## FORTH FORUM MATHEMATIQUE DIDEROT

### Mathématic and Music

### Mathematical logic and musical logic in XX siecle.

Paris, Ircam, 3-4 december 1999.

Mathematical logic, musical logic.
One notices a double evolution in the principle of the 20th century:
logic has become more mathematical, thereby loosing its ancient status of
philosophical discipline; the question of a musical logic has become explicit
as a search for a "coherence" specific to music. This musical
evolution is contemporary with the end of tonality and of thematism which
were the principles that until then, ensured the coherence of musical works.
Tonality and thematism certainly implicated specific logic, which could
possibly be formalised, but which were often based on natural rather than
axiomatic foundations: tonality was based on physics, and thematism was
based on psychology. The "logical" dimensions therefore stayed,
for a large part, dependant on ontological foundations, or foundations in
terms of musical being (key and theme). In the 20th century, composers found
themselves before an ontological void. They had to make decisions that didn't
come from physics and psychology in an obvious way anymore, but which remained
all the same the starting or articulation points of symbolical calculations,
expressing an internal logic of the musical form and material. As for logic,
it looks at progressions that are universally valid due to not being attached
to such or such position of existence. By becoming more mathematical it
has also acquired added calculation power, trying with success formal reasoning
on itself, one emblematic result of which is the non-fulfilment theorem
by Gödel. Mathematics (or its theories) itself could start where axioms
of existence intervene, such as those of the theory of ensembles. We may
then ask ourselves the following questions: do formalisms built on musical
"reasoning" (the mixed ensemble of its rationality, decisions,
aims, and of its calculations, deductions and progressions) have anything
to do with such or such formalism of logic? What formal coherence, described
in a logical way, can exist in a work, beyond the arbitrary of singular
esthetical decisions? Can logic, expressed mathematically help musicians
to clarify the specificity of their way of reasoning? Taking the problem
the other way round, can musical logic be a source of inspiration for mathematicians?

From a text by François Nicolas.