What then is Mathematics? Truth, objectivity, metaphor, and the embodiment of axiom systems

Mathematics is a unique body of knowledge. The very entities that constitute what mathematics is are idealized mental abstractions that cannot be perceived directly through the senses (e.g., a Euclidean point is dimensionless and cannot be actually perceived!). So, what kind of thing is then mathematics? In this talk I will address this question from the perspective of the embodied mind. I want to show how the inferential organization of mathematics emerges from everyday cognitive mechanisms of human sense-making and imagination, realized via mechanisms such as metaphor. I will concentrate on the fundamental concept of axiom, and through the analysis of hypersets—a specific branch of contemporary set theory—and I will show how the quintessential abstract conceptual system we call mathematics (1) emerges from embodied cognitive mechanisms for imagination such as conceptual metaphor; (2) that truth and objectivity comes out of the collective use of these mechanisms; (3) that it can have domains that are internally consistent but mutually inconsistent, (4) and that these domains built on corresponding axiom systems that while grounded in embodied meaning provide different “truths” and inferential organization. Finally, I will show that these properties are not unique to mathematics but that they exist in everyday abstract conceptual systems as well. I will illustrate this point with empirical observations from my investigation contrasting spatial construals of time in the western world with that in the Aymara culture of the Andes’ highlands. I’ll defend the idea that everyday conceptual systems posses elementary embodied forms of “truth”, “axioms”, and “theorems” that are “objective” within the communities that operate with them. These properties of ordinary human imagination serve as grounding for developing more complex and refined forms of abstraction, which find the most sublime form in mathematics.