Aristotle on Mathematics and Abstraction

One of the papers I enjoyed writing most in my short time studying philosophy at the University of Cincinnati was on Aristotle's philosophy of mathematics: Aristotle on Mathematics and Abstraction, written under the direction of Dr. Lawrence Jost.

Although I'm not a great mathematician, mathematics is a fascinating topic to study, and "El Maestro" had a lot to say about the fundamental question of what numbers are and why/how they are capable of accurately representing relationships and entities in the physical world:

Mathematics has been studied by philosophers for millennia, in an attempt to answer two fundamental questions:

What is the nature of mathematical objects? (The Ontological Question.)
What is mathematical reasoning about? (The Epistemological Question.)

Mathematics, in an attempt to answer these two questions, has been broadly subdivided into two disciplines: arithmetic and geometry. Arithmetic deals with the concepts of units, numbers, and the manipulations of these concepts, while geometry deals with the concepts of points, lines, and solids in one or more dimensions.

The purpose of this paper is to show that Aristotle's view of mathematics, in relation to the whole of the mathematical sciences, can be described as a series of descriptive conceptual frameworks abstracted directly from human sensory perceptions of the physical world.

…

Aristotle… recognizes that number terms, which are based on this process of unit-abstraction, do not exist as actual properties of objects in the world. Rather, units (and the numbers enumerated by such units) do exist as useful fictitious concepts that, while they do not exist qua ("as a") unit or number, are valid because they are derived from objects or properties that do actually exist in the physical world. This object and property-independent conception of unit enumeration explains why it is possible to count entities of many different substances and natures in the same process; the unit-perspective is object and property-neutral. Thus, you can validly count any number of objects that actually exist in reality as different substances, like gold molecules, cars, sharks, apples, kangaroos, and clouds in the same manner because each object is considered qua indivisible, and the remaining properties of these objects can be completely ignored.

This same recognition of fictitiousness applies to geometry as well. When the geometer considers an object qua solid, he or she is not positing that this solid actually exists apart from the existence of the object of study. Rather, the geometer is merely ignoring all properties of the object that are not necessary to consider the object qua solid, and focusing only on those that do: the object qua solid does not need to exist separately, as the geometer merely needs to abstract from the physical world that lies before his or her eyes.

Aristotle's conception of mathematics as abstractions from physical objects and their properties was a major advance in the human understanding of reality: it explained how mathematical principles are able to relate directly to the natural world and how humans, as "systematic understanders of the world," are able to grasp them.