The Magnificent Disaster
This chapter discusses a programme for restoring certainty in mathematics. It addresses the establishment of meta-mathematics through which one could try to demonstrate that mathematics is without contradictions. Meta-mathematics should establish a critical mathematical investigation of mathematics itself. This strategy for a mathematical self-critique was formulated by Hilbert, who did not, as Frege and Russell had suggested, try to establish a secure foundation for mathematics in logic. Hilbert advocates a different way out of the foundational crisis caused by the appearance of logical paradoxes in the heart of mathematics. The occurrence of such paradoxes indicates that one has allowed oneself too much: drawn conclusions that are not valid, used axioms that give rise to inconsistencies, or the like. Hilbert wants to analyse the mathematical axiomatics and proof methods in order to ensure that they, when properly chosen, cannot lead to contradictions.
This programme appears extremely ambitious, for how can one ensure that one cannot, sometime in the future, end up in deducing contradictions in some mathematical theories? Mathematics is continually developing, and one can imagine that in the future, one may come to prove theorems that contradict what is proved today. One thing is to guard against already known contradictions, but how can one guards against future and new contradictions? The programme is not only ambitious; it turns out to be unrealistic as well. This unrealism is shown by Gödel, who demonstrates the impossibility of proving the consistency of a mathematical system that includes a certain degree of complexity. With the insight provided by Gödel’s two incompleteness theorems, the meta-mathematical programme becomes revealed as an illusion.
KeywordsAxiomatisation Completeness Consistency Decidability Finitary methods Foundational crisis Gödel’s first incompleteness theorem Gödel’s second incompleteness theorem Independence Meta-mathematics
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