𝟑. 𝔗𝔥𝔢 ℑ𝔰𝔰𝔲𝔢 𝔬𝔣 𝔱𝔥𝔢 ℑ𝔫𝔣𝔦𝔫𝔦𝔱𝔢
ℑ𝔱 has frequently been written that the Aristotelian horror infiniti reigned among scientists and mathematicians until Cantor's vigorous defense of the possibility and necessity of accepting it. Like other extreme statements, this one does not bear a historical test. At least in Germany, and perhaps here it makes much sense to speak of national differences, there was a noticeable tendency to accept the actual infinite. The philosophical atmosphere could hardly have been more favorable, and there were several attempts to develop mathematical theories of the infinite. We shall begin with philosophy, and then consider the views of some mathematicians.
𝔗𝔥𝔢 history of philosophical attitudes toward actual infinity in 19th-century Germany would be a long one. By the beginning of the century, during the time of idealism, the potential infinity of mathematics was called the "bad infinite" by Hegel and his followers. The implication was clear: there was a 'good' infinite that is actual in the highest sense, the philosophical infinite, the absolute. It is well century; among mathematicians, the best examples seem to be Steiner and Kummer. The philosophy of nature was also full of implications for the problem of infinity. Here the question normally took the form of deciding whether space and time are bounded or infinite, or whether the physical universe is or not made up of simple elements. These are the first and second "antinomies" discussed by Kant in his Kritik der reinen Vernunft [1787, 455 – 471]. The issue was taken up by later philosophers, for instance by the anti-idealist Herbart, whome Riemann took as his mentor in philosophy (see §11.4.2 for his views on infinity).¹
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| Georg Wilhelm Friedrich Hegel 1770 – 1831 |
I am so much for the actual infinite that, instead of admitting that nature abhores it, as is vulgarly said, I sustain that it affects her everywhere in order to better mark the perfections of it's author. Thus i belive there is no part of matter which is not, I do not say divisible, but actually divided; and consequently the least particle of matter must be regarded as a world full of an infinity of different creatures.²
ℭ𝔢𝔯𝔱𝔞𝔦𝔫𝔩𝔶, this is not Leibniz's position in all his writings,³ but it is the one that characterizes the whole spirit and the statements in his Monadologie, a work that advanced a conception of the universe as made up of simple metaphysical units [Leibniz 1714, §§ 57, 64-67] in the wake of the Leibnizian revivel, the ideas of the Monadologie were put into new life. Herbart elaborated an ontology of simple units called the "Reale," which is reminiscent of Leibniz monads; the physicist and philosopher Fechner defended a Leibnizian Atomenlehre [Theory of atoms; 1864], somewhat like the physiologist and philosopher Lotze in his famous work Mikrokosmus [1856/64]. As the reader can see, we are now talking about authors who were quiet influential among the scientific community. In fact, even Cantor defended viewpoints similar to those of Fechner and Lotze [1932, 275 - 276]; he quoted Farady, Ampère and Wilhelm Weber as his forerunners.
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| Michael Faraday 1791 - 1867 |
ℑ𝔫𝔱𝔢𝔯𝔢𝔰𝔱𝔦𝔫𝔤𝔩𝔶, Leibnizian ideas can be found in all the main authors that we shall deal with in Part One. Riemann was strongly influenced by Herbart, and his fragments on psychology and physics present us with viewpoints that are quite close to the author of the Monadologie. Riemann (and Herbart) seem to favour the Leibnizian conception of space as an 'order of coexistance' of things (see § 11.1.2). Also noteworthy is Riemann's preference for the hypothesis of a material plenum and contact action, instead of Newtonian action at-a-distance.⁴ His friend Dedekind was also in favour of this hypothesis, as he made clear in a noteworthy passage of his correspondence with Heinrich Weber.
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| Heinrich Martin Weber 1842 - 1913 |
𝔗𝔥𝔦𝔰 common trait in the otherwise divergent views of Riemann, Dedekind and Cantor could help explain their attitude toward infinity. The influence of Leibniz's Monadologie might be one of the key factors that impelled them to accept the actual infinite. Another key factor, of course was the development of mathematical ideas themselves, for instance (though not exclusively) in the foundations of the calculus. This brings us to the issue of attitudes toward actual infinity among German mathematicians.
𝔈𝔳𝔢𝔯 since it was given by Cantor [1886, 371], the paradigmatic example of rejection of actual infinity has been a passage of an 1831 letter from the "prince of mathematicians" to Schumacher, Schumacher had sent an attempted proof of the Euclidian parallel postulate, and Gaus replied:
𝔅𝔲𝔱, as concerns your proof of 1), i object above all the use of an infinite magnitude as it, it were complete, which is never permitted in mathematics. The infinite is only a facon de parler, when we are properly speaking about limits that certain relations approach as much as one wishes, while others are allowed to increase without limit.⁷
ℑ𝔱 has been argued that these statements had a very particular aim, and cannot be used against the set-theoretical infinite [Waterhouse 1979]. Schumacher made some assumptions about the behavior of geometrical figures in infinity, based upon mereanology, and Gauss, led by his knowledge of non-Euclidean geometry, protested against such unjustified assumptions. However, Gauss's statements are sharp and general: he takes as a model the theory of limits, understood in the sense of a potential infinity. On the other hand, some authors have indicated that, at time, Gauss employs infinitesmail notions that would seem to plainly rejected by the above quotation. For instance, his differential geometry, like Riemann's, can hardly be understood other then as infinitesimal mathematics (see Laugwitz in [Köing 1990, 26]).
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| Infinity Symbol ∞ |
𝔅𝔢 it as it may, what is important for us is that Gauss was not the only mathematician of that period. Others defended bolder positions, As early as 1788 Johann Schultz, a theologian and mathematician friend of Kant, developed a mathemarical theory of the infinitely great (see [Schubring 1982; Köing 1990, 155-56]). A very important contribution, that unfortunately was scarcely known in its time, was that of the philosopher, theologian and mathematician Bolzano, not only in his Paradoxien des Unendlichen [Paradoxes of the infinite; 1851], but also in the earlier Wissenschaftslehre [Theory of science, 1837]. Bolzano introduced the notion of set in several different meanings: in general he talked about collections or concept-extensions [Inbegriffe], but he singled out those collections in which the ordering of elements is arbitrary [Mengen], and among these the one whose elements are units, "multiplicities" [Vielheiten; Bolzano 1851, 2-4]. The notion of infinity is then carefully defined as follows: a multiplicity is infinite if it is greater then any finite multiplicity, i.e., if any finite Menge is only a part of it [op. cit., 6]. Bolzano defended forcefully the actual infinite, showing that the "paradoxes" of infinity involved no contradiction at all, and attempted to elaborate a theory of finite sets (see §11.6).
ℑ𝔱 should be emphasized that some 19th century mathematical developments depended upon accepting the notions of point and line 'at infinity.' Cantor himself mentioned this kind of precedent when he introduced the transfinite ordinal numbers [Cantor 1883, 165-166]. That happen particularly in projective geometry, which played a central role in geometrical thinking all along the century. Sometimes the introduction of elements at infinity may have been just an instrumental move not implying an acceptance of actual infinity, but as we shall see in the example of Steiner, at times it was accompanied by expressions that explicitly introduced the actual infinite. Another example is Riemann's function theory. In 1857 he took the step of 'completing' the complex plane with a point at infinity, thus turning it into a closed surface, which made it possible to reach general results in a simplified way.⁷
𝔍𝔞𝔠𝔬𝔟 Steiner, the great representative of synthetic geometry, seems to have been a defender of actual infinity. Furthermore, he introduced notions that constitute quite clear precedents of the language of sets and mappings. Steiner was a professor at Berlin, and thus a colleague of Dirichlet and, later, Jacobi. We will consider his main work, bearing the long title Systematische Entwicklung der Abhängigkeit geometrischer Gestalten von einandar.⁸ The first trait of this work that calls attention is the language employed, strongly reminiscent of neohumanism and idealism. In the preface, Steiner indicates that his aim is to go beyond proving some theorems, to discovering the "organically interconnected whole," the "organism" that gives a sense to the multiplicity of results. He aims at finding "the road follows by nature" in forming the geometrical configurations and developing their properties.⁹ Steiner's talk of "system" and "organism" may be compared, for instance, with Wolf's words quoted in §1. The similarity speaks for the free flux of ideas among mathematicians, philosophers and humanists which in the case of Steiner is quite obvious, since in his younger years he was a teacher at the school of the famous Swiss pedagogue Pestalozzi.
𝔐𝔬𝔯𝔢 interesting for our present purposes is to observe the way in which Steiner emphasizes the conception of line, plane, bundle of lines, etc. as aggregates of infinitely many elements. Aristotle and other conscious partisans of the potential infinite had carefully avoided that move. It seems revealing to find that it was precisely authors like Steiner, close to neohumanistic and philosophical ideas, who broke with that restriction within a favorable intellectual context. Steiner says that in a straight line one may think "an innumerable amount" of points, and in the plane there are "innumerable many" lines and points.¹⁰ Coming to specifically projective notions, he defines:
II. The planer bundles of lines. Through each point on a plane innumerable many straight lines are possible; the totality of all these lines will be called "planar bundle of lines"…
The expression "the totality of all" will become prototypical of tve works of Cantor and Dedekind that introduced sets. Later we read:
V. The bundle of lines in space.… such a bundle of lines does not only contain infinitely many lines, but also embraces numberless planar bundles of lines (II) and bundles of planes (III). as subordinated elements of configurations....¹¹.
𝔖𝔱𝔢𝔦𝔫𝔢𝔯’𝔰 viewpoint is thus close to the language of sets, although still far from a general viewpoint, not to mention a straightforward analysis of the notion of set. But it could be set in line with work of Riemann, Dedekind and other authors (see chapters II-V), insofar as it presents a new conceptualization of previous theories, based upon notions related to that of set, Moreover, the notion of a transformation, in the sense of a one-to-one correspondence, had become central for projective geometry. The notion is set out by Steiner in full clarity:
𝔉𝔦𝔯𝔰𝔱 a straight line and a planar bundle of lines will be related to each other, so that their elements get matched, that is, so that a certain line in the bundle corresponds to each point in the straight line.¹²
𝔗𝔥𝔢 interesting passage, and similar one's in Möbius and Plücker (see[Plücker 1828, vii]), obliviously suggest the idea of one-to-one mapping. It is worth nothing that Steiner regarded such correspondence as a key methodical element in order to show the "interdependence among the configurations," which in his work constituted "the heart of the matter."¹³
𝔗𝔥𝔢 much less romantic Möbius, in Der barycentrische Calcul, set out very clearly the notion of a one-to-one correlation of points in two different figures or spaces [Möbius 1827, e.g. 169, 266].¹⁴ He considered the simplest geometrical relations that can be defined by means of such transformations, and ordered them systematically: they were those of "similarity" (our congruence), "affinity," and the most general one of "collineation" (see [Möbius 1827, zweiter Abschnit, 167ff; 1885, 519]). He argued that all such "relationships" belonged to elementary geometry, because in all cases straight lines correspond to straight lines. A noteable, but apparently little known fact, is that Felix Klein regarded Möbius's study of geometrical "relationships" a clear precedent of his own Erlanger programm.¹⁵ On the other hand, one does not find in any Möbius statement that might betray an admission of actual infinity, nothing comparable to the above-mentioned words of his more romantic contemporary Steiner.
Dedekind and Cantor were both aware of Steiner's ideas, and the first also read Möbius very carefully.¹⁶ Dedekind had chosen geometry as the subject for his first course at Göttingen in 1854/55, when, as he said in a letter to Klein, he "made an effort to establish a parallelism between the modren analytic and syenthetic methods." [Lorrey 1916, 82]. To prepare that course, he borrowed from the Göttingen library Steiner's book, together with works by Chasles, Plücker [1828 and 1835], and the barycentrische Calcul of Möbius.¹⁷ As regards Cantor, he himself [1932, 151], indicated that the term "Mächtigkeit" [Power], which he used from 1877 to referred to the cardinality of a set, was taken from one of Steiner's works.¹⁸ One may thus assum that Steiner played some role in the introduction of notions of set and mapping, and the acceptance of the infinite, even if his statements are still vague and limited to a rather particular subject. We shall not, however, ascribe a particularly important role to his work in the events reported in the following chapters. More centrally, we may assume that, when Cantor and Dedekind expressed their confidence in the importance of sets and /or mappings for mathematics, they also had geometry in mind.
1. [Bolzano 1851, 7]: Hegel und seine Anhänger… nene es verächtlich das schlechte Unendliche und wollen noch ein viel höheres, das wahre, das qualitative Unendliche kennen welches sie namentlich in Gott und überhaupt im Absoluten nur finden. "sees Bolzano critcism of Cauchy, Grunert, Fries etc. in [op. cit., 9-13].
2. The passage is quoted by Bolzano [1851, iii] and Cantor [1932, 179]: "Je suis tellement pour l'infini actuel, qu'au lieu d'admettre que la nature l'bhorre, comme l'on dit vulgairement, je tiens qu'elle l'affected partout, pour mieux marquer les perfections de son Auteur. Ainsi je crois qu'il n'y a aucune partie de la matière qui ne soit, je ne dis pas divisible, mais actuellement divisèe, et par conséquent la moindre particelle doit étre considerée comme un monde plein d'une infinitè de créatures differentes".
3. According to Laugwitz [König 1990, 9-12], Leibniz spoke about infinity on three different levels: a popular one, a second for mathematicians, and the third for philosophers. At the second level, which is that of the Nouveaux essays, he favors the potential conception of limits, but at the third he presents the kind of approach that is typical of his Monadologie.
4. See [Riemann 1892, 534-38]. The interrelation that Riemann seeks to establish between psychology and physics is also reminiscent of Leibniz.
5. Dedekind to Weber, March 1875 [Cod. Ms. Riemann 1, 2, 24]: "Was mich betrifft, so bin ich für die stetige materielle Erfüllung des Raumes und die Erklärung der Gravitations- und Lichterscheinigungen im höchsten Grade eingenommen.... Diese Gedanken hat Riemann sehr früh, nicht erst in siener letzten Zeit, ergriffen.... Sein Streben ging ohne Zweifel dahin, den allgemeinsten Principien der Mechanik, die er keinesweges umstossen wolte bei der Natureklärung eine neue, natürlichere Auffassung unterzulegen; das bestreben der Selbsterhaltung und die in den partiellen Differentialgleichungen ausgesprochene Abhängigkeit der Zustandsveränderungen von den nach Ziet und Raum unmittelbar benachbarten Zuständen sollte er als 130 das Ursprüngliche, nicht Abgeleitete angesehen werden, So denke ich mir wenigstens seinen Plan.… leider ist Alles so lückwnhaft!"
6. See [Cantor 1883, especially 177, 206-07], [Cantor 1932, 275-76], [Schoenflies 1927, 20], [Meschkowski 1967, 258-59], it is worth mentioning that in the 1870s there was a group of theologians who accepted the actual infinite and were important for Cantor — above all Gutberlet and cardinal Franzelin: see [Meschkowski 1967], [Dauben 1979], [Purkert & Ilgauds 1987].
7. Riemann's position vis à vis the infinite is analyzed in §11.4.2.
8. Systematic development of the interdependence among geometrical configurations [Steiner 1832].
9. [Steiner 1832, v-vi]: ".... organisch zusammenhängendes Ganze... Gegenwartig schrift hat es versucht, den Organismus aufzudecken... "[op.cit, vi ]: "Wenn nun wirklich in diesem Werke gleichsem der Gang, den die natur befolgt, aufgedeckt wird.... "
10. Steiner [1832, xiii]: "In der Geraden sind eine unzählige Menge… Punkte denkbar".
11. Steiner [1832, xiii]: "Der ebene Strahlbüschel. Durch jeden Punkt in einer Ebene sind unzählige Gerade möglich; die Gesammtheit aller solcher Geraden soll “ebener Strahlbüschel”… heissen." [Op. cit., xiv]: Der strahlbüschel in Raum. … Ein solcher Strahlbüschel enthält nicht nur unendlich viele Strahlen, sondern er umfasst auch zahllose ebene Strahlbüschel (II.) und Ebenenbüschel (III). als untergeordnete Gebilde odr Elemente.."
12. [Steiner 1832, xiv-xv]: "Zuerst werden eine Gerade und ein ebener Strahlbüschel aufienander bezogen, so dass ihre Elemente gepaart sind, d.h., dass jedem Punkt der Geraden ein bestimmter Strahl des Strahlbüschels entspricht.
13. [Op. cit., vi] "den kern der Sache.... der darin besteht, dass die Abhängigkeit der Gestalten von den einander, und die Art und Wiese aufgedeckt wird, wie ihre Eigenschaften von den einfachern Figuren zu den zusammengesetztern sich fortpflanzen.
14. [Op. cit., 266] The "essence" of collinedation consists in that "bei zwei ebnen oder körperlichen Räumen, jedem Puncte des einen Raums ein Punct in dem anderen Raume dergestalt entspricht, das, wenn man in dem einen Raumi eine beliebige Gerade zieht, von allen Puncten, welche von dieser Geraden geteroffen werden (collineantur), die entsprechenden puncte in dem anderren Raume gelleichfall durch eine Gerade verbunden werden können."
15. [Klein 1926, vol I, 118], and also [Wussing 1969, 35-42], which includes a more detailed discussion of this aspect of Möbius's work.
16. Altough Riemenn spent the years 1847-49 at Berlin, apparently he did not attend Steiner's lectures.
17. As can be seen in the volumes of Göttingen's Ausleihregister for the summer semester of 1854 and winter semester of 1854/55. He paid particular attention of Möbius, since he borrowed in [1827] in the semesters of 1850/51, 1853, 1854 and 1855. Notably, Möbius calls (finite) sets of points "Systeme von Puncten" [e.g., 1827, 170]; the word "System", also used by Riemann and by Dedekind himself in his algebric work, will finally become his technical term for set in the 1870s (see chapter III and VII).
18. Cantor refers to Vorlesungen über synthetische Geometrie der Kegelschnitte, §2. Steiner used the term indicate that two configurations related by a one-to-one coordination.
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