Abstract This contribution is a short postscriptum to The Topos of Music (Mazzola, 2002) and deals with those background strategies which have profiled the book’s overall concern. This one relates to what Kant (1957) qualifies as the mathematical knowledge by reason from concept construction. Our approach to this constructivist perspective shares the nature of Galois theory: to understand new concepts as extensions of given concept frameworks via specific >equations< and their Galois groups. The »formal ontology« of our approach is described in the language of form semiotics over given topoi as described in Mazzola (2002).

1 Introduction

The switch from the original title »Geometrie der Töne« of (Mazzola, 1990) to the title »The Topos of Music« of (Mazzola, 2002) is not only due to the topos-theoretic generalization of the structural setup of mathematical music theory, it more radically testifies a change of methodological principles. The former book was meant as a mathematical description of spaces, models, and theorems about musical objects and facts.

Here, modules, categories and geometric spaces were just a language to restate musical subjects in precise terms of prefigured mathematics. In this framework, the construction of conceptual entities was not a theme per se, but a prescientific activity as it is standard in other applications of mathematics to determined fields of knowledge, such as physics, economics, or psychology.

The progressive infiltration of what they now call mathematical music theory (à défaut de mieux, but what in fact is much more than just a special type of music theory, namely a geometric logic of concepts, theory, and performance) by conceptual construction issues was induced by the programming work in the context of the RUBATO project from 1992 to 2002 as conducted by the author in collaboration with Oliver Zahorka, Thomas Noll, Jörg Garbers, Stefan Göller, Stefan Müller, and Gérard Milmeister. This research taught us that music needs universal concept architectures. We learned this on the level of object-oriented programming, but such engineering procedure is everything less than a proof, and this is why the theoretical counterpart in Mazzola (2002) had to be developed to build the mathematical formalism backing those programming activities.