Hilberts Programme

Above we referred to the reductionist view of a hierarchy of sciences, with the higher levels related to the lower levels through deductive logical reasoning. The structure of mathematical logic itself therefore becomes an interesting issue even for empirical science.

In order to answer such questions as whether the set of all sets was included in itself or not, Russell and Whitehead in 1910-13 published their monumental work Principia Mathematica. The book attempted to be an encyclopaedia of the whole body of mathematical logic.

David Hilbert then proposed the even more grandiose research programme that bears his name: To prove that the contents of the Principia were both internally consistent, and provided a set of tools by which all true mathematical theorems could be proved, if only in principle.

The idea was not to provide a proof machine which could replace the inventive ingenuity of human scientists, only in all modesty to provide a complete set of tools for their use. Whether anybody would be ingenious enough to reconstruct the original "wonderfully simple" proof of Fermat's Last Theorem after 300 years is still an open question, even once Wiles has published his much admired but enormously elaborate proof.

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