The Topos of Music: Geometric Logic of Concepts, Theory, and Performance

by Guerino Mazzola

Man kann einen jeden BegrifJ, einen jeden Titel, darunter viele Erkenntnisse gehoren, einen logischen Ort nennen. Immanuel Kant [258, p. B 324] This book's title subject, The Topos of Music, has been chosen to communicate a double First, the Greek word "topos" (r01rex; = location, site) alludes to the logical and transcendental location of the concept of music in the sense of Aristotle's [20, 592] and Kant's [258, p. B 324] topic. This view deals with the question of where music is situated as a concept­ and hence with the underlying ontological What is the type of being and existence of music? The second message is a more technical understanding insofar as the system of musical signs can be associated with the mathematical theory of topoi, which realizes a powerful synthesis of geometric and logical theories. It laid the foundation of a thorough geometrization of logic and has been successful in central issues of algebraic geometry (Grothendieck, Deligne), independence proofs and intuitionistic logic (Cohen, Lawvere, Kripke). But this second message is intimately entwined with the first since the present concept framework of the musical sign system is technically based on topos theory, so the topos of music receives its top os-theoretic foundation. In this perspective, the double message of the book's title in fact condenses to a unified to unite philosophical insight with mathematical explicitness.

Genres: Music, Mathematics, Theory, Philosophy, Geometry

1335 pages, Hardcover

First published January 1, 2003

Reviews

[J C]

Honestly speaking, I have no idea what this is about, but I became interested in algebraic geometry after reading about a series of papers developing an inscrutably abstract reformulation of algebraic geometry, dubbed 'inter-universal Teichmüller theory' by its creator, Shinichi Mochizuki, who later quite audaciously (but perhaps justifiably) compared the mathematical world's bewilderment at his work with the general public's own bewilderment with mathematics.

I love a good abstraction, which is why I hate linear algebra proofs, and was pleased to find out that the whole field of algebraic geometry was apparently built around highly abstract notions, largely due to an fiery, irreverent and brilliant young mathematician named Grothendieck, so much so that these notions in some sense are more primal than sets (or something like that). Which is why these notions could be applied in fields like logic, and thus computer science.

Apparently 'Homotopy Type Theory' is such an application, which is a reformulation of the foundations of mathematics, borrowing language from algebraic geometry and category theory and skirting around the notion of 'sets'. I got the impression that it was a variant of intuitionism, a movement to make mathematics more rigorous by allowing only constructive proofs, (i.e. forbidding the 'excluded middle', the notion that things, if not true, must be false, and vice versa) which fell out of favour with the mathematical community by the 1930s or so. Apparently, this way of formulating mathematics lends all theorems to a purely computational method of proof.

I love fundamentals, of anything, because one can truly approach it as a beginner and ask (and even start answering) expert questions immediately. There isn't prerequisite knowledge to make you feel inexperienced or stupid. On the other hand, fundamentals are both fun and important; fun because you get to see how things unravel from psychological and philosophical roots into a rich formal system; important because where you begin limits where you can end up.

I have also been interested, since before I can remember, in what music IS. No matter how many rather experienced musicians I've asked about what harmony IS (well just nice ratios of waves whose lack of discordance your brain likes) they've just looked at me with puzzlement... I'm sure there are many other questions about music that've occurred to me which have very interesting answers.

So when I was looking for something interesting to read, I was suprised that this book popped up on my google search 'mathematics of music' uniting, apparently, all these interesting intellectual activities. I guess I want to read it, but only after I've done enough algebraic geometry, which is only after I've done commutative algebra, after I've done abstract algebra, complex analysis, analytical geometry...

Plus it costs ~200$, weighs in at 1500 pages (normally that wouldn't stop me) and I can't get it second-hand. So it will have to wait a while I guess...