Dedekind and Weber’s Theorie der algebraischen Funktionen einer Veränderlichen 

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19th century arithmetic in practices

Michael Potter explains in the introduction of his book Reason’s Nearest Kin. Philosophies of Arithmetic from Kant to Carnap that:
In popular parlance the word ‘arithmetic’ means the study of calculations involving numbers of all sorts, natural, integral, ra-tional, real, and complex. Mathematicians, on the other hand, often use the word to mean only the study of the properties of the natural (i.e., counting) numbers 0, 1, 2, etc. ([Potter, 2002], 1)
This view is largely embraced by philosophers of mathematics. However, the term took various meanings in different historical contexts. It will be useful for my purpose to examine them. This section will present parts of the mathematical context in which Dedekind developed his number theoretical works. It will also allow to un-derline that “arithmetic”, for 19th century mathematicians, such as Gauss and later Dedekind, does not designate the properties of natural numbers as a the sequence but number theory in a relatively large sense.59 In fact, interpretations can go as far as José Ferreirós, who considers that “arith-metic” was a generic word for all pure mathematics.60 According to him, “arithmetic” for Dedekind “is understood (. . . ) in a broad sense that em-braces algebra and analysis”.61 I will suggest that arithmetic had a rather clear meaning for Dedekind. This quote, immortalized in an etching of Gauss and W. Weber by A. Weger appearing in Zöllner’s Wissenschaftliche Abhandlungen (1878, vol. 2, part I), is linked to Gauss’s conviction that arithmetic was at “the top” of the edifice of science. As Ferreirós underlines in the beginning of his paper titled “The Rise of Pure Mathematics as Arithmetic with Gauss” ([Ferreirós, 2007]), the theological assertion behind the motto is somewhat “less interesting than the way in which his adaptation of Plato’s words reflects changing perceptions of mathematical knowledge” (ibid., 237). Indeed, one can observe, in the 19th century, the development of many ‘arithmetical’ approaches to mathematics. As Hilbert put it, the mathematics of this century developed “under the sign of number”.

A note on Riemann’s inequality

An important notion introduced by Riemann is that of boundary cuts (or cross-cuts, Querschnitte) and the connectivity of the surface related to it. Boundary cuts correspond to the idea of cutting or breaking a surface into simpler patches. The boundary cuts themselves are simple curves, not inter-secting themselves, which connect two points on the boundary of the surface. A surface is said to be connected if any two points can be connected by a continuous curve, and if consequently any boundary cut would split the surface in two. A simply connected surface is a connected surface which, roughly speaking, does not have any ‘hole’98. Such a surface is equivalent (one-to-one) to the plane disc.
Riemann, using boundary cuts, transforms a surface into a simply con-nected surface on which the functions admit a unique integral. This al-lows him to define the “order of connectivity” (Zusammenhangszahl): if by the system of n1 boundary cuts, the surface is split into m1 patches, and if by the system of n2 boundary cuts, it is split into m2 patches, then n1 − m1 = n2 − m2. In general, n − m is a constant, the “order of connec-tivity”. This number is independent of the way in which the boundary cuts have been made, and is consequently a characteristic property of the surface ([Bottazzini and Gray, 2013], 263).
Riemann claimed that for the surface of an algebraic function of equa-tion F (x, y) = 0, the order of connectivity is of the form 2p − 1 (this integer p will be called the genus by Clebsch).
This integer p is notably involved in the first important result of the 1857 paper on Abelian function, which we now call the Riemann inequality, Riemann’s part of the Riemann-Roch theorem.
To prove Riemann’s inequality, Riemann starts with an irreducible equa-tion in s of degree n, whose coefficient are polynomials in z of degree m. This equation is associated to a n-sheeted surface T spread over the com-plex plane. This surface is “without boundary”, which means that it can be considered as a surface whose boundary is “rejected to infinity” (infinitely far away) or as a closed surface. Moreover any rational function of s and z is obviously also a single-valued function of the locus the surface T and therefore possesses the same mode of ramification as the function s, ans we will see later that the converse is also true.100 ([Riemann, 1857], 95-96).

Receptions and developments of Riemann’s works

Riemann’s inaugural dissertation was only printed as a separate thesis and never as a research paper (e.g. in Crelle’s Journal), which didn’t help an early adoption of Riemann’s ideas. More significantly, the ideas proposed here are mathematically relatively vague “at certain crucial points and its often murky language”, which opposed an immediate acceptance of the many novelties of the work, according to Bottazzini and Gray ([Bottazzini and Gray, 2013], 277). If the response to Riemann’s works took a few years, and has often been presented as sparse, Bottazzini and Gray draw up a list of works following Riemann which, from the 1860s to the beginning of the 1880s, counts as much as 24 items, from Clebsch and Roch to Christoffel, Kraus and Dedekind and Weber.104 This suggest, for Bottazzini and Gray, in the light of the young age at which Riemann died and the few students he had, that the reception of Riemann’s works was rather important. In fact, if some of his ideas were difficult to develop due to their novelty and to a certain “mathematical vagueness”, his works were recognized immediately as valuable by the mathematical community – suffice it to mention that Weierstrass himself proposed his nomination at the Berlin Akademie der Wissenschaften in 1859.
The density and richness of Riemann’s works in complex function theory leads to a multifaceted reception of his works. Indeed, many aspects were to be considered problematic and/or fruitful, and several different readings and adaptations of his ideas emerged during the following decades. Not only were Riemann’s works vast and fruitful, each point considered will raise problems. The amount of problems to investigate and of possible directions that could be taken after Riemann’s “visionary presentation” of his function theory is very wide, as was already alluded to above and is well illustrated by the variety of authors studied in works such as [Chorlay, 2007], [Houzel, 2002] or [Bottazzini and Gray, 2013].

Heinrich Weber (1842-1913)

Heinrich Weber was a very versatile mathematician. He is well-known for his works in algebra (Galois theory and group theory, as well as the first elements of what is now called class field theory111). His textbook Lehrbuch der Algebra (first published in 1895-96) was one of the most influencial of the late 19th and early 20th centuries (E. Noether, van der Waerden, etc. learned algebra with the Lehrbuch112). But he was also a recognized special-ist of Riemannian function theory, mathematical physics, number theory and wrote elementary mathematics textbooks. Able to understand thoroughly and deeply new ideas in many areas of mathematics, Weber was very active and respected throughout his career, but with a tendency to stay in the shadow of other mathematicians or institutions.
Associated with many mathematicians, from Kronecker to Frobenius to Dedekind or again Hilbert, Weber studied and taught in many different places. From 1860 to 1863, he studied at Heidelberg, where he obtained his doctorate under Otto Hesse. Hesse had been Jacobi’s student and according to Cremona was a very “elegant” algebraist which probably had an impor-tant influence on Weber.114 While he was a student, Weber went to Leipzig in 1861-62, where he followed classes by Möbius and Scheibner. After his doctorate, he went to Königsberg and fully took advantage of the rich scien-tific activities around the mathematics and physics seminars (mathematisch-physikalisches Seminar) founded by Jacobi and Neumann, and studied with Franz Neumann and Richelot (a student of Jacobi as well). In 1866, We-ber went back to Heidelberg and obtained his Habilitation.115 From 1869 to 1870, Weber was Privatdozent and ausserordentlicher Professor in Hei-delberg. In 1870, Weber took Prym’s position at the ETH Zürich (where Dedekind taught for a few years a decade earlier). During these years, he worked with Dedekind on the publication of Riemann’s works.

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More investigations about polygons

So far, only polygons with a positive power have been considered. This im-plies that only positive order numbers have been taken into account, since the poles have a negative order number. Following their arithmetical ap-proach and the correspondence between polygons and ideals, Dedekind and Weber propose a representation of the functions of Ω as quotients of poly-gons. The denominator of the quotient will serve for the description of negative order numbers.
Why is this important? We saw that functions in Ω are completely char-acterized by their order numbers: if η in Ω takes the value 0 with multiplicity r at P, its order number is r; if η takes the value ∞ with multiplicity r at P0, its order number is −r; if η takes any other (constant) value, its order number is 0. There is only a finite number of points at which η has an order number different from 0. In addition, “the sum of all these order numbers is 0, and hence the sum of positive order numbers equals the sum of negative order numbers, and indeed it equals the order of the function η” (ibid., 103). In addition
If the order numbers of a function η are known for each point P, then the function η is determined up to a constant factor. Because if η0 has everywhere the same order number as η then ηη0 has order number zero everywhere and hence is a constant.90 (ibid., 103).
So, if one were to construct a polygon A with each point at which η has a positive order number, and a polygon B with each point at which η has a negative order number, then these two polygons are of order the order of η and characterize the function completely up to a constant. But the definition adopted so far does not include polygons whose points have a negative power. Rather than to propose the definition of an ‘arbitrary’ polygon with arbitrary (positive or negative) powers, Dedekind and Weber “symbolically set” the function η as quotient of two polygons A and B = BA. In accordance with the names used for quotients of ideals, A is called the upper polygon and B the lower polygon. This allows Dedekind and Weber to introduce negative powers – those of the lower polygon – without adopting an approach which would not agree with the arithmetic of ideals as developed in the first part. In fact, by adopting the presentation of negative powers as the powers of the denominator of a quotient, they lean even more towards an arithmetical treatment of the theory.

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 

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