The second, symmetries, and also fractals were used in musical composition, they appear also in nature and play a crucial role in mathematics as well as in physics.

The third, the Yoneda philosophy, says that in order to understand an object, just walk around it. This means understanding by changing perspectives. In mathematics, the Yoneda Lemma has important implications in homological algebra, algebraic topology and algebraic geometry just to name a few. It says that a mathematical object can be classified up to isomorphism by its functor. In music, the score is only its first view and together with all the interpretations constitutes its identity. What a marvelous point of view for both the interpreter and the audience. It takes away all sterile competition from art and science as if they were an Olympic game.

Recently, in his Status Quo 2000 paper (c.f. Mazzola, 2000, 2004, in this volume, which we highly appreciated when presented to the world in México during a splendid plenary exposition), he explains why his geometrical theoretical approach of that time evolved into a framework that is suitable for many musical problems. His new framework is based on much more sophisticated mathematics such as topos theory.

As to performance, Guerino Mazzola says in his articles, » Music performance has been studied from the aesthetical and psycho-physiological point of view« . He worked in developing a performance theory that describes the structures and processes defining a performance, »that without proper tools, performance theory would remain« (and I love this phrase) »a branch of literature in the spirit of music criticism« . »But the possibility of exhibiting algebraic grammatical varieties has a profound consequence for the classification problem of performances. So, comparative criticism becomes a field of precise research and no longer a sector of literature« .

Very recently, Mazzola produced a classification of musical objects, that is, »there is an algebraic scheme whose rational points represent certain isomorphism classes of global compositions« . »Classification means the task of totally understanding an object. This is Yoneda’s Lemma in its full philosophical implication« . »Understanding works of art means a synthesis of all their interpretative perspectives« .

6 General Aesthetical and Ethical Considerations on Mathematics and Music.

And well, what relationships are there between music and mathematics? Or equivalently, what connection or correspondence is there? We have seen how mathematical concepts have been applied several years ago and recently (coming after all from nature or from man’s abstract thought, etc.) to the entertainment with a game of dice, to aesthetics, to musical composition and to creating a precise language for musicology and music among others. Certainly, there are many more music fields where mathematics appear in order to have understanding like in the