More investigations about polygons

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Table of contents

Introduction 
0.1 State of the art and our expectations about Dedekind
0.1.1 Philosophical readings of Dedekind’s works
0.1.2 The historians’s way(s)
0.2 Aims of this study
0.2.1 Some methodological considerations
0.2.2 The argument of the thesis. The corpus and its limits
I Through the side-door 
1 Elements of contextualisation 
1.1 Remarks on the “conceptual approach”
1.1.1 Before Riemann
1.1.2 Riemann’s and Dedekind’s mathematics
1.1.3 Diffuse ideas in space and time?
1.2 19th century arithmetic in practices
1.2.1 Gauss’s Disquisitiones Arithmeticae
1.2.2 After Gauss
1.3 Riemannian function theory
1.3.1 Outlines of Riemann’s function theory
1.3.2 Receptions and developments of Riemann’s works
1.4 Heinrich Weber (1842-1913)
1.4.1 Weber as a Riemannian mathematician
1.4.2 Weber and algebra
1.4.3 Lehrbuch der Algebra
1.5 A note about Dedekind’s theory of algebraic numbers
2 Dedekind and Weber’s Theorie der algebraischen Funktionen einer Veränderlichen 
2.1 Returning to Riemann’s epistemological ideals?
2.1.1 Criticisms and the need of a sound basis for Riemann’s theory
2.1.2 Dedekind and epistemological requirements
2.1.3 Weber’s role?
2.2 Fields of algebraic functions of one complex variable
2.2.1 Algebraic functions, integral algebraic functions
2.3 Ideal theory and the laws of divisibility in the field
2.3.1 Modules
2.3.2 Ideals
2.4 Further investigations on function fields
2.4.1 Fractional functions
2.4.2 Invariance by rational transformations
2.5 The point and the Riemann surface
2.5.1 Points of the Riemann surface
2.5.2 Order numbers of the points
2.5.3 Polygons
2.5.4 The Riemann surface
2.5.5 More investigations about polygons
2.5.6 On the Riemann-Roch theorem
2.6 On the “arithmetical” rewriting
2.6.1 An arithmetical definition of the Riemann surface?
2.6.2 Arithmetic as the science of numbers, from Dedekind’s viewpoint?
2.6.3 What is not arithmetical for Dedekind
2.6.4 From the science of numbers to arithmetic of polygons?
2.6.5 Arithmetical methodology in Dedekind’s mathematics, a first view
II Early years 
3 Dedekind’s Habilitationsvortrag, in 1854 
3.1 Science, an activity of the human thought
3.1.1 Dedekind’s idea of science
3.2 Development of mathematics according to the Habilitationsvortrag
3.2.1 The particular nature of mathematics
3.2.2 Development of arithmetic
3.2.3 In the case of definitions in less elementary parts of mathematics
4 Dedekind’s first works in number theory 
4.1 Galois’s Galois theory
4.1.1 Galois’s 1831 “Mémoire sur les conditions de résolubilité des équations par radicaux”
4.2 Dedekind’s Galois theory
4.2.1 Cayley’s Galois theory
4.2.2 Dedekind’s 1856-58 “Eine Vorlesung über Algebra”
4.3 An approach to higher congruences “rigorously tied to an analogy with elementary number theory”
4.3.1 Outlines of Dedekind’s 1857 paper on higher congruences
III A strategical use of arithmetic? 
5 Theory of algebraic integers in 1871 
5.1 Ideal numbers
5.1.1 Gauss’s generalized arithmetic
5.1.2 Unique factorization in primes and ideal numbers
5.1.3 An example, explained by Dedekind
5.2 A new framework: Fields
5.2.1 An algebraic concept for arithmetical investigations
5.2.2 The concept of field in 1871
5.3 Algebraic integers, a new and more general notion of integer
5.3.1 Algebraic integers in 1871
5.3.2 The redefinition of primality
5.4 The ideal numbers’s new clothes
5.4.1 Module theory in 1871
5.4.2 Ideals in 1871
5.4.3 The proofs of the arithmetical propositions
5.5 A short comparison with Kronecker’s approach
5.5.1 Kronecker’s algebraic magnitude and the Rationalitätsbereich
5.5.2 An overview of Kronecker’s integral algebraic magnitudes and divisors
5.5.3 Dedekind on Kronecker’s reading of ideal theory
5.6 Role and status of ideals
5.6.1 The last paragraphs of Dedekind’s Xth Supplement
5.6.2 Auxiliary theories
5.7 Remarks on the reception and criticisms
5.7.1 Later criticisms
5.8 Conclusion. From ideal numbers to ideals to arithmetic of ideals
6 Towards a more arithmetical theory of algebraic numbers? 
6.1 Ideals and arithmetic of ideals
6.1.1 A note about Avigad’s article
6.1.2 The definition of ideals and ideals as objects
6.1.3 Arithmetic of ideals
6.2 In 1877, a “more arithmetical” version of algebraic number theory?
6.2.1 Properties of algebraic numbers
6.2.2 The ideals’s makeover
6.2.3 Outlines of possible applications
6.2.4 . . . and in 1879
6.3 Arithmetical strategies?
6.3.1 From 1871 to 1877
6.3.2 1877 and 1882
6.3.3 Conclusion: What role and status for arithmetic?
IV Numbers, arithmetic and mathematical practice 
7 Arithmetic, arithmetization and extension of the number concept 
7.1 Arithmetization of Analysis in the 19th century
7.1.1 Klein’s idea of arithmetization
7.1.2 Examples of “Arithmetization of Analysis”
7.1.3 Kronecker’s “arithmetization”
7.2 Dedekind’s Stetigkeit und irrationale Zahlen
7.2.1 A rigorous and arithmetical foundation for continuity
7.2.2 The incomplete domain of rational numbers
7.2.3 Definition of irrational numbers
7.2.4 Order and operations for real numbers
7.3 A Dedekindian arithmetization?
7.3.1 Extension of arithmetic?
7.3.2 Reduction to earlier concepts and rigor
7.3.3 A quest for rigor?
7.3.4 A strategy of arithmetization in Dedekind’s works?
7.3.5 Conclusion. What arithmetic for Dedekind’s arithmetization?
8 The natural number concept in perspective 
8.1 Defining the natural numbers
8.1.1 Laws and operations of pure thought
8.1.2 Defining numbers and rigor, one last time
8.1.3 Arithmetic and the act of thinking
8.2 Was Sind und Was Sollen die Zahlen?
8.2.1 The “simply infinite systems”
8.2.2 Properties of natural numbers and operations
8.3 Peano vs.(?) Dedekind
8.3.1 Peano’s axioms
8.3.2 Peano’s conception of logic
8.3.3 Peano on numbers
8.3.4 Definitions according to Peano
8.3.5 What Peano thought Dedekind was doing
8.4 Mathematics, arithmetization and number concept
8.4.1 A need for a definition of natural numbers?
8.4.2 Arithmetic operations as epistemic tools
8.4.3 Could the uses of arithmetic have influenced the definition of natural numbers?
8.4.4 Some closing remarks
8.5 Epilogue, the 1894 version of algebraic number theory
9.1 Between mathematical explorations and foundational investigations
9.1.1 Mathematical works and arithmetic
9.1.2 On characterizing arithmetic
9.1.3 Arithmetic and foundational researches
9.2 Further on up the road
9.2.1 Investigating the differences deeper
9.2.2 About rewriting Riemann, again
Bibliography