Mathematical Internal Realism
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Tim Button
Abstract
In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of proof, and hence is incomplete. Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. This also allows us to understand why “Models are not lost noumenal waifs looking for someone to name them,” but “constructions within our theory itself,” with “names from birth.”
Abstract
In “Models and Reality” (1980), Putnam sketched a version of his internal realism as it might arise in the philosophy of mathematics. Here, I will develop that sketch. By combining Putnam’s model-theoretic arguments with Dummett’s reflections on Gödelian incompleteness, we arrive at (what I call) the Skolem-Gödel Antinomy. In brief: our mathematical concepts are perfectly precise; however, these perfectly precise mathematical concepts are manifested and acquired via a formal theory, which is understood in terms of a computable system of proof, and hence is incomplete. Whilst this might initially seem strange, I show how internal categoricity results for arithmetic and set theory allow us to face up to this Antinomy. This also allows us to understand why “Models are not lost noumenal waifs looking for someone to name them,” but “constructions within our theory itself,” with “names from birth.”
Chapters in this book
- Frontmatter I
- Contents V
- List of Abbreviations VII
- An Introduction to Hilary Putnam 1
- Introduction to this Volume 47
- Putnam’s Proof Revisited 63
- Language, Meaning, and Context Sensitivity: Confronting a “Moving-Target” 89
- Externalism and the First-Person Perspective 107
- Putnam on Trans-Theoretical Terms and Contextual Apriority 131
- Mathematical Internal Realism 157
- The Labyrinth of Quantum Logic 183
- Fulfillability, Instability, and Incompleteness 207
- Putnam’s Aristotle 227
- Davidson and Putnam on the Antinomy of Free Will 249
- Putnam on Radical Scepticism: Wittgenstein, Cavell, and Occasion- Sensitive Semantics 263
- Natural Laws and Human Language 289
- Balance in The Golden Bowl: Attuning Philosophy and Literary Criticism 309
- Bibliography 331
- Contributors 349
- Index 353
Chapters in this book
- Frontmatter I
- Contents V
- List of Abbreviations VII
- An Introduction to Hilary Putnam 1
- Introduction to this Volume 47
- Putnam’s Proof Revisited 63
- Language, Meaning, and Context Sensitivity: Confronting a “Moving-Target” 89
- Externalism and the First-Person Perspective 107
- Putnam on Trans-Theoretical Terms and Contextual Apriority 131
- Mathematical Internal Realism 157
- The Labyrinth of Quantum Logic 183
- Fulfillability, Instability, and Incompleteness 207
- Putnam’s Aristotle 227
- Davidson and Putnam on the Antinomy of Free Will 249
- Putnam on Radical Scepticism: Wittgenstein, Cavell, and Occasion- Sensitive Semantics 263
- Natural Laws and Human Language 289
- Balance in The Golden Bowl: Attuning Philosophy and Literary Criticism 309
- Bibliography 331
- Contributors 349
- Index 353
