1. Plato and Aristotle on Mathematics Kareem Khalifa Philosophy Department Middlebury College

2. Overview 1. Plato 2. Aristotle

3. 1. Plato 1.1. Ontology 1.1.1. Argument

4. Argument on p.55 P1.Geometrical statements are about points that have no dimensions, lines that can be perfectly straight and have no breadth, and circles that are perfectly round. P2.In the physical world (i.e. the world of Becoming), points have dimensions, lines cannot be perfectly straight and have no breadth, and circles are not perfectly round. C.  Geometrical statements are not about the physical world (from P1, P2).

5. 1. More Plato 1.2. Semantics 1.3. Epistemology 1.3.1. Argument 1.4. Applicability

6. 2. Aristotle’s ontology 2.1. Ontology 2.1.1. Realist/abstractionist 2.1.2. Fictionalist 2.1.3. Platonist challenges

7. 2. More Aristotle 2.2. Semantics 2.3. Epistemology 2.3.1. Frege vs. abstractionism 2.4. Applicability

8. Frege’s argument P1.If abstraction is the means by which we achieve arithmetic knowledge, then to count is to count objects that have no distinguishing properties whatsoever. P2. For all objects x and y, if x and y have no distinguishing properties, then x and y cannot be differentiated from each other. P3.For all x and y, if x and y cannot be differentiated from each other, then it is possible but unknowable that x = y. P4.For all x and y, if it is possible but unknowable that x = y, then it is unknowable whether x and y are one or two objects. P5. If, for all x and y, it is unknowable whether x and y are one or two objects, then counting is impossible. P6. Counting is possible. C1.  Abstraction is not the means by which we achieve arithmetic knowledge. (from P1-P6)