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Mathematical intelligence, infinity and machines: beyond Godelitis

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Abstract:

We informally discuss some recent results on the incompleteness of formal systems. These theorems, which are of great importance to contemporary mathematical epistemology, are proved using a variety of conceptual tools provably stronger than those of finitary axiomatisations. Those tools require no mathematical ontology, but rather constitute particularly concrete human constructions and acts of comprehending infinity and space rooted in different forms of knowledge. We shall also discuss, albeit very briefly, the mathematical intelligence both of God and of computers. We hope in this manner to help the reader overcome formalist reductionism, while avoiding naive Platonist ontologies, typical symptoms of Godelitis which affected many in the last seventy years.

Keywords: Mathematics; cognitive foundation; combinatorial incompleteness; conceptual constructions; incompleteness of formal systems; informal insight; proof-methods

Document Type: Research Article

Affiliations: CNRS and Dept. De Mathematiques et Informatique, Ecole Normale Superieure, Paris, France.

Publication date: 1999-11-01

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