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Sets and Maps as a Foundation for Mathematics

  • José Ferreirós
Part of the Science Networks. Historical Studies book series (SNHS, volume 23)

Abstract

According to Plutarch, the great philosopher Plato said: άεί ό θεòς γεωμετρει, God eternally geometrizes. The sentence was remembered in 19th-century Germany, and it was subject to changes that reflect the changing conceptions of mathematical rigor and pure mathematics. During the first half of the century, one of the greatest German mathematicians said, άεί ό θεòς άριθμητίζει, ‘God eternally arithmetizes;’3 geometry had lost its privileged foundational position to arithmetic. Gauss was of the opinion that, while space has an outside reality and we cannot prescribe its laws completely a priori, number is merely a product of our spirit or mind [Geist; Gauss 1863/1929, vol. 8, 201]. Dedekind essentially agreed, and his most important foundational work, Was sind and was sollen die Zahlen? [1888], bears the motto: άεί ό άνθρωπoς άριθμητίζει, ‘man always arithmetizes.’ It seems that, in Dedekind’s view, numbers are not made by God, but by men;1 mathematics has nothing to do with a world of essences or a Platonic heaven, it is a free creation of the human mind [Dedekind 1888, 335, 360].
Figure 7

Title page of Dedekind’s What are numbers and what could they be? [also: ... and what are they for?] [1888]. Notice the Greek motto: “man eternally arithmetizes.”

Keywords

Natural Number Number System Pure Mathematic Cardinal Number Mathematical Induction 
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References

  1. 1.
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    ’Arithmetische Grundlagen’ [Cod. Ms. Dedekind, III, 4, II]. One should differentiate these manuscripts from the ones on M and mentioned in §1.1.Google Scholar
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    The recursive definitions of the operations are only mentioned expeditiously at one point in the draft [Dedekind 1872/78, 303].Google Scholar
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    [Leibniz 1704, book IV, chap. 7]. But Leibniz is not necessarily the source; Ohm, e.g., did it the same way (see [Bekemeier 1987, 167]).Google Scholar
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    As translated in [van Heijenoort 1967, 99–100]. The German text, and a French translation, can be found in [Sinaceur 1974, 2721. The letter was first noticed by Wang [1957].Google Scholar
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    Dedekind then goes on to introduce addition, with some theorems, to define substraction, and to consider very briefly the introduction of the integers.Google Scholar
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    See [Dedekind 1888, 345], an 1888 letter to Weber [Dugac 1976, 273], and the manuscript `Gefahren der Systemlehre’ [Sinaceur 1971], where we can read that in 1888 he planned to write an announcement of the book, where he would have discussed this delicate point, as he does in the letter to Weber.Google Scholar
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    [Dedekind 1888, 345]. The reasons may have had to do with simplifying the proofs, or with his decision to begin the number sequence with 1, regarding 0 as an integer.Google Scholar
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    [Dedekind 1888, 348]: “Unter einer Abbildung cp eines Systems S wird ein Gesetz verstanden, nach welchem zu jedem bestimmten Element s von S ein bestimmtes Ding gehört, welches das Bild von s heisst und mit (p(s) bezeichnet wird; wir sagen auch, dass cp(s) dem Element s entspricht.” Google Scholar
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    Frege did so in [1884], but Dedekind read this book for the first time in 1889 [Dedekind 1888, 342; Sinaceur 1974, 275].Google Scholar
  32. 1.
    Model theory is a branch of mathematical logic that considers the interplay between formal axiom systems and their models (i.e., mathematical structures that satisfy the axioms). Properly speaking, it consolidated around 1950, although one can find model-theoretic arguments in the 1920s.Google Scholar
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    Letter to Keferstein, 1890, as translated by Wang and Bauer-Mengelberg in [van Heijenoort 1967, 100–01]. The original text can be found in [Sinaceur 1974, 273–75].Google Scholar
  34. 1.
    [Dedekind 1888, 354–55]: “Satz der vollständigen Induction. Um zu beweisen, dass die Kette A o Teil irgendeines Systems E ist — mag letzteres Teil von S sein order nicht —, genügt es zu zeigen,/p. dass A 3 E, und/e. dass das Bild jedes gemeinsamen Elementes von A o und E ebenfalls Element von E ist.“Google Scholar
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    Hilbert [1922], in [Ewald 1996, vol.2, 1121].Google Scholar
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    As a matter of fact, it was only after the publication of Dedekind’s booklet that Cantor began to give definitions of finite and infinite, without naming him [Cantor 1887/88, 414–15; 1895/97, 295].Google Scholar
  37. 2.
    As translated in [van Heijenoort 1967, 101], though with a small change.Google Scholar
  38. 1.
    [Landau 1917, 56]: “Die Existenz unendlicher Systeme, auf der seine Theorie der Zahlenreihe beruht, will Dedekind, anstatt sie einfach axiomatisch zu postulieren, auf das Beispiel unserer `Gedankenwelt’, d.h. die Gesamtheit alles Denkbaren, begründen.... Aber es hat sich doch später (durch Russell u. a.) gezeigt, dass diese Gedankenwelt nicht als System im gleichen Sinne gelten kann. Doch ist diese mehr philosophische als mathematische Begründung seiner Annahme für die weiteren Entwicklungen durchaus unerheblich.”Google Scholar
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    Interestingly, in recent times there have been philosophical contributions that are reminiscent of Dedekind’s viewpoint, see [Benacerraf 1965; Parsons 1990].Google Scholar
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    Weber was among the first mathematicians who showed an interest in Dedekind’s work; eventually, he even published a paper on `elementary set theory,’ in connection with Dedekind [Weber 1906].Google Scholar
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    [Dedekind 1888, 360]: “In Rücksicht auf diese Befreiung der Elemente von jedem anderen Inhalt (Abstraktion) kann man die Zahlen mit Recht eine freie Schöpfung des menschlichen Geistes nennen.”Google Scholar
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    The notion of categoricity was first introduced by Veblen and Huntington in the 1900s.Google Scholar
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    AC is employed when he uses the mappings a n to define new mappings yin that are extensions of each other. On that basis, one can finally define a mapping z: RI->E by the condition x(n) — Wn(n) Google Scholar
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    See [Dedekind 1893, §161, 456–57]. In this work, Dedekind presented Galois theory as dealing with groups of automorphisms.Google Scholar
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    Hilbert noted down, when he visited Berlin in 1888, everybody was talking about the booklet and mostly in critical terms (see his paper of 1931 in [Ewald 1996, vol. 2, 1151]).Google Scholar
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    `Die Voraussetzungen der reinen Geometrie der Lage and deren Beziehungen zur Zahlen-Wissenschaft’ [Cod. Ms. Dedekind XII, 3]. The dating (late 1870s or 1880s) is my own, on the basis of the terminology employed, which suggests that Dedekind had already developed the elements of his theory of sets in the draft [1872/78]. His use of the expression ‘Geometrie der Lage’ may indicate that, at this time, he was influenced by von Staudt or by Reye.Google Scholar
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    The manuscript was found by Cavaillès in Dedekind’s Nachlass, and published in [Dedekind 1930/32, vol.3, 447–48]. Cantor’s conjecture is in [1883, 201] and Bemstein’s proof in [Borel 1898].Google Scholar
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    [Cantor & Dedekind 1976, 261] or [Dedekind 1930/32, vol. 3, 448]: “und stutzte ein wenig, als ich meine Überzeugung aussprach, dass derselbe mit meinen Mitteln (Was sind und was sollen die Zahlen?) leicht zu beweisen sei.” Berstein’s visit had been motivated by Cantor himself, see §VIII.8.Google Scholar
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    Zermelo had independently found the same proof and published it in [1908, 208–09].Google Scholar
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    Regarding deductive structure and equality, compare Leibniz’s `Non inelegans specimen’ [1966, 122–30], first published in 1840. For the rest, see §I.3 and below.Google Scholar
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    Letter to Frobenius, February 1895, in [Dugac 1976, 283]: “die Meisten halten Alles, was ihnen unbewusst durch lebenslängliche Uebung mechanisch geläufig geworden ist, auch fir einfach, und sie erwägen gar nicht, wie lang oft die hierbei ins Spiel kommende Gedanken-Kette ist.”Google Scholar
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    Of course, this idea involves a confusion, well-known to some cognitive scientists, that is typical of the 19th century: to think that logic is somehow `in our brains,’ in our (real, psychological) thinking, and believe that the real thinking process must reproduce the steps into which the task in question can be logically analyzed.Google Scholar
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    [Dugac 1976, 179]: “perpetuo ordine et certa quadam ratione sequens praeceptum antecedentibus innitatur.”Google Scholar
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    For this reason, we have no direct evidence for a Kantian Dedekind, and one should be carefill with treatments like [McCarty 1995]. I myself started twice to write papers on Dedekind and Kant, and then Dedekind and Leibniz, until I became convinced that available evidence was too scanty to warrant conclusions.Google Scholar
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    It was also the orientation of contemporaries like Pasch and Weierstrass, and later Hilbert.Google Scholar
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    [Dedekind 1888, 335]: “Indem ich die Arithmetik (Algebra, Analysis) nur einen Teil der Logik nenne, spreche ich schon aus, dass ich den Zahlbegriff für gänzlich unabhängig von den Vorstellungen oder Anschauungen des Raumes und der Zeit, dass ich ihn vielmehr für einen unmittelbaren Ausfluss der reinen Denkgesetze halte.”Google Scholar
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    This was not uncommon, though the overall discipline was sometimes called analysis [Gauss 1801, xvii] or algebra [Hamilton 1837, 6]. Pasch [1882, 164], Kronecker [1887, 253] and Schröder [1890/95, vol. 1, 441] agree with Dedekind in calling it arithmetic.Google Scholar
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    This would be supported by a free or imprecise reading of Kant’s Kritik der reinen Vernunft, where by (transcendental) logic one understands the doctrine of the a priori, non-intuitive contents of the understanding [Kant 1787]. Imprecise because it would not take into account the distinction between formal and transcendental logic.Google Scholar
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    [Bolzano 1851, 13]: “Es gibt schon im Reiche derjenigen Dinge, die keinen Anspruch auf Wirklichkeit, ja nur auf Möglichkeit machen, unstreitig Mengen, die unendlich sind. Die Menge der Sätze und Wahrheiten an sich ist, wie sich sehr leicht einsehen lässt, unendlich.”Google Scholar
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    Connections between theorem 66 and traditional logic, particularly the analysis of propositions into the form subject—predicate, are emphasized in `Ober den Begriff des Unendlichen,’ an unpublished article in response to Keferstein that can be found in [Sinaceur 1974].Google Scholar
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    If we call `my own ego’ y, the chain cp0(v) is a simply infinite set and could be identified with ICY, on the basis of Zahlen.73. Google Scholar
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    This position was more Leibnizian (see, e.g., [Leibniz 1704, book IV, chap. 7]), while Dedekind’s would appear to be Kantian.Google Scholar
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    By contrast, the possibility of forming such deductive chains was for Dedekind convincing proof that those truths are not gained by inner intuition.Google Scholar
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    Dedekind to Klein, April 1888 [Dugac 1976, 189]: “Und was wird der geduldige Leser am Schlusse sagen? Dass der Verfasser mit einem Aufwande von unsäglicher Arbeit es glücklich erreicht hat, die klarsten Vorstellungen in ein unheimliches Dunkel zu hüllen!”Google Scholar
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    Although he published four years after Frege [1884], Dedekind worked independently and did not even know that book until 1889 (see [Dedekind 1888, 342–43; van Heijenoort 1967, 101]). Thus, Frege’s work could not make him feel secure or motivate him to publish (he was motivated by papers of Kronecker and Helmholtz, see [1888, 335]).Google Scholar
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    There are references to [Schröder 1877] in the last pages of [Dedekind 1872/78].Google Scholar
  69. 2.
    See [Gana 1985]. Peirce’s accusation stumbles upon the fact that Dedekind formulated his definition in 1872.Google Scholar
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    On the Peano school, see [Borga, Freguglia & Palladino 1985] and also [RodriguezConsuegra 1991] for a discussion of its influence on Russell.Google Scholar
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    The sense given to `arithmetic’ by Schröder is the same general sense in which Dedekind used the word [Schröder 1890/95, vol.1, 441 footnote]. Schröder’s logicism has been analyzed in detail by Peckhaus [1991; 1993], though without emphasizing the role of Dedekind’s work in his conversion to logicism.Google Scholar
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    Schröder generalized the theory of chains further, observing that it does not strictly require mappings, but can be applied generally to binary relations.Google Scholar
  73. 1.
    Frege criticized Dedekind’s notion of set in the introduction to his Grundgesetze [Frege 1893], and Cantor’s theory in reviews (see [Dauben 1979, 220–28]; [Frege 1895] criticizes Schröder in a similar vein). A lengthy and sophisticated defense of the extensional viewpoint can be found in [Schröder 1890/95, vol. 1, 83–101].Google Scholar
  74. 1.
    A more detailed and focused discussion of these issues can be found in [Ferreirós 1996], and an analysis of the fall of logicism in [Ferreirós 1997].Google Scholar
  75. 1.
    [Hilbert 1930, 4]: “Wir denken drei verschiedene Systeme von Dingen: die Dinge des ersten Systems nennen wir Punkte.” For his reaction in 1888, see [Dugac 1976, 93; Ewald 1996, vol. 2, 1151].Google Scholar
  76. 2.
    For set theory has to be axiomatized, and then its formal develoment must presuppose the `intuition of iteration’ and the number sequence [Weyl 1918, 116–17]; see §X.1.Google Scholar
  77. 1.
    [Weyl 1918, 35; see also 36]: “Durch Tradition eingesponnen in jenen ja heut in der Mathematik zur unbedingten Herrschaft gelangten Gedankenkomplex, der vor allem an die Namen Dedekind und Cantor anknüpft, habe ich fair mich den aus diesem Kreise herausführenden Weg gefunden und durchmessen, den ich hier abgesteckt habe.”Google Scholar
  78. 2.
    See Noether’s comments in [Dedekind 1930/32, vol. 3, 390–391], and [Moore 1978; 1982].Google Scholar
  79. 3.
    See, e.g., his articles in [van Heijenoort 1967, 375] and [Ewald 1996, vol. 2, 1119, 1121, 1151]; or the appendixes to [Hilbert 1930].Google Scholar

Copyright information

© Springer Basel AG 1999

Authors and Affiliations

  • José Ferreirós
    • 1
  1. 1.Dpto. de Filosofía y Lógica, Avda. San Francisco Javier, s/nUniversidad de SevillaSevillaSpain

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