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Measuring aesthetics

IN a series of three public lectures delivered in 1932, the American mathematician, George David Birkhoff, made a daring ttempt to found a mathematical theory of aesthetics. He came up with a formula to measure the relative importance of aesthetic factors involved in an aesthetic experience.

The aesthetic value or measure M, according to Birkhoff, depends on the elements of Order O (symmetry in sculpture, melody in music, to name a few) and complexity C of an aesthetic object. By complexity he means that which increases a ``feeling of tension'' or ``effort of attention'' involved in the perception of an aesthetic object. The sides of a polygon and notes in melody are cases in point. The beautiful as defined by Hemsterhuis, a Dutch philosopher of the 18th century, is that which gives the greatest number of ideas in the shortest space of time suggested to Birkhoff the formula M = O/C, which he asserted to be analogous to the economic measure of success given by P/I, where P is the profit and I the investment. The formula is thus based on the tacit assumption that beauty increases as complexity decreases.

Birkhoff applied the formula to various polygonal forms, poetry and music, taking into consideration only the formal elements of art. This is because connotative elements ``seem to defy classification since they touch our experience at so many points and in an entirely undefinable way.'' The results convinced him of the formula's validity.

The American poet, Edgar Alan Poe, had perceived mathematical relations in works of art, but the possibility of a mathematical treatment had not occurred to him or anyone with the possible exception of the Swiss mathematician, Leonhard Euler. Euler had almost hit upon the formula with the following description of the aesthetic experience: ``The more easily we perceive the order which characterises the objects contemplated, the more easily and joyfully shall we acknowledge them. But an order which costs trouble to discover, although it will indeed please us, will associate with that pleasure a certain degree of sadness.'' However, Birkhoff's attempt was the first of its kind. He went on to formulate a mathematical theory of ethics as well.

Over 70 years have gone and the formula has been laid on the shelf. Birkhoff's interesting and ambitious attempt was bound to fail as it dispensed with the connotative side of art, which, of course, does not lend itself to formal analysis. Even otherwise, the formula does not rest on a secure foundation. Beauty does increase with order, but to say that it decreases as complexity increases is to assume too much.

Is there not the saying ``more tough the battle, more glorious the victory''? There can be no absolute measure for art. To claim the contrary is to reduce it to a mere set of definitions. And to reduce it to a set of definitions is to seize the poetic licence of the artist!


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