𝓑. 𝔄 𝔑𝔢𝔴 𝔉𝔲𝔫𝔡𝔞𝔪𝔢𝔫𝔱𝔞𝔩 𝔑𝔬𝔱𝔦𝔬𝔫: ℜ𝔦𝔢𝔪𝔞𝔫𝔫’𝔰 𝔐𝔞𝔫𝔦𝔣𝔬𝔩𝔡𝔰

 𝔖𝔥𝔶𝔫𝔢𝔰𝔰, a natural consequence of his earlier sheltered life [as a child], ... never left him completely ... and frequently moved him to abandoned himself to soltiude and to his mental universe, in which he unfolded [his thoughts with] the greatest boldness and lack of prejudices.¹ 

ℑ𝔫 mathematics, the art of posing questions is more consequential then that of solving them.² 

 ℑ𝔫 this chapter we trace back the first influential appearance of a set-theoretical viewpoint to the work of Riemann. Of course, by speaking of "a set-theoretical viewpoint" I do not mean to suggest that Riemann reached technical results that we would classify today as belonging to set theory ... only that he introduced set language substantially in this treatment of mathematical theories and regarded set as a foundation of mathematics. This comes out in a public lecture given in 1854, on the occasion of his Habilitation as a professor at Göttingen. When he proposed a general notion of manifold - the famous 'On the Hypothesis upon which geometry is Founded,' published posthumously by Dedekind in 1868. We shall refer to it as Riemann's Habilitationsvortrag. 

𝔐𝔢𝔫𝔱𝔦𝔬𝔫𝔦𝔫𝔤 Riemann in connection with the history of sets is still quite uncommon, but there are indications that he played an important role in the early phases of development of the notion of set. From 1878 to 1890, his most creative period, Cantor referred to set theory as Mannigfaltigkeitslehre, the 'theory of manifolds.' employing the same word that Riemann had coined in his lecture of 1854. Notably, the 1878 paper in which Cantor first employs the word addresses a problem that is directly related to Riemann's Habilitationsvortrag, the characterization of dimension. And Dedekind understood Cantor's terminology to be related to the work of his close friend Riemann. In a letter of 1879, Dedekind proposed to replace the clumsy word 'mannigfaltigkeit' by the shorter 'Gebiet' [domain], which, he said, is "also Riemannian" [Cantor & Dedekind 1937, 47]. A decade later Dedekind kept mentioning 'Mannigfaltigkeit' as a synonym for set [Dedekind 1888, 334]. So it seems clear that both Dedekind and Cantor interrupted Riemann's notion of manifold as the notion of set. This suggests that it may be important to analyze carefully the origns, scope and implications of Riemann's new concept, as will be done in the present chapter. It is actually a basic thesis of this work that Riemann's ideas, and above all his new vision of mathematics and its methods, influenced both Dedekind and Cantor (see chapters III, IV and VI). 

ℑ𝔫 order to properly understand of origins of Riemann's new notion, we shall discuss the contributions of the two men that he regarded as his predecessors in this respect: Gauss and Herbart. We will also consider the traditional definition of mathematics as a theory of magnitudes [Grössenlehre], for Riemann explicitly presents his manifolds within this context. But the general definition of manifold, as given in the Habilitationsvortrag of 1854, is particularly difficult to interpret. In §2 we shall deal with a necessary prerequisite for satisfactory understanding, namely the ideas of traditional logic; in fact, that section constitutes an important background for much of the present book, since it explains how the notion of set was related to logic from 1850 to the early decades of the 20th-century. 

Benhard Riemann (1826-1866) in 1863 

ℌ𝔞𝔳𝔦𝔫𝔤 done that, it will become possible to analyze Riemann's contribution. §3 considers the mathematical context in which his new notion was forged, and the way in which it seems to have emerged. Then, we will consider Riemann's indications about how arithmetic and topology would be founded upon the notion of manifold, and, finally, the ways in which these contributions were influential upon the history of sets. The appendix discusses the impact of Riemann's ideas on his colleague and friend Dedekind, constituting a bridge to chapter III. 

𝟏. 𝓣𝓱𝓮 𝓗𝓲𝓼𝓽𝓸𝓻𝓲𝓬𝓪𝓵 𝓒𝓸𝓷𝓽𝓮𝔁𝓽 ︓ 𝓖𝓻𝓸𝓼𝓼𝓮𝓷𝓵𝓮𝓱𝓻𝓮, 𝓖𝓪𝓾𝓼𝓼  ℰ 𝓗𝓮𝓻𝓫𝓪𝓻𝓽

𝔓𝔞𝔯𝔱 ℑ, of Riemann's Habilitationsvortrag, which sets out the notion of an n-dimensional manifold, begins by asking for indulgence, since the author lack's practice "in such tasks of a philosophical nature" [Riemann 1854, 273]. The immediate motivation for this incursion into the alleged realm of philosophy is the need of a deeper understanding of "multiply expanded magnitudes," the need to derive this concept from "general concept of magnitudes" [op. cit., 272]. Riemann thus concieves of his manifolds as intimately related to the notion of magnitude, in fact he establishes the new notion of manifold as the basis for a general, abstract theory of magnitudes [op. cit., 274]. On the other hand, the only brief indications on how to confort his task can be found in the work of Gauss and Herbart, two Göttingen professors. We produced to analyze the elements of this historical context. 

𝟏.𝟏. Mathematics as Grössenlehre. One should keep in mind that, by the mid 19th-century, it was still common to define mathematics as the science of magnitudes. That was the customary definition upto that century, following the ancient Greek conception. Aristotle, for instance, distinguished two kinds of magnitudes, discrete and continuous, including number among the former, line, surface and body among the latter (Categories, 4b, 20). In his view, mathematical propositions deal with magnitudes and numbers (Metaphysics, 1077b, 18-20). This viewpoint offered a satisfactory overview of elementary mathematics, since it included the historical roots of this discipline, arithmetic and geometry, under a common conceptions. As is well known, beginning around 1600, with the work of Stevin and many other authors, magnitude and (real) nunber became coextensive [Gericke 1971]. 

𝔗𝔥𝔦𝔰 venerable definition of mathematics can be found again in Euler's Algebra, which begins with the following words: 

𝔉𝔦𝔯𝔰𝔱, everything will be said to be a magnitude, which is capable of increase or diminution, or to which something may be added or substracted. ... mathematics is nothing more than the science of magnitudes, which find methods by which they can be measured.³ 

𝔗𝔬 give a couple more examples taken from German works of the 19th-century, we may refer to mathematical dictionaries. In the first decade of the century, Klügel, a professor at the University of Halle, defined "magnitude (quantitas, quantum)" as "that which is compound of homogeneous parts": everything in reality or in imagination, that possesses the property of being such a compound is an object of mathematics, which is thus properly called the "theory of magnitudes, the science of magnitudes, which may be numerical magnitudes or spatial magnitudes," corresponding to the distinction between the discrete and the continuous With Hoffmann, a magnitude is again that which may be augmented or diminished.⁴ 

Georg Simon Klügel 1739 - 1812

𝔗𝔥𝔦𝔰 traditional definition was not only common in dictionaries, but kept being employed by research mathematicians. We shall see that Gauss still spoke of a theory of magnitudes is connection with numbers and pure mathematics. We will see, however, that in emphasizing the possibility of an abstract theory of magnitudes, and the need for topological investigations, he was going beyond the traditional viewpoint, stretching it to include radical novelties. The same happens to a far greater extent with Riemann, but similar moves can also be found in authors such as Bolzano, Grassmann and Weierstrass. In fact, reconcieving the idea of a magnitude seems to have been one of the ways in which  19th-century mathematicians introduced novel abstract viewpoints and advanced toward the notion of set. 

ℜ𝔦𝔢𝔪𝔞𝔫𝔫 meant his manifolds to become a new, clearer, and more abstract foundation for mathematics, which is consistent with his strong interest in philosophical issues and with his conception of mathematical methodology (§1.4). Unfortunately, this has normally not been taken into account by historians,⁶ probably because they find difficulties in interpreting his – for us – obscure definition of a manifold (see §2), and because one can comfortably resort to modern concepts of differential geometry while trying to interpret the Habilitationsvortrag. Nevertheless, Riemann's contemporaries had no option but to understand his own definitions, and at a time when the foundations of mathematics were unclear, his comments on the issue should have caught the attention of at least some readers. 

𝟏.𝟐. Gauss on complex numbers and "manifolds." In part I of his Habilitationsvortrag on geometry, Riemann [1854, 273] mentions that the only previous work of some relevance that he has had access to is a few short indications of Gauss, and some philosophical investigations of Herbart. 

𝔚𝔥𝔢𝔫 Riemann quotes Gauss's works in his Habilitationsvortrag, he clearly differentiates thos linked to differential geometry, and those related to the general notion of manifold [Riemann 1854, 273]. According to him, some indications that are relevant to the issue of manifolds can be found in an 1832 paper of bioquadratic residues, in the 1831 announcement of that paper [Gauss 1863/1929, vol. 2, 93-148 and 169-78], and in the 1849 proof of a fundamental theorem of algebra, read by Gauss on the occasion of his doctorate golden jubilee [op.cit., vol. 3, 71-102]. The common trait of these works is that all of them deal with the complex numbers. It seems likely that Riemann had carefully studied them already by the time of his disassertion, 1851. In §3.1 we will se Gauss indicating the need of a theory of topology, in the context of his 1849 proof: part I of Riemann's lecture was actually devoted a discussion of fundamental concepts of topology on the basis of the notion of manifold (see[Riemann 1854, 274]. 

ℑ𝔱 is well known that Gauss played an important role in the full acceptance of the complex numbers, with the above-mentioned 1831 and 1832 papers.³ The immediate motivation for this contribution was a question in number theory, biquadratic residues, where Gauss find it necessary to expand the field of higher arithmetic and study the number theory of Gaussian integers, 𝑎 + 𝑏𝒊 with 𝑎,𝑏 ϵ ℤ (§III.3). Anticipating criticism of this step, which might seem "shocking and unnatural" to some [1863/1929, vol. 2, 174], he decided to defend the full acceptability of complex numbers as mathematical objects. As we shall see in §3.1, Gauss seems to have relied on the idea of the complex plane since 1799.⁴ 

𝔊𝔞𝔲𝔰𝔰 regarded the interpretation of complex numbers as points in a plane as a mere illustration of a much more abstract meaning of complex numbers. He argues that some physical situations afford an occasion for employing a particular kind of numbers, and some not. It suffices that there be situations where fractional parts or opposites occur, to make full sense of a theory of fractions or of negative numbers. The same happens with complex nunbers, which in his view, only find application when we are not dealing with substances, but with relations between substances [Gauss 1863/1929, vol. 2, 175-76]. The use of real and complex units for measurement is required. 

if the objects are such that they cannot be ordered into a single unlimited series, but only into a series of series, or, what comes to the same, if they form a manifold of two dimensions; and if there is a relation between the relations among the series, or between the transitions from one to another, which is similar to the already mentioned transitions from one member of a series to another were belonging to the same series ... In this way, it will be possible to order the system doubly into series of series.

The mathematicians abstract entirely from the quality of the objects and the content of their relations, he only occupies himself with counting and comparing their relations to each other.⁵ 

𝔚𝔢 can here observe in some detail what Gauss ment by an abstract theory of magnitudes. In the 1832 tretise on biquadratic residues Gauss again uses the expression 'manifold of two dimensions.'⁶ 

ℑ𝔫 ths previous quotation, Gauss understands by a manifold a system of objects linked by some relations, the dimensionality of the manifold depending on properties and interconnections of relations. Though this is not the way in which Riemann conceived of manifolds in unpublished manuscripts of 1852/53 (§§1.3 and 3.2), Gauss was calling to the attention to the properties that a physical system must have in ordered to be regarded as a 2-dimensional manifold, and this is part of what Riemann tried to analyze. Gauss suggested the terminology, the topological point of view and some embryonic ideas on dimensonality. 

𝔗𝔬𝔴𝔞𝔯𝔡 the end of his 1831 paper, Gauss mention the possibility of relations, among things that give rise to a manifold of more than two dimensions [1863/1929, vol. 2, 178]. In lectures of the 1850s [Scholz 1980, 16-17] one can find indications of the possibility of n-dimensional manofolds, though without making explicit a satisfactory foundation. Actually a move to notion of n-dimensional geometry was not infrequent in the early 19th-century, with in the context of analytic or algebraic problems involving n variables [Scholz 1980, 15-19]. Several mathematicians — Lagrange, Cauchy, Jacobi, Gauss – found it advantageous to employ geometrical language in order to clarify the analytical relations under study. By the mid-century, the British algebraists Cayle, Sylvester, and Salmon made a similar move in their studies of homogeneous functions. In all of those cases n-dimensional language was introduce only by analogy, and, so to say, metaphorically a different case is that of Grassmann's attempt at a pure n-dimensional geometry [Grassmann 1844], since the move is here meant literally, just like in Riemann's lecture; but there is no think that Riemann knew of Grassmann's work. 

𝟏.𝟑. Herbart on objects as 'complexions of properties'. Originally Riemann matriculated at Göttingen in theology and philology, although with in a year he shifted to mathematics. Interest for philosophical topics, however, never abandoned him, and in the early 1850s he studied closely the work of Johann Friedrich Herbart, a professor at Göttingen until 1841. Reference to that work in Part I of the Habilitationsvortrag suggests how highly he valued Herbart's ideas; a manuscript note that he left is clear enough. 

𝔗𝔥𝔢 author is a Herbartian in psychology and the theory of knowledge (methodology and eidology), but for the most part he cannot embarce Herbart's philosophy of nature and the metaphysical disciplines that are related to it (ontology and synechology).⁷ 

ℜ𝔦𝔢𝔪𝔞𝔫𝔫 refers here to the peculiar names given by Herbart to the different parts of his doctrine. It should be noted, following Scholz [1982a, 415], that the part of Herbart philosophy that he adhered to conforms a kind of epistemology. Among Riemann's philosophical texts, those on psychology and epistemology develop Herbartian viewpoints [Riemann 1892, 509-25]. 

ℌ𝔢𝔯𝔟𝔞𝔯𝔱 was a disciple of the idealist Fichte, but by the end of his student time he had become very critical of Fichte's ideas. To mark his opposition to the idealist trend, so powerful in Germany, he always defined himself as a "realist," although some idealist remnants can be found in his psychological theories [see Scholz 1982a]. On a more positive note, he regarded himself as a follower of Kant, but not an orthodox Kantian, since he tried to avoid some aprioristic traits that were still present in the Köingsberg philosopher. Many details in his doctrines were inspired by Leibniz, so that he became a link between the speculations of the great mathematician-philosopher and those of Riemann. 

ℑ𝔫 a discussion of the possible influence of Herbart's ideas upon Riemann's geometrical thought, Erhard Scholz has denied that they may have affected the precise content of the notion of manifold.⁸ More precisely, Scholz mention some key elements of Riemann's notion that are absent from any related ideas of Herbart's: multidemensionally, separation between topological and metrical aspects, and the opposition between a simple local behavior and a complex globel one [Scholz 1982a, 423-24]. Herbart's influence would have been more on the level of general epistemology and, most importantly, of a conception of mathematical research. Riemann transformed some characteristic traits of Herbart's philosophy into guiding principles for his mathematical work [op.cit., 428]. 

ℌ𝔢𝔯𝔟𝔞𝔯𝔱 thought that mathematics is, amoung the scientific disciplines, the closest to philosophy. Treated philosophically, i.e., conceptually, mathematics can become a part of philosophy.⁹ According to Scholz, Riemann's mathematics cannot be better characterized than as a "philosophical study of mathematics" in the Herbartian spirit, since he always search for the elaboration of central concepts with which to recognize and restructure the discipline and its different branches, as Herbart recommended [Scholz 1982a, 428; 1990a]. 

𝔒𝔫𝔢 can certainly grant the general correctness of Scholz's detailed analysis of the interrelations between the ideas of the both authors, and still claim that there are a couple of more direct connections. Herbart's conception of space is developed in his theory of continuity [Synechology], which explains the emergence of the notions of space, time, number and matter, all of which involve continuity. Herbart proposes a more or less psychological explanation of continuity, which emerges from the "graded fusion" [abgestufte Verschmelzung] of some of our mental images [Vorstellungen; Herbart 1825, 192]. His preferred examples were those of the tones, which gives rise to a line, and the colors, which produce a triangle with blue, red and yellow at the vertices, and mixed colors in between [op.cit., 193]. As the quotation above makes clear, Riemann rejected the details of Herbart's theory of continuity. But he seems to have adopted some quite general aspects of Herbart's approach (compare [Scholz 1982a, 422-23]). 

𝔏𝔦𝔨𝔢 Leibniz, Herbart rejects the Newtonian (and Kantian) conception of space as an absolute receptacle for physical phenomena; rather it seems to be an "order of coexistence" of things (see[Leibniz & Clarke 1717; Herbart 1824, 429]). Space does not have an independent reality, but is a form which arises in our imagination as a result of specific traits of the mental images which we gain in experience. As a result, all kinds of mental images may give rise to continuous serial forms, and in all such cases the conception of space arieses. This suggests that anything can be geometrized, and explans why Herbart offered a unified treatment of time, matter, number and space, since all of these cases spatial forms arise. Herbart [1824, 428-29] made it explicit that spatial forms apply to all aspects of the physical world, and even to any domain of mental representations, including the unobservable. The conception of space as linked to the properties of physical objects is characteristic of Riemann's Habilitationsvortrag. From a historical point of view it is quite interesting to find that, apparently, there was a connection between Leibniz's and Riemann's proposals of such a conception, the link being Herbart's doctrines. 

𝔗𝔴𝔬 other interrelations between the ideas of Herbart and Riemann are linked with the latter's general definitions of the notion of a manifold. Here we shall analyze the influence of one aspect of Herbart's thoughts about objects and space, and in §2 we shall consider that of his treatment of logic. 

ℑ𝔫 1853, Riemann explained his idea of a manifold (see §3.2) by making reference to the totality of all possible outcomes of a measuring experiment in which the values of two, or perhaps n, physical magnitudes are determined for a given physical system. We may understand this as, essentially, the notion of the space of states for the given system. This is quite different from the Gauss's explanation of manifolds in the 1831 paper, but it happens to be quite similar to a key idea of Herbart's. 

𝔄𝔠𝔠𝔬𝔯𝔡𝔦𝔫𝔤 to him, any object has to be considered as a bundle or "complexion" of properties, each of which can be regarded as located in a different qualitative continum. The idea is natural given his approach to continuity outlined above, and it is found in two passages that seem to have attracted the attention of Riemann, since he excerpted them [Scholz 1982a, 416, 419]. Ready by a person who was immersed in physical thinking, as Riemann was, that could only suggest the notion of a space of states, at least when we are talking about magnitudes that vary continuously, such as temperature and wieght. 

𝔄𝔭𝔭𝔞𝔯𝔢𝔫𝔱𝔩𝔶, then, Riemann's 1853 explanation of the notion of a manifold may have been suggested by Herbart, just like his 1854 definition must have been influenced by philosophical reading (see§2). Incidentally, it is worth mentioning that the word "manifold" [Mannigfaltigkeit] is extremely frequent in Herbart's writings, although he employs it in the common sense, not in any technical sense. 

1. [Dedekind 1876, 542]: "die schüchtemhiet, ... eine natürliche Folge seines früheren abgeschlossenen Lebens, ... hat ihn auch später nie gänzlich verlassen und oft angetrieben, sich der Einsamkeit und seiner Gedankenwelt zu überlassen, in welcher er die grösste Kühnheit und vorurtheilslosigkeit entfaltet hat".

2. [Cantor 1932, 31]: "In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi". 

3. The wide diffusion of the geometrical representation only took place around 1830, with the publication of some treatises in France and England, and then with the contribution of Gauss. See [Nagel 1935, 168-77; Pycior 1987, 153-56; Scholz 1990, 293-99]. Some interesting comments can be found in [Hamilton 1853, 135-37]. 

4. The treatment of complex functions as conformal mappings, given by Gauss in 1825 (see §3.1), was also depended on the idea of the complex plane [Gauss 1863/1929, vol. 2, 175]. 

5. This text is somewhat reminiscent of Cantor's work [1932, 420-39] on n-ply ordered sets. [Gauss 1863/1929, vol. 2, 176]: "Sind aber die Gegenstände von solcher Art, dass sie nicht in Eine, wenn gleich ubegernzte, Reine geordent werden können, sondern nur in Reihen von Reihen ordnen lassen, oder was dasselbe ist, bilden sie eine Mannigfaltigkeit von zwei Dimensionen; verhält es sich dann mit den Relationen einer Reihe zu einer andern oder den Uebergängen aus einer in die andere auf eine ähnliche Weise wie vorhin mit den Uebergängen von einem Gliede einer Reihe zu einem andern Gliede derselben Riehe, so bedarf es offenbar zur Abemessung... ausser den vorigen Einheiten +1 und -1 noch zweier andern ... +𝒊 und -𝒊. ... Auf diese Weise wird also das System auf eine doppelte Art in Reihen von Riehen geordnet werden können. /Der Mathematiker abstrahirt gänzlich von der Beschaffenheit der Gangenstände und dem Inhalt ihrer Relationen; er hat es bloss mit der Abzählung und Vergliechung  der Relationen unter sich zu thun."

6. [Op.cit., vol. 2, 110]: "varietates duarum dimensionum." It is perhaps convenient to remind the reader that 'varietates' is the most adequate Latin translation of 'manifold;' accordingly, the name for this notaion is 'variété' in french, 'variedad' in spanish. 

7. [Riemann 1892, 508]: "Der Verfasser ist Herbartianer in psycologie und Erkenntnistheorie (Methodologie und Eidologie), Herbart's Naturphilosophie und den darauf bezüglichen metaphysischen Disciplinen (Ontologie und Synechologie) kann er meistens nicht sich anschliessen."

8. In this he distance himself from Russell and Torretti [Scholz 1982a, 414]. 

9. Philosophisch behandelt, wird sie selbst ein Theil der Philosophie... [Scholz 1982a, 437]. 

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