Kant & Mill on Mathematics
Kareem Khalifa
Philosophy Department
Middlebury College
Overview
1. Historical background
(let’s skip this)
2. Kant
3. Mill
2.1. Kant: Key Concepts
2.1.1. Analytic vs. synthetic statements
2.1.2. Concepts vs. intuitions
2.1.3. 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:
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• Auxiliary construction
– Step 1: ||||||||
– Step 2: |||||||||
…
– Step 5: ||||||||||||
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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. Ontology
2.4. Semantics
2.5. Applicability
3. Mill
3.1. Epistemology
3.2. Ontology
3.3. Semantics
3.4. Applicability