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

system of axioms written in a pure formula language that, although having a standard
semantic interpretation, could be manipulated according to the mechanical rules of pure
syntax (much like a computer program of today). Hilbert's program consisted in finding a
system of primitive formulas called axioms from which, according to fixed rules of
proofórules of syntaxóone could derive all the theorems of the given mathematical
domain. Two features of such a formal system were crucial: consistency and completeness.
As a prophylactic against unwelcome surprises, a formal system had to be consistent: two
theorems that contradict each other should not be able to be derived from the axioms.
And the system should be complete, in the
sense that all true statements expressible within the system (under a suitable
interpretation) should be derivable from the axioms. To prevent circularity, the system in
which consistency is to be proved must not itself employ any mathematically suspect or
controversial procedures that could render its own consistency suspect. It must be, to use
Hilbert's invented term, not exactly finite but rather "finitary," in the sense that its proofs
must be in principle surveyable by sense experience and must not at any point appeal to an
abstract, completed infinity of the kind proposed by Cantor.
Hilbert's formalism was just one exampleóthe most rigorous, mathematical oneóof the
spirit that lies at the very heart of the twentieth century. At its core it involves the
dominance of form over content, syntax over semantics, proof over truth. It is no surprise
that the principal embodiment of a formal system, the computer, a pure syntax machine,
would become the century's dominant mechanical device. But the computer was still just
one element of the Zeitgeist. In art, science, philosophy, mathematics, music,
architecture and linguistics, formalism in its most general sense became the dominant
theme. In painting, for instance, Cezanne's realism was a hidden case of the free play of
geometrical forms, a fact his subjects came increasingly to appreciate as they realized
that the geometrical constraints of his canvases dominated any attempt to capture the
shape or spirit of those who sat before him. This paved the way for the explicit rule of the
play of free forms by the cubists, led by Gris, Braque, and Picasso. The Cezanne of music
was Brahms, whose post-Romantic chromaticism hid the dominance of pure logical
structure at its core. Wittgenstein, in whose Vienna home Brahms performed, put the
matter darkly: In Brahms, he said, "I can begin to hear the sound of machinery." This
hidden formalism, this logical machinery, was not lost on Brahms's admirer Schoenberg,
who would soon champion the freely constructed forms of serial music, the most explicitly
conventional, mathematical method ever undertaken in music. The principal standardbearer
of Schoenberg's piano music, Glenn Gould, would speak of producing a structural "xray"
of the
score in his performances. As the high priest of bones without flesh, Gould made it clear
that his first god was Bach, whom Schoenberg also worshipped, his gospel, the fugue.
Nor was physics left behind. On the contrary, it was in the vanguard. In special relativity,
Einstein had abandoned the Kantian intuitions of space and time for the mathematical
formalism of space-time, constrained only by the formal requirement of Lorentz invariance
and the physical postulate of the limiting value of the speed of electromagnetic signals.
General relativity, the more inclusive theory, would yield an abstract structure governed
by yet more general logical constraints. And the cognitive and social sciences would follow
physics' lead. Later in the century, Noam Chomsky would re-create linguistics as a