THE AESTHETIC DIMENSION OF SCIENCE

JEAN LADRIÈRE

HARMONY AND NUMBER

          The question of the aesthetic dimension of science is noted in the text of Hermann Weyl from his magnificent book on symmetry: "We still share the belief of a mathematical harmony of the universe. It has withstood the test of ever-widening experience. But we no longer seek this harmony in static forms like the regular solids, but in dynamic laws."¹ The regular solids to which Hermann Weyl refer are the five regular polyhedrons used by Kepler in his famous Mysterium Cosmographicum (published in 1596), in order to reconstruct a priori the structure of the solar system.

          The idea of using the regular polyhedrons in a cosmological context was not at all new. They had been used already by Plato, in the Timaeus, to explain the composition of the "body" of the cosmos and to give a mathematical expression to the old theory of the five elements. What is quite remarkable in Plato's idea of Plato is that, by transposing the classical theory of elements into a mathematical form, he introduced necessity into that theory. Indeed Plato knew the theorem concerning the regular polyhedrons and stated that there exist five regular polyhedrons and not more. This theorem is proposition 18 of the 13th Book of the Elements by Euclid. It can be considered as a very profound theorem because it establishes a negative property: it is not possible to construct regular polyhedrons other than those with four faces, six faces, eight faces, twelve faces or twenty faces. This means that among all the possibilities, which could be considered as infinite, only these five correspond to real, that is, effectively constructable, mathematical objects.

          It is highly significant for our question that Weyl's statement introduces the word "harmony", which evokes an aesthetic property and formulates a thesis, presented under the form of belief rather than as scientific knowledge, regarding the aesthetic dimension of science. It is true that Weyl refers to science not as a whole but to the science of nature. His thesis can be considered as a thesis of natural philosophy. The question of the extension of that thesis to other parts of scientific knowledge must, of course, be raised, but what we can learn from physics could in any case be suggestive for the other sciences. The thesis of Weyl contains three parts:

          a) the adequate scientific representation of the universe is a mathematical one,

          b) there is a specific harmony in mathematics,

          c) this harmony reflects an intrinsic property of the universe and that property can thus be called a "mathematical harmony."

          The first proposition is an epistemological principle and can be radicalized under the form of an ontological principle that the internal structure of the universe, taking into account all properties, is actually a mathematical one.

          This ontological version of the proposition is the grounding principle of the Timaeus, probably reflecting Pythagorean concepts. As reinterpreted by Plato, this proposition becomes the key to the solution of a fundamental speculative problem, namely, that of mediation. The aim of the work is to construct a theory of the cosmos which process is explained as if it was the work of a supreme artist, the demiurge, according to a fundamental analogy between nature and art.

          The demiurge is good, and his aim is to build a world as perfect and as beautiful as possible. He takes as a model for his work a realm of pure forms which is the paradigm of excellence. The cosmos must be concrete and therefore visible and tangible. The problem then is to create a model compatible with this concrete status. In other words, the forms which give it its configuration must be imprinted in a principle of receptivity, conceived as a receptacle, the chôra. The problem then becomes one of mediation. In order to ensure the relation between the forms and the chôra some intermediate entity, participating in the perfection of the forms and also in a certain way of the opacity of the chôra, must intervene. For Plato, mathematics plays the role of that intermediate entity.

          The cosmos must be like a work of art and by its own resources must provide for its constitution and appearance. To ensure its unity, that is, the full integration of its constitutive parts, it must be a living entity, a great living being. Like any living being, it must have a soul and a body, which two constitutive principles are the two levels of the mediation between the purity of the model and the lack of determination of the receptacle. The "how" of that mediation is explained by the mathematical structures of the soul and body of the cosmos. These two mathematical structures provide the principles of an astronomical theory, on the one hand, and the principles of a theory of the elements on the other hand. This is the spirit of mathematical physics, but developed very visibly from an aesthetic point of view. This is particularly clear in the platonic reconstruction of the constitution of the soul of the cosmos.

          The leading idea of that reconstruction is that the soul is structured by numbers, according to fundamental ratios. The demiurge makes a mixture of two primary components. This duality accounts for the difference between the absolutely regular movement of the sky and the movement of the planets (the sun included) which presents irregularities. The demiurge divides the mixture in numerical ratios, the core of the theory of the soul being the construction of a numerical series defining the intervals according to which the mixture is divided. That construction starts from the two numbers which immediately follow the unity, two and three. The first step is to build a series obtained by composition of two geometric progressions, a progression of ratio, starting from 1 and limited to four terms, 1, 2, 4 and 8, and a progression of ratio, starting also from 1 and limited to four terms, 1, 3, 9 and 27. By taking the two together, in the natural order of their terms, we obtain the basic series 1, 2, 3, 4, 8, 9 and 27. Then the intervals are "harmonized", by equalization of the ratios, thanks to the insertion between those integers of rational numbers, obtained by the use of the arithmetic and the harmonic mean.

          The result of that construction gives the relative distances between the planets. But it gives also exactly, and unexpectedly, the Greek musical scale. Thus it appears that at the same time the same mathematical structure gives account of the principles of music and of the constitution of the solar system. This shows that the astronomical world, which corresponds to the global structure of the cosmos, is organized according to the same numerical properties as those by which the virtue of human art creates musical beauty. That effect is due to the structure of the soul, formed by the series constructed by mathematical operations. Finally with the structure of that formal numerical series we can find the reason of the beauty of the cosmos as well as of the beauty of the musical works. By generalizing that conclusion of the reading of the Timaeus, we could say that for Plato nature is beautiful because the cosmos has a mathematical structure and because mathematics itself is beautiful, or perhaps more exactly, because mathematics itself discloses the formal configurations which make the beauty of what appears beautiful.

ANTIQUITY: NUMBER AS THE A PRIORI OF TIME

          Particularly significant here is that number is the a priori of time. After the explanation of the structure of the soul in the Timaeus comes the explanation of the constitution of the heavens and of the apparition of time, which is the ordered motion of the heavens. According to a current interpretation, time is conceived by Plato as "the mobile image of eternity." Remy Brague, in his book Du temps chez Platon et Aristote,² has criticized that interpretation and proposed another one, which seems to conform much more to the thought of Plato. It is that time is indeed linked with motion, actually the motion of the heavens, but this motion occurs "according to the number," and that number is the number of the soul. It is not time, but the heavens which image eternity, and this image is going kat'arithmon. It is precisely because the motion of the heavens is conformed to number that it is the image of the eternal reality: number is the mediation between the model and the heavens. Time, then, according the interpretation of Remy Brague, "is the ordered motion of the heavens, which manifests the numerical structure of the soul of the world. Conceived in this way, the soul produces time rather than taking consciousness of it." That conception of time orders the priority to number as the structuring condition, not created by human thought but inscribed in the very constitution of the cosmic reality.

          That same speculative strategy is pursued in the Platonic analysis of the body of the world, that is to say, in the study of the elements and of the dynamics of cosmic transmutations. In the context Plato makes use of the theorem about the five regular polyhedrons. He establishes a correspondence between those mathematical objects and the traditional four basic elements: fire is identified with the tetrahedron, earth with the cube, air with the octahedron, and water with the dodecahedron. Concerning the eicosihedron, the polyhedron with twenty faces, Plato explains that the demiurge has used it for the figure of the whole, that is, for the envelope of the cosmos as a whole, which is a sphere. That polyhedron is indeed, among the five regular bodies, the one nearest approximation of the sphere.

          But the real scope of that theory of the elements lies in the dynamics of transmutations, based upon the analysis of the polyhedrons. Plato shows that those can be constructed with rectangular triangles which are really the elementary constituents of the cosmos. These triangles differ only from each other by the ratios of their longer side with respect to their shorter side; they are of the same nature, and this homogeneity entails that each one can be used for the construction of any one of the five polyhedrons. A transmutation can then be understood as a recomposition, and the possible processes of decomposition and recomposition determine a priori the transmutations which are admitted in the cosmos. Thus, two particles of water, being polyhedrons of twelve faces, can give way to three particles of air, which are octahedrons: 2 x 12 = 3 x 8. In qualitative language: water can become air, or, in more modern terms, water can pass into a gaseous state. Again, that geometrical structure of the regular polyhedrons, conceived as by themselves constituting the elements of what we call matter, provides a priori the reason for all the kinds of transformations we witness in the world. The explanatory power of mathematical interpretation matches its elegance and allow us to understand how the demiurge has succeeded in his project.

          In the Greek tradition the relationship between the properties of some mathematical objects and the constitution of the cosmos is completely analogous to the relationship between some mathematical properties and the works of art which give a prominent role to proportions, music and architecture. They have a close analogy to each other as shown for example by the myth of Amphion. Paul Valéry, in his famous poem, Cantique des Colonnes, has celebrated the connection between architecture and music and has evoked the role of mathematics as providing the principle of the concrete harmony of the constructive art when he wrote the columns that they are "daughters of the golden numbers, resting on the laws of heavens."

          Plato saw in certain arithmetical properties the key at the same time of music and of cosmology. It would be interesting to take into consideration also the case of architecture. In his Ten Books of Architecture, Vitruvius explains "ex quibus rebus architectura constet." Vitruvius puts a particular emphasis upon what he calls "proportion" which he defines in the following terms: "Proportion is the relation that all the work has with its parts, and the relation which the parts have separately, comparatively with the whole, according to the measure of a certain part."³ The precise expression of those relations is, of course, given by mathematical relations.

          A good example is given by the famous "golden number," also known as the "golden section," it is defined as the ratio between two quantities submitted to the following order: the ratio of the sum of those quantities to the greater one is equal to the ratio of this greater one to the other one. If a is greater than b, that condition can be written: a + b / a = a / b. The geometrical version of this concept is the solution of the following problem: given a segment AC, find a point B on this segment such that the above condition applies to the segments AB and BC. We must thus have the following equation: AB + BC / AB = AB / BC. Let us denote by the letter g the numerical value of this ratio. A simple calculation shows that g has two possible values: g = 1,618..., and g = ‑ 0,618. . . .

          That number n has remarkable arithmetical and geometrical properties. For example, if we take the successive powers of g, we obtain a series, 1, g, g2, g3, . . . where each term is the sum of the two preceding ones. This number g appears in different properties of the polygons and of the polyhedrons. For example, in the case of the dodecahedron, which has twenty summits, we have the following property: those summits are also the summits of four regular pentagons, equal two by two, placed in parallel planes cutting the dodecahedron. The lengths of the sides of the two greater pentagons, are to the lengths of the sides of the two smaller pentagons exactly in the ratio g.

          From the point of view of its architectural use, the most interesting intervention of the golden number is in the definition of the so‑called "rectangle of modulus g." It is a rectangle such that the ratio of the longer side to the shorter side is equal to g. That rectangle has the curious property that if we divide the longer side in two segments having between themselves the ratio g, we can draw a new rectangle, inscribed in the given one, which is also a rectangle of modulus g. This process can, of course, be continued up to infinity. The series of those rectangles has the marvelous property that it is possible to draw a logarithmic spiral passing by three summits of decreasing rectangles of the series, this being the case for the whole series.

          We have here purely geometrical properties. The striking fact is that the architectural use of those properties, as determining proportions, creates harmonious effects which to give to architectural works those qualities of equilibrium, of clarity, of rhythmical composition, and of perfect correspondence between the parts and the whole, which are characteristic of classical monuments. Thus what is apparent in works of art is already present intrinsically in mathematical objects. In the case of the golden number, what produces an aesthetic impression is not only the quality of the proportions determined by that number, but what could be called the fruitfulness of its definition. That fruitfulness appears, on the one hand, in the fact that the number g can be found again in many geometrical properties, and, on the other hand, in the fact that it gives way to processes of iteration, in the case of the series of powers of g and in the case of the rectangles of modulus g.

          In the ancient speculation which finds its inspiration in the Platonistic tradition, the beauty of the cosmos is grounded in the properties of the numbers. And the numbers themselves have this virtue of producing the harmonious texture of the visible world because they belong themselves to the invisible world of the pure intelligible. In the sixth book of the sixth Ennead, Peri Arithmôn, Plotinus undertakes a long discussion to demonstrate that the number exists by itself, independently of the beings which are determined by it, independently also of our thinking, because it belongs to the intelligible, which exists before thought.⁴ That realm of the intelligible is organized according to a hierarchy of degrees, corresponding to the movement of release of the One, which is also a process of generation, the procession from the One. The first degrees following the One are the Being, the Intelligence and the Living in itself. Plotinus shows that number is before the Intelligence and the Living in itself, and that it is thus in the Being. It is produced by the Being inside itself, and the Being in itself produces the finite beings according to the numbers, which are the rules of every generation. There is thus an internal procession of the numbers, which is the condition of possibility and the exemplary cause of the procession of the physis. The numbers which are in the Being are finite in "that nothing can be added to the intelligible number,"⁵ but they are infinite in "that there is nothing above them which could limit them and that they determine themselves . . . by a movement interior to themselves. . . . The infinite number is like the First Number, which is to say the one which contains, realized, the model of all the possible numbers; it designates all the properties, ratios or proportions  which can exist between numbers. It is like an eternal arithmetic."⁶

          Saint Augustine has reinterpreted that neo-Platonic conception of the numbers in his own conception of the eternal harmonies, developed in the second part of the sixth book of his De Musica, "The eternal harmonies and God their source". He explains the sensible harmonies by numerical proportions and he relates those to the properties of pure numbers, like Plotinus. However he places those eternal harmonies in God himself, by reinterpreting the Greek theory of the Intelligible in a theory of the divine Ideas, exemplary causes of creation.

MODERNITY: SYMMETRY AND FINALITY

          The modern science of nature follows the idea of the fruitfulness of the mathematical representation without trying to justify it by speculative considerations. It continues to raise an actual philosophical problem, namely, the aesthetic aspect of science. In order to approach that question in its modern version, it would be useful to evoke two concepts which play a central role in physics and which illustrate significantly our question, the concepts of symmetry and of extremality.

Symmetry

          As Hermann Weyl explains in his book on symmetry, there is an intuitive notion of symmetry and a precise geometric one. According to the intuitive idea, symmetry is the quality of what is well proportioned; namely, the "concordance of different parts by the virtue of which they integrate themselves in a whole." The precise geometric idea is the notion of bilateral symmetry: a spatial configuration is symmetric with respect to a plane if it can be transported in itself by a reflection with respect to this plane. A reflection with respect to a given plane is an application of the space in itself which transfers an arbitrary point of that space into a point which is its "image" with respect to that plane (and can be defined by a simple geometric construction). That notion can be generalized by deletion of the condition "with respect to a plane." We have then the simple notion of "application": it is a correspondence, many‑one, which associates to each point of the space a point‑image. If the correspondence is one‑one, the application is called a transformation, for example, a translation or a rotation around an axis. What is characteristic in a geometrical object is what is not affected by transformations, or in other terms is invariant with respect to specific transformations. It could be said that such an invariance reflects something intrinsic to the object studied, which appears the same from all the points of view, which can be taken by submitting the space to transformations of a certain kind. The notion of symmetry expresses precisely this idea of invariance and thus of intrinsicality. An object can thus be characterized by the symmetries that it exhibits under given transformations.

          The geometric analysis makes use in particular of a kind of transformation which does not change the structure of the space. Such a transformation is usually called "similarity," but Weyl prefers to call it "automorphism." It appears that the automorphisms form a group, endowed with the operation of composition of two transformations, the one following the other. The identity, which applies each point on itself is an automorphism, the composition of two automorphisms is an automorphism, and to each automorphism corresponds an inverse, the composition of an automorphism with its inverse being the identity. Now, given a spatial configuration, the automorphisms let this unchanged configuration form a group which is actually a subgroup of the group of automorphisms of the space. A geometric object can thus be characterized by the group of automorphisms which transform it into itself with respect to which it is symmetric. For example the sphere is symmetric with respect to the spatial rotations, the five regular polyhedrons are symmetric with respect to rotations around a point in space, and so on.

          That notion of symmetry has been transposed from geometry to physics and thereby generalized. As the physical situations and events occur in space and time, their representation must make use of a geometric frame, endowed with an appropriate structure. Such a frame can be considered as a space, this term being taken in an abstract sense. A "space" in this sense includes the properly geometric space and time: it is actually a "space‑time." The idea of symmetry can be used therefore to characterize the geometrical aspects of the physical phenomena. But it can be extended to other aspects of physical reality. It is indeed possible to define transformations applying to the properly physical magnitudes, like the sign of an electric charge: this sign can be changed into its opposite.

          The laws governing the physical phenomena are expressed by mathematical relations, or "equations," connecting the variations in time of the different parameters characterizing the kind of phenomenon which is studied. Those parameters are dependent upon the spatial coordinates defining a point in the space‑time. This means that they can take different values when measured at different points of space‑time. Thus a spatial transformation can transform the values of the parameters. The relation between different parameters can be conserved under spatial transformations, and more generally under certain sets of transformations, including those which are not spatial transformations. Given a set T of transformations, if a given relation R remains the same when the transformations of T are applied, that relation expresses a symmetry with respect to T. A symmetry, in quite general terms, is thus an invariance, which is to say an independence with respect to the particular point of view under which the phenomenon is observed, and even with respect to particular aspects of the phenomenon (submitted to some of the transformations of the set T), for example the sign of the electric charge of the particles involved. Again symmetry is the symptom of intrinsicality. It appears that a property of symmetry entails the conservation in time of certain dynamical characteristics of the system studied and also certain rules of selection, which stipulate that in such or such experiment, implying a transformation, only such or such effects are possible. This explains that the search of symmetries is a very important heuristic tool in the exploration of the physical world. A representative example of the usefulness of this concept is that the introduction of the new principles of symmetry led to the models unifying the so‑called "weak" and "strong" electro-magnetic forces.

          Like numerical proportions, symmetry plays an important role in art, at least in that part of the artistic tradition which can be called "classical" and to which the rejection of symmetry makes indirect reference. In any case, that role is probably connected with some properties of perception, which could explain why it can have a positive or negative affective impact. The study of symmetry in physics leads to very abstract formulations. Something from the perceptive properties of symmetry is conserved in the physical concept of symmetry. The kind of intellectual pleasure which we experience when we discover how highly sophisticated mathematical theories, like the theory of representation of groups, correspond so magnificently with what nature reveals of its internal principles of organization. This is not so different from that kind of aesthetic emotion which we can experience in the presence of a musical piece or of an architectural work where we discover that the parts answer to each other as images of each other, giving thereby an intrinsic intelligibility to the work as a whole.

Extremality

          The idea of extremality plays also a fundamental role in theoretical physics. The law governing a dynamic process can usually be expressed by means of a differential equation. Such an equation says how the system under consideration passes from one state to an infinitely proximate state, and thus gives the key point by point of a reconstruction of a trajectory between any two points. A differential equation expresses a local property. Apparent in the main physical theories, it is possible to deduce the fundamental differential equations of the theory from what is called a "principle of extremum," which is an integral one expressing a global property. It stipulates that the system under consideration, passing from a state A to a state B, must follow a line of evolution in such a way as to give to a certain expression L, which is related with the energy of the system, an extremal value, the least possible or the greatest possible one (according to the cases). An extremal principle imposes thus a constraint upon the trajectories or the lines of evolution of a system; in a sense it contains the complete specification of the dynamic possibilities of the system. The formulation of a dynamic theory on the basis of such a principle is thus extremely elegant and provides a high degree of intelligibility, as it reduces all that can be known about the behavior of a system to one simple condition.

          The interesting fact, for our problem, is that this clearly has an aspect of finality which could be formulated in the form of an injunction: "In your passage from state A to state B, follow the path for which the expression L has the greatest value [or, according to the case, the least value]." That injunction could be in turn reformulated as follows: "In passing from state A to state B, adjust your behavior so as to give the highest [least] possible value to the expression L." Here the condition of extremality is presented as a final cause, which explains the behavior of the system. What is actually imposed upon the system is precisely to adopt as maxim of its behavior the realization of the end represented by the condition of extremality.

          That example is particularly significant because it introduces the idea of finality, which plays a fundamental role in Kant's doctrine of judgement. In his Critique of Judgement, Kant introduces a fundamental distinction between two kinds of judgement. In general, judgement is the faculty which enables us to think the particular in the general. It can function according to two modes: by going from the general to the particular or the other way around. In the first case, judgement determines the particular object by subsuming it under the apriority of a law. In the second case, it is in search of an a priori principle which would be able to unify the different empirical laws concerning a particular realm of reality. The "principles of pure understanding" give way to "determining judgements," thanks to which we can obtain a certain rational knowledge of nature. But that determination is quite general. It does not give a real understanding of the diversity of the natural world and in particular of the living forms. It is only by "reflecting" upon the conditions of a rational knowledge of nature in all its details that we can find the concept adequate for such a task. This is the concept of finality, in which we think the unity of all the particular laws of nature such as it would be conceived by a superior understanding. The judgement in which the objects of nature are thus understood under this concept is therefore only a "reflecting judgement," and it has only a subjective value. Kant gives the following definition: "The concept of an object, in as much as it contains also the reason of the reality of that object is named end."⁷

          For Kant it is this concept of finality that forms the basis of the aesthetic judgement as well as the hermeneutic understanding of nature and which gives thus a foundation to the kind of mutual reflection between art and nature. This same concept of finality discloses a certain parallel between the domain of art and mathematics. There are two kinds of representation of the finality of nature, on the one hand, the aesthetic representation and, on the other hand, the "logical" representation: the first is subjective, the second objective. "What, in the representation of an object, is simply subjective, that is to say what constitutes its relation to the subject, not to the object, is its aesthetic nature; but what in the representation is serving or can be serving to the determination of the object (for knowledge) is its logical value."⁸ What is proper to the aesthetic representation is the affective character: "The subjective element of a representation, what cannot become knowledge, is the pleasure or the pain which are attached to it, because they do not give anything to know of the object of the representation, although they can be the effect of some knowledge."⁹ The simple apprehension of the form of an object of intuition, without the intervention of a concept, can give way to a reflective judgement which, at least, "compares that apprehension with its own power to relate intuitions to concepts".¹⁰ "If, in that comparison, imagination (as faculty of a priori intuitions), is in accordance with understanding, as faculty of concepts, . . . and if that agreement produces pleasure, the object must be considered as final for the reflecting judgement. Such a judgement is an aesthetic judgement about the finality of the object which is not founded upon any existing concept of the object and does not provide any one."¹¹

          In the aesthetic representation of finality, this one is represented "only subjectively, by reason of the agreement of its form, when it is apprehended (apprehensio) prior to any concept, with the faculties of knowledge in order to unite intuition and concept in a general knowledge."¹² In the logical representation of finality, this one is represented "objectively, by reason of the agreement of its form with the possibility of the thing itself according to a concept of that thing which precedes and contains the cause of that form."¹³ The first kind of representation "depends upon the immediate pleasure produced by the form of the object, in the pure and simple reflection on that form", whereas the second kind of representation "brings back the form of the object, not to the faculties of knowledge of the subject in apprehension, but to a determined knowledge of the object under a given concept." It therefore has "nothing to do with a feeling of pleasure in the presence of the things, but addresses itself to the understanding for the judgement to be passed on them ".¹⁴

          To this distinction between two forms of representation of finality corresponds a distinction between two kinds of reflecting judgement, the aesthetic judgement, which is "the faculty of judging the formal (subjective) finality thanks to the feeling of pleasure or of pain," and the teleological judgement, which is "the faculty of judging the (objective) finality of nature thanks to understanding and reason."¹⁵ The aesthetic judgement expresses itself in the concept of beauty which contains an intrinsic relation to finality. This is expressed in its third characteristic according to Kant from the point of view of relation: "Beauty is the form of the finality of an object as perceived in it without the representation of an end."¹⁶ He gives the following example: "A flower, for example a tulip, is regarded as beautiful because while perceiving it we encounter a certain finality which, judged as we do, does not relate to any end."¹⁷

          Now a new distinction is introduced in the realm of objective finality: this one can be purely formal or material. The formal objective finality is the kind of finality which we find in mathematics, the material objective finality is the kind of finality which we find in nature. The difference is the difference which separates what is only a determination of space, a pure form of intuition, from what is given in empirical experience. The kind of finality which we find in some mathematical objects is their property of giving the solution of a multitude of problems by a purely a priori necessity without any reference to possible applications. There is a striking contrast between the simplicity of their definition and of their construction and the fecundity of their explanatory and unifying power, which can be seen in the case for example of the circle or of conic sections. It is this capacity of unifying a great variety of particular cases and of disclosing thus the internal connections which make of a mathematical domain an organic whole which produces that harmony which suscitates our admiration and which induces us to attribute to those objects the quality of beauty. But the mathematical objects, being only determinations of space, are not "constitutive properties of external things" but "simple interior forms of representation." The finality which I recognize in a geometrical figure is introduced in it by myself, by drawing it "according to a concept, that is to say, according to my proper form of representation of an external object, which in itself can be any thing."¹⁸ The finality of mathematical objects is thus "not in need of an end as principle nor, consequently, of a teleology."¹⁹

          By contrast, the objective material finality is that kind of finality which can be recognized in existing things, given in the empirical intuition. The unity exhibited by the object, in this case, unifies different particular rules which are synthetic in character and "do not derive from a concept of the object."²⁰ What unites them is an external principle, "distinct from our faculty of representation;"²¹ in the empirical datum. To characterize what can be recognized as "an end of nature," Kant introduces the idea of self‑organization. Comparing a product of nature with a work of art, he underlines the fact that in a natural object each part not only exists for all the others as an instrument or organ -- which could be the case for a work of art -- but must be considered as an organ generating all the others and reciprocally. It is thus "as organized and organizing itself that a natural being can be named end of nature."²²

          We find another suggestive point of view about mathematics in the section of the Critique of Judgement dedicated to the Analytic of the sublime. This analysis constitutes the second part of the theory of the aesthetic judgement. Kant introduces the distinction between the concept of beauty and the concept of sublimity by opposing the character of limitation connected with the idea of finality to the character of infinity connected with the idea of unformedness. "The beauty of nature concerns the form of the object and that form consists in the limitation; but the sublime is encountered also in an unformed object in as much as infinity is represented in this one or thanks to it, while adding moreover by thought the totality of infinity. And this entails that the beautiful seems to serve as presenting an undetermined concept of understanding, but that the sublime serves as presenting an undetermined concept of reason."²³

          The sublime is characterized by an emotion and this one is related by imagination or to the faculty of knowing or to the faculty of desiring. It can thus be attributed to the object either as a mathematical disposition or as a dynamic disposition. The dynamic sublime is what is attributed to nature as a power of arousing fear. The mathematical sublime is defined by Kant as "absolutely large."²⁴ That characterization corresponds exactly to the mathematical concept of infinity. As absolutely large, infinity is beyond any finite determination. An infinite magnitude cannot be "entirely apprehended, but is nevertheless judged . . . as given in its entirety, it demands the totality, thus a comprehension in an intuition."²⁵ The possibility of such a judgement demands "a faculty of the soul exceeding every sensible measure."²⁶ Visibly, we have to do here with the actual infinite, which cannot be effectively given in all its terms but which can be conceived by that faculty in the human mind which is precisely able to pass from a given conditioned to the totality of its conditions. That faculty, reason, is by itself the exigency of the thinking of the totality, in every field of conditioning.

          The history of mathematics shows that it is not only possible to introduce infinity in reasoning in a coherent way, for example by using a process of passing to the limit in a converging series, but to speculate in mathematical terms about what could be called the internal structure of the infinite. The path has been opened by the works of Cantor about the so‑called "transfinite numbers". The decisive step, apparently, has been the discovery by Cantor of the famous diagonal argument, which shows that the cardinality -- discrete measure of size -- of the set of all the parts of a given set, called the "power‑set" of that given set, is greater than the cardinality of that given set. This entails that, once we have a set of a given cardinality, we can pass to higher cardinalities by the operation of taking the power‑set of the given set. So, starting with the set of natural numbers, which is the first type of infinity which we know, and whose cardinality is denoted by the term "denumerable," we can pass to the set of all the parts, or sub‑sets, of this set and obtain thus a higher cardinality called "the continuum." This process can be continued. But after the continuum it becomes purely formal and apparently uninteresting. Cantor has built a theory which opens the possibility of a constructive ascent through an infinite hierarchy of determined infinities.

          The medium of that construction is the "ordinal number." We can consider a natural number as an ordinal: it characterizes the type of order of the sequence made of the preceding ones. For example the number 3, taken as an ordinal, is the type of order of the sequence 0, 1, 2. The different natural numbers, taken as ordinals, characterize thus the types of order of the successive finite sequences of natural numbers (taken themselves as ordinals) belonging to the whole sequence of the natural numbers. They are called "the finite ordinals," and they form a class, called "the first class of ordinals." If we take the whole sequence of the finite ordinal numbers, in their natural order, we can attribute to it a type of order, which is denoted by the Greek letter omega. The cardinality of that sequence is evidently the denumerable. Let us denote that cardinality by the letter aleph with the suffix zero, aleph‑0. Now we can build a new sequence by adding to the sequence of the finite ordinals the new ordinal omega, which is an infinite ordinal (as type of order of an infinite sequence). Let us denote the type of order of that new sequence of ordinals "omega + 1". It is easy to show that the cardinality of that new sequence is again the denumerable. It is thus possible to generate an infinity of infinite sequences of ordinal numbers, each one of those sequences having the cardinality of the denumerable. But they have different types of order and we can thus associate with them different ordinals. We shall say that those ordinals are of the cardinality of the denumerable (aleph‑0), which is the cardinality of the first class of ordinals.

          Those ordinals, having the cardinality of the denumerable, form a class, called "the second class of ordinals." It can be shown that the cardinality of that class is greater than the cardinality of the denumerable. Let us denote that new cardinality by the letter aleph with the suffix 1, aleph‑1. The second class of ordinals, itself of cardinality aleph‑1, is thus formed by all the ordinals of cardinality aleph‑0, each one of those ordinals representing a possibility of ordering a denumerable set of ordinals of the first class, that is, of finite ordinals. That construction can be iterated. We can consider the different possibilities of ordering infinite sets of cardinality aleph‑1, made of ordinals of the second class, that is to say of ordinals of cardinality aleph‑0. We can attach to each of those possibilities a new ordinal. We shall say that it is of cardinality aleph‑1. Those ordinals form a new class, called the third class of ordinals whose cardinality is greater than aleph‑1. We shall denote that cardinality by the letter aleph with the suffix 2, aleph‑2, and so on.

          In general let us say that an ordinal belonging to the nth class, whose cardinality is aleph‑(n ‑1), is of the cardinality of the preceding class, aleph‑(n ‑2). Each ordinal belonging to that nth class represents a possibility of ordering a set of cardinality aleph‑(n ‑2), made itself of ordinals of cardinality aleph-(n ‑3). We can thus say that the cardinality aleph‑n is the cardinality of the class formed by all the ordinals of cardinality aleph-(n ‑1). It is thus possible to generate an indefinitely extensible class of cardinalities, a new cardinality being formed by the ordinals of the preceding cardinality. As a cardinality is denoted by the symbol aleph with an ordinal number as suffix, and as each new cardinality gives birth to a new infinity of ordinal numbers, we have a construction in zigzag, the cardinality sending back to the ordinality and vice versa.

          There is something beautiful in a theory like the general theory of symmetry; there is something sublime in the cantorian theory of the transfinite. Whereas the beautiful "induces in the mind a state of calm contemplation and maintains it," as Kant says,²⁷ the sublime arouses an "emotion"²⁸ which is at the same time one of admiration and of surprise, the breathtaking ascent toward the higher levels of infinity opening ever larger horizons and revealing ever new possibilities for the thought. But precisely what is thus proposed is of the order of possibilities. Those possibilities must be explored and give way to the creation of new objects, which at their turn will manifest new aspects of mathematical beauty. So the beautiful and the sublime are not really separated. They are the two forms of the aesthetic dimension of mathematics.

THE AESTHETIC DIMENSION OF SCIENCE IN GENERAL

          We must now come back to the question of the aesthetic dimension of science in general. Could we say simply that the aesthetic aspect of a scientific theory is due entirely to its mathematical apparatus? First it must be recalled that the role of mathematics is only partial in many theories and highly problematic in the field of the so‑called "human sciences." We could nevertheless consider the case of theoretical physics as paradigmatic for our problem. A scientific theory, in general, is made of a conceptual system endowed with an explanatory power and applicable to a given empirical domain. Such a system can by itself manifest an aesthetic quality, even if it is not expressed in mathematical terms, and even if there are arguments to show that, by its very nature, it is not expressible in mathematical terms. It could be suggested, in such cases, that the foundation of their aesthetic quality lies in some implicit structural properties which are not presented as such, but which operate implicitly quite analogously to the way in which the explicit mathematical structures of theoretical physics operate openly.

          In the case of physical theories we must take account of the fact that a mathematical structure cannot be fruitful unless interpreted in physical terms. This means that there is in such a theory a conceptual apparatus underlying the mathematical representation. It is through that apparatus that a correspondence can be established between the mathematical terms and propositions, on the one hand, and the terms and propositions in which the experimental data can be expressed, on the other hand. The role of the mathematical representation is, as Kant says, to provide "the construction of the concept," it is to say the presentation of the concept in a kind of intuition, which makes it visible and able to be submitted to different kinds of operations. That intuition is very probably not the a priori intuition of Kant, but what could be called a formal intuition, the capture of the intrinsic meaning of formal expressions, for example of the definition of an abstract object and of the operators acting upon such objects.

          The aesthetic quality of a physical theory is the aesthetic aspect of its conceptual frame. As this one is presented in a mathematical apparatus, that aesthetic aspect is actually made visible in that apparatus, and as such it becomes identical with the aesthetic aspect of that apparatus itself. This could justify the idea that it is, indeed, the aesthetic dimension of the mathematical structure used by a theory which confers its aesthetic dimension to that theory. It is must be added that the fruitfulness of a mathematical structure, from the point of view of the applications of the theory in which it is used, contributes to the aesthetic aspect of that structure. There is very often something marvelous and worthy of aesthetic admiration in the way in which a very abstract structure fits the empirical facts. It is told that Einstein said about the equation of Dirac, which is at the basis of the relativistic theory of the electron, "It is a true miracle!". This connection of the theory with an empirical domain is what is proper to a physical theory as such and makes the difference between pure mathematics and mathematical physics. This entails that there is something more, from the point of view of the aesthetic dimension, in a physical theory as such than in the pure mathematical theory which gives the representation of its underlying conceptual frame.

CONCLUSION

          It is time to come to a general conclusion, not in order to close the reflection but on the contrary in order to open it to possible developments. The case of theoretical physics must be considered as particularly suggestive, but from what it suggests we must try to bring to light the fundamental disposition which, in the very texture of the scientific discourse in general, confers to it an aesthetic dimension.

          In this perspective, it would be useful to take support in the analogy of the poem. There are, in a poem, two components which react upon each other to produce the specific kind of resonance proper to the poetic language. On the one hand, there is a dimension of discursivity, according to which the poem conveys an informative meaning. On the other hand, there is a dimension which gives to that meaning a kind of sensible manifestation, which inscribes it in the materiality of the words of which the poem is made. There are two ways of understanding that specific status of the meaning in poetic language. There is a formalist understanding, which retains only that materialization of the meaning achieved by the poetic language and pays particular attention to the musical resources used by that kind of language. On the other hand, there is what could be called a speculative understanding, which attributes to the concretization of the meaning the capacity to give access to an originary word that reveals something of the secret heart of reality and of the ultimate meaning of existence.

          Analogously, there is in the scientific discourse a dimension of discursivity and a dimension which gives to it a similarity with the poem, and which we could call the dimension of poematicity. In its discursive dimension, the scientific discourse makes manifest the intrinsic intelligibility of the world, showing that there is an immanent logos in the cosmos and in ourselves. There is certainly an inadequateness in our representations, but at least they resonate to that intrinsic understandability of the real, and they reveal themselves as attuned, partially in any case, to that intelligibility of the world, capable of producing limited but authentic agreement between themselves and what is given in the field of our experience. That discursive aspect of scientific discourse ought to be interpreted, from the metaphysical point of view, in the light of the great idea of exemplarity. That immanent intelligibility of the world, whereof our mind is able to give account in its proper means of representation, is the trace of a constituting Idea from which it derives by participation. We could remember in this context the concept of vestigium in the thought of Saint Thomas Aquinas.

          But there is also in scientific discourse a poematic dimension in which respect it is susceptible to the two kinds of interpretation of poetic language. On the one hand, it falls within the province of formal interpretation in the measure which it gives, in its models, a concrete form to theoretical ideas. This is particularly the case when the theoretical models are mathematical structures which illumine the meaning of the underlying concepts by focusing on abstract objects, presented in the concreteness of formal representation. On the other hand, scientific discourse, in its poematic dimension, falls also within the province of the speculative interpretation of poetic language. It expresses an originary word, in the measure in which it makes sense of the visible world through the invisible presence of the Word giving it its reality and intelligibility.

          Whereas discursivity reveals the intelligible, poematicity is celebration. In the scientific poem we hear, so to say, the voice of Nature, which is itself celebration. Nature celebrates, by the way it reveals its proper being as partaking of its Source, and thus celebrates the Source itself.

                                                                          NOTES 

          1. Hermann Weyl, Symmetry (Princeton University Press, 1952), p. 77.

          2. Remy Brague, Du temps chez Platon et Aristote. Quatre études, (Coll. Epiméthée) (Paris: Presses Universitaires de France, 1982). See also "Le temps, image mobile de l'éternité" (Platon, Timée, 37 d), pp. 11‑71.

          3. Quotation according to the French translation of Perrault: Vitruve, Les Dix Livres d'Architecture. Traduction intégrale de Claude Perrault, 1673, revue et corrigée sur les textes latins et présentée par André Dalmas (Paris: Balland, 1979), p. 31.

          4. Plotin, Ennéades, VI, 2ème partie, Texte établi et traduit par Emile Bréhier, (Collection des Universités de France), Deuxième édition (Paris: "Les Belles Lettres", 1954).

          5. E. Bréhier, Notice. In: Plotin, op. cit., pp. 15‑16.

          6. E. Bréhier, op. cit., p. 16.

          7. Quotation according to a French translation: Emmanuel Kant, Critique du Jugement, Traduit de l'allemand par J. Gibelin (Paris: Librairie philosophique J. Vrin, 1946), p. 21.

          8. Kant, op. cit., in the Introduction, VII, De la représentation esthétique de la finalité de la nature, p. 28.

          9. Idem.

          10. Ibid., p. 29.

          11. Ibid., p. 21.

          12. Ibid., p. 31.

          13. Ibid.

          14. Ibid.

          15. Ibid., p. 32.

          16. Ibid., p. 67.

          17. Ibid., footnote 1.

          18. Ibid., p. 173.

          19. Ibid., p. 172.

          20. Ibid., p. 173.

          21. Ibid.

          22. Ibid., p. 181.

          23. Ibid., p. 74.

          24. Ibid., p. 77.

          25. Ibid. , p. 83.

          26. Ibid.

          27. Ibid., p. 76.

          28. Ibid.