Philosophy of Mathematics

The first incompleteness theorem, demonstrated by K. Gödel in 1931, reveals a hiatus between the concepts of truth and proof. Two divergent philosophical interpretations of this hiatus can be proposed, depending on whether prevalence is given to the model-theoretic or the proof-theoretic approach. The model-theoretic approach suggests that the concept of proof fails to properly capture the concept of mathematical truth. There are statements which are true according to the standard interpretation of natural numbers and yet cannot be proved. The proof-theoretic approach suggests that the concept of proof reflects the intrinsic ambiguity of the concept of mathematical truth. There are statements which, although true according to the standard interpretation of natural numbers, cannot be considered as absolutely true because of the existence of non standard interpretations. The course analyzes the first incompleteness theorem and its philosophical consequences for the relationship between truth and proof in mathematics.