Course Description
Probability and inductive inference seem so closely tied that it is almost impossible to discuss one without the other. This course develops the provocative, alternative view that the underlying structure of the problem of induction and the true path to its epistemic justification are topological rather than probabilistic. Whereas the usual motive for probabilism since Hume is coherence (conclusions “fit together” in an extended logical sense), topology is more intimately connected with reliability and learnability, an alternative foundational perspective in epistemology. The idea is that empirical verifiability, refutability and convergence to the truth are all ultimately topological concepts. That is directly the case when inquiry is literally driven by propositional information about the world under study. Surprisingly, perhaps, it is also true within a broadly frequentist statistical framework for inductive inference.
- Lecturer: Kevin Kelly (Carnegie Mellon University)
- TA: Dong Huanfang (董焕防)
- Time: 9:50-12:15, 26 June – 30 June
- Venue: 一教201
Schedule
1. Some Philosophy of Science
1.1. Coherentism and Reliabilism
1.2. Contextualism
i. Information
ii. Question
iii. Similarity
1.3. Optimalism
1.4. Verifiability and Hume’s Problem of Induction
2. Why Topology?
3. Worlds and Propositions
3.1. Distribution and DeMorgan Laws b. Duality
4. Propositional Structures
4.1. Homeomorphism
4.2. Types of Propositional Structures
4.3. Operations on Propositional Structures
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Sequential Outcomes
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Empirical Context
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Topological Operators
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Interior = “will be verified”
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Exterior = “will be refuted”
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Closure = “will never be refuted”
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Boundary = “will never be decided”
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Frontier = “false but will never be refuted”
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Hume’s Problem of Induction
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Popper’s Problem of Metaphysics
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Topological Complexity
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Open = “verifiable”
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Closed = “refutable”
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Clopen = “decidable”
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Locally Closed = “is a model”
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Sigma-constructible = “is a paradigm”
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More Examples
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Asynchronous Outcomes
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The Sleeping Scientist
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Continuous Variables
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Laws
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Methods
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Concepts of Learning
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Topological Characterizations of Learnability
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Examples
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The Negative Induction from the History of Science
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Ockham’s Razor
6.1. Empirical Simplicity
6.2. Ockham Methods
6.3. Justification
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Scientific Contexts
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The Golden Triangle
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Metaphysical Similarity
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Metric Topology and Metrizability
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Realism, Empiricism, and Instrumentalism
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Underdetermination
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Topological Separation Axioms
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Theoretical Identifications
8.1. Morning and Evening Stars
8.2. Light and Electromagnetic Waves
9. No-Miracles Arguments
10. Ideals and Sigma-ideals
11. Nowhere Density and Meagerness
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Anything Goes
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Statistical Coherentism and Reliabilism
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Stochastic Contexts
3.1. Sigma Fields b. Parameters c. Chances
4. Statistical Learning
4.1. In Chance
4.2. Almost Sure
5. The Topology of Statistical Verifiability
6. Virtual Information States
7. Reduction of Statistical Learnability to Propositional Learnability
8. Ockham’s Statistical Razor
9. Examples
Lecture Slides
Please find the slides here
Recordings
Please find the recordings here