Redefining Reality - The Intellectual Implications of Modern Science

is true, then it can’t be proven, but if it is false, then it would be
provable. It is a contradiction if we adopt Russell’s view that
mathematical truth and provability are the same thing.
Because sentences about mathematics are mapped onto
mathematical equations, then there would be truths of
mathematics that are unprovable. No mathematical system
could be both complete—contain all truths—and sound—
contain only truths. Gödel showed that logic wasn’t enough
to justify mathematical truth.
Turing took Gödel’s work and crossed it with what Blaise Pascal
and Wilhelm von Leibniz had done in the 17 th century with
computing machines. Turing believed that if there was a rule, then
a machine could be built to follow the rule. In this way, he came up
with the idea for what we now know as .
If Gödel had succeeded in translating talk about math into
mathematical equations and we had machines that could solve
mathematical equations, then we could develop machines that
could do proofs. On the basis of this reasoning, Turing believed we
could have machines do the abstract work for which we think we
need human mathematicians.
Just as this question became interesting, the British declared war
on the Germans. The British were able to intercept German radio
transmissions, but the Germans had developed the most complex
encryption code ever created, the Enigma code. It was widely
thought to be unbreakable.
The code, however, was a translation problem similar to what
Turing had seen in Gödel. Turing had worked on the question in
terms of abstract machines. All he had to do was actually build one
task, but he built it, and it worked. Turing’s machine cracked the
German code.
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