In this contribution, we describe the presently most complete version of this mathematical formalism. It has two main components: the first of them is a basis of mathematical objects which we take for granted from classical mathematical theory. To guarantee all necessary structural and logical constructions required in musical conceptualization, this basis must be a topos . Typically, this topos is the topos of sets, or the topos of presheaves over the category of modules and diaffine maps. The latter category is sufficiently powerful to englobe all usually envisaged structures in Mathematical Music Theory and is therefore central to the exposition in Mazzola (2002). However, performance theory in its generalization to gestural dynamics needs topoi that are related to differentiable structures. Therefore the limitation to a single topos is neither preconized from practice nor mandatory in theory.

The second component provides us with the mechanism for concept construction. This mechanism is a recursive one in the sense that already construed concepts are used to build new ones by virtue of universal tools, such as limits, colimits, and power objects. It is not clear whether other than topos-theoretic »universal« constructions are required for future developments. We admit however a fundamental extension of classical recursion: Our recursive construction process includes circular concepts, i.e., the objects being under construction are defined by use of universal tools when applied to--among others--these same objects .

Although in such a context, existence theorems are crucial, we do not stress the mathematical aspect. Rather it is our concern to communicate the fundamentally philosophical relevance of this methodology. Kant characterizes the mathematical method by its constructivist nature: A concept must be understood from its construction and not by pure philosophical meditation (Kant, 1957, A713-B741). This construction is of a significant nature: A new concept is defined as a solution of a defining >equation<. Here, equation means that we have to solve functional correspondences between determined domains of concepts, coupled with equations of topos-theoretic nature: limits, colimits, and power objects.

The point is that such solutions do not exist within the given concept domains, they generate proper conceptual extensions, such as the extension of the real numbers to the complex numbers, stemming from a solution of the equation , which over the reals is impossible. We have learned to handle such extensions as plain solution spaces, i.e., algebraic field extensions. But they are in fact conceptual extensions which were brought to life under hard existential struggles. We claim that any fundamental mathematical progress is due to conceptual extensions which enable solutions of hitherto unsolvable >equations<. Recall that one of the more recent dramatic events in this development was Deligne’s solution of the Weil conjectures in view of the generalized concept of a topological space as proposed and developed by Grothendieck.

We contend that conceptual extensions are precisely what Grothendieck in his autobiographic Récoltes et semailles (Grothendieck, 1985) alludes to when explaining his method of smoothening and eventually dissolving the hard surface of a coconut in tepid water under the sun’s patient warmth. What happens is that a manifest identity--the coconut’s firm shape--is being dissolved under the osmose of its negation, not under the brute force destruction of its identity. Let us look