Article Discussion: Re: "Can One Hear the Sound of a Theorem"

Can One Hear the Sound of a Theorem?  Rob Schneiderman
Notices of the American Mathematical Society August 2011
http://www.ams.org/notices/201107/rtx110700929p.pdf

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In this blog, I believe I find myself generally having a positive tone. Normally I am writing about musicians and concerts, and have been having a great time, and, as I understand it, the general practice of critiquing music, perhaps more so in North America than in Europe, but nonetheless, is to acknowledge what has been given and achieved, and only then perhaps state which elements one might take exception to.

However, from my philosophical studies, I've learned also that the world of Acadamia is happy to enter battles of opinions rather more overtly, without mincing words.

Rob Schneiderman's article is of this nature, and my response to it is equally so.

I originally sent my response to the editor of the AMS, who wrote back to say I would have been invited to submit it for publication, but that I'd written too late. I blame summer/fall musician travels - the mind is elsewhere.

In any case, here goes. I will dig in. Oh, and yes, I include my credentials at the end.

Dear AMS Editor,

I am mystified as to what the point of Schneidermann's rather contrarian article "Can One Hear the Sounds of a Theorem" in the August issue might be, other than that it is very difficult to prove a rigorous correlation between mathematics and music. Certainly, his own examples don't hold up to his kind of scrutiny, any more than the examples of others he takes such pains to shoot down. The article reads more like a rant than a discussion or constructive criticism, and lacks a basic knowledge in the philosophy of aesthetic theory.
Take for example his attempt to distinguish between music/mathematics from other arts and sciences, via assigning "intrinsic meaning". Schneiderman realizes that this is a problematic concept (without a meaning-giver, ie. a person, is there actually meaning?) - and tries to get out of it by whittling down "intrinsic meaning" to "degrees of intrinsic meaning" ... but if a thing doesn't exist, how can you have degrees of it? As anyone who has studied the problems surrounding both formalism and hermeneutics knows, the concept of meaning is very difficult if not impossible to extract from human qualities and experiences. Equally problematic is his assertion regarding reference, that "both mathematics and music frequently do refer to the natural world". Reference, as philosophy of language tells us, is as troublesome a concept as meaning, and in taking the viewpoint that music and mathematics can refer, Schneiderman suddenly adopts a hermeneutic viewpoint that is completely unsupported by anything else in his article, and actually contradicted by his formalist stance elsewhere. Just as unsupported is his opinion that music is somehow more abstract than other art forms, which he defends solely on the basis of his own intuition, "I stand by the claim," and an attempt to describe his definition as "what is special to mathematics and music is that their content is capable of being expressed entirely in terms of their own raw material, namely, logical thought and audible sound". However, this latter sentence is merely a definition of abstraction, which exists in all the other art forms - painting as expressed in terms of interplays of light, dance as expressed in terms of motion, poetry as expressed in uniqueness of form, which may contain jibberish - the Jabberwocky poem of that great mathematician/writer Lewis Carroll comes to mind. Schneidermann tries to get out his dilemma by stating that abstracted poetry is music - but this is just an easy way out: "abstracted anything is mathematics or music", and takes us to a circular definition.
In the end, it appears that he is making the claim that "patterns that do not rely on sensory perception" are the core of his special categorization of mathematics and music. But again, this definition contradicts what he wrote earlier, having included "audible sound" as a "raw material" of music. Setting aside that discrepancy (and the floodgate of the artforms that the "raw materials" admission allows, as discussed above), it is unclear how he is establishing that there is anything that is especially non-sensory about music, other than his claim that it is somehow mathematical.

Another distinction he is missing is that between analysis and creativity, two opposites in terms of attitude. One similiarity between creating/discovering a musical piece and developing a theory along a branch of mathematics is that one sees how far an idea can go. The difference is that mathematical rules are less subjective, most will agree with what is proper or improper procedure, whereas in music one can decide at any moment to do anything - to create - and the result will be either more or less pleasing to various audiences, or up for argument. He dimisses all music that is inspired by mathematics - but where is the mention of some of the greatest "mathematical" composers, for example, Milton Babbitt, Iannis Xenakis, or even Pierre Boulez? There is a distinction between being inspired by and using mathematical constructs in musical composition, and creating new forms utilizing them (as so many 20th-and 21st-century composers do and with great success), and adhering to a rigorous one-to-one mapping between a mathematical concept and some (arbitrarily) chosen correspondence in the sound world, as appears to be the thing Schneiderman is railing against. The beauty and elegance of a composer's meaningful and convincing message using complicated mathematical constructs, is precisely that, on the basis of what appears to touch people, it need not be via one-to-one correspondence, and that we can feel a strong message even if the mathematical idea utilized is hidden. His example of Bach's crab canon is apt here - where the correspondence is too strong, the message we feel is that it is a little too perfect: it works, it tells us something, but there is something not entirely natural-feeling about it. And that is a difference between mathematics and music - we are supposed to be able to work out and understand mathematics to the last degree, and perfection of symmetry can be an end, but music is meant to leave something inexplicable, and also tell us something about ourselves - why do we find ourselves attracted to one form or another?

Perhaps his thesis is actually, that correlation between mathematics and music is merely metaphorical - for precisely the reason that mathematics does not take "raw material" to fill its place-holders, and music does. But if he wishes to claim that music is as pattern-oriented as mathematics, in order to support his claim that music is of an abstract nature, then he has again taken the leap of faith he criticizes in the other authors. Yet perhaps it is precisely this leap of faith that we must take in order to get the "what, if anything, do-music-and-math-have-in-common" project off the ground at all. That would be an interesting idea to develop further - and amazingly, would lead us not to need to reform the school curriculum anywhere as violently as he suggests, for his currently narrow view of what he considers "correct" in this regard would, I posit for the reasons I've mentioned above, open up to many more possibilities.

Sincerely,

Claudia Schaer

Faculty, Bloomingdale School of Music, New York City

D.M.A. Stony Brook University, Violin Performance
Graduate Certificate Equivalent - Philosophy and Music

Aesthetics and Logic studies at Columbia University
MM, BM, The Juilliard School, Violin Performance