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Parallel to the Vienna Circle, or maybe somewhat precursor and behind the scenes to it, was the foundations of mathematics work that went on. Gottlob Frege published several editions of "Foundations of Mathematics." Just before he published his last edition, he got a letter from a then young Bertrand Russell (who was like a father of the positivism of the Circle). It contained the Barber's paradox. Frege had failed to put all of mathematics on a single foundation. So Russell and Whitehead tried it with a theory of types. In their turn, they got a pair of theorems by Kurt Godel, proving the project pointless. In any mathematics complicated enough to have the integers, there will be true facts that cannot be proven. Worse yet, there will be no way to determine which true facts they are.

Oddly, it is the concept of zero -- sunya in the original -- that makes the integers possible.

It seems like you are familiar with fuzzy sets, from the vocabulary you used? I think the 'vagueness' of neural nets is more akin to complexity/chaos -- the notion that you would need infinite precision in a completely deterministic world to be able to predict outcomes with arbitrary precision far enough from the point from which they evolved (known as sensitive dependence on initial conditions).

Great set of essays. Fascinating.

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