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Only art has the power of a form. Mathematics is an exercise for monks.

Alain Badiou

Art and Mathematics

Übersetzt von Steven Corcoran

PDF, 12 Seiten

Taking as his starting point a juxtaposition of Plato and Aristotle, Badiou explores in this article the relationship between art and mathematics. While for Plato both realms present independent orientations for thought, in Aristotle the relation to being and to the idea of both mathematics and art is withdrawn. Two problematic questions arise from this: That of knowing whether art and mathematics are distinct but nevertheless connected in being oriented toward the idea, and that of knowing whether the problem of form is the point at which art and mathematics intersect. Badiou establishes that there are four different contemporary tendencies that present an answer these problems: The Platonic, the Nietzschean, the Aristotelian, and the Wittgensteinian.

We all know that the relationship between mathematical activity and artistic creation is a very old one. We know that for a start the Pythagoreans tied the science of number not merely to the movements of the stars but to musical modes. We know that Babylonian and Egyptian architecture presupposed elaborate geometrical knowledge, even if the notion of demonstration had still not been won. Further back still, we find formal, or abstract, outlines mixed in with animal representations, in the great prehistoric decorations, without our knowing precisely to what it is that these mixtures refer. 


For the philosopher that I am, or that I believe I am, the entry into our question, as so many others, passes through the contrasting disposition between Plato and Aristotle. 


For Plato, mathematics is fundamental in the sense that it mediates between, on the one hand, experience, or the relation to the sensory world, and, on the other, pure intellection, or dialectical movement. Plato exalts mathematics from a point of view that relates it to being in itself, the form of which is what he calls the Idea. He nonetheless sees its impurity, which comes from its having to affirm its hypotheses – we say its axioms – without being able to infer them from a supreme general principle. Whence its inferiority relative to the dialectic. Yet this should not conceal its superiority over all forms of empirical knowledge. And, especially as mathematics is more structural, less bound to unverifiable intuitions. Plato would have surely admired the refined constructions of Galois, or of Grothendieck. He would have applauded the objectivist, ontological vision of mathematics, that of Kronecker, for example. In the order of the contemporary philosophy of mathematics, he would have rallied to Gödel’s simple realism, or to Albert Lautman’s dialectical realism.


Art, as for it, Plato holds in suspicion, due to its very own readiness to imitate natural objects. Plato is the first formalist, in the very precise sense of a theory of forms. For him, every movement of genuine thought aims at a Form, which is snatched from the real and transcends it. The imitative arts, descriptive poetry or painting, remain captive to an immediate form, at the edge of the formless, instead of separating themselves from it to exhibit a pure form, of which immediate forms are only weak consequences. Plato stigmatizes the purely decorative, or purely melodramatic, effects of trompe-l’œil painting or blood-soaked tragedies. Plunging people into a sort of illusory stupor or cultivating in them sensational affects, turns them from the idea, and in fact accustoms them to the formless. And this is a way of corrupting them, so as to be able to dominate them better. 


However, in what is best about them, the sensible world and its artistic imitation can also play the role of mediation toward the Idea, not from the point of view of Thought, but from that of Affect. The loving admiration for the beauty of a body or of an elegiac poem can serve as entries to the no less admiring grasp of the Idea of the Beautiful. Art, if restricted to a sort of severity, if held closer to form than to its supposedly natural or psychological origin, can at least point to where one must look to find the exit from the Cave. Plato would have certainly recognized the greatness of Corneille, or of Poussin, but also of Mondrian, of Kandinsky, or of Boulez. He would have indeed gladly come here on the occasion of the Soulages exhibition. He would have also appreciated the didactic will of Bertolt Brecht, his idea that theatre is made to dispel the public’s illusions and to show the underside of the social and political décor. 


Fundamentally, for Plato, mathematics plays, in the order of the True, the role that purified art can play in the order of the Beautiful. And this role is that of mediation, of an introduction to the emancipatory powers of the dialectic. What links art and mathematics to one another is that both teach us, at the edge of the empirical but without reducing to it, what a form is, thereby enabling us to start our ascension to the purely formal seizing of Ideas.


Let’s say that for him, under the generic idea of form, art and mathematics can be points of departure for authentic thought. Art, by devoting itself to beautiful forms in the sensible, pushes our Affect in the purified direction of the Idea. Mathematics, by devoting itself to structures that one can extract from the real, pushes our Thought in the same direction. In the end, the difference of both processes is concentrated in the dialectic of two Ideas, Beauty and Truth. You could say that art leads off toward the possible Truth of the Beautiful, and mathematics leads off toward the possible Beauty of the True. 


For Aristotle, things are entirely different. First, mathematics falls directly, and in entirely explicit fashion, within aesthetics. A demonstration is not properly speaking true; it is essentially beautiful. ­Aristotle develops this conviction in books Bêta and Mu of the Metaphysics. The conclusion is irrevocable: after having said, and I cite, that “the highest forms of the beautiful are order, symmetry, the definite,” Aristotle claims that “the beautiful is the main object of mathematical demonstrations.” 


This conclusion is set in a context absolutely opposite to that of Plato. There is no question in all this of advancement toward the truth of the Idea. If, for Aristotle, mathematics is a pure aesthetics of thought, this is quite simply because it has no relation with being, with the real. For Aristotle, first mathematical objects, the matematika, as he says, have, despite their name, no independent, objective reality, no purely intelligible existence (as is the case for Plato). I quote him: “it is plainly impossible that mathematical objects should exist separate from sensible being.” But, second, it is just as impossible that these mathematical objects exist in the sensible, without being able to be separated from it. So they have no empirical existence either. As Aristotle says, mathematical objects are, with regard to our sensible experience, neither separated, nor unseparated. In truth, they do not exist in actuality. Having neither physical existence nor metaphysical existence, they are pure fictive constructions, which consist in pretending to affirm the separate existence of that which precisely cannot be really separate. For example, the mathematical sphere exists in separating sphericality from every real sphere. People will say that this separation is impossible. And well, precisely, Aristotle tells us, this impossible is the mathematical fiction itself. And its norm is the transparent beauty of the simple relations that it constructs on the basis of objects that do not exist. Aristotle would have liked Hilbert’s pure axiomatic formalism. He would have taken delight in Russell’s famous statement, according to which, basically, in mathematics one knows neither what one is talking about, nor if what one says is true. He would have applauded Bourbaki’s presentation of mathematics as a pure game of writing. 


On the side of art properly speaking, Aristotle is just as anti-Platonic. He excludes from artistic processes any vocation to the Idea or to a revelation of being. Just as mathematics, being neither physical nor metaphysical, is an aesthetics, so too art, not being in the least a form of knowledge [connaissance], in fact comes under anthropological practice. What counts is the artistic act, its effects on the Affect of the spectator, or reader, or witness. And Aristotle’s general Idea is that this effect is one of purification; today one would say of sublimation. Art relates us momentarily, through artificial means, to situations of exception. And in doing so, it divests us of the troublesome dreams and various inhibitions that the real of these situations generally provokes in us. Art is a subjective and social intervention. It is in reality a dimension of the collective Ethic. 


Aristotle, from his grave, would applaud contemporary movements that assign art functions, as it were, of critically sublimating all that is violent or even repugnant. He would identify with Artaud’s theater of cruelty, the morbidity of German expressionism, choreographies of bared, tortured, and sullied bodies. He shares with many contemporary creators the conviction that what counts in artistic activity is not form but effect, not truth but expressive sincerity, not separation but immanence, not the differed and the eternal but action here and now. Aristotle would cast a benevolent eye at performances and installations on which Plato would hardly spend any time at all. Little matter the precariousness of the montage and the deliberate poverty of everything: long live trash ripping up, excrement, cadaverous horror, if it all effects, on its witnesses, a new subjective chemistry. So we are therefore at a complex crossroads.


First choice: Do art and mathematics define truth processes that are no doubt distinct, but that touch on the very being of that which is, be it in the pure dimension of the Idea? Or are they both in registers that bear no relation to the true knowledge of what is, which is certainly physical, perhaps metaphysical, but neither mathematical nor artistic?


Or again: Do the Beautiful and the True designate itineraries that are totally distinct in their point of departure and their means, but that have a common direction? Or does the Beautiful designate a domain of fiction, whether mathematical or artistic, separated from the True, and whose import is aesthetic or ethical but in no way theoretical?


Second choice: Is the notion of Form the real point at which art and mathematics converge? In this case, we will have a sort of, at least local, entanglement of mathematics and of certain arts. Or it is a matter of a pure homonymy? In this second case, the form will designate in mathematics the linguistic crystal of structures, and in art the appropriate means to seduce or bring about subjects, and even the unconscious of these subjects, by isolating, by reformalizing, and by exposing, fragments of the sensible real.


Each of the options offered in the first choice again divides by two, which at this stage of the examination generates, for this fundamental choice, four options, fixing for always, just like beacons in a port, the great epistemological and aesthetic discussion on the relation or the non-relation between mathematics and the arts. 


If, for the first choice, one adopts the Platonic perspective, according to which mathematics and art, Truth and Beauty, can and must be both at the level of the Idea, and organize the same dialectic, it remains to be known what the extent of their difference is, and which of the two has precedence. The issue can be put thus: mathematics and art both have possible didactic functions for a Subject that is meant to be oriented toward the Idea (or oriented by the Idea). What difference is there between these two didactics? The philosophical projection of this question is very clear. For Plato, Descartes or Spinoza, for Husserl or for myself, it is mathematics that, first and foremost, saves thought. For Schelling, Nietzsche, and Heidegger, for Wittgenstein, or for Deleuze, it is art that opens the way. We will therefore distinguish, under the joint sign of an Idea able to break with the alienation of ordinary life, two tendencies:


Tendency A will say: it is mathematics that, in the history of humanity, opens thought to the reign, at once rational and suprasensible, of the Idea. In this respect, it provides us with a paradigm by which to judge what an art worthy of this name is, that is to say, an art that affirms, on the basis of sensory materials, that Subjects can desire the Idea. We will name this tendency Platonic. It is without a doubt the one furthest removed from contemporary democratic preoccupations.

Tendency B will say: it is art that is the first to deliver, under its sensible form, the power of the Idea. It is art that elevates Subjects above resentment, art that enables the glorious affirmation of that which is. As Schopenhauer maintained, without music, life is not worth the trouble of living. Mathematics sometimes goes in the same direction, notably during the radical, and even violent, theoretical events by which it asserts itself against vulgar and alienating opinions. This is so, when it affirms that the measure of a segment can be irrational, or that there exists an infinity of different infinities. Yet in its ordinary existence, which is academic, it remains at best an inferior game, at worst a discipline enslaved to the ravages of technique. We will call this tendency Nietzschean.


Suppose now that we adopt the Aristotelian perspective according to which arts and mathematics are not at all on the plane of the Idea or of being, but have limited anthropological functions, aesthetic in one case, ethical in the other. The question that arises is to know whether, from within a general anthropology, these two functions can or cannot be unified. Can mathematical fiction itself be situated on the same plane as artistic creation? Is the mathematician an artist?


Tendency C will say, ultimately, yes. When Aristotle himself evokes as criteria order, symmetry, and the definite, it is clear that he could be speaking of architecture as much as of mathematics. We know the criteria of mathematical aesthetics. There is the principle of economy: an axiomatics is more beautiful if one reasonably limits the number of axioms; a demonstration is more beautiful if it is shorter, or if it dispenses with complicated means so as to arrive at a sort of elementary simplicity. There is also a principle of rational totalization: A new theory is magnificent if it integrates a host of previously scattered results, and shows their coherence. There is a principle of fecundity: A theorem is especially admirable for entailing significant consequences, including ones that are very far removed from its immediate context. All the above is applicable by and large unchanged to all sorts of artistic activities. You will be able to perform the exercise yourself: take a look at the rules of classical theater as used by Racine, at novelistic narration in the works of James, at the construction of Hölderlin’s great didactic poems, at the function of leitmotiv in Wagner, at the synthetic power of Tintoretto’s immense religious paintings, at the function of silence in Webern, at the treatment of space in Nicolas de Staël, or at the principle of series in Anselm Kiefer. It can thus be concluded that mathematics is a branch of aesthetics. We will call this tendency C Aristotelian. 


Lastly, tendency D will proclaim a strong dissymmetry between art and mathematics. In the same initial orientation, wherein art and mathematics have a purely anthropological value, it will be considered that mathematical aesthetics differs radically from all others. This tendency, similar to the Nietzschean one, tendency B, will in fact consist in reducing the value of mathematical aesthetics on account of its subjective poverty, of the fact that, even within the space itself of anthropology, it does not manage in the least to serve humanity in that which really matters to it. The games of mathematical writing will be considered too arbitrary and abstract to attain the point at which is actually decided the only important thing, which is the meaning of life, and the critical expression of that which hinders the free creative expression of the human subject. Mathematical aesthetics will be held to be cold, impersonal, perhaps even devoid of all sense. In any case, it will be underlined that it has no deep relation to the interiority and to the unconscious of the subject, which it is the function of art to shake, to touch upon, to express, and to sublimate. Art’s critical and ethical function will be held to be essential. We will call this tendency D 
Wittgensteinian. This is without a doubt the dominant tendency today, because it befits contemporary victimary humanism. 


It is from the viewpoint of the four tendencies – Platonic, Nietzschean, Aristotelian, and Wittgensteinian – that we ought now to raise the crucial question implied by the second choice mentioned above, and which bears on the concept of Form. Has the word, or can it have, the same sense in both of the disciplines that concern us today? 


That the question of art is one of displacing the border between that which has form and that which is held to be formless is stating the obvious. After all, the history of modern and contemporary art can be read as the progressive inclusion of a growing part of the formless into apparatuses that are formal at least insofar as they are separated, were it by almost nothing. With Duchamp, as we know, and have done since the beginning of the last century, the separation of anything at all, for example of a urinal or of a bicycle wheel, is reduced to its exhibition and to its nominal label. This is enough for any object whatsoever to take the function of an artwork and to be signed as such. Art, today, is perhaps the site where the endlessly displaced border between the immediately formless and the formal is tested out in infinite fashion. This also implies the exploration of different modes of separation and of in-separation between what one or some subjects decide, and what is already given. To go as close as possible to in-separation, to reduce exhibition, which validates the signature, to almost nothing, is the explicit aim of theater without theater, of performance included in the fabric of everyday life, of objects exhibited anywhere at all, of noise recorded as music, and so on and so forth. At bottom, contemporary art asks in act what a form is by exploring its minimal differential possibilities. 


Moreover, that forms are at issue in mathematics is also a dominant aspect of modern and contemporary mathematics. Following the primitive notion of mathematical objects, such as figures of geometry and numbers of arithmetic, came the reign of structures and of the constitutive relations of these structures. All mathematical schools have presented themselves, since Hilbert, as formalist. And certainly, the formal paradigm can change. It is certain that, dominated by algebra in the 1930s, then by topological or differential geometry at the end of the same century, today the paradigm is constructivist and algorithmic – under the pressure, it really ought to be admitted, of external circumstances. We have the mathematics that the obsession with financial calculation and the overwhelming domination of IT merits. But in all cases, it is really the nature of formal relations that defines these infra-mathematical paradigms, and not the existence, supposed natural, of such-and-such a type of object.


Does the word “form” mean the same thing in both cases? A Platonist, a person of tendency A, will doubtless answer “yes.” If “form” means that which orients us toward the Idea, a work of art will merit being admitted only if, regardless of its pomp and sensorial seduction, what its form, or forms, affirms is, in the last instance, of a purely intellectual nature. One will even be able to say that a work of art is the articulated movement of its specific forms. By this I mean to say that what constitutes it is actually, as in mathematics, a system of relations. The difference is that it proposes to activate these relations directly in the sensible, between the different blocks of legible, sonorous, and visual objectivity that it extracts from the real to exhibit to the public. I will go so far as to say that from this point of view, there will not be, for the Platonist, any difference of nature between a work of art and a theorem. It will be objected that the work of art is a singularity that must be seen, received, understood. But a theorem must also be received and understood, for an idea to be had of what its existence as a theorem is. Is a theorem more opaque, as it moves towards the plane of Ideas, than a Boulez sonata or a Pollock painting? Does it ask of its witnesses efforts of a really different nature? In my view, not at all. A poem by Mallarmé or by Wallace Stevens demanded of the reader of the epoch an attention to the novelty of the relations put into play of the same type as the last quartets of Beethoven demanded from the listener at the beginning of the 19th century, or as understanding Galois’ theory demanded, in terms of mathematical, of the informed amateur. 


The reader will have grasped that I am a Platonist; that I speak in the name of tendency A. The price to be paid for this is a strict discrimination in the domain of art. For the above identity between theorem and work to be discerning, it presupposes a determined artistic orientation and that the border between form and formlessness is not exaggeratedly obscured. In mathematics, even when you talk about the formless, when you create a theory of Chaos, you are inside transparent formalisms. The Platonist will demand of art that, as close as it holds to the formless, to trash, to the obscene, the formal distance it keeps is perceptible and affirmed. This means that he or she will maintain the primordial concern with the relations between blocks of sensible real, without sacrificing it to the spontaneous movement of expression, or to a concern with violent effect. Everyone can easily find examples of what this position entails in terms of adoption and exclusion. 


A person of tendency B, a Nietzschean, will not reason in the same way. For Nietzsche, the form of art is a projection of vital energy, a creative outgrowth of that which ties us to the great terrestrial affirmation. As he writes, “the dying Zarathustra holds the world embraced.” Art proceeds from the body. Certainly, it breaks with the ordinary uses of resentment and guilt, but only so as to discover better the saintly affirmation that constitutes our belonging to powerful inorganic life. There is, by contrast, something stunted and grayish in mathematics. Even if science is, above all for the young Nietzsche, an emancipatory discipline, this is due not to its formalist dimension, but due to its critical power concerning in particular Christian prejudice. Art, in truth, has nothing to do with mathematics; it must rather liberate us from them, liberate us from formal relations to enable the discovery of the infinite multiplicity of interpretations, of variations. Art ought to remain as savage as possible, a luminous savagery, which by no means sacrifices the Idea, but exalts it. It is perhaps in our times that its manifesto is given by dance, the direct mobilization of bodies, and there is a reason why Zarathustra is a “mad dancer.” To know what is at play in the Nietzschean orientation, that of a body wholly transported and transfigured, of an Idea in-separated from the body, let us look at the choreographic interpretations of Stravinsky’s Sacre du Printemps, from Nijinski to Pina Bausch including Béjart and all the others! Here one will see the history of the Form-body, of form-life, such as it rids art, precisely, of all possible mathematics.


But the person of tendency C, the Aristotelian, will hold yet another view. Form, for this person, is an abstraction common to the arts and to mathematics, because it accepts general norms that stand above them both. Order, symmetry, measure of effects …. Why is this the case? Because we are no longer on the plane of the Idea, but on the search for a reasonable anthropology. It is anthropology that distributes the ethical norm, the most important one, and this norm promotes equilibrium, personal blossoming, each person’s arrival in the place proper to him or her. If mathematics is a positive aesthetics, it is because it refrains from the monstrous, from the bizarre exception, from shaky relations. Even when it encounters obstacles that are apparently pathological, such as irrational geometrical relations, or continuous functions without a derivative in any point, or infinite sets of points whose measure is nil, it ends up by integrating them into systematic theories. Mathematical aesthetics calms the fictive eccentricities of reason, just as theatrical aesthetics purges us of all harmful passions. The Aristotelian is a humanist, quite simply. He or she is one who, in the name precisely of form, will distrust the exaggerated formalism of set theory as much as the abstraction of integral serialism of 1950s Boulez. He will denounce the theory of categories as “abstract nonsense” while simultaneously seeing contemporary art’s attraction to the morbid and the repugnant as a detestable exaggeration. The Aristotelian is and remains, as far as forms are concerned, the person of the happy medium, in mathematics as well as in art: neither too close to the Idea, and therefore opposed to formalism: nor too close to the formless, and therefore opposed to sensualism. On this condition, mathematics and arts share what could be called the good form. 


The person of tendency D, lastly, the Wittgensteinian, will propose yet another way to look at things. For this person, supposed mathematical form is itself formless. What is in point of fact the Idea? Or what is – it amounts to the same – the moral salvation of the Subject? It is that which exceeds both the world and ordinary language, and manages to give meaning to existence beyond the language games to which empirical reality constrains us. This beyond of ordinary experience is of an aesthetic or ethical nature, and for Wittgenstein himself, aesthetics and ethics are the same thing, the same form, which he calls the mystical element. This element is affective, ethical, and aesthetic; it contrasts with mathematical form, which is a monotonous succession of equalities, a calculation without thought, a hollow form that one can treat, as he does, as a “joke.” Wittgenstein will say, and I quote: “Feeling the world as a limited whole – it is this that is mystical.” And this here is truly that toward which artistic form tirelessly works: to form in each person the sentiment of the frontiers of the world. This is why artistic form is always lacunary, unforeseeable, unstable. It exhibits the critical uncertainty of our belonging to the world; it is nomadic and fleeting. The Wittgensteinian will like the short pieces of Schumann’s folly, Picasso’s drawings, Cage’s pieces, minimalist art, René Char’s poetic aphorisms, performances. Wittgenstein’s life itself is a succession of obscure performances. When all is said and done, arts and mathematics do not share any common concept of form. The form of art concerns the mystique of a meditation on the poverty of the world. Mathematical form is the repetitive writing of possible tautologies. 


So there is a discordant quartet as regards the question concerning us today, and which in the last instance is that of form, in mathematics and in art. 


The first violin, tendency A, the Platonist, tells us that form, insofar as it is always a pathway toward the Idea, ultimately falls within the same orientation of creative action, or of thought, whether it is a matter of a work of art or a theorem of mathematics. 


The second violin, tendency B, the Nietzschean, tells us that true form is always born of earthly and corporeal life, that it is accomplished in the existential dance which opens us to Dionysian affirmation. Only art has the power of such a form. Mathematics is an exercise for monks; it has the smell of a guilty obsession. Only exceptionally does it rediscover the vigor of its birth, the eternal return of its Greek violence. 


The alto, tendency C, the Aristotelian, tells us that every form is a production situated in anthropological space. It thus falls under a humanist and measured norm, and on this point it makes no difference whether it is mathematical or artistic. In any case, form places a fiction in the serene dimension of an order. 


The violoncello, tendency D, the Wittgensteinian, tells us that every form is mystical, and that the mystical element is precisely that which mathematics is absolutely incapable of. In truth, in order to go beyond the enclosure of the world, to explore its boundaries, through sentiment, it is necessary to invent forms, aesthetic and ethical, that absolutely exceed the tautology into which mathematical calculation resolves. 


One might say: choose your tendency! And you have probably already chosen it.


I will add, however, that there is sometimes a mysterious harmony from our quartet. Suddenly, our four soloists agree on what a form is – whether mathematical or artistic no longer makes any difference. This is when is produced, in one field or another, a mutation so major, a novelty so luminous, that no creator of good faith can resist its power. This is the power proper to that which is given, not as the continuation of a tendency or of a school, but as an event so strong that it sweeps aside the subjective oppositions concerning the relation of art and science. We see this in the Renaissance and at the start of the 17th century, when everyone circulated between the staggering novelties of science and the no less radical mutations of the pictorial, musical, theatrical arts … Think of Leonardo de Vinci. We see this also at the beginning of the last century, before the war of 1914, when a vertiginous movement spread to painting as well as algebra, to physics as well as music, to axiomatics as well as the novel, to formal logic as well as poetry. So, suddenly, something makes itself heard, as to the form, as to its relation with old forms as well as to its relation with the formless, something that passes through every subject and through all the domains in which truths are in question. Think of Joyce, of Freud, of Einstein …


This is what is called an event of forms, or an event of the relation between forms, on the one hand, and, on the other – whatever the register of these forms – the Idea that orients them. 


In times like these, without thinking too much about their former stance – A, B, C, or D – everyone can say, along with René Char in The Lightening lasts me [L’éclair me dure]:


How to say my liberty, my surprise, at the term of a thousand detours: there is no bottom, there is no upper limit.

In the movement of forms, between, on the one side, “severe mathematics,” as Lautréamont calls it, and, on the other, music, theater, the poem, painting, the novel, architecture, sculpture, cinema, dance, video, performance, installation, and also all that has come and will come, we are, today, rather in the closed and confused space of the thousand detours. This certainly prepares, to our own surprise, the freedom whose unknown possibility only an event creates: a world where there is neither bottom [fond], nor upper limit [plafond].


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Alain Badiou

Alain Badiou

ist Philosoph, Mathematiker, Dramatiker und Romancier. Seine politischen Aktivitäten drücken sich in der von ihm mitbegründeten »Organisation politique« aus. Er lehrte Philosophie an der Universität Paris VIII-Vincennes, der École normale supérieure und dem Collège international de philosophie.

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