Paradise Recovered? Some Thoughts on Mengenlehre and Modernism. José Ferreirós

Transcription

1 Paradise Recovered? Some Thoughts on Mengenlehre and Modernism José Ferreirós (Draft Jan 2007, slight revisions Sept 2008) The topic of modernism and its connection with modern mathematics and its foundational crisis was studied in a pioneering work by Herbert Mehrtens: Moderne Sprache Mathematik (1990). Mehrtens thesis was that one can better understand the foundational crisis not just as having to do with problems of rigor and foundations, but as an expression of social readjustements within the discipline, of conflicting cultural definitions of the figure of the mathematician himself, the subject of formal mathematical systems. It was a new configuration of the discipline, not just in the seemingly objective sense of the theories and methods of modern mathematics, but especially the disciplining of the subjects who were going to bring it forward. Hence his subtitle, a history of the fight for the foundations of the discipline and the subject of formal systems. According to Mehrtens, the different attitudes and options expressed in the foundational debate correlate with the modernism or counter-modernism of the personalities involved. From the late 19 th century, set theory or Mengenlehre (as Cantor named it) became a quintessential element of so-called modern mathematics. One often hears about it as a foundational discipline, a sophisticated special branch of mathematical logic, and a core element of the foundational debate which took place from about But indeed, around 1900 set theory was more than anything else a key ingredient of the most advanced inroads into uncharted mathematical territory. In an obituary talk of 1910, the legendary David Hilbert refered to it as: that mathematical discipline which today occupies an outstanding role in our science, and beams out [ausströmt] its powerful influence into all branches of mathematics, namely, the theory of sets [Mengentheorie]. (Hilbert 1910, 466, obituary of Minkowski)

2 To remain in the belle époque, , one can mention the influence of set theory on: the modern theory of [algebraic] numbers (Dedekind, Hilbert) algebra and algebraic geometry (Dedekind, Weber, Hilbert) modern analysis, measure theory (Peano, Borel, Lebesgue, Baire) general topology (Fréchet, Hausdorff) new researches on geometry (Hilbert, Peano, Pieri, Padoa). The mathematicians just named were mainly centred around Göttingen, Torino, and Paris, but it was especially the Germans (seconded by the Italians) who promoted the new viewpoint most forcefully. The influential group of Göttingen mathematicians around Hilbert became the main promoters of this neue Mathematik, which they paradigmatically identified with set-theoretic methods. For Hilbert and his colleagues, the new, daring theoretical framework elaborated by Cantor and Dedekind was at the core of the new mathematics. Not surprisingly, Mehrtens has presented Hilbert as the prototypical modernist, alongside Dedekind and Cantor, while his or their opponents, the critics of modern methods and set theory, are (almost by definition) non-moderns. The catchwords for the new attitudes of the modern mathematicians were freedom, productivity, fruitfulness, and it was also frequent to talk about abstraction and about formal language (Mehrtens 1990, chap. 2). The countermoderns favoured instead the concrete, and intuition or the given; they expressed concerns about the integrity of mathematics, and they explicitly promoted restrictions on the way mathematics was to be conducted, having to do with meaning and truth (op. cit., chap. 3). A not very careful reader of Mehrtens will thus extract the idea that there were merely two sides the modernists or progressives promoting so-called modern mathematics, and the conservatives calling for reactionary reform. Such an overly simplistic scheme would be untenable. It has to be said on Mehrtens behalf that his proposal in Moderne Sprache Mathematik is more sophisticated and faithful to the events. He employs three main categories: modernism (Moderne), counter-modernism (Gegenmoderne), and antimodernism (Antimoderne). Only the anti-moderns are depicted as reactionary traditionalists, while both the moderns and the counter-moderns are regarded as opposite

3 expressions of the broader cultural trend of modernism. (Obviously, the terminology employed is inconvenient, because it promotes the careless reading that I mentioned above.) The scheme has the merit of acknowledging that cultural modernism was a broad movement with strong tensions between diverging tendencies and interpretations. As Herf wrote (1984, 12) modernism was not a movement exclusively of the political Left or Right; we may add that it was not a movement of cultural pessimism or optimism, nor simply a movement of technological aversion or enthusiasm. Even in the prototypically optimistic belle époque we find some modernists that are cultural pessimists. This is well known with respect to writers and artists (e.g. the contrasting attitudes toward technology and the new industrial world in H. Broch and R. Musil), but the same tensions find natural expression also inside the mathematics community (e.g., Brouwer and Hilbert, in that same order). And that is of course quite reasonable. Europe witnessed in the late 19 th century the full impact of two great Revolutions of the previous century: industrialisation and democratisation (or reactions thereto, as in Germany). Indeed, a second industrial revolution was under way, related to technological, science-based industries exemplified by the electrical and chemical companies; and it might not be inappropriate to say that a second political revolution was also happening (extension of voting to all of society, the rights of women, etc.). The most obvious understanding of modernism is that it consisted in an explicit and often high strung cultural reaction to that historical experience. Rather obviously, there is a broad palette of possible cultural reactions to an intensive transformation of life and society, ranging from fully optimistic embracement to an equally extreme and pessimistic rejection. 1 But what is most characteristic of the modernists is their emphasis on the new and their call for further transformation, or even open revolution. 1 Full classification, it seems to me, would be unbearably complicated: anti-modernism (Kronecker, Hermite), reactionary modernism (Cantor, Brouwer?), proactive or progressive modernism (Hilbert and others), revolutionary modernism (Hausdorff, Brouwer, Weyl), and probably some other labels.

4 The first third of the 20 th century was a time when many were expecting full arrival of the new, if not fighting to promote the revolution. It could be a new world of technological marvels, but also new social ways, new kinds of human relations; or a new art, music, literature, painting, or a new science, in radical break with the past; or most important it could be a new man having the traits defined by the Marxists, or those proposed by the Fascists, but also by many other groups that have been more successful in their later sociocultural reception. In what follows, my principal aim is to emphasize the complexity of the ensuing picture, warning against simplistic interpretations of the period. Generally speaking, cultural movements are never homogeneous and monovalent, but rather they are marked by tensions. And this complicates any attempt to relate scientific developments with the cultural atmosphere that made them possible. We shall find partisans of the modern methods in mathematics that show clear signs of cultural traditionalism, and opponents of the new math that may be counted among the best examples of modernist mathematicians. Along the way, I shall not refrain from indulging in some nice cultural analogies, comparing my mathematicians with influential figures in the arts; but this is always done in a sceptical mood, and merely because it helps make my narrative more lively and easier to understand. Perhaps my most important argument will be a vindication of the figure of Egbertus Brouwer as a modernist which has the side-effect that, in my view, an identification of modernism with so-called modern mathematics is not tenable. 1. Hilbert: productive specialist and master of method. In the 1900s David Hilbert became one of the most reputed mathematicians in the whole world. He was a leader within the institution that led international mathematics at the time, the University of Göttingen, an institution that contributed significantly to the modernisation of the mathematicians work. As we have seen, Hilbert was a decided promoter of set theory and the new math. Ernst Zermelo, the axiomatiser of set theory, said that it was only the influence of D. Hilbert which made him realize the importance and deep significance of its fundamental problems. Hermann Weyl, Hilbert s most prominent disciple, spoke of the way in which his Göttingen education in the 1900s had confined

5 him in set-theoretic ways of thinking. Hilbert s attempt to lift the veil of the new century, his famous 1900 conference on the future problems of mathematics, began with two questions intimately related to set theory (Cantor s continuum problem, and the existence of the set of real numbers, i.e., the consistency of arithmetic). 2 This emphasis has to be seen, I believe, as a conscious attempt to influence the direction of mathematics. It was Hilbert s bet for the future dominance of the set-theoretic orientation, in contrast to the severe criticisms voiced by powerful members of the older generation. Indeed, Hilbert was willing to invest all of the influence and respect he had accumulated over the years, in the attempt to preserve the new set-theoretic methods. This is exactly what he did during the famous foundational crisis of the 1920s, and in this way he became a scientific icon, the name most directly associated to the modern in mathematics. Another reason why Hilbert became an icon of the modern was the strong link established between his name and the idea of axiomatisation. He contributed a more sophisticated understanding of the mathematics of axioms than previously available, especially with the celebrated treatise Grundlagen der Geometrie (published in 1899, with significant revisions in 1902 and later), and he was the champion of axiomatics in all public arenas. But in the present context it is important to notice that his understanding of axiomatics was profoundly linked with the use of set-theoretic methods. 3 In the 1920s, already more than 60 years old, Hilbert was still inventing methods based on axiomatisation for grounding mathematics anew proof theory and metamathematics. But he was also deploying highly charged rhetoric: Cantor was depicted as the prophet of the new mathematical paradise, Hilbert himself as his vindicator; the intuitionist Brouwer was presented as a leader of the reactionary party, a follower to that villain figure of Kronecker. 4 The following sentence is well known, but perhaps it deserves to be quoted again: 2 Even though his public presentation was limited to 10 of the 23 problems in his list, those two were among the chosen ones. 3 On this topic, see Kanamori & Dreben, Ferreirós forthcoming. 4 Arthur Schoenflies did at the time historical work that established Kronecker s reputation as a fierce, malevolent enemy of Cantor, who was a major cause of his mental illness. Hilbert s rhetoric aligned Brouwer with Kronecker, and Hilbert himself with Cantor. (I should add that Hilbert had obtained an impression of Kronecker s ways of promoting his enmity to the new mathematics from his friends Minkowski and Hurwitz, who knew well the Berlin master.)

6 We shall carefully investigate those ways of forming notions and those modes of inference that are fruitful; we shall nurse them, support them, and make them usable, whenever there is the slightest promise of success. No one shall be able to drive us from the paradise that Cantor created for us. (Hilbert 1926, ) Some years before, he had answered Weyl s salute to Brouwer as the Revolution, saying that he was not a revolution at all, but rather an attempted Putsch that merely repeated the previous failed attempt by Kronecker. Clearly it was a fight between Progress and Reaction; one could even say, without abandoning the biblical language that Hilbert liked to employ: Good vs. Evil (words that sadly remind us of recent political events). Both the Promethean connotations of such rhetoric, and the abstractness of the axiomatic and set-theoretic mathematical style and developments, suggest the possibility of finding links between the new math and cultural modernism. Set theory helped to establish the new mathematical developments on foundations that not only seemed to guarantee methodological rigor, but especially a freedom of thought that was strongly emphasized by Cantor, Dedekind and Hilbert. Cantor went as far as saying that freedom is the essence of mathematics, while Dedekind liked to repeat that numbers and other mathematical objects are free creations of the human mind. This is the ultimate referent for Hilbert s defiant cry of 1925, the portrait of himself as a rebellious Adam fighting to remain in the Garden. And all of this was part and parcel of a clear move towards the autonomy and selfcontainedness of mathematics. Dedekind had moved in this same direction even more clearly than Cantor, as his contributions were purely mathematical, free from the influence of broader considerations from natural science (let alone metaphysics). And Hilbert of course did the same, raising the tendency to a peak with his (unsuccessful) attempt to make mathematics itself answer to key philosophical problems regarding its foundations. Hilbert s Beweisstheorie or Metamathematik was self-containment at its utmost (and it is tempting to see its failure as one of the many failures of extreme modernist tendencies shortly before the Second World War). Likewise, it is the case that the active group of mathematicians at Göttingen was instrumental in modernising considerably the practice of mathematics. Work at Göttingen became much more collective and oral than was usual in the past, when mathematicians worked in isolation on the basis of books and journals. There was a great number of

7 students around, and the weekly meetings of the Göttingen Mathematical Society played a decisive role, with many visitors coming by. 5 Even more important, the puristic values of German mathematics were tempered thanks to the efforts of Felix Klein, leading to the forging of new links with engineering and the natural sciences. There is little doubt that Hilbert can be dubbed a modern man. He was far from the old habits of German University mandarins, to the point of being criticised for his careless way of meddling with students. There are clear signs of his progressive stance in matters of culture and society, like his promotion of social democrat philosopher L. Nelson or his way of defending that Emmy Noether should be appointed a University professor ( Meine Herren, the faculty is certainly not a public bath ). He had the attitudes of an enterprising man of science, his life fully devoted to his specialised business, which he conceived as an autonomous enterprise. In fact, one of Hilbert s contributions in connection with set theory and foundational studies was to free their discussion from the philosophical and metaphysical elements that had figured prominently in Cantor and other members of the older generation. A detailed analysis of his foundational views and the development of his metamathematics would show this in full clarity. But here it may suffice to indicate a simple but clear symptom: as I indicated in the quotation given above, on the first page, in 1910 Hilbert wrote Mengentheorie and not Mengenlehre. In doing so he was avoiding the traditionalistic overtones so frequent in Cantor s work, opting for a straight denomination that already underscores the specialized and autonomous nature of set theory. But perhaps one should ask for more when talking about modernism. In fact, it seems to me that an attempt to consider seriously the links of the transformations in the sciences from 1900 to 1940 with the contemporary modernisms requires a distinction between modernism and modernisation. The mere fact that the socio-cognitive and institutional conditions of mathematical work were significantly modernised in Göttingen does not imply that the relevant actors were modernists. Indeed, historical evidence pointing to the distinction is very close at hand. Felix Klein was more important than Hilbert for the modernisation of the enterprise of mathematics at Göttingen, but Mehrtens himself (1990, 206ff) has 5 For a thorough discussion of this topic, see Rowe (1985) and (2004).

8 emphasized the ambivalence of Klein s attitudes and the difficulty of classifying him although he finally puts him in line with the counter-moderns. Which, by the way, already calls into question the parallels between modernism and modern math: historically Klein was a central figure in the promotion of the modern methods (associated with the names of Riemann, Dedekind, Cantor, Hilbert) in very many ways, through his editorial activities in the Mathematische Annalen, his work in the Enzyklopädie of the mathematical sciences, his activity as leader of a mathematical school, and not least his promotion of the rising young star Hilbert during the 1890s. Now, if we have to differentiate between modernism and modernisation in the case of Klein, the same must surely apply to Hilbert. And in fact, it seems to me that the figure of the latter is not sufficiently associated with a strong and explicit cultural position, to deserve the use of the adjective modernist. Of course, if we are to apply such criteria strictly, not so many names will be left. The best examples of modernist mathematicians that I know are Felix Hausdorff (b. 1868), L. E. J. Brouwer (b. 1881), Hermann Weyl (b. 1885), and Alfred Tarski (b. 1902) 6 perhaps Russell and Whitehead might be other good candidates. But of those four names, two are strongly linked with the critique of modern mathematics and the proposal of alternative methodologies: Brouwer and Weyl were counter-moderns, to use Mehrtens rather unsatisfactory label. Unfortunately, I shall only discuss one of those names, partly because of my limited knowledge of the rest, and partly because I intend to remain close both to the issue of Mengenlehre and the material treated by Mehrtens. 2. Dedekind: the quiet revolutionary. We now come to consider the issue of modernism in connection with the pioneers of Mengenlehre. During the 1900s, Hilbert and his younger colleague Zermelo promoted the view that Cantor and Dedekind were the creators of set theory. 7 As we have seen, both were important influences upon Hilbert s outlook on mathematics and his understanding of its new methods, including axiomatics. As both figure prominently in Mehrtens account of modernism and the foundational crisis, they deserve a closer look here. Regarding the 6 See respectively the volumes that are coming out of the Hausdoff Edition, van Dalen s biography, the book by E. Scholz, and Feferman & Feferman. 7 See Zermelo 1908, and Hilbert s lectures Logische Grundlagen des mathematischen Denkens of 1905.

9 time frame, one might consider that Cantor published his first path-breaking paper on set theory the same year that the impressionists held their first exhibition (1874), and he gave birth to the stairway to heaven formed by the transfinite numbers in the year that construction of the first skyscraper began in Chicago (1883). 8 According to Herf, the central legend of modernism was the free creative spirit who refuses to accept any limits and who advocates what Daniel Bell has called the «megalomania of self-infinitization». In this central inspiration a romantic motif lingers on. Both Cantor and Dedekind seem to have received the cultural impact of romanticism early in their lives, but of course ideas and trends associate rather freely in different minds: flexibility or plasticity is the rule. Dedekind preserved an emphasis on the free creative spirit and on the absence of limits for mathematical creation (concept formation), but in a classicist reading that reinforced more and more the idea of the logical limits and laws at play. Cantor, by contrast, remained closer to Herf s description, much more of a romantic throughout his life, and even combated explicitly the customary depiction of the human mind as finite. 9 It will be useful in the following to keep these broad motives in mind. The figure of Richard Dedekind will hardly raise the impression of a modernist, at least when judged merely by his lifestyle. As Klein remarked, he was a contemplative nature, leading a quiet life away from the centres of scientific power, a bachelor in Brunswick, living in the company of his mother and sister. The first impression one gets from knowledge of his life is, too much, that of a provincial man of the Biedermeier type. 10 On the other hand, there are very intriguing features in his actual mathematical style, discussed below, which suggest that there may have been more than met the eye below the quiet surface. Furthermore, it is safe to say that he was not a reactionary: there are clear traces that he opposed Prussian nationalism and the associated political trends, and this may be 8 Even though he was clearly older, Dedekind s milestone dates fall in those same years: he published important contributions in 1871, 1872, 1879, 1882, and Although Cantor stayed within the bounds of a rather traditional morality, and not in the amoral and aestheticizing orientation (typically exemplified by Nietzsche) that Herf had primarily in mind his aim being to understand the social, moral, and political orientations that ended up in nazism. 10 On the contrast between Biedermeier Germans and fin-de-siècle culture, see Schorske (1981).

10 related to his failure to accept University positions in Prussian cities. 11 For all of these reasons and more, the man Dedekind remains a bit of a mystery to me. In fact, Dedekind can profitably be compared with Paul Cézanne ( ), often considered the father of modern painting. (The following passages are intentionally written so as to suggest the points of comparison, though without going explicitly into them.) He was a man of orderly habits, leading a quiet, retired life. He consecrated his existence to the rather new mathematical problems that he posed for himself, problems to which he applied the sternest criteria. Living an unproblematic life in the external, he was devoted to a passionate fight in order to realize in his mathematical work an ideal of perfection. This explains, for instance, the constant rewriting of the theory of algebraic numbers that has to be seen as his lifework. 12 As mathematician and chess-player Lasker said, Dedekind s writings are a true oasis where one can rest oneself (which reminds us of the clarity, tranquillity and sense of equilibrium provided by Cézanne). And Emmy Noether, the Picasso of modern algebra, used to say: es steht alles schon bei Dedekind, it s all in Dedekind already. Without aiming to be revolutionary in any overt way, Dedekind was led by his deep convictions to a rupture with traditional work. Again, number theory offers us the clearest example: traditionally it had been the theory of the properties and relations of concrete numbers, the integers or more recently some algebraic integers; 13 Dedekind transformed it into the theory of some infinite sets of integers (called ideals ) within what he called bodies or fields of algebraic numbers. The transformation was so radical, that it took twenty years for other mathematicians to follow his lead; Hilbert was one of the first with the work that is probably his masterpiece, published in He was also far from enjoying the new German music trends, even though he was a very musical man. Coming out of a concert where Wagner was played, he remarked that he had understood everything (probably meaning the harmonic structure of the thing) but that he found it quite boring. 12 Three different versions (1871, 1879, 1894), leaving aside the first, not quite successful attempt (around 1860). 13 The algebraic integers are numbers (which can be real or complex numbers) that constitute solutions to algebraic equations of the form: x n + a n 1 x n a 1 x + a 0, whose coefficients a 0, a n 1 are integers.

11 Even though Dedekind was not inclined to long theoretical digressions, his aphoristic remarks reflect crisp ideas that are well worth detailed consideration. He seems to have written them with the same care that one must devote to the formulation of a mathematical theorem (or to a poem). Thus for instance, when he emphasizes: arithmetic must be developed out of itself, autonomously (without any recourse to foreign notions like that of measurable quantity). The topics of autonomy and purity of method were all-important to him. In his view, arithmetic and all of pure mathematics was a pure product of the mind, perfectly independent of external conditions or features of the physical world. Thus, measuring is foreign to arithmetic, and has to be seen as a mere application of the science of numbers. Another foreign notion was that of form or formula, which had become the very keystone of mathematical knowledge during the 18 th century. Arithmetic was to be the study of relations between number-objects, and those had to be captured in a novel way, directly, leaving behind the inherited thicket of formulae, polynomials, and the like. Dedekind s strenuous efforts to purify the methods of arithmetic, algebra and analysis, to eliminate the foreign elements and rebuild the whole thing from scratch, were a key to the extremely novel concepts and methods he applied. However, in his mind none of that was forced or unnatural. On the contrary, traditional ways of doing were unnatural since they gave too much room to familiar methods familiar only because they were taught generation after generation, because they were customary, but not due to any special accord with the nature of the topic itself. The comparison between this trait and contemporary events in painting and other arts is so obvious, that it suffices to mention it. Dedekind saw himself returning to the nature of things, to a more natural and direct stance, making an enormous effort to be faithful to this naïve standpoint (as he called it himself sometimes). Such was, for him, the deeper sense of his constant use of set-theoretical means. Cézanne fought to unify the new discoveries and novel methods of impressionism with his longing for the classical sense of harmony, solidity and equilibrium. Here, too, one could find a parallel, as Dedekind was led by the classical ideals of architectonic simplicity, harmony, and solid construction behind the paradigmatic works of Euclid and Gauss. He unified them with the radically novel methods of set theory, creating architectures that he

12 regarded as perfectly constructed in all of their parts, unshakable, (quote letter to Keferstein). One can argue that the very keystone to modern art was the conscious decision to regard questions about the represented or the real as foreign to painting, concentrating on art in and of itself, on the peculiar processes and means of the art. Dedekind did not go so far as a mathematician (again like Cézanne!), but he took a central part in the 19 th century purification of mathematics, its growing separation from mathematical physics and other so-called applications, which were never his main concern. Another aphorism of Dedekind has to do with free creation. His saying is famous: to the question, what are numbers?, he replies: numbers are free creations of the human mind (1888). (Already in 1872 he explained that the concept of a continuum does not come from the external world, but rather is developed by the mind and imposed on our representations of the external; and in this connection he spoke of the creation of points to make space continuous in thought. ) The freedom that Dedekind allowed himself was justified, in an interesting letter to his friend H. Weber, with a sober reflection on the technological marvels of the industrial world: we certainly possess creative abilities, not only in material affairs (railroads, telegraphs), but very especially in matters of the mind. 14 It was put to work very clearly in the novel concepts and methods employed by Dedekind, and it was emphasized by Dedekind s terminological choices in a way that deserves special emphasis. Dedekind loved literature and liked to read daily (often in the company of his sister Julie, who was a minor but successful writer herself). He also wrote very elegantly and clearly, and he allowed himself unparalleled freedom in choosing mathematical language. Having 14 I should add that this text is introduced by an appeal to the Bible: We are of divine lineage, and we certainly possess. This is almost the only explicit reference to established religion in Dedekind s writings or letters, but it may be the case that he was a believer (although that remains another mystery). One reason to think so is that his sister Julie was deeply religious: Schon 1850 hatte Julie aus tiefer Religiosität heraus mit ersten Schritten auf dem Gebiet der Wohlfahrtspflege begonnen. Sie unterrichtete in der elterlichen Wohnung junge Mädchen (Bibelstunden und Handarbeit), um sie vor Bettelei und Müßiggang zu bewahren. (Gerke & Harborth, But of course this is far from conclusive. Actually, it may find its counterpart in the fact that Dedekind transformed Gauss s motto: God is always doing arithmetic, to write: man is constantly doing arithmetic (1888). At any rate, and in stark contrast to Cantor, it is clear that religious issues were totally foreign to his views on mathematics.

13 thought about it for some time, I am unable to find another mathematician who was so colourful in his choice of terms. It suffices here to remind you of the fields or bodies [Körper] that Dedekind regarded as the core object of algebra, and the ideals [Ideale] that were at the center of his algebraic number theory. In both cases, the referent is an infinite sets of numbers, and there was a deep methodological point behind the terminological freedom. Dedekind imposed on himself the task of analysing the mathematical concepts employed so that all of their characteristic features would be made explicit. This is why he is one of the most important originators of modern axiomatics, and for that reason, a well constructed edifice must be such that all technical terms can be replaced by expressions without a meaning (quote to Lipschitz). The pictorial language employed by Dedekind served to reinforce that point, making it clear that all depended on the logical interrelations, and nothing on connotations of the terms employed. (Incidentally, that belongs to the background of Hilbert s famous claim that in geometry one should be able to speak of tables, chairs, and beer mugs. ) The discovery of the antinomies of set theory, between 1897 and 1903, was a hard blow to Dedekind s logico-philosophical convictions. For all of his life, at least since 1858, he had been convinced that pure mathematics was an outgrowth of the laws of thought; this included the basic principles of set theory, which for him was just a part of logic. When Cantor presented him with arguments establishing the paradoxes of set theory, he felt it as a death blow. However, in 1911 he expressed his confidence in the inner harmony of our logic, hoping that a stern investigation of the creative power of the mind... will certainly lead to laying the foundations of my work in an irreproachable manner. This time, the creative power of the mind was identified with our ability to form out of given elements a set, a thing that is necessarily distinct from those elements. Lastly, it is intriguing that one of Dedekind s main innovations was the theory of Abbildungen, representations or mappings. Abbildung is the relation between an object or a set, the original, and another object or set that represents it, the image (these are his own technical terms). And in a letter of 1890, Dedekind explains that the Abbildung is the painter who paints. The novel ingredient here was to thematize in an explicit form what

14 previous mathematicians had merely used and relied on, to concentrate one s attention on an element that is indispensable for mathematics (even for thinking in general, Dedekind remarked) and to explore the very novel possibilities opened by its explicit use. A clear and deep example is the very novel presentation of an algebraic topic like Galois theory, by means of groups of automorphisms (the automorphism being an Abbildung of a field or body into itself). 3. Cantor: new forms and the divine. Georg Cantor is celebrated as the creator of a whole new line of mathematical research with epoch-making contributions such as his introduction and pioneering study of the transfinite numbers, and the very formulation of the continuum problem (no. 1 in Hilbert s famous list of 1900). Although his unique position in mathematical history has been exaggerated, there is no denying the very idiosyncratic and highly original nature of his contributions. And of course there are many elements in his mathematics that are clear examples of modernity. As a matter of fact, Cantor was and remains a controversial figure: celebrated by key moderns like Hausdorff and Hilbert, his work has been regarded with dislike by many others (including modernist philosophers), due I surmise to the reactionary or backward-looking elements in the views of this visionary thinker. In my opinion, these ambivalent reactions are natural, since Cantor s work is marked by ambivalent traits, a mixture of modernist and anti-modern elements. You recall, of course, that the Garden from which Hilbert fought to avoid expulsion was the Cantorian paradise, a paradise of freedom and fruitfulness provided by the powerful settheoretic methods and ways of forming notions. Hilbert chose Cantor as his paradigmatic founding figure because Cantor himself had had to fight for the Garden. Criticised severely by a powerful mathematician, Leopold Kronecker, who was in a position to create opinion and (not irrelevantly) to establish university chairs, Cantor made in 1883 a profound and deeply-felt plea for free mathematics. He emphasized that the mathematician is entirely free from considerations of the empirical or the metaphysical, enjoying a conceptual freedom that is tempered only by logical consistency and the fruitfulness of the ideas themselves. As he said emphatically:

15 the essence of mathematics lies precisely in its freedom. Kronecker was motivated by (rather traditional) considerations about the way in which arithmetic or geometry respond to features of the real world, but Cantor emphasized its independence and theoretical autonomy. That was clearly a move in the modern direction of the autonomy and self-containedness of mathematics. Surprisingly, in order to justify that viewpoint Cantor felt the need to introduce heavily metaphysical considerations (1883, ). He spoke about the absolutely realistic, but at the same time no less idealistic foundations of his philosophico-scientific views, and about the unity of the All, to which we ourselves belong, in order to conclude that the conceptual consistency and coherence (the immanent reality ) of our mathematical theories ensures that they are also realized in some respects, and even in infinitely many respects, in the outer world. It was on this basis that he derived the conclusion that mathematics in its development only has to consider the immanent reality of its concepts, and is totally free from ontological considerations. That was the anti-modern standpoint from which he proposed to rename pure mathematics, calling it free mathematics. All of this was published in the Mathematische Annalen, to the surprise of his fellow mathematicians, in the context of what is arguably Cantor s most original and revolutionary paper, the Grundlagen [Foundations for a general theory of sets], featuring the introduction of the transfinite numbers. Some of his colleagues, not least his former close friend H. A. Schwarz, could not help to mock: remember that this happened in 1883, in an intellectual atmosphere of positivism. But indeed, Cantor s great mathematical innovations were done in the service of natural science, metaphysics, and theology. His ultimate goal was to promote the harmony between faith and knowledge (as he said in 1886). A short technical aside. The transfinite numbers established in 1883 were the ordinals ω, ω+1, ω 2,, ω ω,, which allowed Cantor to introduce crucial refinements in his theory of infinite sets. They represent certain types of orderings of infinite sets, hence the name ordinals. In what follows, however, when there is talk about the different modes or gradations of the infinite, the reader may simply think of the better known alephs:

16 ℵ 0, ℵ 1, ℵ 2,, ℵ ω, ℵ ω+1, The alephs represent the different sizes of infinite sets, and Cantor s first great discovery in 1874 was that the set of real numbers is of a bigger size than the infinity of the natural numbers. Hence the alephs are also called the transfinite cardinals. They come ordered in a simple (but absolutely infinite ) ascending sequence, so that ℵ 1 is the next bigger size beyond the infinity ℵ 0 of the natural numbers, ℵ 2 is the next bigger than ℵ 1,, ℵ ω is immediately greater than all the alephs of a finite subindex. And so on, ad infinitum et ad maiore Dei gloriam. Cantor s great goal of promoting the harmony between faith and knowledge called for new mathematical ideas on which to base an organicist explanation of Nature. This is actually a typical theme of German romantic thought in the 19 th century, which is found under different forms in the idealistic philosophers and also in late romantic, post-idealistic trends. The organicistic approach would, for the first time, allow science to be fair to the world of living beings and to the human mind. As I understand the matter, the topological aspects of set theory that were at the centre of Cantor s attention from 1872 to at least 1884, were most closely associated with his hopes for a new form of natural science. The link between these topological ideas and intended applications to the biological world are relatively easy to follow. In doing so, the organicistic approach would complement or even replace the cold and reductionistic ideas of the mechanical explanation of Nature. Again literal words from the epoch-making Grundlagen! The reader should recall that the positivism of scientists in this era was only superseded by their strong adherence to mechanicism. 15 Set theory and the organicistic approach based on it would help overcome the sceptical philosophies of empiricism, materialism, and Kantianism (sic in Cantor s writings), finally securing a fluid interaction and coherence between science and religion. As Cantor presented the matter, the great heroes whose work he was following and recovering from 15 For further details, see my (2004), where I studied what I ve called the extra-mathematical motivations behind his contributions to set theory. I argue in detail that one cannot explain the orientation of Cantor s research by considering only the problem-situation in mathematics at his time.

17 criticism were none other than Plato, Spinoza and Leibniz (Cantor 1883). Set theory was called to solve the riddles that blocked the satisfactory development of their metaphysical ideas into full-grown scientific theories. The link was, in his mind, totally intimate. Giving his very first definition of a set, Cantor employed language reminiscent of the Greeks a set or manifold is every Many that can be thought of as a One and with this he believed to have defined something akin to the Platonic eidos or idea. Cantor often presented himself as nothing more than a faithful scribe, an interpreter or mere transmitter of the revelations opened to him by Deus sive Natura (to use the memorable formula of his beloved Spinoza). His very last paper begins with quotations from two of his admired figures, Newton s hypotheses non fingo and Francis Bacon s Latin text: For we do not dictate rules to the intellect or the things in accordance with to our will, but as faithful scribes we receive and copy them from the revealed voice of Nature herself. 16 Incidentally, in order to understand that peculiar way of presenting himself as a scribe, one should take into account that Cantor suffered a manic-depressive illness, and it is relatively natural to interpret the bouts of such an illness as episodes during which one s self is taken control from above, so to say. A related theme is the important role played by theological ideas. One cannot understand the actual development of Cantor s views without this ingredient: from the beginning he conceived of the transfinite as intermediate between the finite and the Absolute or God; and this scheme, together with his strong metaphysical convictions, was crucial to his positive reaction to the discovery of the paradoxes in the late 1890s. Similarly, the metaphysical and theological beliefs eliminated for Cantor, from the very beginning, doubts concerning the meaningfulness and existence of different transfinite cardinalities (correspondingly, of the so-called higher number classes) that were inescapable for authors of more sceptical persuasion. Mathematics cannot be founded [ist nicht zu begründen] without a bit of metaphysics, he once wrote. He was fully convinced that all of the different modes of the transfinite exist in concreto, in Nature, and he regarded those same modes as an 16 Cantor, Abhandlungen, p My translation.

18 ascending ladder, leading as it were towards the throne of God. All of these themes are present in a very relevant note of the Grundlagen (endnote no. 2): I have established in their concept, once and for all, the different gradations of the actual infinite and I consider it as my task not only the mathematical investigation of the relations between the suprafinite numbers, but also to hunt them and reveal them wherever they occur in Nature. There is for me no doubt that, following this way, we shall reach always further, without ever finding an insuperable limit, but also without obtaining a grasp (however approximate) of the Absolute. The Absolute can only be acknowledged, but never be known, not even approximately known. to each suprafinite number, however great it may be, there follows a collection of numbers and number classes that is not in the least reduced It happens with this something similar to what Albrecht von Haller said of eternity: I withdraw it [the immense number] and you [eternity] remain entire in front of me. Therefore the absolutely infinite succession of numbers seems to me, in a sense, an adequate symbol of the Absolute As one can see, Cantor was greatly motivated by his diverse interests in philosophy, science, and religion. He was also anxious to make his novel ideas known, because he was convinced they would contribute to an epochal shift in worldviews. In fact one cannot make historical sense of Cantor s work without considering questions like the above, which show up clearly in his writings, but have been ignored by historians of mathematics due to an effect that one might call disciplinary blindness. At the same time, however, Cantor s researches incorporated and developed further many of the crucial traits of the modern mathematical methodology that was then being developed by a small group of pioneers, and which as a result were beginning to be criticised. In previous work (1999, chap. 1) I have traced back this methodology to what I called the Göttingen group formed by Dirichlet and above all Riemann and Dedekind. There I also review the most important of Cantor s contributions, which to give a most brief summary helped transform mathematical analysis, turned set theory into an autonomous mathematical discipline, and helped launch a prototypically modern branch of mathematics, topology. Now, the criticisms voiced by Berlin mathematicians, in particular Kronecker, were instrumental in convincing Cantor of the need to justify the new methods. He perceived this as a necessary propaedeutics for the radical move of introducing the transfinite numbers, and it was for this reason that the Grundlagen became a mathematicophilosophical treatise, showing the remarkable traits we have been indicating. As one can see, the new and the old, modern mathematical methods and traditional metaphysical ingredients, can go together in the peculiar views of an original thinker.

19 Because of the way in which Cantor combined romanticism with mathematics, modern methods with a vindication of rationalist metaphysics and theology, recent scientific trends with Platonism and an emphasis on the soul, it is not far fetched to label his orientation a reactionary modernism. Paraphrasing Thomas Mann, one might say that it was a highly mathematical romanticism. 17 Such mixtures of the old and new, in fact, are by no means unheard of among avant-garde artists at the time: a well known example would be Kandinsky. But since I find interesting analogies between Cantor s work and Art Nouveau, I am naturally led to a comparison with Catalan architect Antoni Gaudí ( ). For Gaudí too can be regarded as a reactionary modernist. Some year ago, while visiting the École de Nancy Museum during a pause in an interesting congress, I realized that there are interesting parallels between Cantor s motivations and those of Art Nouveau. So now my concept of modernism is most concrete, having to do with that new trend in architecture, arts and crafts during the period Art Nouveau reacts to industrial life, urban styles, and the new technologies; it rejects the neat geometrical shapes of the past, looking for inspiration in the intricate forms of Nature, in particular the living beings. Cantor too looked for inspiration in Nature, trying to capture the forms of living beings, and thus he opened the way to extremely intricate forms and shapes with his ideas in point-set topology (dense sets, nowhere-dense sets, isolated sets, perfect sets, Cantor sets) 18. These links can be followed in surprising detail, for instance when Cantor employed a theorem proved in connection with isolated sets, to argue that the number of cells in the 17 Mann spoke of a highly technological romanticism in connection with nazi modernism, and J. Herf quotes it approvingly (1984, 2). The concept of reactionary modernism was coined by Herf in his important study Reactionary Modernism: Technology, culture, and politics in Weimar and the Third Reich (Cambridge UP, 1984), to describe a German right-wing trend in the 1920s and 30s, which managed to combine nationalism and romanticism with technological modernisation. This combination of Kultur and technology is quite alien to German trends in the 19 th century, but it seems to me natural to use the label reactionary modernism in a broader way, in particular for the combination of romanticism, traditionalism, and advanced scientific knowledge represented by Cantor. 18 A beautiful example is the famous ternary set of Cantor. It is the collection of all real numbers given by the formula c1 c c! z = L + + L where the coefficients c ν can only take the values 0 or 2 (one is working 2! with the real numbers in ternary representation). This is a perfect set, i.e., it contains all of its points of accumulation, but it is nowhere dense, and yet it has the power of the continuum.

20 Universe is none other than ℵ 0 (see Ferreirós 2004). He proposed to refine the then-usual physical views on the ether and atoms, with the set-theoretic hypothesis that the number of atomic particles is ℵ 0 and the number of ether particles is ℵ 1. I have also suggested that the perceived links between point-sets and the forms of micro-organisms revealed by 19 th century microscopy, offers the best explanation for the definitions of continua and semicontinua that Cantor offered (again in the 1883 Grundlagen). Interestingly, here too one can find a link very indirect, to be sure with Art Nouveau: the German biologist and speculative thinker Ernst Haeckel produced a famous work in which the biological shapes were presented as art-forms, very much in Art Nouveau style (Kunstformen der Natur, ). Below we shall find again the name of Haeckel, famous for his fierce evolutionism and materialism, which he transformed into the metaphysics of Monism. While many of their contemporaries were simply scared and frightened by the modern technologies, Art Nouveau artists used the new materials and technologies creatively, establishing a new style and approach to problems like those of building (architecture). Similarly, Cantor s displeasure with materialism and mechanicism did not lead him simply to reject the scientific outlook and mathematics, but rather to transform them creatively. Set theory provided scientists in general with new conceptual materials and procedures, with which they could recreate mathematics and science in the pursue of some very novel problems. Coming back to the link between Cantor and Gaudí, I would like to add one remark. Cantor believed that his new set-theoretic concepts merely reproduced what exists out there, in Nature, and his strenuous scientific efforts were ultimately in the service of God. For both reasons, Gaudí s astonishing temple, the Sacred Family in Barcelona with the paradox of its modernist motives inspired in natural forms, ultimately to form a modern cathedral, its stylised figure rising high in an attempt as it were to reach the heaven could serve as a fitting symbol for the unbounded ascending ladder, the stairway to God s throne that the transfinite numbers formed, according to Cantor s vision. (Fitting symbol, that is, within the severe limitations of the architectonically feasible, which seems like nothing compared to the amazing field of play of mathematical conception.)