Marr’s levels are not levels of abstraction

Abstract: The relationship between mathematical descriptions and the material systems they describe is sometimes framed in terms of abstraction, e.g., some claim that we “abstract away” to a computational description (or model). This frames the relationship between mathematical entities and the physical world in terms of a stepwise ladder that seamlessly links material descriptions with mathematical descriptions through the process of detail omission. However, this tendency to draw the relation between the material world and abstract descriptions in terms of abstraction is a mistake because the process of abstraction does not simply “reveal” the mathematics. To make this case, I use Marr’s levels of analysis to provide a framework for my argument while using color vision as a case study. Finally, I offer a “bridging” theory that may help to connect abstract mathematical descriptions with the material systems that they purport to describe.