c Springer-Verlag 2000 Arch. Hist. Exact Sci. 54 (2000) 375–402.

The Poincar´e-Volterra Theorem: From Hyperelliptic Integrals to Manifolds with Countable Topology∗ Peter Ullrich Communicated by U. Bottazzini ¨ The talk of Adolf Hurwitz (1859–1919) “Uber die Entwickelung der allgemeinen Theorie der analytischen Funktionen in neuerer Zeit” (“On the development of the general theory of analytic functions in recent times”) [39] before the First International Congress of Mathematicians at Z¨urich in 1897 is generally considered as the “official recognition” of set theory by the mathematical community, in particular, by complex analysts.1 In the present article we will study the history of one of the results which led to this recognition and which Hurwitz also mentions in his lecture.2 The text deals with a development which took place mainly between the years 1888 and 1925 and during which set theory or, strictly speaking, set theoretic topology appeared within the theory of functions of one complex variable or, seen from a different point of view, crystallized at problems of complex analysis. The “Poincar´e-Volterra theorem” was formulated for the first time in connection with the definition of an analytic function that was promoted by Karl Weierstraß (1815– 1897).3 One starts off with convergent power series of one complex variable. If two of these power series with different centers of expansion attain the same values on the intersection (of the open interiors) of their discs of convergence, then, by the identity theorem for power series, the two series resulting from re-expanding the given series with respect to the same point in the intersection coincide. Hence, one can join together

* Partial results of this article have been published under the title “Georg Cantor, Giulio Vivanti ¨ und der Satz von Poincar´e-Volterra” in the Tagungsband des IV. Osterreichischen Symposions zur ¨ Geschichte der Mathematik, Christa Binder, Ed. Osterreichische Gesellschaft f¨ur Wissenschaftsgeschichte: Wien 1995, pp. 101–107. My thanks go to the Handschriftenabteilung der Nieders¨achsischen Staats- und Universit¨atsbibliothek G¨ottingen for the permission to publish from the letter of Georg Cantor to Giulio Vivanti dated June 26, 1888 (Cod. Ms. Cantor 16, draft No. 84, pp. 180–181) and to the Akademiearchiv of the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin, for the permission to quote from the letter of Karl Weierstraß to Hermann Amandus Schwarz dated March 14, 1885 (Schwarz estate, No. 1175). 1 cf. [27, pp. 213–214], [28, pp. 471–472], [64, p. 100], also [23, p. 247], [30, pp. 362–363]. 2 [39, pp. 99–100]. 3 e.g., [80, pp. 93–97].

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such discs with non-empty intersection step by step, thus forming the Weierstraßian “Kreisketten” (“chains of discs”). Now, an analytic function in the Weierstraßian sense is defined by analytic continuation along these chains of discs. Each value at a point c in the complex plane ⺓ which can be obtained in such a way is called one value of the analytic function at c. Here “multi-valued” functions are entirely admitted, i.e., functions for which continuation along different chains of discs leads to different values at c.4 The question now arises: How large can the cardinality of the set of values be which are attained at the point c? An obvious – nowadays, to be sure – translation of the problem to the language of Riemann surfaces reads as follows: Let an (always connected) concrete Riemann surface be given, i.e., a Riemann surface together with a ⺓. How many points of the surface can non-constant analytic map to the Riemann sphere b at most lie over a fixed point c ∈ b ⺓? The answer to this question is that the cardinality can only be countable. Proofs of this fact were published independently by Henri Poincar´e (1854–1912) [52] and Vito Volterra (1860–1940) [73] in 1888, who gave their names to this theorem. The problem to consider the “largeness” – measured in some vague sense – of the set of values of a multi-valued analytic function had, however, appeared much earlier in mathematics, long before the first investigations on denumerability by Georg Cantor (1845–1918) in 1873 [10] and even before the first definition of an analytic function by Weierstraß. This prehistory will be considered in Sect. 1 of the present article. The temporal coincidence of the proofs given by Poincar´e and Volterra of that theorem was by no means by pure accident, but a reaction of them on the explicit posing of the problem by Giulio Vivanti (1859–1949) in a note [69] in the very year 1888, cf. Sect. 2. An inquiry of unpublished letters in Sects. 3 and 4 reveals, however, that Cantor and, after him, Weierstraß were in possession of the statement of the “Poincar´eVolterra theorem” years ago, Weierstraß certainly in March 1885, possibly even in 1878. The publication of the “Poincar´e-Volterra theorem” was not too straightforward: At first Vivanti gave a proof [70] which soon turned out to be fallacious (see Sect. 5). The (correct!) proofs of Poincar´e and Volterra are discussed in the Sects. 6 and 7, where for the Volterra part of the story we mainly refer to the study on the Vivanti-Volterra correspondence by Giorgio Israel and Laura Nurzia [41]. In Sect. 8 we investigate the “reaction” of the mathematical community to the result, in particular its representation in standard texts both on function theory of one complex variable and on topology until the 1930ies, its reformulation to the effect that a concrete Riemann surface has a countable basis of topology and its generalization to arbitrary abstract Riemann surfaces. The concluding Sect. 9 then shows why mathematicians soon gave up generalizing the above statement on one-dimensional complex analytic manifolds to other cases: Despite the simplicity of statement and proof of the Poincar´eVolterra theorem there is no further generalization, neither in the real nor in the complex

4

[80, p. 97].

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case. Only in the complex one-dimensional case all analytic (connected) manifolds have a countable basis of topology. 1. Abelian integrals of genus greater than or equal to 2 The elementary functions known at the end of the 18th century pose no problems with respect to the “largeness” of the set of values of a multi-valued function at a point. For algebraic functions, e.g. roots, the sets are always finite. (On the other hand, already these functions prove that all finite cardinalities really can appear.) Furthermore, the logarithm shows that the set can also by countably infinite; in this case the different values differ by integer multiples of 2πi as already Leonhard Euler (1707–1783) knew in 1749 [26]. Also the inverse functions of trigonometric functions behave in this way. However, these examples are “harmless” in the sense that here the set of values attained at a point c always is discrete in ⺓ (and by a result of Cantor from 18835 one knows that discrete subsets of a real number space ⺢n , hence in particular of ⺓, are automatically countable). The story becomes more intricated when one starts considering the integrals Z I (c) :=

c c0

dx √ p(x)

with p(x) a polynomial in x without multiple roots and c0 a fixed point in ⺓. The value of I (c) at a point c depends on the choice of the path of integration from c0 to c. And, as with the logarithm, one gets all possible values by adding to one single value all the linear combinations of the periods of the integral with integer coefficients. In the elliptic case, when p(x) has degree 3 or 4, the integral possesses exactly two periods, which are linearly independent over the real numbers, so that one again gets only a discrete set of values.6 The picture changes drastically if one passes over to hyperelliptic Abelian integrals where the polynomial p(x) has a degree greater than or equal to 5 (so that the corresponding Riemann surface has genus greater than or equal to 2). In fact, Jacobi found out in 1834 that already if the degree equals 5 or 6 (and hence the genus is 2), these integrals have four periods, which in general are linearly independent over the rational numbers ⺡.7 Three linearly independent complex numbers are enough, however, to generate an additive subgroup of ⺓ which has an accumulation point at 0,8 by which fact the set of values I (c) is not discrete in ⺓ for any point c. Even more, this set – though obviously countable – is dense in ⺓: In the situation of a real polynomial p(x),

5

[12, Theorem I]. This fact was clear to Carl Friedrich Gauß (1777–1855) already in 1798 [29, Vol. 3, pp. 433– 435, p. 492, p. 494; Vol. 10.1, pp. 194–206, pp. 274–278] and was published by Niels Henrik Abel (1802–1829) and Carl Gustav Jacob Jacobi (1804–1851) in the 1820ies. 7 [42, §§ 4–6]. 8 [42, §§ 2–3]. 6

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which Jacobi studies, this easily follows from the fact that of the four periods (which are linearly independent over ⺡) two are real and two are purely imaginary.9 Of course, the notations “discrete” and “dense” cannot be found in Jacobi’s publication [42]. But he remarkably clearly points out this difference in quality: quemadmodum integralia elliptica pro eodem sinu amplitudinis numero valorum dupliciter infinito gaudent . . . inter se aequidistantes: ita integralia . . . hyperelliptica . . . tantam multiplicatem [sic!] valorum secum ferunt, ut . . . e numero valorum, quos idem integrale pro iisdem datis limitibus quibuslibet induere potest, semper sint, qui a dato quolibet valore reali aut imaginario minus differant quam ulla quantitate data quantumvis parva.10

By this discovery Jacobi was led to attacking the problem of the inverse function for hyperelliptic integrals in a different manner than for elliptic integrals: The inverse functions of the latter are the elliptic functions, i.e., analytic – with the exception of poles –, doubly periodic, and univalent functions of one complex variable. Contrary to Rc √ this, in the case of degree 5 or 6 Jacobi considered not only the integral (1/ p(x)) dx c0

Rc

√ but also the integral (x/ p(x)) dx and studied the inverse function as a function of c0

two complex variables.11 By this, “in hac quasi desperatione”,12 as he himself wrote, he founded the theory of functions of several complex variables. Speaking in modern terms, in this case, which later on was sometimes called the “ultraelliptic” one, Jacobi embedded √ √ the compact Riemann surface of p(x), where the holomorphic differential dx/ p(x) is defined, into its “Jacobian”, the complex two-dimensional torus (⺓2 modulo its period lattice), by means of the mapping   c Z Zc 1 x dx, √ dx . c 7→  √ p(x) p(x) c0

c0

On the other hand, at two places in his article Jacobi passes the verdict “absurd(um)” on attempting to solve the inversion problem for hyperelliptic integrals by functions of only one variable, which would have to be fourfold periodic.13 Jacobi’s judgement gave rise to a longlasting debate in the 19th century on the inverse functions of hyperelliptic integrals. For example, on February 28, 1856 Bernhard Riemann (1826–1866) wrote with respect to Jacobi’s verdict:

9

[42, § 1, §§ 7–8]. “just as the elliptic integrals attain doubly infinitely many values for the same value of the sine of the amplitude, . . . which are equidistant, the hyperelliptic integrals . . . carry with them such a strong multiplicity of values that . . . among the values that the same integral can attain for the same arbitrary given boundaries there will always be some which differ from an arbitrary given real or imaginary value less than any given quantity, however small it may be.” [42, § 8]. 11 [42, §§ 9–11], cf. [36, pp. 78–80]. 12 “in this so to speak desperate situation” [42, § 8]. 13 [42, § 4, p. 61 resp. p. 32 and § 7, p. 71 resp. p. 43], cf. also [63, esp. pp. 89–91]. 10

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Vielleicht etwas unvorsichtig [von Jacobi zu behaupten], es lasse sich keine mehr als zweifach periodische Funktion von einer Variabeln denken.14

(This remark was only published in 1902 in the “Nachtr¨age” of Riemann’s “Gesammelte mathematische Werke”. Note also that Riemann explicitly speaks about “einwerthige” (“univalent”) functions in his letter to Weierstraß of October 26, 1859 with the “Beweis des Satzes, dass eine einwerthige mehr als 2nfach periodische Function von n Ver¨anderlichen unm¨oglich ist”.15 ) This debate has already been treated several times in the literature.16 Therefore at this place we only discuss the aspect which will turn out to be of importance for the Poincar´e-Volterra theorem in the sequel. In the course of the edition of the second volume of Jacobi’s collected works in the year 1882 Weierstraß commented in a footnote that Jacobi’s verdict could no longer be upheld “[v]om Standpunkte der heutigen Functionenlehre aus”.17 Weierstraß must have found this criticism of Jacobi, whom he highly revered otherwise, easier insofar as “heutigen Functionenlehre” meant nothing else than the theory of, possibly multi-valued, functions of one complex variable which he himself had co-founded to a decisive extent. Namely, his “analytische Gebilde” (“analytic formations”) – or, equivalently, the “Fl¨achen” (“surfaces”) introduced by Riemann – made it possible to define the inverse function of a hyperelliptic integral as an analytic, be it multi-valued, function of one complex variable.18 Encouraged by Weierstraß’ remark,19 Felice Casorati (1835–1890) took up anew his investigations on the inversion of hyperelliptic integrals, which he had laid aside for almost twenty years, and brought them to an end in the form of two articles [19], [20] in “Acta mathematica” in 1886 in which he solved the inversion problem in the language of Riemann surfaces by means of explicit calculation.

2. The formulation of the problem by Vivanti A discussion of Abelian integrals of genus greater than or equal to 2 and their inverse functions is also the starting point of a note [69] by Vivanti, which he finished on June 22, 1888 and which was communicated at the session of the “Circolo Matematico di Palermo” on July 8, 1888. Here one finds the question which cardinalities are possible for the set of values of a multi-valued analytic function at a given point posed in print for the first time.

14

“Maybe a little bit imprudent [of Jacobi to claim] that it is not possible to imagine a function of one variable which is more than doubly periodic.” [62, p. 709]. 15 “Proof that a univalent more than 2nfold periodic function of n variables is impossible” [60, p. 197 resp. p. 326]. 16 [6], [50, pp. 8–21], also [63], [74], and [81, pp. 332–334, Notes 13–17]. 17 “from the point of view of today’s function theory” [43, Vol. 2, p. 516]. 18 For a longer exposition on this theme by Weierstraß himself see [79, pp. 123–144]. 19 [6, pp. 48–49], [50, pp. 13–14].

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At first, Vivanti rightly criticizes20 the fact that the statement of the denseness of values of I (c) in ⺓ – which is precisely formulated in Jacobi’s original paper21 – was reproduced in parts of the (for Vivanti) contemporary literature in an incorrect version, namely, that the integral I (c) would attain each complex number as a value. As examples, he refers to the “Theorie der Abelschen Functionen”22 by Alfred Clebsch (1833–1872) and Paul Gordan (1837–1912) and the second edition of the “Neue Theorie der ultraelliptischen Functionen”23 by Friedrich Prym (1841–1915). Referring to “ricerche di G. C a n t o r” (“G. C a n t o r ’s investigations”), Vivanti substantiates in his article24 that the set of values of I (c) is only countably infinite, hence of “1a potenza” (“first cardinality”) whereas the set ⺓ of all complex numbers is of “2a potenza” (“second cardinality”).25,26 Now Vivanti defines in general that an analytic function (in the sense of the Weierstraßian definition) has “first cardinality” or is “of first class” (“una funzione ha la 1a potenza od e` della 1a classe”)27 if for each fixed point c the set of values of the function at c is of first cardinality, i.e., countable. He immediately proves that this coining of a terminology is – a priori, at least – no futile enterprise: On the one hand he shows by consideration of the associated Riemann surface that each analytic function of first cardinality possesses an inverse function in the sense of the Weierstraßian theory which again is of first cardinality.28 Here he does not waste a word on the existence of the inverse function, but remarks that for the case of Abelian integrals of genus greater than or equal to 2 one can directly read off his result from the explicit representation [19], [20] given by Casorati. On the other hand, Vivanti shows that each analytic function which is defined by an algebraic differential equation has first cardinality.29 This result brings him to the last third of his article, where he discusses Poincar´e’s result of 1883 [51] on the uniformization of analytic functions.30 The task was to parametrize a given multi-valued analytic function y = y(x) by means of a new variable z in such a way that y = y(z) and x = x(z), respectively, are univalent analytic functions of z. Contesting with Felix Klein (1849–1925), Poincar´e had made great progress towards the solution of this problem in his article of 1883, where he made use of countably

[69, no 1]. [42, § 8]. 22 “Theory of Abelian functions”, see [22, p. 134]. 23 “New theory of ultraelliptic functions”, see [55, pp. 1–2]. 24 [69, no 2]. 25 [69, p. 136]. 26 Thus he presupposes the continuum hypothesis as Cantor had stated it at the end of his article [11] in 1878 and also still at the end of the 1884 paper [14]. Cantor himself, on the other hand, had spent the summer and autumn of the year 1884 in a hot-and-cold bath of proofs and rejections of the continuum hypothesis (cf., e.g., [23, pp. 99–101, p. 118, pp. 135–137], [30, pp. 356–357], [49, pp. 139–141], [56, pp. 67–71, p. 81] for this aspect). 27 [69, p. 136]. 28 [69, no 3, A]. 29 [69, no 3, B]. 30 [69, no 4–6]. 20 21

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infinitely multi-valued elliptic modular functions.31 This, however, had the consequence that Poincar´e’s method could only by used in order to uniformise functions of first cardinality as Vivanti was the first to point out, concluding from the results mentioned above: “Dunque la dimonstrazione di P o i n c a r e´ vale solo per le funzioni aventi la 1a potenza.”32 Yet Vivanti abstains in his article [69] from any supposition whether the property of being of first cardinality holds for each analytic function.

3. Cantor’s letter to Vivanti This did not mean that Vivanti was not occupied with this question: Already on May 15, 1888 he had written to Cantor – with whom he was corresponding at least since December 3, 188533 – a letter whose second part was treating this problem. Unfortunately, this letter has not been preserved in the Cantor estate in the Handschriftenabteilung der Nieders¨achsischen Staats- und Universit¨atsbibliothek in G¨ottingen.34 Cantor’s answer, however, can be found in his letter book for the years 1884–1888 under the date of June 26, 1888.35 The relevant part for the question under consideration reads as follows: Geehrter Herr Vivanti. Entschuldigen Sie freundlichst, daß ich erst heute Zeit finde, Ihr werthes Schreiben v. 15 Mai zu beantworten. . . . Was den zweiten Theil Ihres Schreibens betrifft, so haben Sie Recht, daß “jede durch eine algebraische Differentialgleichung definirte Function die Eigenschaft hat, f¨ur jeden Werth der unabh. Variablen nur eine abz¨ahlbare Menge von Werthen zu erhalten.” Dieser Satz ist aber nur ein Spezialfall eines andern, den ich vor mehreren Jahren Herrn Weierstraß, dem er neu war, mittheilte, n¨amlich des Satzes: “Jede anal¨ytische Function (im Weierstraßschen Sinne) hat, wenn sie unendlich vieldeutig ist, . . . [deletions by Cantor] nothwendig eine Vieldeutigkeit nur von der ersten M¨achtigkeit ω.” ¯ Weierstraß, der sich f¨ur diesen Satz sehr interessirte, theilte mir einige Jahre sp¨ater mit, daß auch er einen Beweis dieses Satzes mit H¨ulfe seiner Theorie der Minimalfl¨achen gefunden h¨atte. Ich hatte gehofft, daß er seinen Beweis publicieren w¨urde. Allein dies ist unterblieben, vermuthlich weil er in diesem Falle meine Theorie des Transfiniten h¨atte erw¨ahnen m¨ussen, was aber aus R¨ucksicht auf Kronecker und Helmholtz [subsequently inserted by Cantor: in Deutschland] bekanntlich nicht geschehen darf.

31

[51, p. 115 resp. p. 60]. “Therefore Poincar´e’s proof is valid only for functions which have first cardinality.” [69, no 4, p. 138]. 33 [18, p. 251]. 34 Cod. Ms. Cantor 12 with the letters from Vivanti to Cantor only contains writings from the years 1892 to 1894. 35 Cod. Ms. Cantor 16, draft No. 84, pp. 180–181. 32

382

P. Ullrich Wenn Sie einen Beweis f¨uhren, so lassen Sie sich hoffentlich nicht auch abhalten, ihn zu ver¨offentlichen.36, 37

Some comments on Cantor’s letter seem to be apt: Weierstraß’ esteem of Cantor’s denumerability arguments is well-known. Already on December 22 and 23, 1873, Cantor had reported to Weierstraß on his results concerning the denumerability of the algebraic and the non-denumerability of the real numbers and Weierstraß had “veranlasst” (“caused”) Cantor to publish the article [10], as the latter informed Richard Dedekind (1831–1916) in a letter dated December 25, 1873.38 Shortly after that Weierstraß had used the method to denumerate the rational and even the algebraic numbers, e.g. in lecture courses, in order to give examples of “pathological” functions.39 Furthermore, in a letter in English to Philip Jourdain (1879–1919) dated March 29, 1905 Cantor gives an extremely positive resume´e of his relations to Weierstraß: With Mr. Weierstrass I had good relations . . . . Of the conception of enumerability of which he heared from me at Berlin on Christmas holydays 1873 he became at first quite amazed, but one or two days passed over, it became his own and helped him to an unexpected development of his wonderful theory of functions.40

(Since it does not seem to be known to which result of the Weierstraßian theory of functions Cantor alludes here,41 the letter quoted above offers the tempting suggestion that it is the theorem on the denumerability of the set of values of an analytic function at a point.)

36

“Honoured Mr. Vivanti. Would you kindly excuse that I only find time today in order to answer your esteemed letter of May 15. . . . Regarding the second part of your letter, you are right that “each function which is defined by an algebraic differential equation has the property of attaining only a countable set of values for each value of the independent variable.” This theorem, however, is only a special case of another, which I have communicated to Mr. Weierstraß – for whom it was new – several years ago, namely of the theorem: “Each analytic function (in the Weierstraßian sense) which is infinitely multi-valued necessarily has . . . [deletions by Cantor] a multiplicity only of the first cardinality ω.” ¯ Weierstraß, who was very interested in this theorem, has informed me some years later that he also had found a proof of this theorem by means of his theory of minimal surfaces. I had hoped that he would publish his proof. But this did not take place, probably because he would have had to mention my theory of the transfinite in this case, which, as is known, is not allowed [subsequently inserted by Cantor: in Germany] in deference to Kronecker and Helmholtz. If you give a proof then it is to be hoped that you will not be deterred from publishing it, too.” 37 Joseph Dauben has pointed out in [23, p. 342, Note 37] that this letter contains the first instance of Cantor using the superscripted bar in order to denote the cardinality of a set, in this case ω¯ for the cardinality of the set ω of natural numbers. 38 [17, pp. 16–17] For the positive reaction of Weierstraß on these results see also [64, p. 99, esp. footnote 1]. 39 cf. [66, pp. 155–156]. 40 [30, p. 384] or [31, p. 124]. 41 [31, p. 124, footnote 17].

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Furthermore, in a letter to Sofja Kowalewskaja (1850–1891) dated March 24, 1885, Weierstraß comments upon the problem of Abelian integrals of genus greater than or equal to 2 and their inverse functions: . . . die zu einem und demselben Werthe von c geh¨origen Werthe von I (c) bilden eine abz¨ahlbare Menge, von der Cantor, wie ich u¨ berzeugt bin, in unanfechtbarer Weise bewiesen hat, daß es nicht nur unendlich viele Werthe giebt, die nicht nur nicht darin enthalten sind, sondern eine Menge von h¨oherer M¨achtigkeit bilden . . . 42

(Note that this letter is three years older than the article [69], in which Vivanti uses the same argument.) Besides these aspects which strongly back the statements made in Cantor’s letter there are, however, other arguments which may call for some reservation. First of all, in the Cantor estate at G¨ottingen one can find no evidence for Cantor communicating his result to Weierstraß: In the letter book Cod. Ms. Cantor 16, which is the only one containing drafts of 1888 or earlier, there is only one letter to Weierstraß, dated May 16, 1887, and this deals with the non-existence of infinitely small quantities. And Cod. Ms. Cantor 13 with the letters from Weierstraß to Cantor merely contains three writings, all dating from the years 1881–82, none of which concerns the problem under consideration. One has to admit, however, that the Cantor estate has been preserved in a rather incomplete state and that Cantor may well have communicated the result to Weierstraß during a personal meeting. Actually, in two of the three writings from Weierstraß mentioned above the date of a visit of Cantor is discussed. Furthermore, if Weierstraß was really “very interested in this theorem” as Cantor writes, then one might expect that he would have mentioned the result in his letter to Kowalewskaja of March 24, 1885 quoted above. This, however, is not the case. Even more, in summer 1886 Weierstraß gave a lecture course on “Ausgew¨ahlte Kapitel aus der Funktionenlehre” (“Selected chapters from the theory of functions”). According to the notes taken there [79], he elaborated upon how to repeal Jacobi’s “absurdum” verdict by means of his “analytic formations”.43 By means of logarithms he even constructed examples of analytic functions whose value set at each point lies dense in ⺓ and which are somewhat easier than the hyperelliptic integrals.44 There is, however, no indication in the notes that Weierstraß had mentioned anything about the cardinality of the value set in general. The text has even led Reinhard Siegmund-Schultze to the hypothesis that at that time – 1886 – Weierstraß did not yet know the Poincar´e-Volterra theorem,45 which, however, is in slight contradiction to Cantor’s statement of 1888 that he had informed Weierstraß “several years ago”. Even more serious is the fact that the relation between Cantor and Weierstraß was by no means always as happy and unproblematic as Cantor’s letter to Jourdain seems “. . . the values of I (c) which belong to one and the same value of c form a countable set, of which Cantor, as I am convinced, has shown in an indisputable way that there are not only infinitely many values which are not contained in it, but that [these] form a set of higher cardinality . . . ” [5, p. 71] or [81, p. 329], mathematical symbols adapted to the use above. 43 [79, pp. 123–144]. 44 [79, pp. 125–126]. 45 [79, p. 253, Note 194]. 42

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to indicate. From the beginning 1880ies until the death of Weierstraß Cantor expressed in several letters his feeling that Weierstraß did not pay due credit to him and his set theory, attributing this in part to Weierstraß’ own opinion and in part to his deference to other mathematicians.46 In particular, one is tempted to assume that Cantor’s hypothesis on the reason for Weierstraß not publishing his proof says more about Cantor’s relations to Leopold Kronecker (1823–1891) and Hermann (von) Helmholtz (1821–1894) than about Weierstraß. The notoriously bad relation between Cantor and Kronecker is depicted in any biography of Cantor. The problems between Cantor and Helmholtz are not that obvious since Dedekind had been successful in convincing Cantor to weaken his comments on Helmholtz’ contributions on the foundations of geometry before Cantor’s article [11] was published.47 However, Cantor’s opinion on Helmholtz had not improved in the meantime, as letters from Cantor to G¨osta Mittag-Leffler (1846–1927) dated December 30, 1883 and to Giuseppe Veronese (1854–1917) dated November 17, 1890 reveal.48

4. Weierstraß’ use of the “Theorie der Minimalfl¨achen” The reservations given above against some of the claims in Cantor’s letter were, so to speak, from the point of history of mathematics. However, there is also a formulation in it which might make one purse one’s brow because of a purely mathematical reason: Cantor writes that Weierstraß had given his own “Beweis dieses Satzes mit H¨ulfe seiner Theorie der Minimalfl¨achen” (“proof of this theorem by means of his theory of minimal surfaces”). Of course, Weierstraß had worked on minimal surfaces, even as early as in the 1860ies [75], [76], [77]. But such a proof would seem overly complicated if one looks at the proofs of the theorem given by Poincar´e [52] (cf. Sect. 6) and by Volterra [73] (cf. Sect. 7) and compares them with the analytic machinery involved in minimal surfaces. These doubts, however, are unfounded since Weierstraß made use of the term “theory of minimal surfaces” in a sense different from the one presently current. This becomes clear from a letter of Weierstraß to Schwarz dated March 14, 1885,49 which also proves that Weierstraß was in possession of the statement of the Poincar´e-Volterra theorem on the denumerability of the value set as early as March 1885: Daran kn¨upft sich eine Frage. Kann man f¨ur eine beliebige analytische Funktion f (u) beweisen, daß f¨ur jeden Werth von u die zugeh¨origen Werthe von f (u) eine abz¨ahlbare Menge bilden, also zu einer Reihe geordnet werden k¨onnen? Ohne Zweifel wird diese Frage zu bejahen sein. In einer Minimalfl¨ache, die durch die bekannten Formeln, in denen s, f (s) figuriren, definirt werden, entspricht jedem Werthpaare (s, f (s)) ein Punkt, in der Art, daß in demselben die complexen Gr¨oßen s, f (s) eine bestimmte geometrische

46

For details see [5, esp. p. 73–75], also [23, p. 138, p. 162, p. 164]. Compare, for example, the letter from Cantor to Dedekind dated June 25, 1877 [17, pp. 33– 34] with the published text [11, p. 244 resp. pp. 120–121]. 48 [18, p. 162] resp. [18, p. 330]. 49 Akademiearchiv of the Berlin-Brandenburgische Akademie der Wissenschaften, Berlin, Schwarz estate, No. 1175. 47

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Bedeutung haben. Durch eine solche Minimalfl¨ache mache ich die Gesammtheit der Werthepaare (s, f (s)), also, was ich ein monogenes Gebilde erster Stufe im Gebiete zweier complexer Variabeln nenne, meiner Vorstellung viel anschaulicher wie durch die Riemann’sche Fl¨ache – und mit H¨ulfe dieser Anschauungsweise habe ich mir einen Beweis f¨ur die Bejahung der aufgeworfenen Frage zurecht gelegt. Gewiß wird es auch ohne ein solches geometrisches H¨ulfsmittel gehen. Nimmt man den Satz von Poincar´e als richtig an, daß jedes monogene Gebilde im Gebiete zweier Ver¨anderlichen x, y sich durch die Gleichungen x = f1 (t),

y = f2 (t)

definiren lasse, wo f1 (t), f2 (t) eindeutige Funktionen von t bedeuten, so l¨aßt sich der fragliche Beweis sehr leicht f¨uhren.50

Here, “the known formulas, in which s, f (s) appear” are the nowadays so-called “Weierstraß-Enneper representation formulas”: Z s˜ (1 − s 2 )f (s) ds , x(˜s ) = x0 + < s0

Z y(˜s ) = y0 + < Z z(˜s ) = z0 + <

s˜

s0 s˜

s0

i(1 + s 2 )f (s) ds , 2sf (s) ds ,

where x0 , y0 , z0 are fixed real numbers, s0 is a fixed complex number and < denotes the real part. For each complex analytic function f (s), these formulas parametrize a minimal surface in the real three-dimensional space with coordinates x, y, z as a function of the complex variable s˜ .51

50 “There is a question connected with this. Can one prove for an arbitrary analytic function f (u) that for each value of u the corresponding values of f (u) form a countable set, so that they can be arranged into a series? No doubt, the answer to this question is affirmative. In a minimal surface which is given by the known formulas, in which s, f (s) appear, to each pair of values (s, f (s)) there corresponds one [and only one] point in such a way that in it the complex quantities s, f (s) have a certain geometrical meaning. By means of such a minimal surface I make the totality of pairs of values (s, f (s)), thus what I call a monogenic formation of first grade in the domain of two complex variables, much more visual to my imagination than by the Riemann surface – and by the help of this way of visualization I have figured out for myself a proof for the positive answer to the proposed question. Certainly, it will also be possible without such a geometrical expedient. If one takes Poincar´e’s theorem for granted – that each monogenic formation in the domain of two variables x, y can be defined by the equations

x = f1 (t),

y = f2 (t)

with f1 (t), f2 (t) denoting univalent functions of t – then the proof in question can easily be given.” 51 These formulas were published by Weierstraß in 1866 [75, § 1], but had been known to him already in 1861 [75, p. 39]. At about the same time equivalent representation formulas were also found independently by Alfred Enneper (1830–1885) [25, pp. 107–108] and Riemann [61, § 9].

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So, the use of the “theory of minimal surfaces” that Weierstraß makes is only a representation of the given analytic function (or the “analytic formation”) as an object in the space of three real variables and, by this, a visualization of it.52 Furthermore, the above letter of Weierstraß confirms that he was not content with his proof of the theorem of the countability of the value set. But the reason he gives here is that he would have to use either “a geometrical expedient” or Poincar´e’s uniformization result [51]. By no means does he express any reservation against Cantor’s “theory of the transfinite” as the latter had assumed in his letter to Vivanti. Of course, Weierstraß does not mention Cantor’s name in the above quote, but as Reinhard B¨olling points out,53 Weierstraß hardly ever mentions Cantor in his correspondence; seemingly the problems in their relation were not one-sided. One question that still remains open is, when did Cantor find the theorem and communicate it to Weierstraß? The trivial lower bound is December 1873, of course, when Cantor started his investigations on denumerability. The above letter of Weierstraß gives as upper bound March, 1885, in accordance with Cantor writing in 1888 that it took place “several years ago”. In any case, in the notes of the introductory lecture course of Weierstraß on the theory of analytic functions from summer 1878 which were taken by Hurwitz one finds after the definition of an analytic function g by means of chains of discs with centers c1 , c2 , . . . , cn the following text: Es findet nun der folgende wichtige Satz statt: “Entweder g(x|b) ist vollkommen unabh¨angig von den vermittelnden Stellen c1 , c2 . . . cn oder doch nur verschieden f¨ur eine endliche Anzahl von Werthsystemen c1 , c2 . . . cn .” 54

Admittedly, the statement is vague and somewhat cryptic.55 But it is definitely not a version of the monodromy theorem since this is proven some lectures later for starshaped domains.56 So the above text must be another statement on the size of the value set of the analytic function g at the point b and one might speculate that it is nothing else than a distorted version of the theorem on the denumerability of this set.57

5. Vivanti’s attempted proof Getting back to Cantor and his letter to Vivanti of June 26, 1888, one finds a remarkable illustration of Cantor’s distinction between the reception of his “theory of the 52

For a more detailled analysis of the use of geometric imagination in the mathematics of Weierstraß see [67]. 53 [5, p. 71]. 54 “Now the following important theorem takes place: “Either g(x|b) is completely independent of the intermediate points c1 , c2 . . . cn or just different for a finite number of systems of values c1 , c2 . . . cn .” ” [80, p. 96]. 55 Cf., e.g., the somewhat insecure comments in [80, pp. xviii–xix]. 56 [80, pp. 143–144]. 57 If this would be the case then Hurwitz, who obviously cited the statement verbatim, would be the last to blame for the distortion: Since 1862 Weierstraß did not personally write at the blackboard but had this done by an advanced student to whom he was dictating the text [44, p. 61].

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transfinite” in Germany and abroad.58 Already on July 30, 1888 – only about one month after finishing the note [69] and receiving Cantor’s writing – Vivanti completed another article [70], which was read at the session of the “Circolo Matematico di Palermo” on August 12, 1888. In this note he does not only state the theorem that each analytic function is of first cardinality (i.e., has only a countable number of different values at each point),59 but he also attempts to prove it. As to the origin of this theorem he states in a footnote: Questo teorema, di cui io sospettava l’esistenza, mi fu comunicato recentemente dal ch.mo prof. G i o r g i o C a n t o r , il quale nello stesso tempo mi esortava, a tentarne dal mio canto la dimonstrazione.60

Similar reports on the genesis of the theorem are given by Volterra and Abraham (Adolf) Fraenkel (1891–1965).61 Neither, however, mentions that Vivanti had already conjectured the statement. On the other hand, Fraenkel adds that Cantor had informed Vivanti by means of a letter. Here the specification of the exact creator of the proof given in Vivanti’s article [70] is of importance for an unpleasant reason (at least for Vivanti): To be sure, the statement of the theorem is correct, but he gives a fallacious proof of it! He considers the concrete Riemann surface over ⺓ that corresponds to the given multi-valued analytic function which is defined on a domain in ⺓.62 From the present (mathematical) point of view one might think that this construction was unproblematic and well-known at that time. For example, the Riemann surfaces which one has to use for this purpose are nothing else than the “analytic formations” which Weierstraß used to define in lengthy detail in his introductory lecture course on the theory of functions of a complex variable, including the treatment of the branch points.63 On the other hand, one should keep in mind that the notion of a Riemann surface had by no means been axiomatized at the end of the 19th century.64 Much of the analysis of Vivanti’s argument is therefore left to interpretation since Vivanti gives no definition of the notion of a Riemann surface which he uses and also is rather short in his exposition. In any case, by taking resort to geometrical imagination and the picture of “elicoidale” (“winding stairs”), Vivanti argues that only countably many “fogli” (“sheets”) of the Riemann surface can meet at a branch point.65 Here a “sheet” of a Riemann surface should be interpreted as something like a (maximal) open connected domain of the surface where the function remains univalent. On the other hand, on such a sheet, at

58

cf. also [49, pp. 176–177]. [70, p. 150, Teorema]. 60 “This theorem, whose existence I had conjectured, has recently been communicated to me by Professor G e o r g C a n t o r , who at the same time called upon me to try a proof on my own.” [70, p. 150]. 61 [73, p. 359] resp. [27, p. 254], [28, p. 467]. 62 [70, p. 150]. 63 e.g., [80, pp. 159–165]. 64 Cf. Volterra’s criticism of Vivanti’s argument as decribed in [41, pp. 172–173, pp. 178–182], and briefly here in Sect. 7. 65 [70, a)]. 59

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least on its interior, the branch points lie isolated.66 Then Vivanti can quote Cantor’s result of 1883 that each subset of a number space ⺢n consisting only of isolated points is countable,67 by which he gets that on any sheet of the Riemann surface there are only countably many branch points.68 Up to this point Vivanti’s argument was all right – or can at least be interpreted in a way that evades faults. The really delicate point comes now: Hence, he concludes, each sheet of the Riemann surface is in direct connection with only countably many other ones and therefore, “in base ai principi della teoria degli aggregati”,69 also in indirect connection with only countably many other sheets. The problem here is the argument that each sheet is in direct connection with only countably many other ones. Vivanti’s text is rather laconic: “Dalle due osservazioni precedenti risulta che ciascun foglio . . . e` in comunicazione immediata con un insieme enumerabile di fogli.”,70 but it seems that the argument is meant as follows: There are only countably many branch points on each sheet, only countably many sheets meet at each of the branch points, so each sheet is in direct connection via branch points with only countably many other sheets. Though this line of thought is correct, Vivanti seems to have failed to notice that sheets may have connection with other ones also in a different way than via branch points or, putting things somewhat differently, that also a domain in ⺓ whose complement does not only consist of isolated points, e.g., an annulus, may have a multi-sheeted covering and hence be the domain of definition of a non-univalent analytic function. So the defects of Vivanti’s argument, which, by the way, were immediately pointed out by Hurwitz in his review [37], come from the theory of functions of a complex variable, whereas the set theoretic conclusions are correct. Therefore, in order to complete the proof, it is sufficient to give a class of (connected) open subsets of the Riemann surface in such a way that the analytic function is univalent on each of these sets, that they cover the whole surface and that each of them has a non-empty intersection with only countably many other ones. Then the canonical denumerability argument gives that there are only countably many of these sets at all and that, therefore, the function can attain only countably many values at each point. Poincar´e and Volterra were the two mathematicians who, evidently independently of one another, had the necessary idea for this.

6. Poincar´e’s letter In the meantime, on July 17, 1888, Vivanti’s first note [69] had been printed and published in the “Rendiconti del Circolo Matematico di Palermo”. Since it contained

66

supposing that a “Riemann surface” is defined in the way usual today, e.g., [40, 4. edition, III. 5. § 3, esp. p. 388]. 67 [12, Theorem I]. 68 [70, b)]. 69 “on the basis of the principles of set theory” [70, p. 151]. 70 “From the above two observations it follows that each sheet . . . is in immediate connection with a countable set of sheets.” [70, p. 151].

The Poincar´e-Volterra Theorem

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the proof that Poincar´e’s uniformization theorem [51] holds only for analytic functions of first cardinality – leaving open whether this property holds for each function –, it can be easily imagined that, in Poincar´e’s own wording, the reading of this note “m’a vivement int´eress´e et m’a inspir´e diverses r´eflexions”.71 Poincar´e’s enthusiasm is the more understandable because he was acquainted with set theory since his research on automorphic functions, in which perfect, non-dense sets appear as sets of singularities, and he even knew Cantor personally since the beginning of 1884.72 Poincar´e laid down the abovementioned “reflections” [52] in a letter to GiovanniBattista Guccia (1855–1914), the editor of the “Rendiconti”. The letter dates of October 27, 1888, it was read in excerpts at the session of the “Circolo” at November 11, 1888, and published in the same volume of the “Rendiconti” as Vivanti’s two articles [69], [70]. Here Poincar´e proves that each analytic function is of first cardinality, but does not refer to Vivanti’s attempted proof [70].73 Poincar´e’s exposition is complete, convincing and, yet, only three printed pages long. First, he explains the notion of denumerability and recalls the Weierstraßian definition of analytic functions by means of power series and chained discs. Then he notes that the set of all power series which define a given analytic function is uncountable.74 (At the end of the last Section a certain set of subdomains of the Riemann surface was required. Poincar´e’s remark has the consequence that one cannot satisfy this requirement by taking the set of all subdomains of a Riemann surface on which the function is described by a power series.) Poincar´e’s decisive observation is that one does not need to consider all power series in order to determine the analytic function: If one starts off with a given power series and can attain a value of the function at a point by means of any chain of discs then the same can be performed by means of a chain of discs whose centers have rational real and imaginary parts. Hence it suffices to consider power series whose centers of expansion have rational coordinates.75 For a given power series and a given point in ⺓ the identity theorem implies that there is at most one power series with the given point as center of expansion that coincides with the given power series on the non-empty intersection of the open discs of convergence, i.e., a power series that is an immediate continuation of the given power series. Since there are only countably many points in ⺓ with rational coordinates, this implies that there are only countably many power series with rational points of expansion which are immediate continuations of a given one. By use of the denumerability argument which has already been mentioned in connection with Vivanti’s article [70] it follows then that there are altogether only countably many power series with rational center of 71

“has vividly interested me and has inspired me to diverse reflections” [52, p. 197 resp. p. 11]. [23, p. 280]. 73 As Vivanti’s second note [70] was printed almost a month after the first one [69], it is possible that Poincar´e – who only in 1890 became member of the “Circolo” – was not aware of the second note when writing his letter. 74 [52, p. 198 resp. p. 12]. 75 [52, p. 198 resp. p. 12] Translated into the language of the Riemann surface associated to the analytic function – which Poincar´e does not use, however –, this means that the images of power series with rational centers of expansion cover the whole Riemann surface. 72

390

P. Ullrich

expansion which are continuations of a given one, whether directly or indirectly. Hence, in particular, only a countable number of values can be attained at a given point.76 While developing his line of thought Poincar´e even gives an explicit – and very elegant – proof of the set theoretic argument used, viz., that the set of all sequences of natural numbers of arbitrary finite length is countable: To each sequence (α1 , α2 , . . . , αn ) with n ∈ ⺞, α1 , α2 , . . . , αn ∈ ⺞ arbitrary he assigns the finite continued fraction 1 α1 + . 1 α2 + 1 .. .+ αn This gives rise to a bijection between the set of finite sequences of natural numbers and the set of finite continued fractions. The latter one, however, is nothing else than the set of positive rational numbers, hence countable.77

7. Volterra’s article In that very year 1888, when the articles [69], [70] by Vivanti and [52] by Poincar´e appeared, also Volterra published a proof of the theorem that each multi-valued analytic function attains only countably many values at each point. This proof, however, was not published in the “Rendiconti del Circolo Matematico di Palermo” but in the “Atti della Reale Accademia dei Lincei” [73]. As to the contents of this article, Hurwitz writes in his review in the “Jahrbuch u¨ ber die Fortschritte der Mathematik”: “Die Betrachtungen des Verfassers stimmen im wesentlichen mit denen der Herren Vivanti und Poincar´e u¨ berein” (“The considerations of the author essentially coincide with those of Messrs. Vivanti and Poincar´e”), from these expositions, however, “die Arbeit des Verfassers . . . sich durch Gr¨undlichkeit auszeichnet” (“the author’s work distinguishes itself by its meticulousness”) [38]. Whereas Poincar´e’s note bears the character of a clear but terse notice by letter, Volterra has worked out the theory in all details, just as if he had wanted to publish a corrected version of Vivanti’s second note [70]. He develops the Weierstraßian method of analytic continuation.78 In particular, he studies “dominii di monodromia” (“domains of monodromy”)79 on which the function remains univalent. He lists the properties of these domains in detail,80 noting especially that the power series expansions around centers with rational coordinates give rise to a countable number of “domains of monodromy” which have the property that each value of the function is attained on one of them and that each “punto regolare di diramazione”, i.e., interior branch point, of the function lies on the boundary of such a domain.81 From this he does not only conclude the de-

76 77 78 79 80 81

[52, p. 200 resp. p. 13]. [52, pp. 199–200 resp. p. 13]. [73, pp. 356–357]. [73, p. 357]. [73, pp. 358–359, esp. Teorema I to V]. [73, p. 358, Teorema I and p. 359, Lemma].

The Poincar´e-Volterra Theorem

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numerability of the values at a point but also that the set of (interior) branch points is countable.82 Concerning the relation of Volterra’s article to those of the other two authors, Hurwitz in his review comes to the conclusion that “die Arbeit . . . offenbar unabh¨angig von den Publicationen der Herren Vivanti und Poincar´e entstanden [ist]” (“the work . . . was apparently created independently of the publications of Messrs. Vivanti and Poincar´e”) [38]. Indeed, Volterra does not quote any of the three other articles which treat the problem. Only in a footnote to his statement on the denumerability of the values he makes the annotation Questa propriet`a e` dovuta al prof. G. Cantor, che la comunic`o senza dimonstrazione al dott. G. Vivanti.83

This is, so to speak, the official and published version of the story. The Volterra estate in the “Accademia Nazionale dei Lincei”, however, reveals that Volterra’s part is definitely more complicated. (The following account is based on the study [41] by Israel and Nurzia.) After the printing of his second article [70] Vivanti had sent an offprint of it to Volterra with whom he was in correspondence since 1887.84 Following this, on August 21, 1888 Volterra wrote a letter – not to Vivanti but to Cantor in order to ask for the latter’s opinion concerning his, Volterra’s, objections against Vivanti’s method of proof. The points of criticism have essentially already been mentioned in the discussion of Vivanti’s paper [70]: the possibility that sheets are in connection in other ways than via branch points, the question whether branch points are always isolated and, in general, the problem of constructing the Riemann surface corresponding to an arbitrary analytic function.85 Cantor’s answer to this on August 25, 1888 is rather evasive. But remarkably he again mentions that he had communicated the theorem in question several years ago to Weierstraß who had found a proof of it by means of the theory of minimal surfaces some time later on.86 After this reply, in the time between end of August and mid-October, 1888, Volterra worked on his article [73] and confronted Vivanti with his criticism only on October 13.87 From this a lively correspondence between the two resulted in the course of which Vivanti indeed could solve the problem of the sheets which are in connection otherwise than via branch points. But at last he had to admit his failure to give an exact definition of the Riemann surface corresponding to an arbitrary analytic function.88 Then, on November 18, 1888 Vivanti informed Volterra that Poincar´e’s letter [52] had been read at a session of the “Circolo”,89 whereupon Volterra, who seems to have at first had in mind to publish his article in the “Rendiconti”, submitted it to the “Atti” of the “Accademia Nazionale dei Lincei” on November 20, whose corresponding

82

[73, p. 359, Corollario], [73, p. 359, Teorema V]. “This property is due to Professor G. Cantor who has communicated it without proof to Doctor G. Vivanti.” [73, p. 359, footnote (3)]. 84 [41, p. 171], [41, p. 177]. 85 [41, pp. 171–172, p. 175]. 86 [41, p. 176]. 87 cf. [41, pp. 172–173]; [41, pp. 177–178]. 88 [41, pp. 172–173, pp. 178–182]. 89 [41, p. 183]. 83

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P. Ullrich

member he had become recently.90 The paper was read at the session on December 2, 1888.

8. Further developments After untangling the parsimonious remarks on Cantor and Vivanti in the two proofs of the Poincar´e-Volterra theorem, we turn to the reception of this theorem within the mathematical community. Starting with Vivanti himself, this result is surprisingly not mentioned in his textbook on complex function theory [71] which appeared for the first time in 1901. Of course, one might be tempted to assume that this is a consequence of his negative experiences with that complex of problems. But, on the other hand, one has to take into account a reason coming from within mathematics: Following a general tendency at the turn of the century, Vivanti abandons the Weierstraßian definition of multi-valued analytic functions in his book. When in 1906 August Gutzmer (1860– 1924) revised and translated it into German, the attribute “eindeutigen” (“univalent”) was explicitly added to its title, and Gutzmer writes in connection with the definition of an analytic function: Was man gew¨ohnlich eine n i c h t e i n d e u t i g e oder auch eine m e h r d e u t i g e Funktion nennt, ist nach unserer Auffassung keine Funktion, sondern der Inbegriff von mehreren, bezw. unendlich vielen Funktionen.91

Contrary to this, the first volume of the textbook on complex function theory [3] by Ludwig Bieberbach (1886–1982) still contains that “sch¨onen von Poincar´e und Volterra herr¨uhrenden S a t z”.92 The first editon of this volume appeared in 1921. In the fourth edition, published in 1934, even a complete subsection was devoted to the theorem under the title “Analytische Funktionen sind h¨ochstens abz¨ahlbar vieldeutig.”93 As late as 1962, the book [35] of Einar Hille (1894–1980) discusses the Poincar´e-Volterra theorem in connection with multi-valued analytic functions.94 On the whole, the Poincar´e-Volterra theorem did by no means fall into oblivion either when formulated in terms of a multi-valued analytic function or when formulated in terms of the associated Riemann surface, quite the contrary. In his first note [69] Vivanti had already pointed out the connection with Poincar´e’s uniformization result [51]. Now, when in 1907 the uniformization theorem was proven in full generality by Paul Koebe (1882–1945) and, independently, by Poincar´e, both authors referred in their articles to the Poincar´e-Volterra theorem, Koebe in a footnote without giving a source:

90

[41, p. 172], [41, p. 173], [41, p. 177]. “What usually is called a n o n - u n i v a l e n t or also a m u l t i - v a l u e d function is no function according to our conception, but the collection of several respectively infinitely many functions.” [72, p. 109, footnote 1]. 92 “beautiful t h e o r e m stemming from Poincar´e and Volterra” [3, 1. edition, pp. 203–204]. 93 “Analytic functions are at most countably multi-valued.” [3, 4. edition, pp. 202–203]. 94 [35, p. 14, Theorem 10.3.2]. 91

The Poincar´e-Volterra Theorem

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R i e m a n n sche Fl¨achen mit nichtabz¨ahlbarer Bl¨atterzahl oder nicht abz¨ahlbarer Anzahl der inneren Windungspunkte gibt es bekanntlich nicht.95 ,

Poincar´e referring to his own paper [52] of 1888: Pourquoi cet ensemble doit-il eˆ tre d´enombrable, c’est ce que j’ai expliqu´e au tome 2 des Rendiconti del Circolo Matematico di Palermo.96

Also Bieberbach in his proof of the uniformization theorem in the second volume of his “Lehrbuch der Funktionentheorie”97 mentions this result. Here, by the way, he explicitly talks about the Riemann surface of a (multi-valued) analytic function. The theory of Riemann surfaces had been axiomatized in the meantime, mainly by Hermann Weyl’s (1885–1955) monograph [82], which appeared for the first time in 1913. Here he also proves the von Poincar´e und Volterra ausgesprochene. . . Theorem. . . , daß es in einem analytischen Gebilde h¨ochstens abz¨ahlbar unendlichviele regul¨are Funktionselemente . . . mit vorgeschriebenem Mittelpunkt . . . gibt.98

In his proof Weyl in the main follows the lines of the articles by Poincar´e [52] and Volterra [73] and also cites them. (The names Cantor and Vivanti do not appear.) Theorem and proof are also found – without any mentioning of the authors, however – in the textbook on complex function theory [40] by Hurwitz and Richard Courant (1888–1972) from the second edition of 1925 onwards. Remarkably, they are not in the first part “Allgemeine Theorie der Funktionen einer komplexen Ver¨anderlichen” (“General theory of functions of a complex variable”) which was written by Hurwitz who had been involved in the events of 1888, if only as a reviewer. But they are given in the third part on “Geometrische Funktionentheorie” (“Geometric theory of functions”) by Courant, viz., in the fifth chapter, § 3. (At the beginning of this very section there is a reference to Weyl’s book [82]. Since this reference can also only be found in the second and later editions of the book of Hurwitz and Courant, it seems probable that the theorem had found its way to their book via Weyl’s book [82].) Within the function theoretic setting, the books of Weyl and of Hurwitz and Courant mark the definite shift from the arithmetically defined multi-valued analytic functions of Weierstraß to their geometric counterparts, the associated (concrete) Riemann surfaces. Yet, the method of proof of the Poincar´e-Volterra theorem essentially remained the same as in the original articles by Poincar´e [52] and Volterra [73]: One uses the fact that the countable set of rational numbers lies dense in the set of real numbers and then argues that one can connect two arbitrary points by using only points with rational coordinates as intermediate points.

95

“It is known that there are no R i e m a n n surfaces with an uncountable number of sheets or a non-countable number of interior branch points.” [47, p. 198, footnote 2]. 96 “In volume 2 of the Rendiconti del Circolo Matematico di Palermo I have explained why this set is countable.” [53, p. 4 resp. p. 73]. 97 [4, p. 161 resp. p. 162]. 98 “theorem stated by Poincar´e and Volterra . . . , that in an analytic formation there are only countably infinitely many regular function elements . . . with prescribed center” [82, here 1. edition, pp. 14–15].

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This proof hardly uses any analytic property of the function or its Riemann surface at all. In fact, nowadays the Poincar´e-Volterra theorem is usually considered as a topological statement, namely that each concrete Riemann surface has a countable basis of topology. A topological space is said to have a countable basis of topology (sometimes even in short: a countable topology) if there exists a countable set of open subsets of the space such that for each point of the space arbitrarily small neighborhoods belong to this set. Since each open subset of a real number space ⺢n has a countable basis of topology – just take the balls with rational radii centered at points with rational coordinates which are contained in the open subset –, one only needs to check whether a given (real or complex) manifold, e.g., a Riemann surface, contains a dense countable subset in order to show that it has a countable basis of topology. Therefore, also in the new clothing of the theorem, the idea of proof remains the same even though the wording differs considerably from the versions of 1888. The abstraction of the notions involved in the statement and the proof of the Poincar´eVolterra theorem, i.e., the translation into the language of set theoretic topology began in the beginning of the 20th century. In his monograph [32] of 1914 Felix Hausdorff (1868–1942) already discusses the “zweite Abz¨ahlbarkeitsaxiom” (“second axiom of denumerability”) for topological spaces, i.e., that the topology of the space has a countable basis.99 Surprisingly, however, in [32] Hausdorff does not mention the Riemann surface of an analytic function as an example of a space for which the second axiom of denumerability holds. The same applies for the revised versions of this book which appeared in 1927 and 1935100 and for the monograph by Paul Alexandroff (1896–1982) and Heinz Hopf (1894–1971) of 1935.101 Surely, spaces with a countable dense subset are discussed in these sources, in particular in connection with Paul Urysohn’s (1898–1924) results concerning whether a given topological space carries a metric. But the result of Poincar´e and Volterra is not mentioned. A result connected with their names was mentioned, though, when Bourbaki revised his “Topologie g´en´erale” [7] for the third edition of 1961. There he (or: they) added to Chapitre 1, § 11 a section no 7 entitled “Application: le th´eor`eme de Poincar´e-Volterra”. Of course, the topic is no longer analytic functions or Riemann surfaces: “Corollaire 3 (th´eor`eme de Poincar´e-Volterra)” states that a separated and connected topological space X has a countable basis of topology if there is a locally compact and locally connected topological space Y with a countable topology and a continuous map p: X → Y which is locally homeomorphic on X. (The following seems to be an appropriate dictionary for translating this back to the original situation of a multi-valued analytic function: ⺓, X as the Riemann surface or Take Y as the domain of definition of the function in b “analytisches Gebilde” defined by it and p as the projection to the domain of definition, i.e., the local inverse of the analytic function.)

99

[32, Kap. VIII, §§ 1, 3] The “erste Abz¨ahlbarkeitsaxiom” (“first axiom of denumerability”) states that each point has a countable basis of neighborhoods [32, Kap. VIII, §§ 1, 2]. This axiom applies for Riemann surfaces and, more generally, for real topological manifolds. 100 [33], [34], in both cases § 25, § 30, 4., § 40, III. 101 [2, Kap. I, § 7].

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It should be recalled that the Poincar´e-Volterrra theorem only makes a statement on concrete Riemann surfaces, i.e., Riemann surfaces on which a non-constant analytic ⺓ is given. Already Klein, however, had studied Riemann map to the Riemann sphere b surfaces without such a concretization. And at least since Weyl’s topological definition of a Riemann surface it made perfect sense to study also such abstract Riemann surfaces.102 For the case of concrete Riemann surfaces, i.e., multi-valued analytic functions, Vivanti had noted in his article [69] that the denumerability of the values of a function at a point is a necessary prerequisite for Poincar´e’s uniformization result [51] to hold. Similarly, the problem whether an abstract Riemann surface has a countable basis of topology arose in connection with the question of uniformization. Now, in order to use exhaustion procedures in the construction of uniformising functions, it was necessary to secure that each (connected) Riemann surface possesses a triangulation, i.e., a division into topological triangles. It is immediately clear that a triangulated surface has a countable topology. The converse also holds, but an exact proof is by no means trivial. A short proof that only each compact real 2-dimensional manifold can be triangulated was published as late as 1968 and in no less a journal than “Inventiones mathematicae” [24]. Weyl ridded himself of this problem in his “Idee der Riemannschen Fl¨ache” [82] by raising the existence of a triangulation in the first two editions and the denumerability of the topology in the third edition to an additional axiom for Riemann surfaces.103 Also Koebe a priori required the existence of a triangulation for the “Riemannschen Mannigfaltigkeiten” (“Riemannian manifolds”) which he considered in his article of 1917.104 This solution surely was effective but also not too elegant. Even more, it was unnecessary as Tibor Rad´o (1895–1965) first noted in 1922. He stated the theorem and also sketched a proof that each (abstract!) Riemann surface possesses a triangulation.105 Rad´o’s explanations were sketchy for the only reason that Heinz Pr¨ufer (1896–1934) had communicated his conjecture that each surface, even if only topological, can be triangulated to Rad´o who, therefore, was insecure whether his proof for the special case of a Riemann surface was worth being published at all.106

9. Counterexamples In January 1923 Pr¨ufer, however, revoked his conjecture that each topological surface can be triangulated and informed Rad´o of an example of a surface which cannot be triangulated and hence has no countable topology.107 Following this, Rad´o published 102

It is still subject to debate whether Riemann himself thought of his surfaces as concrete or abstract, cf., e.g., the exposition [59, pp. 184–185]. 103 [82, 1., 2. edition, p. 21], [82, 3. edition, pp. 21–22]. 104 [48, pp. 70–71]. 105 [57, pp. 35–36]. 106 [57, p. 35]. 107 [57, p. 35, footnote 9], [58, p. 102, footnote 1].

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another article in 1925 where he exhibited Pr¨ufer’s counterexample, showed that each surface with countable topology can be triangulated, and finally proved that each Riemann surface has countable topology and hence can be triangulated.108,109 Pr¨ufer’s real 2-dimensional counterexample consists of a halfplane with half a unit disc attached at each point of the boundary line, in the following way: First, in order to add half a unit disc at one point a of the boundary line, think of a as fixed, shift each point of the half plane one unit length away from a (in the same direction in that it lies, as seen from a), and insert half a unit disc in the space thus created. This gives, for each a of the boundary line, a half plane which consists of the shifted points of the original half plane and the inserted half disc. Now, make one manifold out of all these half planes by identifying again the points of the original half plane in the way as they were situated before the respective shifts.110 The resulting manifold is a connected surface with uncountable topology. Even more – although this fact is not stated in [58] – by writing down the construction in a convenient way, one can easily see that this surface is real analytic.111 Therefore, this a priori 2-dimensional example implies that in any real dimension greater than or equal to 2 there are real analytic manifolds which have no countable basis of topology. If Pr¨ufer had only been interested in topological spaces without additional conditions, however, then he could have simply asked Hausdorff concerning his conjecture on the countability of the topology. Indeed, a recent examination of Hausdorff’s mathematical estate in the Universit¨atsbibliothek at Bonn by Egbert Brieskorn and Walter Purkert has brought to light that as early as May 30, 1915 Hausdorff knew an example of a connected topological surface which does not fulfil the second axiom of denumerability, i.e., has no countable topology.112 Contrary to Pr¨ufer’s construction, Hausdorff’s counterexample is the cartesian product of a common real interval with the so-called “long (half)line”. By use of the lexicographical order one “inserts” at each point of the usual real line a copy of the real unit interval. This gives rise to a connected real one-dimensional topological manifold which is “longer“ than the usual line (and, in fact, has no countable basis of topology). The basic idea for this construction can already be found with Cantor in 1883: Die erweiterte ganze Zahlenreihe kann, wenn es die Zwecke fordern, ohne Weiteres zu einer continuirlichen Zahlenmenge vervollst¨andiget werden, indem man zu jeder ganzen Zahl α alle reellen Zahlen x, die gr¨osser als Null und kleiner als Eins sind, hinzuf¨ugt. Es wird nun vielleicht hieran die Frage gekn¨upft werden, ob man, da doch auf diese Weise eine bestimmte Erweiterung des reellen Zahlengebietes in das Unendlichgrosse

108

[58, § 2], [58, § 3], [58, § 4]. Nowadays, Rad´o’s result on the denumerability of the topology of Riemann surfaces is usually proven by considering the surface with a disc removed, constructing a non-constant holomorphic function on the universal covering of this “punched surface” by the help of methods from function theory and then applying the Poincar´e-Volterra theorem, e.g, in the Bourbaki version cited above. 110 For explicit formulas see [58, pp. 107–108]. 111 cf., e.g., [9, p. 337]. 112 Kapsel 31, Fasz. 121, sheet 2, cf. [8, p. 2] and [59, p. 188]. 109

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erreicht ist, nicht auch mit gleichem Erfolge bestimmte unendlich kleine Zahlen, oder was auf dasselbe hinauslaufen m¨ochte, endliche Zahlen definiren k¨onnte, welche . . . sich an muthmaasslichen Zwischenstellen inmitten der reellen Zahlen ebenso einf¨ugen m¨ochten, wie die irrationalen Zahlen in die Kette der rationalen oder wie die transcendenten Zahlen in das Gef¨uge der algebraischen Zahlen sich einschieben?113

Though the description is somewhat vague, one easily recognizes the idea of inserting further numbers, e.g., the unit interval, into the real line. In are more detailed way, Leopold Vietoris (*1891) has treated the “long halfline” in his doctoral dissertation which originates from the years 1913 to 1919 and was published in 1921.114 He, however, says nothing concerning the denumerability of its topology. One should note that Vietoris’ treatment of the “long line”, in particular his giving credit to Cantor, is one of the paradoxes of the history of mathematics. He writes on his construction: Nach C a n t o r s Anleitung interpolieren wir zwischen je zwei aufeinanderfolgende unter einer beliebigen Schranke σ liegende Ordnungszahlen einschließlich σ selbst ein dem Kontinuum [0, 1] a¨ hnliches Linearkontinuum . . . .115

But “Cantor’s instruction”, i.e., his question quoted above, is purely rhetorical and the intended answer is negative. Cantor strictly refused to accept the possibility of infinitely small numbers as they appear in the construction of the “long (half)line”. In his article of 1883116 this is not as clearly expressed as in his paper of 1887117 and in his dispute by letter with Veronese in 1890.118 As mentioned above, Hausdorff had a clear perception of the topological properties of the “long halfline” already in May 1915. However, he did not publish his result, but communicated it to Heinrich Tietze (1880–1964).119 The latter included this as an example120 in a manuscript which he had essentially completed in summer 1922. But

113

“If the ends demand it, the extended series of the integer numbers can be completed without further ado to a continuous set of numbers by adding to each integer number α all real numbers x which are greater than zero and smaller than one. Maybe, the following question will be posed as a consequence to this: Since one has reached a certain extension of the domain of real numbers to the infinitely large in this way, why should one not with the same success insert certain infinitely small numbers or, what may amount to the same, finite numbers which . . . fit into the conjectural intermediate places in the midst of the real numbers, just like the irrational numbers insert themselves into the chain of rational or the transcendental numbers into the fabric of algebraic numbers?” [13, § 4, p. 552 resp. p. 171]. 114 [68, pp. 183–184]. 115 “Following C a n t o r ’ s instruction, we interpolate between any two successive ordinal numbers lying below an arbitrary bound σ , including σ itself, a linear continuum which is similar to the continuum [0, 1] . . . ” [68, p. 183]. 116 [13, p. 552 resp. pp. 171–172]. 117 [15, VI, pp. 407–409]. 118 cf. [18, pp. 326–332], also [5, p. 78], [23, pp. 128–132, pp. 233–238]. 119 [65, p. 217]. 120 [65, pp. 217–218].

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he submitted it only in October 1923 to the “Mathematische Annalen” since he wanted first to get himself informed “¨uber ausl¨andische Literatur”.121 Already on August 1, 1923, however, a manuscript by Alexandroff had reached this journal in which also the “long halfline” was discussed.122 (In fact, Alexandroff had reported on the main results of [1] already in 1922 in Moscow and on June 26, 1923 in G¨ottingen.123 ) Although Alexandroff’s paper [1] was published one volume of “Mathematische Annalen” later than Tietze’s, a tradition stems from these circumstances to denote the “long (half)line” as the “Alexandroff (half)line”.124 Remarkably, this means that the existence of a real one-dimensional manifold without countable basis of topology was documented in print only after the corresponding fact for the 2-dimensional situation, i.e., Pr¨ufer’s example. Rado mentioned it already in his article [57] that appeared in volume 90 of the “Mathematische Annalen”, whereas the articles of Tietze and Alexandroff were published only one and two volumes later, respectively. Furthermore, whereas it is more or less evident from its construction that Pr¨ufer’s example is real analytic, it was not so clear whether the “long (half)line” bears not only a topological but also a real analytic structure. Only in 1957 Hellmuth Kneser (1898– 1973) proved this fact.125 Hence, there are real analytic manifolds of any dimension which admit no countable basis of topology. In the complex analytic situation Rad´o’s theorem says that for (complex) dimension 1 each connected complex analytic manifold has a countable basis of topology. For higher dimensions, however, Eugenio Calabi and Maxwell Rosenlicht have given counterexamples in 1952 [9]. Putting things the other way round: Rad´o’s result – or the Poincar´e-Volterra theorem – deals with the only real or complex analytic case in which one always has a countable basis of topology.126 So, seen from the point of view of mathematics, the Poincar´e-Volterra theorem concerns the exceptional case and it looks like pure accident that mathematicians came upon it. But the history depicted above shows this was by no means the case. The advances of the complex analysis of one variable in the middle of the 19th century had substantiated the hypothesis that the value set of a multi-valued analytic function could not be so large that it would prevent analytic inversion, e.g., of a hyperelliptic integral. Cantor’s concept of denumerability then supplied a language to express this belief. And altogether five mathematicians at least to tried to give proofs of the denumerability of the value set, even if the result nowadays is connected with the names of only two of them.

121

“on foreign literature” [65, p. 224]. [1, p. 295, footnote 2]. 123 [1, p. 301]. 124 cf., e.g., [45, p. 4], [59, p. 189]. 125 [45, Satz 2] There exist even continuously many non-isomorphic real analytic structures on the Alexandroff (half)line as he showed in a joint paper with his son Martin (*1928) in 1960 [46]. 126 For a further discussion of the interplay between countable topology and complex structure on a manifold see [59, esp. pp. 189–191]. 122

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Mathematisches Institut Westf¨alische Wilhelms-Universit¨at 48149 M¨unster Germany (Received May 19, 1999)