof of Pontrjagin’s duality theorem
			8.5.5 Pontrjagin’s use of the words “dual” and “duality”
		8.6 Later developments
			8.6.1 Van Kampen’s work
			8.6.2 Early applications in other fields
			8.6.3 The relation of Pontrjagin’s theory to linear algebra and functional analysis
		8.7 Cohomology
			8.7.1 Kolmogoroff’s contribution to the 1935 Moscow conference on topology
			8.7.2 Alexander
	Chapter 9 Tannaka Tadao‘s 1938 paper on the duality of noncommutative topological groups and its historical background
		9.1 Introduction
		9.2 The structure of the paper on duality
			9.2.1 Outline
			9.2.2 A letter by John von Neumann
			9.2.3 Mid-term reception of the paper
		9.3 Production of mathematical knowledge in early 1930s Japan
			9.3.1 Teaching in mathematical departments
			9.3.2 Exchange outside of university curricula
			9.3.3 Publishing infrastructure
		9.4 From 1935 to 1938
			9.4.1 The annual meeting of the Physico-Mathematical Society of Japan in Ōsaka 1935
			9.4.2 Between the annual conferences in 1935 and 1936: early reception of Pontrjagin duality in Japan
			9.4.3 The annual meeting of the Physico-Mathematical Society of Japan in Tōkyō 1936
			9.4.4 Tannaka’s publications until 1938
		9.5 Conclusion
		9.6 Appendix: Information on relevant Japanese mathematicians
	Chapter 10 Philosophical and mathematical duality in Albert Lautman’s work
		Introduction
		10.1 From pairs of dual notions to duality theorems in combinatorial topology
			10.1.1 Mathematical ideas, concepts and theories
			10.1.2 An overview of dual notions in Lautman’s writings
			10.1.3 Duality as a mode of linking notions: the example of the local and the global
			10.1.4 Duality between intrinsic and induced properties: the case of combinatorial topology
			10.1.5 Sources and elements of knowledge in combinatorial topology called forth by Lautman
			10.1.6 Poincaré’s and Alexander’s duality theorems
		10.2 Symmetries, dissymmetries and duality at the crossroad between mathematics and physics
			10.2.1 The symmetry/dissymmetry duality
			10.2.2 A thematization of duality
			10.2.3 A structural conception of duality via lattice theory
		Conclusion : a variety of dualities

Part III Functional analysis and related fields
	Chapter 11 The development of dual spaces in functional analysis
		11.1 The introduction of Riesz’ Lp spaces
			11.1.1 About Frigyes Riesz
			11.1.2 The Riesz-Fischer theorem (1907)
			11.1.3 Schmidt’s results (1908)
			11.1.4 The classes [Lp] and [L p/p−1 ] (1910)
			11.1.5 Dual pairings
		11.2 Helly’s concept of polarity and the Hahn-Banach theorem
			11.2.1 About Eduard Helly
			11.2.2 Connection to Riesz and Schmidt
			11.2.3 Minkowski’s “Aichkörper” (1896)
			11.2.4 Helly’s concept of polarity (1921)
			11.2.5 Hahn’s “polar space” (1927)
			11.2.6 Dual norms
		11.3 Köthe’s “resolution criterion”
			11.3.1 About Gottfried Köthe
			11.3.2 Köthe’s and Toeplitz’ dual sequence spaces (1934)
			11.3.3 Köthe’s “resolution criterion” for systems of linear equations with infinitely many unknowns (1938 and 1939)
			11.3.4 Dieudonné’s result (1942)
			11.3.5 Dual systems
		11.4 An additional comment: the moment problem
	Chapter 12 Duality in Banach’s 1929 work on functionals
		12.1 Introduction
		12.2 From a constructive to a synthetic view of the space
			12.2.1 The extension theorem
			12.2.2 First consequence on the relation between E and E′
		12.3 Through the looking-glass, and what Banach found there
			12.3.1 A detour into the bi-dual and countable basis
			12.3.2 Sub-linear maps and transfinite sequential topology
			12.3.3 Comparison of topologies in the separable case
		12.4 Application to the extension of the theory of operators on Banach spaces
		12.5 Conclusion
	Chapter 13 Marshall Stone and duality: from differential equations to Boolean algebras
		13.1 Marshall H. Stone: biographical data
		13.2 The adjoint of a differential equation, from Lagrange to Stone
			13.2.1 From Lagrange to the 19th century
			13.2.2 “Expansions”, from Liouville to Birkhoff and Stone
		13.3 Von Neumann and Stone on the spectral theorem
			13.3.1 The adjoint of a differential equation in mathematical physics and quantum mechanics
			13.3.2 Von Neumann’s program
			13.3.3 Adjoint operators
			13.3.4 Stone enters the field
			13.3.5 Von Neumann’s proof of the spectral theorem
		13.4 From sets of operators to Boolean algebras
			13.4.1 Von Neumann 1930 on rings of operators
			13.4.2 Sets of projections as lattices and Boolean algebras
			13.4.3 The emergence of projection-valued measures
		13.5 Stone’s representation theorem for Boolean algebras
			13.5.1 The chronology of Stone’s work on the topic
			13.5.2 The first idea: ideals in Boolean algebras
			13.5.3 From analogy to subsumption: Boolean rings, or a reinterpretation of +
			13.5.4 Symmetric difference: multiple “rediscoveries”
			13.5.5 The joint von Neumann-Stone paper
			13.5.6 Representation as algebras of classes
			13.5.7 Influences on Stone’s theory of Boolean algebras
		13.6 Stone duality and related developments
			13.6.1 Stone’s duality theorem
			13.6.2 The label “duality”
			13.6.3 The Gelfand representation theorem
			13.6.4 The term “spectrum”
		13.7 Concluding remarks
	Chapter 14 Duality à la Bourbaki
		14.1 Bourbaki members on duality, up to around 1940
			14.1.1 Dualities in André Weil’s work, 1935-1940
			14.1.2 Chevalley’s 1936 talk in the Séminaire Julia
			14.1.3 Dieudonné on topological vector spaces
			14.1.4 A project by Henri Cartan
		14.2 Case study: the discussion on chapter II of Algèbre
			14.2.1 The relevant sources
			14.2.2 Duality and solution of linear equations
			14.2.3 The Dieudonné-Chevalley debate
		14.3 Epilogue: parallel and later developments
			14.3.1 Topological vector spaces: the examples of Mackey and Katetov
			14.3.2 Abelian groups: Freundlich Smith, Godement, Raikov
		14.4 Conclusions
	Chapter 15 Duality and Distributions: An Application of Topological Vector Spaces
		Introduction
		15.1 Schwartz’s mathematical education
		15.2 Generalized solutions of differential equations
		15.3 Schwartz’s first generalized functions: Convolution operators
		15.4 Distributions proper
		15.5 The Fourier transform
		15.6 Schwartz’s sources
		15.7 L F Spaces
		15.8 Recognition
		15.9 Conclusion

Part IV The post-war outlook
	Chapter 16 From duality in mathematical programming to Fenchel duality and convex analysis: Duality as a force of inspiration in the creation of new mathematics
		16.1 Introduction
		16.2 Duality in linear programming: from logistic problem to mathematical theory
		16.3 Fenchel duality
		16.4 The significance of Fenchel duality for R. T. Rockafellar’s development of convex analysis in the early 1960s
		16.5 Jean Jacques Moreau and extension of Fenchel duality to infinite-dimensional spaces
		16.6 Conclusion
	Chapter 17 An Historical Perspective on Duality and Category Theory: Hom is where the heart is
		17.1 Introduction
		17.2 Eilenberg & Mac Lane 1945 : the vocabulary
			17.2.1 The dual of a cateogry
			17.2.2 Pontrjagin-type duality
		17.3 Doing Mathematics in Self-Dual Structures
			17.3.1 Mac Lane’s Duality for groups
			17.3.2 Axiomatic duality and the metamathematical duality principle
			17.3.3 Mac Lane’s Bicategories, Abelian categories and Abelian bicategories
			17.3.4 Buchsbaum’s Exact categories and duality
			17.3.5 Buchsbaum’s exact categories
			17.3.6 Grothendieck’s abelian categories and duality
		17.4 Kan’s adjoint functors and duality
		17.5 Chasing Concepts, Theorems and Proofs in Dual Structures in Algebraic Topology
			17.5.1 Eckmann and Hilton’s duality in homotopy theory
			17.5.2 Lifting the heuristic process into the categorical machinery
			17.5.3 Structures and costructures in categories
		17.6 Developing Mathematics using Dual structures
			17.6.1 Morita duality
			17.6.2 Grothendieck’s foundations of algebraic geometry
		17.7 Explaining Dualities from a Higher point of View
			17.7.1 Topology and algebra
			17.7.2 Lawvere’s work and program
			17.7.3 Putting the pieces together: 1965-1972
			17.7.4 The abstract functorial form of concrete dualities
		17.8 Dimensions of Duality

List of abbreviations
Bibliography
Index