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].
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.”
Nothing capable of proof ought to be accepted in science without proof.1
Of all the aids which the human mind has for simplifying its life, i.e., the work in which thinking consists, none is so rich in consequences and so inseparably bound up with its innermost nature as the concept of number. Arithmetic, whose sole object is this concept, is already a discipline of insurmountable breadth, and there is no doubt that there are absolutely no limits to its further development. Equally insurmountable is its field of application, for every thinking man, even if he does not clearly realize it, is a man of numbers, an arithmetician.2
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References
[Dedekind 1888, 335]: “Was beweisbar ist, soll in der Wissenschaft nicht ohne Beweis geglaubt werden.”
Dedekind, undated manuscript [Dugac 1976, 315]: “Von allen Hilfsmitteln, welche der menschliche Geist zur Erleichterung seines Lebens, d.h. der Arbeit, in welcher das Denken besteht, ist keines so folgenreich und so untrennbar mit seiner innersten Natur verbunden, wie der Begriff der Zahl. Die Arithmetik, deren einziger Gegenstand dieser Begriff ist, ist schon jetzt eine Wissenschaft von unermesslicher Ausdehnung und es ist keinem Zweifel unterworfen, dass ihrer ferneren Entwicklung gar keine Schranken gesetzt sind; ebenso unermesslich ist das Feld ihrer Anwendung, weil jeder denkende Mensch, auch wenn er dies nicht deutlich fiihlt, ein Zahlen-Mensch, ein Arithmetiker ist.”
Kronecker [1887, 252] attributes the sentence to Gauss an the basis of apparently reliable evidence. Kline [1972, 104] says it was Jacobi who coined it, and quotes Plato’s dictum in [Kline 1980, 16].
Kronecker is reported to have said, in a lecture to a congress of 1886, that good God made the integers, and all the rest is the work of man [Weber 1893a, 15].
Since the book is divided into numbers corresponding to definitions or theorems, we shall refer to them as follows: Zahlen.66 refers to proposition 66 in Dedekind’s numbering.
[Dedekind 1888, 335]: “selbst bei der Begründung der einfachsten Wissenschaft, nämlich desjenigen Theiles der Logik, welcher die Lehre von den Zahlen behandelt.”
[Dedekind 1854, 430–311: “Aber die so gegebenen Definitionen dieser Grundoperationen genügen der weitern Entwicklung der Arithmetik nicht mehr, und zwar aus dem Grunde, weil sie die Zahlen, mit denen sie operieren lehrt, auf ein sehr kleines Gebiet beschränkt annimmt. Die Forderung der Arithmetik nämlich, durch jede dieser Operationen das gesamte vorhandene Zahl-gebiet jedesmal von neuem zu erzeugen, oder mit andern Worten: die Forderung der unbedingten Ausführbarkeit der indirekten, umgekehrten Operationen, der Substraktion, Division usw., führt auf die Notwendigkeit, neue Klassen von Zahlen zu schaffen, da mit der ursprünglichen Reihe der absoluten ganzen Zahlen dieser Forderung kein Genüge geleistet werden kann.”
The genetic approach, on the basis of the integers, the motivation for the expansion of the number system, and the focus on a rigorous redefinition of the operations, can all be found in Ohm (§§I.3 and IV.1). More specifically, see [Bekemeier 1987, chap. 2, particularly 103ff], or the shorter exposition in [Novy 1973, 83–89].
Preserved under the signatures [Cod. Ms. Dedekind III, 2] and [III, 4]. The viewpoint is rather elementary, and the terminology is not yet that of [1872/78]; the topic is referred to in [Dedekind 1872, 317–18]. Thus, they seem to have been written before 1872
Dedekind borrowed Hamilton’s book from the Göttingen Library in 1857, as evidenced by the Ausleihregister kept at the Niedersächsische Staats-and Universitätsbibliothek.
On this topic, see [Hankins 1976; Mathews 1978, Hendry 1984]. I shall not refer to the 1837 paper, a notable piece of work, since it is very likely that Dedekind did not read it.
Interestingly, Hamilton referred to the possibility of a theory of ordered sets or “systems” [1853, 132], as an extension of those of ordered couples and ordered quadruples (the quaternions). The reader should take into account that Hamilton is properly talking of ordered n-tuples, not sets in the modern sense of the term. But one may assume that this kind of statement may have encouraged Dedekind in his tendency to introduce the abstract notion of set or “system” in arithmetic, algebra, analysis, and number theory.
And to Helmholtz [1887], an article cited in [Dedekind 1888, 335]. A similar criticism can be found earlier in Cantor [1883, 191–92]; it is one of those instances in which one doubts whether they came independently to the same idea, or they discussed it in one of their few meetings.
As late as 1876 Dedekind had not read this work [Lipschitz 1986, 74], which means that his contribution was independent.
From this viewpoint, one has to say that his axiomatization of the real numbers is not adequate, for it does not include an axiom of continuity (completeness). Grassmann kept talking about `magnitudes,’ although in an extremely abstract sense; one should probably blame this reliance on a traditional approach for his inadvertence of the need to enforce a continuous number-domain.
’Arithmetische Grundlagen’ [Cod. Ms. Dedekind, III, 4, II]. One should differentiate these manuscripts from the ones on M and mentioned in §1.1.
The recursive definitions of the operations are only mentioned expeditiously at one point in the draft [Dedekind 1872/78, 303].
[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]).
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].
Dedekind then goes on to introduce addition, with some theorems, to define substraction, and to consider very briefly the introduction of the integers.
[Dugac 1976, 293]: “Die Begriffe des Systems, der Abbildung, welche im Folgenden eingeführt werden, um den Begriff der Zahl, der Anzahl zu begründen, bleiben auch dann für die Arithmetik unentbehrlich, selbst wenn man den Begriff der Anzahl als unmittelbar evident (`innere Anschauung’) voraussetzen wollte.” For the meanings of `Zahl’ and `Anzahl,’ see [op.cit., 300, 303].
Actually, Dedekind eliminated a reference to Cantor that appeared in his 1887 draft of the preface (§VII.2.1 and [Cavaillès 1962, 120]). Passages that are relevant to Cantorian set theory are the footnote to [Dedekind 1888, 387] and also [Zahlen.34 and 63].
When it comes to one-to-one mappings, Cantor speaks of correlating [beziehen] or coordinating [zuordnen] univocally and completely, or else of a correspondence (§VI.2–4); in the context of well-ordered sets, he speaks of “mappings” [Abbildungen], meaning order-isomorphisms (§VIII.4.1); and in another context he talks of “coverings” [Belegungen], which correspond to Dedekind’s mappings (§VIII.7).
[Dedekind 1888, 344–47]: “echter Teil,” “zusammengesetze System.,” “Gemeinheit der Systeme.”
[Dedekind 1888, 344]: “Es kommt sehr häufig vor, dass verschiedene Dinge a, b,c,... aus irgendeiner Veranlassung unter einem gemeinsamen Gesichtspunkte aufgefasst, im Geiste zusammengestellt werden, und man sagt dann, dass sie ein System S bilden; man nennt die Dinge a, b,c,... die Elemente des Systems S, sie sind enthalten in S; umgekehrt besteht S aus dieser Elementen Ein solches System S (oder ein Inbegriff, eine Mannigfaltigkeit, eine Gesamtheit) ist als Gegenstand unseres Denkens ebenfalls ein Ding; es ist vollständig bestimmt, wenn von jedem Ding bestimmt ist, ob es Element von S ist oder nicht.”
[1888, 357]: “Meine Gedankenwelt, d. h. die Gesamtheit S aller Dinge, welche Gegenstand meines Denkens sein können, its unendlich.”
[Cod. Ms. Dedekind, III, 1, III, p. 2]: “Ein System kann aus einem Element bestehen (d. h. aus einem einzigen, aus einem und nur einem), kann auch (Widerspruch) leer sein (kein Element enthalten).... Erweiterung (des Begriffs) im Gegensatz zu Verengerung.”
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.
[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.
[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.”
Frege did so in [1884], but Dedekind read this book for the first time in 1889 [Dedekind 1888, 342; Sinaceur 1974, 275].
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.
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].
[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.“
Hilbert [1922], in [Ewald 1996, vol.2, 1121].
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].
As translated in [van Heijenoort 1967, 101], though with a small change.
[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.”
Interestingly, in recent times there have been philosophical contributions that are reminiscent of Dedekind’s viewpoint, see [Benacerraf 1965; Parsons 1990].
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].
[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.”
[Dedekind 1888, 371]: “Satz der Definition durch Induktion. Ist eine beliebige (ähnliche oder unähnliche) Abbildung 9 eines Systems SI in sich selbst und ausserdem ein bestimmtes Element w in Q gegeben, so gibt es eine und nur eine Abbildung yi der Zahlenreihe N, welche den Bedingungen/I. v(A1) 3 Q,/II. yi(1) = w,/III. v(n) = 9W(n) genügt, wo n jede Zahl bedeutet.”
The notion of categoricity was first introduced by Veblen and Huntington in the 1900s.
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)
See [Dedekind 1893, §161, 456–57]. In this work, Dedekind presented Galois theory as dealing with groups of automorphisms.
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]).
`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.
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].
[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.
Zermelo had independently found the same proof and published it in [1908, 208–09].
[Meschkowski & Nilson 1991, 302]: “Das künstliche System der 172 sich nur um das Elementarste und zum Theil Trivialste drehenden Dedekindschen Sätze scheint mir mehr geeignet, die Natur der Zahlen zu verdunkeln als sie aufzuhellen.”
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.
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.”
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.
[Dugac 1976, 179]: “perpetuo ordine et certa quadam ratione sequens praeceptum antecedentibus innitatur.”
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.
It was also the orientation of contemporaries like Pasch and Weierstrass, and later Hilbert.
[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.”
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.
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.
[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.”
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].
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.
This position was more Leibnizian (see, e.g., [Leibniz 1704, book IV, chap. 7]), while Dedekind’s would appear to be Kantian.
By contrast, the possibility of forming such deductive chains was for Dedekind convincing proof that those truths are not gained by inner intuition.
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!”
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]).
There are references to [Schröder 1877] in the last pages of [Dedekind 1872/78].
See [Gana 1985]. Peirce’s accusation stumbles upon the fact that Dedekind formulated his definition in 1872.
On the Peano school, see [Borga, Freguglia & Palladino 1985] and also [RodriguezConsuegra 1991] for a discussion of its influence on Russell.
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.
Schröder generalized the theory of chains further, observing that it does not strictly require mappings, but can be applied generally to binary relations.
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].
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].
[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].
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.
[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.”
See Noether’s comments in [Dedekind 1930/32, vol. 3, 390–391], and [Moore 1978; 1982].
See, e.g., his articles in [van Heijenoort 1967, 375] and [Ewald 1996, vol. 2, 1119, 1121, 1151]; or the appendixes to [Hilbert 1930].
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Ferreirós, J. (1999). Sets and Maps as a Foundation for Mathematics. In: Labyrinth of Thought. Science Networks. Historical Studies, vol 23. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-5049-0_7
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