Music, Mathematics, Phenomenology
26 February 2010
The formality of music and the aesthetic dimension of mathematics have long been parallel themes. There are many wonderful quotes from the literature of music to this effect. I quoted several of my favorites in Algorithms of Ecstasy, which I encourage the curious reader to peruse.
I have long thought that Die Kunst der Fuge is not unlike L’Art de Penser. Music is a formal language, but it is also more than this. I see in both music and mathematics the same dialectic of the formal and the informal. The sensuousness of music blinds many to its formal elements. When Schopenhauer said that music is the pure language of the will, unmediated by representation, free of the forms which dominate the other arts, he obviously was not thinking of the formal rules of composition. The industriousness with which Bach develops and exploits a theme through counterpoint, inversion, retrograde, and retrograde-inversion has more to do with the intellect than with the will. A more recent example of composition as an intellectual exercise is afforded by serialism.
In contrast to the sensuality of music that often blinds us to its formal elements, the abstractness of mathematics blinds many to its non-formal, intuitive elements ; although, for example, axiomatics (a distinctively mathematical mode of thought at least since Euclid) forces us to recognize the intuitive foundations of any mathematical theory right from the start, with its primitive terms, axioms, and rules of inference which must be accepted in order to begin. However, after this intuitive foundation, all that follows is formal, and it is the formality of the axiomatic method which is widely understood to be it distinctive contribution to mathematical thought.
The Greeks were eminently suited to unfold formal reasoning to the world, given their preoccupation with the virtues of limitation, finitude, order—peras. Indeed, those qualities which shared the right side of the Pythagorean table of opposites with peras—all that is sharply and clearly defined—represent all of the properties upon which mature formal systems have converged. When logical thought at long last began to catch up with logical practices Frege gave eloquent expression to these same concepts necessary to the development of formalism: “rigour of proof, precise delimitation of extent of validity, and as a means to this, sharp definition of concepts.” (Frege, Foundations of Arithmetic, § 1) While intended as an assertion of the demands of formal thinking, it could also serve as a formalist aesthetic manifesto.
Both mathematics and music are developed with the same eye to aesthetic purity, and as such they stand, more than many human endeavors, outside the causal order, a little bit outside the world, outside existence. Thus it ought to be natural to approach mathematics phenomenologically, suspending the world through the epoché, disregarding existence. And yet I have not read anything which considers what I think are the genuine issues of a phenomenological philosophy of mathematics. The heavily ontological nature of most contemporary discussions in the philosophy of mathematics—Are numbers objects? Do they exist? Etc.—is a preoccupation which prevents a phenomenological perspective from being heard.
Suppose we interpret the epoché as the suspension of any consideration of existence, that the natural standpoint naively assumes the existence of familiar objects and these unthinking judgements are precisely those which need to be set aside: what then remains of today’s ontological philosophy of mathematics? Very little, I think. Let us approach a phenomenological philosophy of mathematics taking the epoché seriously, and defining it in some way which does not involve us in disputes as to the possibility of having some kind or other of subjective experience (i.e., we need to avoid allowing the epoché itself to be a problem). By this I mean that we need some kind of formal or quantifiable definition of the epoché, and here I will simply take it as ruling out any reference to existence. Thus traditionally troublesome issues in the philosophy of mathematics, such as whether sets exist, are ruled out from the beginning. The question which has so vexed formulations of the axiom of choice—whether there exists a set which consists of an infinite number of members, each element taken out of an infinite number of sets—cannot even be a question in a phenomenological context.
I think the above suggests a fresh way of thinking about mathematics. Thus by the method of the phenomenological epoché we arrive at a position not unlike that of early analytical philosophy which simply ruled out large classes of traditional philosophical questions by finding them meaningless. (And we should keep in mind in this context the close association of early analytical philosophy with logicism.)
As unlikely as it may seem, this line of thought may have applications to music as well. Recently in Another Kind of Auditor I mentioned my enjoyment of and experience of late medieval and early renaissance vocal polyphony, specifically in relation to Monteverdi’s madrigals. I recently found an amusing quote that demonstrates apparently incommensurable differences in taste:
“The vigor of the new age was not found everywhere. Music, still lost in the blurry mists of the Dark Ages, was a Renaissance laggard; the motets, pslams, and Masses heard each Sabbath — many of them by Josquin des Pres of Flanders, the most celebrated composer of his day — fall dissonantly on the ears of those familiar with the soaring orchestral works which would captivate Europe in the centuries ahead, a reminder that in some respects one age will forever remain inscrutable to others.” (William Manchester, A World Lit Only by Fire, p. 88)
William Manchester judges this music to be “laggard” while I consider it to be one of the high points of civilization, even a symbol of civilization. Well, we all know the old Latin line, de gustibus non est disputandum. In any case, in Another Kind of Auditor I made the claim that this music that I love, and that William Manchester believes to be “laggard,” refuses any participation of the listener. I further expanded on this observation:
One cannot “sing along” with a Monteverdi madrigal. One cannot even tap one’s toe in time to the music, or sway one’s body to the rhythm. Such gestures are futile and inappropriate. One must listen only. One must become an ear, nothing but an ear — a pure auditor. Monteverdi’s madrigals hold the auditor at a distance even while enveloping him in layers of vocal textures.
I could have said that Monteverdi’s late madrigals lack any instrumentality whatsoever. There is nothing that we can do with them, no “purpose” (in the vulgar sense) to which they can be used. With such music there is a complete absence of readiness-to-hand (to employ a Heideggerian turn of phrase — and hopefully more on this at a later date).
Another way to formulate this unique character of the music would be to say that the character of the music itself forces the auditor into an intellectual position not unlike the phenomenological epoché. Indeed, the music itself could be taken as a method of the epoché, a particularly systematic and thorough method for attaining to a consciousness in which the music that one hears “disappears” in terms of any utilitarian or instrumental presence and becomes something that can only be beheld. To hear such music is to forget the name of the thing one hears.
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