Kant argued that arithmetic is synthetic. He claimed that the predicate "=12" is not contained in the subject "7+5" of the statement "7+5=12."

The formalists in mathematics—especially David Hilbert and Gottlob Frege, and all who followed in their wake—rejected that claim about arithmetic and mathematics, holding mathematics and arithmetic to be either formal (Hilbert) or reducible to logic and set theory (Frege). The empiricists, especially Hume and the Logical Empiricists (Logical Positivists) and their followers (most of whom considered themselves to be intellectual descendants of Hume) rejected the claim that there are any possible statements of any form that are synthetic a priori. So for the empiricists and logical positivists, there are only two kinds of statements, analytic ones and synthetic ones; moreover, they claimed, all analytic statements are a priori and all synthetic statements are a posteriori, so analytic = a priori, and synthetic = a posteriori.

The notion that arithmetic can be a formal system was refuted, however, by the work of Czechoslovakian mathematician-logician Kurt Gödel (1906 – 1978). In what has come to be known as Gödel's Proof he showed that the axiomatic method, when applied to the arithmetic of cardinal numbers, cannot show both the consistency and the completeness of the axiomatized system. In other words, simple arithmetic cannot be reduced to or comprehended in an axiomatic system. Given any set of axioms for arithmetic, there are true statements of arithmetic that cannot be derived from those axioms. In addition, no proof of the formal consistency of such a set of axioms is possible.

Whether or not Gödel's Proof refutes the claim that arithmetic is analytic can be debated. But at the least Gödel's Proof shows that arithmetic cannot be reduced to or comprehended in a formal axiomatic system. Although it may not prove it, this does tend to lend support to Kant's claim that the statements of arithmetic are synthetic propositions (assuming that the analytic-synthetic distinction can be maintained; an assumption that is suspect after the work of Quine.)

]]>Colossus and Guardian are not trying to formally prove anything, just communicate. I am sure if the book was written now there would be technobabble about unbreakable codes and such.

Besides, I am sure that if Godel had entered into the movie's plot it would be to say that Colossus and Guardian had proved him wrong!

]]>I'm afraid that Godel Incompleteness wouldn't have any bearing on the story that I can say. It *doesn't* say, for example, that the computers become omniscient, or that they learn all of mathematics in its entirety.

There might be a problem though in that a purely mathematical language working from "first principles" would only be able to make extremely generic statements about mathematics itself. The computers wouldn't be able to talk about the U.S. or the USSR, or each other or their creators or anything regarding history or politics or biology or really anything concrete, specific, or physical.

https://en.wikipedia.org/wiki/Colossus:_The_Forbin_Project

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"Colossus requests to be linked to Guardian. The President allows this, hoping to determine the Soviet machine's capability. The Soviets also agree to the experiment. Surprising everyone, Colossus and Guardian begin to slowly communicate using mathematics. Even more surprising, the two systems' communications quickly evolve to complex mathematics far beyond human comprehension and speed, whereupon the two machine complexes become synchronized using a communication protocol that no human can interpret."

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Math-type people — wouldn't Gödel's incompleteness theorem prevent this (or anything derived from "first principles") from actually happening? (Looking for a "yes" answer here (!)).

At least two smart phones would not have access to the nuclear arsenal, or would they? ]]>