Sometimes it is the people no one imagines anything of who do the things that no one can imagine - Alan Turing

The "computable" numbers may be described briefly as the real numbers whose expressions as a decimal are calculable by finite means. ...According to my definition, a number is computable if its decimal can be written down by a machine. ...I show that certain large classes of numbers are computable. They include, for instance, the real parts of all s, the real parts of the zeros of the Bessel functions, the numbers π, e, etc. The computable numbers do not, however, include all definable numbers. ...[C]onclusions are reached which are superficially similar to those of Gödel. ...[I]t is shown ...that the Hilbertian can have no solution. In a recent paper ... reaches similar conclusions...

The view that machines cannot give rise to surprises is due, I believe, to a fallacy to which philosophers and mathematicians are particularly subject. This is the assumption that as soon as a fact is presented to a mind all consequences of that fact spring into the mind simultaneously with it. It is a very useful assumption under many circumstances, but one too easily forgets that it is false. A natural consequence of doing so is that one then assumes that there is no virtue in the mere working out of consequences from data and general principles.

"One day ladies will take their computers for walks in the park and tell each other, "My little computer said such a funny thing this morning"." ~ Alan Turing