Gödel's Theorems from the Point of View of Physicalist Philosophy

• What is logic? What makes  the rules of logic "correct"? What makes a mathematical statement "true"? Mathematical truth vs the truth in physics.

• The formalist philosophy of mathematics vs. mathematical platonism, etc.

• Physicalism in general. The physicalist philosophy of mathematics.

• Introduction to the first order predicate logic: language, axioms, derivation rules, proof, etc. Interpretation and model. Meta-theory.

• Examples for first order axiomatic systems: group theory, Euclidean geometry (Tarski axioms), Peano arithmetic, set theory.

• Gödel's numbering. Representation of meta-theoretic sentences in the object theory. Gödel's first incompleteness theorem (with proof). Gödel's second incompleteness theorem (with proof).

• The usual interpretation of the theorems and their philosophical relevance. Related similar topics: halting problem and computability, self reference and endophysics.

• Criticism of the usual interpretations from a formalist/physicalist point of view.

Grading criteria, specific requirements:

Oral exam from the material of the lectures. Video records and the slides of the lectures will be available.

Required reading:

• J. N. Crossley, et al., What is Mathematical Logic?, Dover Publications, New York, 1990.

• L. E. Szabó: Formal Systems as Physical Objects: A Physicalist Account of Mathematical Truth, International Studies in the Philosophy of Science, 17 (2003) 117. (preprint: PDF)

Suggested further reading:

• K. Gödel: On formally undecidable propositions of principia mathematica and related systems, Oliver and Boyd, Edinburgh, 1962.

• E. Nagel and J. R. Newman: Gödel's Proof, New York Univ. Press, 1958.

• Mathematical facts in a physicalist ontology, Parallel Processing Letters, 22 (2012) 1240009 (12 pages), DOI: 10.1142/S0129626412400099 [preprint]

• A. G. Hamilton: Logic for mathematicians, Cambridge Univ. Press, 1988

• L. E. Szabó: Meaning, Truth, and Physics, In G. Hofer-Szabó, L. Wroński  (eds.),   Making it Formally Explicit, European Studies in Philosophy of Science 6. (Springer International Publishing, 2017) DOI 10.1007/978-3-319-55486-0_9. (Preprint: http://philsci-archive.pitt.edu/14769/)