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.

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The lecture slides will be available in a pdf file.


Suggested readings

• 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.

• A matematika filozófiája a 21.század küszöbén. Válogatott tanulmányok, Szerk. Csaba Ferenc, Osiris, Bp. 2003

• 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)

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

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

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