A well-known dualism in cognitive science is the distinction between top-down and bottom-up processing. These features of cognition are often linked to concepts and emotions, or the systems of dual process theory (System 1 and System 2).
In relation to the thesis of this book, bottom-up processing is encapsulated by the mathematics of set theory, and top-down is encapsulated by mereology. The first is the science of making sets from elements, and the later is the science of making parts from wholes. In both cases, sets and parts are understood in terms of some mental or physical space (that space is an essential and a priori feature of mind has been argued by scholars such as Longchenpa and Kant).
From a mathematical perspective, either bottom-up or top-down would be sufficient, since either framework (mereology or set theory) is provably complete in a mathematical sense. In fact, mereology and set theory were formulated independently at about the same time in history, and set theory “survived” in mathematics because it was felt that only one of the two was necessary. From a psychological perspective, however, it is clear that cognition relies heavily on both bottom-up and top-down processing. Therefore, if we wish to bring mathematics in line with cognitive science, we must give an account of the top-down process.
Cognitive Set Theory is a first attempt to do so. In analogy to the point, the starting point of point-set topology, we have posited “everything”, the building block of mereotopology. In that attempt, bottom-up processes were assumed to operate “prior” to top-down processes, but that is probably disputable. The significant distinction seems to be that the top-down process is conceptual, and as such, is better modeled by a non-transitive structure such as sets (whereas the parallel-processing bottom-up structure requires a transitive structure: parts).
In a nutshell, that is how bottom-up and top-down processing might be modeled as mathematical spaces. Excuse the lack of formality of this essay, but I have been intrigued by recent work in mathematics and psychology, and I believe it is important to clarify from the start of such an endeavor how mathematics is both suitable and unsuitable in its current formulation.