Carnap’s Aufbau was an attempt to link the phenomenological analysis of human sense objects to the scientific conception of the world. More precisely, its aim was to come up with a method to translate every scientific sentence into a structured one consisting solely of (i) logical symbols and (ii) terms that refer to the objects “the given” such that (iii) the defined expression and the defining expression of each of the definitions necessarily have the same truth conditions. It is clear from the mention of logical signs and sentences that Carnap’s translated sentence would involve connectives, quantifiers and primitive terms denoting concepts related to human sense experience. Although Carnap’s Aufbau is widely considered to have failed, much of analytic philosophy proceeding Carnap is focused on creating the tools for such a translation. This is clearly seen the in the study of modal logics. Modal logics attempt to extend the standard logical vocabulary of \(\wedge,\vee,\neg,\forall,\exists\) with modal operators that capture certain abstractions used in everyday language. An example of this is epistemic logic, where propositions can be embedded in operators representing “a knows that” and “a believes that” (where a is an epistemic agent) in a manner similar to the classical operator \(\forall\). Carnap’s Aufbau parallels the development of model theory, a methodology for studying mathematical theories. However, since 1945, a new methodology for studying mathematical theories has emerged called Category Theory. Category Theory has its advantages over model-theoretic methods in achieving the Aufbau’s goal, because the fundamental abstractions in category theory are objects (analogous to physical objects) and morphisms (analogous to conceptual relations between objects) whereas quantifiers, connectives, and modal operators are not so intimately connected with the scientific conception of the physical world.

Ideally, I would like to think of the universe of objects apparent to my experience as lying in the (pseudo-mathematically precise) category of human sense objects. For which analyzing aspects of these physical objects require functorially mapping this category to some category of mathematical objects. To sketch this vision: I have a coffee mug sitting in front of me available to me through my visual sense. Suppose I would like to analyze this coffee mug’s deformation properties, then I would map this coffee mug to the category of topological spaces that describes the topology of the coffee mug. Using the tools in the category of topological spaces, I would find that the topological space of the coffee mug is homeomorphic to the torus. Now I can conclude that given the resources to make a coffee cup deform continuously, I can deform the coffee mug to a donut.

Of course, the topological properties of the mug are not the only properties of the mug I might care about. Suppose I would like to calculate the volume that this mug could hold. Volume is a statement about space, so to do this I would consider the functor from the category of human sense objects to the category of measure spaces. Specifically, the space around the mug would correspond to the Lebesgue measure that measures subsets of Euclidean space \(\mathbb{R}^{3}\) (for which the mug is a definable subset of Euclidean space).

In this view, these functors from the category of human sense objects, capture Carnap’s original idea of a translation from an imprecise sentence to logio-mathematical sentence. In presenting such a functor, the connection between a physical object and a mathematical model for such an object is clearly stated.

Edited by: Hunan Rosotmyan