Kant & Mill on Mathematics

Kareem KhalifaPhilosophy Department

Middlebury College

Overview

1. Historical background (let’s skip this)

2. Kant3. Mill

2.1. Kant: Key Concepts

2.1.1. Analytic vs. synthetic statements2.1.2. Concepts vs. intuitions2.1.3. The a priori

Example of “conceptual analysis” & analytic truths

1. All vixens are foxes.2. For all x, x is a vixen if and only if x is a female

fox.3. All female foxes are foxes.• For all x, if x is F and x is G, then x is G.

2.2. Kant’s epistemology

2.2.1. How is mathematical

knowledge synthetic?

Intuition and counting

• Enunciation: 7+5 = 12• Ecthesis/setting-out:

||||||| |||||• Auxiliary construction– Step 1: |||||||| ||||– Step 2: ||||||||| |||

…– Step 5: ||||||||||||

2.2. Kant’s epistemology, continued

2.2.2. Why is mathematical knowledge a priori?2.2.3. What are objective structures of space &

time?2.2.4. Why isn’t mathematical knowledge

analytic?

The rest of Kant’s philosophy of math

2.3. Ontology2.4. Semantics2.5. Applicability

3. Mill

3.1. Epistemology3.2. Ontology3.3. Semantics3.4. Applicability