## Aesthetics

In my opinion it is much more reasonable to regard optimization in science as the result of application of an aesthetic rather than of a metaphysical principle.

Few scientists explicitly admitted the influence of aesthetic principles. However, for instance the great mathematician Hermann Weyl in a frank interview admitted that he preferred the beautiful even to the true (probably in terms of what he liked to publish, maybe an elegant conjecture rather than a messy proof). In the same spirit the physicist Paul Adrien Maurice Dirac, inventor of the Dirac Function which greatly facilitated to widen the class of tractable partial differential equations in Physics, wrote that it was more important to "have beauty in one's equations than to have them fit experiment".

In this context it also is appropriate to recall that Poincare considered aesthetics to be a more important element in scientific creativity than even logic.

The optimality issue is a little more confused in social science, where conscious individuals are considered, and to whom optimising behaviour is imputed or could be imputed. Often social scientists leave it open whether their theoretical agents optimise or only act as if they optimise something. This is a comfortable position, and the issue is not simplified when supra-individual optimisers such as the invisible hand also come into play.

Anyhow, aesthetics does not involve any kind of danger for leading science astray, as long as empirical check of the consequences is there. Before formulating his three famous laws of planetary motion, Johannes Kepler tried to organize the planetary orbits on spheres inscribed in and inscribing the five regular polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron, and the ikosahedron. As there for geometrical reasons only exist these shapes with triangular, square, and pentagonal facets, he was particularly satisfied with the fact that the sequence of supporting shells for these polyhedra corresponded to the number of known planets. Unfortunately, this beautiful theory did not fit the facts, and Kepler himself discarded it in favour of his later laws for the elliptic orbits, which as we know led to the development of Newton's theory of gravitation.

In 1990 "The Mathematical Intelligencer" announced an outright beauty contest for mathematical theorems among its readership. Among the official candidates were two theorems by Euler:

where V is the number of vertices, F is the number of faces, and E is the number of edges in a polyhedron, and the series formula:

22 32 42 6

Neither of them did win the prize, which was awarded to:

There have been several attempts to define beauty in terms of various characteristics pertaining to the aspects of simplicity and complexity. We should note that none of those would capture what is really at issue here, because proof is an integral part of the theorem, and, for instance the Euler series

Fig. 5.6: Kepler's first attempt at cosmology, "Mysterium Cosmographicum"of1596. A cube is incribed in a sphere, which inscribes another spehre, in its turn inscribing a tetrahedron. The sphere inscribed in this further inscribes an dohekahedron (with twelve pentagonal faces), and so it goes, until all the Platonic solids have been used. Kepler was delighted that the total number of spherical shells equalled the number of known planets, and tried to organize the planetary system in terms of orbits on these shells. The outmost shell was supposed to contain Saturn's orbit (Pluto being unknown as yet), and the next nested one would contain Jupiter's. Kepler was struck by the fact that the ratio of the radii of these two shells so well approximated the ratio of the actual orbit diameters for these planets. Unfortunately, the rest was not recocileable with facts, so Kepler abandoned this beautiful theory to favour of the elliptic orbits.

Fig. 5.6: Kepler's first attempt at cosmology, "Mysterium Cosmographicum"of1596. A cube is incribed in a sphere, which inscribes another spehre, in its turn inscribing a tetrahedron. The sphere inscribed in this further inscribes an dohekahedron (with twelve pentagonal faces), and so it goes, until all the Platonic solids have been used. Kepler was delighted that the total number of spherical shells equalled the number of known planets, and tried to organize the planetary system in terms of orbits on these shells. The outmost shell was supposed to contain Saturn's orbit (Pluto being unknown as yet), and the next nested one would contain Jupiter's. Kepler was struck by the fact that the ratio of the radii of these two shells so well approximated the ratio of the actual orbit diameters for these planets. Unfortunately, the rest was not recocileable with facts, so Kepler abandoned this beautiful theory to favour of the elliptic orbits.

would lose all its attraction if it was just a definition of n in terms of a series.

Such a "dynamic" interpretation of the concept of beauty in science also perspires through the discussion of this topic in the remarkable book "The Mathematical Experience" by Davis and Hersh. The feeling of beauty is attributed to the alternation between feelings of tension and relief related the complex and the simple, and to the realization of expectations. Note how close the experience of following mathematical reasoning comes to listening to a piece of music with its alternation between dissonance and consonance.

In general, it is true that, as long as science is allowed to apply its idealization procedures and to use its theoretical concepts, as long is it admissible for the scientists to incorporate either aesthetic or metaphysical principles to their taste. As already stressed, other procedures of science guarantee the relation to the facts of reality.

The internal development forces of science, however immaterial they may seem to its purpose, have proved extremely important. They save science from the dull fate of becoming mere collection and classification of facts, and they are responsible for almost all scientific development up to now.

There is no guarantee at all that a systematically planned research programme, such as we nowadays, since the successful experience of spacecraft research planning, are more and more aiming at, will in general be as successful as the traditional.

Maybe even planning fits in the list aesthetic principles, admittedly given an absurdly bureaucratic feel for beauty. Here, however, is an answer from a responsible Chinese "main stream" mathematician from the days of the Cultural Revolution to inquiries from a visiting committee of distinguished US mathematicians:

"There is no theory of beauty that people agree upon. Some people think one thing is beautiful, some another. Socialist construction is a beautiful thing and stimulates people here. Before the cultural revolution some of us believed in the beauty of mathematics but failed to solve practical problems; now we deal with water and gas pipes, cables, and rolling mills. We do it for the country and the workers appreciate it. It is a beautiful feeling."

The quote again comes from the most enjoyable "The Mathematical Experience" by Davis and Hersh, referred to already.

Fig. 5.7: Symmetric icon designed by Clifford Pickover. Many recent pictures of fractal geometrical objects emphasize the aesthetic properties of pictures to a degree that makes the borderline between science and art floating; The intriguing detail of many fractal images and the infinite variation between the regular and the irregular has a strange attraction, especially when complex patterns are the outcome of very simple algorithms. The reverse of this is the recent science of image compression which attempts at finding the appropriate algorithms needed to reproduce any given picture. The driving force behind this is the attempt to save storage space for picture files, as it takes much less to store the "recipe" for a picture than to store the picture itself, but it cannot be excluded that something can be found out about aesthetic properties through the algorithms for their generation.

Fig. 5.7: Symmetric icon designed by Clifford Pickover. Many recent pictures of fractal geometrical objects emphasize the aesthetic properties of pictures to a degree that makes the borderline between science and art floating; The intriguing detail of many fractal images and the infinite variation between the regular and the irregular has a strange attraction, especially when complex patterns are the outcome of very simple algorithms. The reverse of this is the recent science of image compression which attempts at finding the appropriate algorithms needed to reproduce any given picture. The driving force behind this is the attempt to save storage space for picture files, as it takes much less to store the "recipe" for a picture than to store the picture itself, but it cannot be excluded that something can be found out about aesthetic properties through the algorithms for their generation.

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