Mathematical Intuition

1. The Concept of Intuition in Mathematics.- 1. Introduction.- 2. Knowledge, Evidence, and Intuition.- 3. Intuition "of" and Intuition "that".- 4. Some Recent Views of Mathematical Intuition.- 5. Hilbert and Bernays.- 6. Parsons.- 7. Brouwer.- 8. Some "Extended" Proof-Theoretic Views.- 9. Godel on Sets.- 10. Platonism and Constructivism.- 11. Mathematical Truth and Mathematical Knowledge.- 12. Principal Objections to Mathematical Intuition.- 2. The Phenomenological View of Intuition.- 1. Introduction.- 2. Intentionality and Intuition.- 3. Intuition of Abstract Objects.- 4. Acts of Abstraction and Abstract Objects.- 5. Acts of Reflection.- 6. Types and Degrees of Evidence.- 7. Comparison with Kant.- 8. Intuition and the Theory of Meaning.- 3. Perception.- 1. Introduction.- 2. Sequences of Perceptual Acts.- 3. The Horizon of Perceptual Acts.- 4. The Possibilities of Perception.- 5. The "Determinable X" in Perception and Indexicals.- 6. Perceptual Evidence.- 7. Phenomenological Reduction and the Problem of Realism / Idealism.- 4. Mathematical Intuition.- 1. Introduction.- 2. Objections About Analogies Between Perceptual and Mathematical Intuition.- 3. Objections Based on Structuralism.- 4. Objections About Founding.- 5. A Logic Compatible With Mathematical Intuition and the Notion of Construction.- 6. Is Classical Mathematics to be Rejected?.- 5. Natural Numbers I.- 1. Introduction.- 2. The Concept of Number Cannot Be Explicitly Defined.- 3. The Origin of the Concept of Number.- 4. Intuition of Natural Numbers.- 5. Ordinals.- 6. Ordinals and Cardinals.- 7. Constructing Units and the Role of Reflection and Abstraction.- 8. Syntax and Representations of Numbers.- 6. Natural Numbers II.- 1. Introduction.- 2. 0 and 1.- 3. Numbers Formed by Arithmetic Operations.- 4. Small Numbers and Singular Statements About Them.- 5. Large Numbers and Mathematical Induction.- 6. The Possibilities of Intuition.- 7. Summary of the Argument for Large Numbers.- 8. Further Comments on Mathematical Induction.- 9. Intuition and Axioms of Elementary Number Theory.- 7. Finite sets.- 1. Introduction.- 2. A Theory of Finite Sets.- 3. The Origin of the Concept of Finite Set.- 4. Intuition of Finite Sets.- 5. Comparison with Godel and Wang.- 6. Unit Sets, the Empty Set, and Mereology vs. Set Theory.- 7. Large Sets and a Hierarchy of Sets.- 8. Illusion in Set Theory.- 9. Concluding Remarks.- 8. Critical Reflections and Conclusion.- 1. Introduction.- 2. Summary of the Account.- 3. Areas for Further Work.- 4. Platonism, Constructivism, and Benacerraf's Dilemma.- Notes.