Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
Yourgrau P. A world without time.. the forgotten legacy of Goedel and Einstein (Basic Books, 2005)(ISBN 0465092934)(176s)_PPop_
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system <strong>of</strong> axioms written in a pure formula language that, although having a st<strong>and</strong>ard<br />
semantic interpretation, could be manipulated according to <strong>the</strong> mechanical rules <strong>of</strong> pure<br />
syntax (much like a computer program <strong>of</strong> today). Hilbert's program consisted in finding a<br />
system <strong>of</strong> primitive formulas called axioms from which, according to fixed rules <strong>of</strong><br />
pro<strong>of</strong>órules <strong>of</strong> syntaxóone could derive all <strong>the</strong> <strong>the</strong>orems <strong>of</strong> <strong>the</strong> given ma<strong>the</strong>matical<br />
domain. Two features <strong>of</strong> such a formal system were crucial: consistency <strong>and</strong> completeness.<br />
As a prophylactic against unwelcome surprises, a formal system had to be consistent: two<br />
<strong>the</strong>orems that contradict each o<strong>the</strong>r should not be able to be derived from <strong>the</strong> axioms.<br />
And <strong>the</strong> system should be complete, in <strong>the</strong><br />
sense that all true statements expressible within <strong>the</strong> system (under a suitable<br />
interpretation) should be derivable from <strong>the</strong> axioms. To prevent circularity, <strong>the</strong> system in<br />
which consistency is to be proved must not itself employ any ma<strong>the</strong>matically suspect or<br />
controversial procedures that could render its own consistency suspect. It must be, to use<br />
Hilbert's invented term, not exactly finite but ra<strong>the</strong>r "finitary," in <strong>the</strong> sense that its pro<strong>of</strong>s<br />
must be in principle surveyable by sense experience <strong>and</strong> must not at any point appeal to an<br />
abstract, completed infinity <strong>of</strong> <strong>the</strong> kind proposed by Cantor.<br />
Hilbert's formalism was just one exampleó<strong>the</strong> most rigorous, ma<strong>the</strong>matical oneó<strong>of</strong> <strong>the</strong><br />
spirit that lies at <strong>the</strong> very heart <strong>of</strong> <strong>the</strong> twentieth century. At its core it involves <strong>the</strong><br />
dominance <strong>of</strong> form over content, syntax over semantics, pro<strong>of</strong> over truth. It is no surprise<br />
that <strong>the</strong> principal embodiment <strong>of</strong> a formal system, <strong>the</strong> computer, a pure syntax machine,<br />
would become <strong>the</strong> century's dominant mechanical device. But <strong>the</strong> computer was still just<br />
one element <strong>of</strong> <strong>the</strong> Zeitgeist. In art, science, philosophy, ma<strong>the</strong>matics, music,<br />
architecture <strong>and</strong> linguistics, formalism in its most general sense became <strong>the</strong> dominant<br />
<strong>the</strong>me. In painting, for instance, Cezanne's realism was a hidden case <strong>of</strong> <strong>the</strong> free play <strong>of</strong><br />
geometrical forms, a fact his subjects came increasingly to appreciate as <strong>the</strong>y realized<br />
that <strong>the</strong> geometrical constraints <strong>of</strong> his canvases dominated any attempt to capture <strong>the</strong><br />
shape or spirit <strong>of</strong> those who sat before him. This paved <strong>the</strong> way for <strong>the</strong> explicit rule <strong>of</strong> <strong>the</strong><br />
play <strong>of</strong> free forms by <strong>the</strong> cubists, led by Gris, Braque, <strong>and</strong> Picasso. The Cezanne <strong>of</strong> music<br />
was Brahms, whose post-Romantic chromaticism hid <strong>the</strong> dominance <strong>of</strong> pure logical<br />
structure at its core. Wittgenstein, in whose Vienna home Brahms performed, put <strong>the</strong><br />
matter darkly: In Brahms, he said, "I can begin to hear <strong>the</strong> sound <strong>of</strong> machinery." This<br />
hidden formalism, this logical machinery, was not lost on Brahms's admirer Schoenberg,<br />
who would soon champion <strong>the</strong> freely constructed forms <strong>of</strong> serial music, <strong>the</strong> most explicitly<br />
conventional, ma<strong>the</strong>matical method ever undertaken in music. The principal st<strong>and</strong>ardbearer<br />
<strong>of</strong> Schoenberg's piano music, Glenn Gould, would speak <strong>of</strong> producing a structural "xray"<br />
<strong>of</strong> <strong>the</strong><br />
score in his performances. As <strong>the</strong> high priest <strong>of</strong> bones <strong>without</strong> flesh, Gould made it clear<br />
that his first god was Bach, whom Schoenberg also worshipped, his gospel, <strong>the</strong> fugue.<br />
Nor was physics left behind. On <strong>the</strong> contrary, it was in <strong>the</strong> vanguard. In special relativity,<br />
<strong>Einstein</strong> had ab<strong>and</strong>oned <strong>the</strong> Kantian intuitions <strong>of</strong> space <strong>and</strong> <strong>time</strong> for <strong>the</strong> ma<strong>the</strong>matical<br />
formalism <strong>of</strong> space-<strong>time</strong>, constrained only by <strong>the</strong> formal requirement <strong>of</strong> Lorentz invariance<br />
<strong>and</strong> <strong>the</strong> physical postulate <strong>of</strong> <strong>the</strong> limiting value <strong>of</strong> <strong>the</strong> speed <strong>of</strong> electromagnetic signals.<br />
General relativity, <strong>the</strong> more inclusive <strong>the</strong>ory, would yield an abstract structure governed<br />
by yet more general logical constraints. And <strong>the</strong> cognitive <strong>and</strong> social sciences would follow<br />
physics' lead. Later in <strong>the</strong> century, Noam Chomsky would re-create linguistics as a