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: 9:5012:15, 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

Sequential Outcomes

Empirical Context

Topological Operators

Interior = “will be verified”

Exterior = “will be refuted”

Closure = “will never be refuted”

Boundary = “will never be decided”

Frontier = “false but will never be refuted”


Hume’s Problem of Induction

Popper’s Problem of Metaphysics

Topological Complexity

Open = “verifiable”

Closed = “refutable”

Clopen = “decidable”

Locally Closed = “is a model”

Sigmaconstructible = “is a paradigm”


More Examples

Asynchronous Outcomes

The Sleeping Scientist

Continuous Variables

Laws


Methods

Concepts of Learning

Topological Characterizations of Learnability

Examples

The Negative Induction from the History of Science

Ockham’s Razor
6.1. Empirical Simplicity
6.2. Ockham Methods
6.3. Justification

Scientific Contexts

The Golden Triangle

Metaphysical Similarity

Metric Topology and Metrizability

Realism, Empiricism, and Instrumentalism

Underdetermination

Topological Separation Axioms

Theoretical Identifications
8.1. Morning and Evening Stars
8.2. Light and Electromagnetic Waves
9. NoMiracles Arguments
10. Ideals and Sigmaideals
11. Nowhere Density and Meagerness

Anything Goes

Statistical Coherentism and Reliabilism

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