Richard Dedekind 1888: Das Auflösevermögen des Arithmetikers

1888 veröffentlichte Dedekind Was sind und was sollen die Zahlen? (Zahlen), das gemeinhin von Mathematikern und Mathematikhistorikern zusammen mit Peanos Arithmetices Principia als der Grundstein zur Axiomatisierung der Arithmetik angesehen wird. Dabei wurde der für eine mathematische Schrift ungewöhnliche Titel entweder übersehen oder miss- bzw. umgedeutet. Interessanterweise lässt sich das an den unterschiedlichen Übersetzungen festmachen, zum Beispiel bei André Weil („Que sont et que représentent les nombres?“, also „Was sind und was stellen die Zahlen dar?“), dessen Übertragung dann vom Dedekind-Spezialisten Pierre Dugac übernommen wurde. Im folgenden Beitrag wird die Geschichte dieser Interpretationen rekonstruiert sowie nach Gründen gesucht, weshalb Dedekinds Vorwort unserer Ansicht nach unterschätzt wurde. Das führt uns zur These, dass Zahlen kein rein mathematisches Traktat war, sondern eine Kampfschrift, und dass sich Dedekind von seinem Abbildungsbegriff weit mehr erhoffte, als man vom rein mathematischen Standpunkt aus vermuten möchte. Schlüsselwörter: Abbildung, axiomatische Methode, unendliche Mengen, Zahlbegriff, Dilthey, Helmholtz, Kronecker, Abbildungsvermögen, Einbildungskraft, Schöpferkraft, Weltanschauung.

What Is a Number?

An examination of and plea for a time-honored answer to the title question, that answer being, “A number is one principal result among others of a process of converting magnitudes drawn from a continuum, via a scheme of measurement, into arithmetic quantities.” Ideas on this subject of Paul du Bois-Reymond, Richard Dedekind, and Otto Hölder are subjected to detailed statement and close analysis. At the very center lies du Bois-Reymond’s demonstration that the Cantor-Dedekind Axiom–that an intuition into the nature of continuous magnitude shows that the geometric line is isomorphic to the array of Dedekind real numbers–is not merely unprovable but wholly false.

Frege’s Relation to Dedekind: Basic Laws and Beyond

Among all of Frege’s contemporaries, Richard Dedekind is arguably the thinker closest to him in terms of their general backgrounds and core projects. This essay provides a reexamination of Frege’s critical reactions to Dedekind, in Grundgesetze and some related texts. The reexamination includes documenting their interaction in some detail and putting it into a broader context, both philosophically and systematically. It also involves separating Frege’s less compelling criticisms of Dedekind from those that have deeper, more lasting significance. The essay ends with a suggestion for how to reconcile Fregean and Dedekindian forms of logicism, based on distinguishing two distinct but complementary kinds of abstraction principles.

A CONSTRUÇÃO DO SISTEMA DOS NÚMEROS REAIS POR WEIERSTRASS

Apesar de já na Antiguidade Clássica se ter reconhecido a existência de grandezas incomen-suráveis, não seria antes do século XIX que se estabeleceriam definições rigorosas do conceito de número irracional, sem recurso a intuições geométricas. O conceito mais geral de número real era apenas percebido intuitivamente e a sua existência apenas assegurada por considerações de natureza geométrica e algébrica. A partir do início do século XIX surgiu uma preocupação crescente em colocar a Análise sobre bases aritméticas sólidas; reconhecia-se que a falta duma teoria dos números reais tornava incorretas (ou, pelo menos, incompletas) as demonstrações de certos resultados. Desta forma, uma etapa importante do processo de aritmetização da Análise seria a elaboração duma teoria da reta real sobre fundações puramente aritméticas. Dos três nomes que devem referenciar-se neste contexto – Charles Méray, Karl Weierstrass e Richard Dedekind – destacaremos o de Weierstrass que, contrariamente aos outros dois, não se limitou a construir os reais a partir duma pressuposta construção dos racionais. Weierstrass parte da noção mais geral de número e das operações fundamentais da Aritmética; introduz inicialmente o con-ceito de número natural e, de seguida, o de número racional positivo; considerando “agregados” destes números obtém então grandezas para além das racionais. Por esta razão, na teoria dos números reais de Weierstrass, não se podem dissociar as naturezas dos números naturais, racionais e reais. Weierstrass constrói a sua teoria de modo inteiramente analítico, dotando-a dum rigor muito característico de toda a sua obra matemática e elaborando a teoria dos números reais mais completa do século XIX. Palavras-chave: Aritmetização da Análise; construção dos números reais; Weierstrass.

The Mathematics of Continuous Multiplicities: The Role of Riemann in Deleuze's Reading of Bergson

A central claim of Deleuze's reading of Bergson is that Bergson's distinction between space as an extensive multiplicity and duration as an intensive multiplicity is inspired by the distinction between discrete and continuous manifolds found in Bernhard Riemann's 1854 thesis on the foundations of geometry. Yet there is no evidence from Bergson that Riemann influences his division, and the distinction between the discrete and continuous is hardly a Riemannian invention. Claiming Riemann's influence, however, allows Deleuze to argue that quantity, in the form of ‘virtual number’, still pertains to continuous multiplicities. This not only supports Deleuze's attempt to redeem Bergson's argument against Einstein in Duration and Simultaneity, but also allows Deleuze to position Bergson against Hegelian dialectics. The use of Riemann is thereby an important element of the incorporation of Bergson into Deleuze's larger early project of developing an anti-Hegelian philosophy of difference. This article first reviews the role of discrete and continuous multiplicities or manifolds in Riemann's Habilitationsschrift, and how Riemann uses them to establish the foundations of an intrinsic geometry. It then outlines how Deleuze reinterprets Riemann's thesis to make it a credible resource for Deleuze's Bergsonism. Finally, it explores the limits of this move, and how Deleuze's later move away from Bergson turns on the rejection of an assumption of Riemann's thesis, that of ‘flatness in smallest parts’, which Deleuze challenges with the idea, taken from Riemann's contemporary, Richard Dedekind, of the irrational cut.

“A voice unknown and mysterious” : Emmy Noether / Jean Cavailles, Exchanges Cantor-Dedekind (1937) – a warning translated, and the birth of set theory.

Translation of the forewords to Emmy Noether and Jean Cavailles’1937 Briefwechsel Cantor-Dedekind, an edition of exchanges between GeorgCantor and Richard Dedekind, the foundations for set theory.

Logicism

The term ‘logicism’ refers to the doctrine that mathematics is a part of (deductive) logic. It is often said that Gottlob Frege and Bertrand Russell were the first proponents of such a view; this is inaccurate, in that Frege did not make such a claim for all of mathematics. On the other hand, Richard Dedekind deserves to be mentioned among those who first expressed the conviction that arithmetic is a branch of logic. The logicist claim has two parts: that our knowledge of mathematical theorems is grounded fully in logical demonstrations from basic truths of logic; and that the concepts involved in such theorems, and the objects whose existence they imply, are of a purely logical nature. Thus Frege maintained that arithmetic requires no assumptions besides those of logic; that the concept of number is a concept of pure logic; and that numbers themselves are, as he put it, logical objects. This view of mathematics would not have been possible without a profound transformation of logic that occurred in the late nineteenth century – most especially through the work of Frege. Before that time, actual mathematical reasoning could not be carried out under the recognized logical forms of argument: this circumstance lent considerable plausibility to Immanuel Kant’s teaching that mathematical reasoning is not ‘purely discursive’, but relies upon ‘constructions’ grounded in intuition. The new logic, however, made it possible to represent standard mathematical reasoning in the form of purely logical derivations – as Frege, on the one hand, and Russell, in collaboration with Whitehead, on the other, undertook to show in detail. It is now generally held that logicism has been undermined by two developments: first, the discovery that the principles assumed in Frege’s major work are inconsistent, and the more or less unsatisfying character (or so it is claimed) of the systems devised to remedy this defect; second, the epoch-making discovery by Kurt Gödel that the ‘logic’ that would be required for derivability of all mathematical truths can in principle not be ‘formalized’. Whether these considerations ‘refute’ logicism will be considered further below.