Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_

words: "It is not true, as Kant urged, that intuition is a pure a priori means of knowledge,
but rather that it is a force of habit rooted in psychological inertia." And the patron saint,
as we have seen, of the Vienna Circle was Wittgenstein, its bible, the Tractatus, which
declared that the only real facts are of the empirical variety. "The essence of this view,"
Godel noted, "is that there exists no such thing as a mathematical fact." Mathematics, for
Wittgenstein, consists merely of the transformations of formulas to obtain mathematical
identities. In particular, it is not derived from mathematical facts, whereas physics is
derived from facts about the physical world.
Between Hilbert and Wittgenstein, it seemed, the positivists had finally laid to rest the
ghost of mathematics, which had seemed to resist incorporation into their fold. The ghost,
however, was hardly dead, as Godel would demonstrate from within the heart of
Wittgenstein country, Vienna. His incompleteness theorem was a grenade aimed at Hilbert
that landed in the very laps of the positivists. Vienna's circle could not after all be
completed.
4 A Spy in the House of Logic
Every spy's life has ended in ignominious death. ANAIS NIN
Russell's Paradox, communicated to Frege in 1902, engendered widespread fear in
mathematical circles. As Godel put it, in retrospect, "[Russell] brought to light the amazing
fact that our logical intuitions . . . are self-contradictory." On pain of contradiction, it
could no longer be assumed that every property determines the class of things that have
that property. It could still be trusted that the property of redness sufficed to determine
the set of all red things, but such confidence was no longer justified for every property.
The concept of class or set, in particular, which had assumed increasing importance in
mathematics and logic and which seemed intuitively clear, turned out to be so poorly
understood that in Frege's epoch-making formulation of modern logic it led to a
straightforward contradiction. The concern was not a communist under every bed but a
paradox asleep on every mathematical bedspread.
There was no place to hide. Frege had built a contradiction into the very foundation stone
of the mighty edifice of logic. Cantor, the creator of modern set theory, had also
constructed, if not contradictions, then at least paradoxes aplenty in his dizzying hierarchy
of transfinite numbers. Even Euclidean geometry, the ancestor of all models of
mathematical rigor and certainty, the soft pillow for two thousand years of sweet
mathematical dreams, had become suspect due to
the recent demonstration of the logical consistency of non-Euclidean geometries. If
Euclidean geometry could be rejected without contradiction as the truth about space (as
arithmetic is the truth about natural numbers), wherein did its truth lie? Everywhere,
intuition was under suspicion. If the house of logic was not itself secure, where else could
safety be found? Something had to be done.