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 19th 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 1900. 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)

To remain in the “belle époque,” 1890-1914, 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 counter-

moderns 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 anti-

modernism (Antimoderne). Only the anti-moderns are depicted as reactionary

traditionalists, while both the moderns and the counter-moderns are regarded as opposite

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 19th 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.

The first third of the 20th 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 socio-

cultural 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

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.)

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, 375–76)

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 self-

containedness 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

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).

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.

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 1888. 9 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).

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 (1839–1906), 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 1897.

11 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: xn + an–1xn–1 + … + a1x + a0, whose coefficients a0, …an–1 are integers.

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 18th 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

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 19th 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, http://www.studsem-bs.de/2/ausbild/mathe/html/history/dede.htm) 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.

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

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 set-

theoretic 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:

“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, 181-182). 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, … ω2, …, ωω, …, 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:

ℵ0, ℵ1, ℵ2, …, ℵω, ℵω+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 ℵ1 is the next bigger size

beyond the infinity ℵ0 of the natural numbers, ℵ2 is the next bigger than ℵ1, …, ℵω 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 19th 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.

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. 282. My translation.

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 mathematico-

philosophical 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.

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 (1852–1926). 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 1890–1914. 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 19th 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 LL ++++=

!

!

3332

21ccc

z where the coefficients cν can only take the values 0 or 2 (one is working

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.

Universe is none other than ℵ0 (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 ℵ0 and the number of ether particles is ℵ1. I have also suggested that the

perceived links between point-sets and the forms of micro-organisms revealed by 19th

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,

1899–1904). 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.)

The traditionalist modernism of Cantor can be illuminated by a comparison with the anti-

traditionalist modernism of one of his fellow set theorists, Felix Hausdorff. Both Cantor

and Hausdorff were deeply interested in philosophy, an interest that is not separable from

their respective pathways into set theory. Hausdorff published several “literary” works

under a pseudonym around 1900, among them a radical philosophical work, The Chaos in

Cosmic Interpretation (Das Chaos in kosmischer Auslese, 1898), which was inspired by the

philosophy of Nietzsche and contained interesting ideas concerning the theory of

knowledge. According to the experts of the Hausdorff Edition, this work, and in particular

Hausdorff’s discussions of the concepts of time and space, was what put him in contact

with Cantorian set theory (again, the links between Cantor’s new ideas and natural science

become crucial). Subsequently he became one of the leaders within the next generation of

set theorists.

Hausdorff knew Cantor personally; both could interact easily as they lived in Leipzig and

Halle, respectively; he dedicated to him his important textbook Grundzüge der

Mengenlehre (1914). Nevertheless, Hausdorff avoided discussing his philosophical ideas

with the enthusiastic and opinionated man from Halle, because he knew too well of their

incompatibility. Cantor’s philosophy was marked from beginning to end by his belief in

God, his referents were Plato, Spinoza and Leibniz, Catholic theologians were among his

correspondents – and Nietzsche was among his bêtes noires. In 1900, Cantor congratulated

his colleague Friedrich Loofs, professor of Church history at Halle, for a new book entitled

Anti-Haeckel:

I think it is very valuable that from now on the impudent appearance of a scientific character will be taken away, in front of the widest circles, from Haeckel’s shameless attacks against Christianity. … I have only recently found an occasion to form a more precise image of the so-called Nietzschean philosophy and its dependence on Haeckel’s monistic evolutionary philosophy. It finds among us an uncritical acceptance because of its stylistic appeal, which seems highly questionable to me considering the perverse contents and the Herostratic-antichristian motives.19

19 To Loofs, 24 Feb. 1900 (photographically reproduced in Meschkowski 1983, 292–93).

4. Brouwer: the ambivalence of modernism.

L. Egbertus J. Brouwer (1881–1966) belonged to a later generation which falls squarely

within the high time of modernism: he was 36 years younger than Cantor, 19 younger than

Hilbert. And he was one the best mathematicians in his time, with very strong philosophical

interests, and an eccentric lifestyle. His high achievements in the field of topology made

him internationally famous and brought him offers of chairs at Berlin and Göttingen in

1919 – and remember, Göttingen was still the world centre for mathematics. The Berlin

professors stressed that Brouwer had provided “the firm foundation” for topology, “long

sought in vain” (van Dalen 1999, 300), and they opined that he “is equalled in the

originality of his methods by none of the mathematicians of the younger generation”.

Hilbert too seems to have regarded Brouwer as the most brilliant among his generation, and

their relations were still very good. But there was an enormous difference between the

scientific and personal outlooks of both men. Hilbert’s talks of the 1900s (especially his

famous conference of 1900 on mathematical problems) are examples of “belle époque”

optimism, transmitting the message that the future is ours, full of freedom, and the received

concepts and methods are basically all right, perhaps requiring minor corrections. But the

young Brouwer’s philosophical works are as pessimistic as Spengler, and his dissertation of

1907 is a call for rethinking and reforming mathematics deeply. He came to regard his ideas

of this period as the “first act” of intuitionism, with a “second act” after the Great War,

when he devoted himself fully to an even deeper reform of pure mathematics, that would

poison his relations with Hilbert. The big fight between both in the 1920s has stamped

these differences so much, that it seems almost impossible for most mathematicians to

regard Brouwer as “modern.”

Mathematics was not the only major theme in Brouwer’s life. According to his main

biographer, the great themes throughout his intellectual life were mysticism, topology,

intuitionism, and the philosophy of language. The philosophico-religious themes were

connected to the intuitionistic foundations he came to propose. From very early age,

Brouwer was an individualist to the point of defending ontological solipsism, conceiving of

the world as essentially a spiritual unity, and tending to romantic, mystic conceptions of the

unity of everything. He felt deeply distrustful of established social views, bourgeois life and

the like, but also of socialist proposals for amendment. These ideas were expressed in his

1905 monograph Life, Art and Mysticism. In the early days, Brouwer had strong qualms

about becoming a mathematician, as he was deeply engaged with philosophy and mystical

ideas. The thought of becoming an expert seemed to him a temptation of the will, a way of

falling into attitudes that would separate him from the true perspective offered by

mysticism. During the 1900s he was strongly critical of science, technology, and modern

civilisation; and much of this was retained throughout his life. Causal thinking, language,

science, and technology were viewed as negative forces; an admirer regarded this work as

the most formidable accusation against “our ‘civilisation’.”

The question of language is also particularly relevant, for Brouwer would continue to

emphasize that mathematics does not depend on language or logic, being prior to language

and logic. Language is merely an instrument of social domination, and it makes impossible

a real communication: “nobody has ever communicated his soul to someone else by means

of language.” In 1908, this line of thought would derive into a denounce of classical logic

and of axiomatic systems, which obviously from his standpoint cannot be the real

foundation of mathematics.

I shall begin with some remarks about Brouwer’s foundational ideas and his views on set

theory. In 1919, precisely the year when he received calls to the Universities of Berlin and

Göttingen, Brouwer published in the Jahresbericht of the German Association of

Mathematicians (DMV) a paper entitled ‘Intuitionistische Mengenlehre.’ The year before he

had started a series of papers on this same topic (in German, but published in the

transactions of the Dutch Academy of Sciences), with the goal of revising set theory

thoroughly and developing it “independently of the logical principle of excluded middle.”

This was not a minor reform, it was actually one of the crucial traits of intuitionism, that

make it strongly deviant from “modern” (also called “classical”) mathematics. After all, the

application of excluded middle to infinite sets is one of the crucial traits of the modern

mathematical methodology, and one that the constructivists came to reject. In his paper for

the DMV Brouwer stressed that he had been elaborating these ideas since 1907, before his

involvement with topology, and mentioned how (in his opinion) they also forced him to

disagree with Hilbert’s conviction that all mathematical problems are solvable. He

emphasized that the foundations for set theory provided both by the logicists and Zermelo

were to be rejected, and that a new form of “constructive set theory” was required.

His proposals were the most radical in this age of radicalism: the set theory of Cantor and

Dedekind was to be completely revised or overturned, the classical ideas about the real

numbers had to be abandoned, and with them classical analysis, replaced by a very novel

intuitionistic analysis. The world of mathematics was to be constructed anew, from scratch,

in such a way that the meaningless features of traditional approaches would be erased, and

a new pure edifice would be erected, full of sense. Applause came from unexpected sides,

especially when Hilbert’s most brilliant student, now a fully mature mathematician among

the very best in the time, saluted him with the famous words: “and Brouwer – that is the

Revolution” (Weyl 1921).

In his dissertation of 1907, Brouwer had actually explained how he could accept some of

Cantor’s ideas, including his transfinite numbers ω, ω+1, … up to a certain point (as long

as they are denumerable and in a certain sense constructible) but not the further concepts of

a totality of all such denumerable numbers. Cantor introduced this totality in the 1883

Grundlagen, and called it the “second number class;” thesis 13 of Brouwer’s dissertation

states: “The second number class of Cantor does not exist” (van Dalen 1999, 113). And it

was not the set-theoretic paradoxes that caused his reaction. As he remarked in 1923,

an incorrect theory, even if it cannot be checked by any contradiction that would refute it, is none the less incorrect, just as a criminal policy is none the less criminal even if it cannot be checked by any court that would curb it. (Hesseling 2003, 62)

The point for the intuitionists is that mathematics is a mental construction erected freely by

the mind. It is simply an illusion to conceive of mathematics as dealing with independently

existing objects, with an objective reality somehow external to the mind. But this is what

modern mathematics does: the objects of the theory are conceived as elements of a totality

or set that is regarded as given, totally independently of the thinking subject. This feature is

deeply embedded in the methods employed in mathematics, and (following Bernays, a key

collaborator of Hilbert) it is often called the “Platonism” of modern mathematics.

Meanwhile, the constructivists’ treatment of mathematics – exemplified by intuitionism – is

based on careful consideration of the processes by which numbers, etc., are defined or

constructed. Each and every thing that a mathematician can legitimately talk about must

have been explicitly constructed in a mental activity.

As time went by, Brouwer realized that it was better to avoid talking of “sets” at all, and he

introduced new terminology (“species” and “spreads”). But the core of his intuitionism, the

novel ideas that he was developing and making public among other places in the

Mathematische Annalen (edited by Hilbert and others), are a direct continuation of the

views on “intuitionistic set theory” proposed in 1919. On their basis, Brouwer submitted

mathematical analysis to a deep revision: the concept of the real numbers was affected, and

the usual development of analysis had to be abandoned. (A technical point that is relatively

easy to grasp is the following: the basic theorem that an infinite, bounded set of real

numbers always has a least upper bound, has to be abandoned.20) As Brouwer’s

reconstruction of mathematics developed in the 1920s, it became more and more clear that

intuitionistic analysis was extremely subtle, complicated and foreign. Brouwer was not

worried, for “the spheres of truth are less transparent than those of illusion,” as he remarked

in 1933. But Hermann Weyl, who saluted Brouwer as “the Revolution” in 1921, even

though convinced that he had delineated the domain of mathematical intuition in a

completely satisfactory way, would remark:

the mathematician watches with pain the largest part of his towering theories dissolve into mist before his eyes (Weyl 1925, 534).21

As for Brouwer the man, a close friend of his, the Dutch poet and socialist thinker Adama

van Scheltema, regarded him as

20 This is a feature not only of intuitionism, but more generally of constructivism; the development of analysis is severely affected by having to circumvent that simplifying principle. 21 Soon Weyl abandoned intuitionism, although he remained a constructivist. See Mancosu 1998.

in all respects the paradigm of a man of exceptional genius – for being a genius he lacks the connection between his own mind and the world around him (van Dalen 1999, 26).22

That seems to have been said with the romantic, 19th century idea of genius in mind: a

stormy personality, a soul capable of rising to heaven and reaching the depths of misery.

Brouwer could be a high-nosed man, extremely demanding with himself and others,

scornful towards most of his fellows in mankind, fully imbued of an elitist conception. In

his youth he thought of himself as a “king,” one of the best and noblest, who worked for the

redemption of things and of their fellow men (van Dalen 1999, 37-38), i.e., to “put

somewhat in the right position” the “grand totality” (op. cit., 34). Unsurprisingly, he soon

abandoned the socialist convictions of Scheltema: “I rarely think about politics, but my

political sympathies are in the liberal, anti-democratic direction,” he wrote in 1909 (van

Dalen 1999, 35).

Our ‘genius’ was a high-strung and nervous person, described as uncompromising, and

indeed he could act as a “justice fanatic” (in the words of Bieberbach, later notorious as the

most important nazi mathematician). His 1905 monograph Life, Art and Mysticism displays

chapter titles as expressive as ‘The sad world’ and ‘Man’s downfall caused by the intellect.’

He criticized severely the so-called outer world, the intellect (related with means-end

rationality, and contrasted with the soul or ego) for being the source of evil, and also social

activities; he called for a return to nature and for placing confidence in the self and its

will.23 Intuitive introspection and mystical views were given preference, with the names of

Meister Eckehart, Jakob Böhme and the Bhagavad Gîta emerging as important references.

And the book ends with a Schopenhauerian exaltation of nihilism.

That nihilism was a natural consequence of the philosophical solipsism and the mystic

inclinations of Brouwer. As early as 1898, being only a boy of 17, he wrote: “the only truth

is my own ego of this moment, surrounded by a wealth of representations in which the ego

believes, and that makes it live” (van Dalen 1999, 18). Recall that, according to Daniel Bell,

modernist spirits had a definite tendency to the “megalomania of self-infinitization;” one

22 That’s something Scheltema sought to remedy in their student years, endeavouring “tirelessly … to make him come closer to the material world.” 23 Hesseling 2003, 30–34. Van Stigt 1996.

may read from this viewpoint the words “you feel almighty,” that Brouwer wrote in his

1905 booklet. Consequently with this worldview, he believed in a God and devoted himself

to the tasks imposed by God.24 But he also managed to find ways to reconcile his deep

beliefs with the ways of life.

A life, by the way, that was unconventional, far from the established manners of the

bourgeois. It was centred on a hut in Blaricum, a village not too far from Amsterdam, from

which Brouwer could pay regular (but scarce) visits to the University. There he enjoyed life

among the artists, the vegetarians, and the health seekers.25 He led a most Spartan lifestyle,

with peculiar diets and habits (62ff), and he became notorious for his sexual freedom, never

hidden from his wife. Surprising facts in the life of someone who was convinced – at least

in his passionate youth, as expressed in Life, Art and Mysticism – that “the illusion of

woman” burdens your soul’s karma, and that the ultimate goal should be to sacrifice

everything, neglect everything, in order to reach the highest perfection: “the world of

freedom, of painless contemplation, of – nothing.”26

Brouwer could also be extremely perceptive, e.g. in his denunciation of the “human cancer”

with its will for power and domination, and its way of turning the beautiful world into

barren land (van Dalen 1999, 68). This happened at a time when it was not easy to figure

out the destructive power of mankind, although it happened in the Netherlands, a man-

made land. Science and technology were to him clear expressions of such a will, and so is

mathematics most often. But the pure cultivation of mathematics can become independent

from all that, it can be a pure activity worthy of the human mind. It was in this way that he

reconciled himself with the idea of exploiting his great mathematical talents. But it was

quite a fight to try accommodating the philosophical insights with his worldly passions and

24 I say consequently because it is precisely the “utter powerlessness” experienced by the ego at times, that reveals all around him the presence of a “higher power.” 25 As he described it himself around the turn of the century: people of both sexes in vest with bare black feet and blue nails, the sunbathing of bare backs, the gnawing of raw turnips and carrots (see van Dalen 1999, 28). 26 Or maybe not so surprising. The solipsist who regarded most men as unpleasant gnomes that showed up in the Garden of his self, is likely to have regarded many women as pleasant appearances that deserved to be cultivated. And as regards his wife, Life, Art and Mysticism makes clear the expectations of submission that Brouwer had at the time of choosing her (see van Dalen 1999, 73).

temptations. And in these trials of the soul, the options were romantically presented as a

matter of all or nothing:

as one cannot stop the growth of one’s beard, so one cannot stop the growth of the philistine tissue through one’s soul. Then let me be great as a philistine! … And so leave my trace on the thus melancholy world. That is, Ambition is born within me, perhaps! … I will have to remain obscure for a few more years, then my grasp will be felt. Just because I feel the futility of all worldly things, no detraction or fear will disturb my course. (letter to Scheltema, Jan 1904; in van Dalen 1999, 35-36).

And so be it: his grasp was felt, especially after the Great War, and his life was full of

fights and quarrels. “It still strikes me as curious that a person can get involved in so many

disputes,” writes van Dalen (1999, ix).

Brouwer’s reconciliation with mathematics as a pure activity worthy of the human mind

depended on the proviso that it be kept apart from its “applications,” even those in pure

science. The tendency to purification was strong in the 19th century, although not exactly

for Brouwer’s reasons.27 It started in northern Germany, and with the great success of the

German mathematicians it spread out all over Europe and into the United States, leaving its

mark well into the 20th century. Brouwer was not the only modernist who was interested in

mathematics precisely because of its purity and autonomy from worldly concerns. Another

important example is Robert Musil, who had been an engineer and admired mathematics

greatly:

It is understandable that an engineer is preoccupied by his specialty instead of coming into the freedom and space of the world of ideas, even if his machines are delivered to the furthest corners of the world; because he does not need to be able to apply the daring and novel aspects of the soul of his technique to his personal soul … About mathematics one cannot say this; it is the new method itself, the spirit itself, in it lay the sources of time and the origin of an immense transformation. (Der Mann ohne Eigenschaften, 1930)

By contrast, an important part of the modernization brought about by Klein and Hilbert in

Göttingen consisted precisely in moderating the purifying tendencies of German

mathematics, countering them with the creation of strong ties between the mathematicians,

physicists, and engineers.28 Little wonder, then, that in 1919 Brouwer manifested a

27 See e.g. Goldstein, Gray & Ritter (1996), Bottazzini & Dalmedico (2001). 28 Witnesses abound: the Göttingen Association for the Promotion of Applied Physics (1898), created by Klein in association with industrialists; his efforts to hire Ludwig Prandtl, who became head of the Institute for Technical Physics in 1905 and did pioneering work on aerodynamics; the professorship of applied mathematics created in 1904 for Carl Runge; the important contributions that Hilbert and Minkowski made to physics, their deep involvement with the subject in the 1900s and 10s; the close links with the physicists, which Max Born for instance describes. See Rowe (1985).

preference for Berlin instead of the leading center, Göttingen: his whole worldview was

antagonistic to the Göttingen ways, but much more in line with the purism and idealism

represented by Berlin.

All of those are, of course, reasons why people who identify modernism with

modernisation have no doubts: Brouwer can only be classified as a conservative, an

inheritor of 19th century idealism, a man who turned against the modern world, an anti-

modern. Thus, for instance, van Dalen writes that he “was basically a conservative” (1999,

72) and finds it hard to understand why others think of him as a revolutionary innovator.

But this seems to me an ill-informed judgement, ignoring as it does how strongly

modernism was linked with the idealistic and romantic legacy, how clearly represented are

all of Brouwer’s cultural themes among the more pessimistic and revolutionary modernists.

Mehrtens, as usual, is subtler and better informed: he has catalogued Brouwer as the

prototypical “counter-modern” (Mehrtens 1990, ).

I insist again: we must distinguish modernism from modernisation, for otherwise – to speak

now about people in the arts – we would have to reclassify many of the modernists, people

like the writer Hermann Broch, the painter Wassily Kandinsky, and so on. Kramer explains

that

the emergence of abstraction early in the second decade of this century represented for its pioneers a solution to a spiritual crisis. … The conception of this momentous artistic innovation entailed a categorical rejection of the materialism of modern life; and … abstraction was meant by its visionary inventors to play a role in redefining our relationship to the universe.29

Properly revised, those are indeed fitting words for Brouwer’s intuitionism, its origins and

its goals. Thiking about a mathematical counterpart to abstraction in painting, it seems to

me that intuitionism is clearly the best. And if you consider again Herf’s characterisation of

modernism (quoted above), it should be obvious that Brouwer fits it much better than any

of the other figures we have studied so far. Modernism was not a movement of the political

left or right, it could well be anti-democratic and elitist, it was often against the myths of

progress and modern civilisation.30 Its “central legend,” says Herf (1984, 12), was “the free

29 How that relationship was to be redefined was explored by Kandinsky not only in his paintings but in his influential treatise Concerning the Spiritual in Art (1911). 30 Notice that Herf is not defining reactionary modernism, here, but modernism in general.

creative spirit, at war with the bourgeoisie, who refuses to accept any limits” – and how

well does this fit Brouwer! (Among the three other men we have considered, only Cantor

comes close, but not so much in connection with the impulse to “reach beyond” – at least

not if this is thought to involve centrally the idea of going beyond morality, but indeed in

other directions.)

Brouwer’s outlook may seem reactionary, and yet there is nothing in his proposals that

resonates with the traditionalistic tendencies of a Cantor. It is true that Brouwer opposed

technological progress, and he also rejected socialism, but because of his fierce

individualism – is this not a modernist trait? And one should also remember the

aestheticizing tendencies of modernism, which again fit Brouwer’s life and work quite well.

His truly rooted mathematics, based on intuition, is highly deviant from modern

mathematics, but this can hardly constitute an argument for classifying him from the

cultural and intellectual point of view. The great difference between math and art, at this

point, is that the former – unlike the latter – has seen a dominant orientation during the 20th

century. But if we let this fact influence too much our cultural judgements about the past,

we shall commit an anachronism.

I should add that, if the reader is pondering the possibility of using the label “reactionary

modernism” for Brouwer, there are some reasons why this runs into trouble. Herf’s

reactionary modernists were not only cultural traditionalists, but also nationalists, and they

managed to combine some form of romanticism and cult of the soul with advanced

technology. Brouwer was clearly not a traditionalist, he was against nationalism, and

against modern technology. The point concerning the latter has already been done, and as

regards the former suffice it to say that, in the middle of the Great War, he wrote

approvingly of anti-nationalism, denouncing “words-of-power” such as ‘fatherland’ as a

“means of defense of injustice” (van Dalen 1999, 248-249). One wishes that his views on

this topic were better known in the world today. There have been attempts to align Brouwer

with reactionary political ideas, largely because he was a great admirer of German culture,

and he fought against the decision of the allied scientists to exclude Germans from

international meetings and scientific organisations, during the 1920s. But the historical

circumstances of these events were far more complex than this oversimplifying picture

suggests (see vol. II of van Dalen’s biography).

I think we have to admit Brouwer’s modernism and with it the ambivalent nature of this

cultural trend. Let me try to summarize the point in a simple and quick way, by adapting

the words of Ad Reinhardt that the organizers have brought to our attention: The key point

of Brouwer’s modern mathematics was awareness of mathematics of itself, of math

preoccupied with its own processes and means, with its own identity and distinction, math

preoccupied with its own unique statement, mathematics conscious of its own evolution

and history and destiny, toward its own freedom,31 its own dignity, its own essence, its own

reason, its own morality, and its own conscience.

In my opinion, it is not only this cultural trend that turns out to be ambivalent, but quite

generally cultural movements of the kind are rather complex and ambivalent (think of the

enlightenment, of romanticism, of post-modernism).32 Historical epochs are marked by

conflicts and tensions, because the new is not monovalent, and since the new and the old

may combine in an endless variety of ways. This complicates significantly the task of

historians who want to trace the links between cultural movements and the sciences, and

makes it very difficult to formulate them in textbook-style summary form. But it is only by

tackling that complexity, that we shall first begin to confront the difficult question of the

status of science as a form of culture.

References:

van Atten, M. 2003, On Brouwer, Belmont (CA): Wadsworth.

Bottazzini, U. & A. D. Dalmedico, Changing images of mathematics. London: Routledge,

2001.

31 Not formal freedom devoid of sense, but full of sense, imbued in sense – and at the same time completely independent from questions of nature (naturalism) or reality (realism). 32 I have made this same point for romanticism in Ferreirós (2003).

Brouwer, L. E. J. 1905, Leven, Kunst en Mystiek [Life, Art and Mysticism], Delft,

Waltman. English translation in Notre Dame Journal of Formal Logic 37 (3).

––– 1907, Over de grondslagen der wiskunde [On the foundations of mathematics],

Dissertation, Amsterdam. English translation in Collected Works, North-Holland.

––– 1919, Intuitionistische Mengenlehre, Jahresbericht der DMV 28. English translation in

Collected Works, North-Holland.

––– 1925/27, Zur Begründung der intuitionistischen Mathematik, I. II. III. Math. Annalen.

Cantor, G. 1874, Über eine Eigenschaft des Inbegriffes aller reelen algebraischen Zahlen,

Math. Annalen. In Cantor 1932.

––– 1883, Grundlagen einer allgemeinen Mannichfaltigkeitslehre, Leipzig, Teubner. Also

as an article in Math. Annalen. In Cantor 1932.

––– 1932, Abhandlungen mathematischen und philosophischen Inhalts, Berlin: Springer.

van Dalen, D. 1990, ‘The War of the Frogs and the Mice, or the Crisis of the

Mathematische Annalen’, Mathematical Intelligencer, 12(4): 17-31.

––– 1999, Mystic, Geometer, and Intuitionist: The life of L. E. J. Brouwer, Oxford Univ.

Press, vol. I: The dawning Revolution. [Vol. II, 2005: Hope and Disillusion.]

Dauben, J. 1979. Georg Cantor: His mathematics and philosophy of the infinite. Harvard

Univ. Press.

Dedekind, R. 1888, Was sind und was sollen die Zahlen?, Braunschweig, Vieweg.

––– 1872, Stetigkeit und irrationale Zahlen. Braunschweig, Vieweg.

––– 1890, Brief an Keferstein, in J. van Heijenoort, From Frege to Gödel. NO: la otra.

Dugac, P. 1976, Richard Dedekind et les fondements des mathématiques, Paris, Vrin.

Feferman, A. B. & S. Feferman, 2004. Alfred Tarski, life and logic. Cambridge Univ. Press.

Ferreirós, J. 1999, Labyrinth of Thought: A history of set theory and its role in modern

mathematics, Basel, Birkhäuser (2nd edition, 2007).

––– 2003, Del neohumanismo al organicismo: Gauss, Cantor y la matemática pura, in:

Ciencia y Romanticismo (eds. J. Montesinos, J. Ordóñez & S. Toledo), Tenerife,

Fundación Canaria Orotava de Historia de la Ciencia, 2003.

––– 2004, The Motives Behind Cantor’s Set Theory: Physical, biological and philosophical

questions, Science in Context 17, nº 1/2 (2004), 1–35

––– forthcoming, Hilbert, Logicism, and Mathematical Existence, to appear in Synthese.

DOI: 10.1007/s11229-008-9347-1.

Goldstein, C., J. Gray & J. Ritter, L'Europe mathematique / Mathematical Europe:

histories, myths, identities, Paris, Editions Maison des Sciences de l'Homme, 1996.

Hausdorff, F. 1898, Das Chaos in kosmischer Auslese – Ein erkenntniskritischer Versuch,

Leipzig, C. G. Naumann.

––– 1914, Grundzüge der Mengenlehre, Leipzig, Veit & Co.

Herf, J. 1984. Reactionary Modernism: Technology, culture, and politics in Weimar and

the Third Reich, Cambridge Univ. Press.

Hesseling, D. 2003, Gnomes in the Fog. The reception of Brouwer’s intuitionism in the

1920s. Basel, Birkhäuser.

Hilbert, D. 1899/1902, Grundlagen der Geometrie. Berlin, Teubner (1899); Paris,

Gauthiers-Villars (1900); Chicago, Open Court (1902).

––– 1900, Sur les problémes futurs des mathematiques / Mathematische Probleme, .

––– 1905, Logische Principien des mathematischen Denkens, lecture course.

––– 1910, [obituary of Minkowski]

––– 1926, Über das Unendliche, Math. Annalen.

Kanamori & Dreben.

Mancosu, P. (ed.), 1998, From Hilbert to Brouwer: The debate on the foundations of

mathematics in the 1920s, Oxford Univ. Press.

Mehrtens, H. 1990, Moderne – Sprache – Mathematik. Eins Geschichte des Streits um die

Grundlagen der Disziplin und des Subjekts formaler Systeme, Frankfurt, Suhrkamp.

Meschkowski, H. 1983, Georg Cantor, Leben, Werk und Wirkung. Mannheim:

Bibliographisches Institut Wissenschaftsverlag.

Purkert, W. & H. J. Ilgauds, Georg Cantor, Basel,Birkhäuser.

Rowe, D. 1985, Klein, Hilbert, and the Göttingen mathematical tradition, Osiris.

––– 2004, Making Mathematics in an Oral Culture: Göttingen in the Era of Klein and

Hilbert, Science in Context (2004), 17: 85–129.

Scholz, E. (ed). Hermann Weyl's Raum - Zeit - Materie and a General Introduction to his

Scientific Work. Basel: Birkhäuser (Series Oberwolfach Seminars, Vol. 30).

Schorske, Carl E. 1981, Fin-de-siècle Vienna: Politics and culture, Cambridge University

Press, 1981.

van Stigt, W. P. 1996, Introduction to Life, Art, and Mysticism. Notre Dame Journal of

Formal Logic 37 (3): 381–387.

Weyl, H. 1921, Über die neue Grundlagenkrise der Mathematik, Math. Zeitschrift 10. In

Mancosu (1998).

––– 1925, Die heutige Erkenntnislage in der Mathematik, Symposion 1. In Mancosu

(1998)..

Zermelo, E. 1908, Untersuchungen über die Grundlagen der Mengenlehre, Math. Annalen.