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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What Godel discovered, however, was that not only are <strong>the</strong> Peano postulates in fact<br />
incomplete, any system <strong>of</strong> axioms or postulates (even if infinitely large) from which<br />
arithmetic can be derived that satisfies any reasonable ma<strong>the</strong>matical criteria <strong>of</strong><br />
surveyability by a finite mind is <strong>of</strong> necessity incomplete. (An infinite mind, like God's,<br />
which can grasp all <strong>the</strong> numbers at once, presumably has no need <strong>of</strong> axioms.) So <strong>the</strong><br />
simplest <strong>and</strong> most basic domain <strong>of</strong> ma<strong>the</strong>matics, <strong>the</strong> arithmetic <strong>of</strong> <strong>the</strong> natural numbers,<br />
<strong>the</strong> rock on which <strong>the</strong> gr<strong>and</strong> edifice <strong>of</strong> ma<strong>the</strong>matics st<strong>and</strong>s, turns out to be, from a formal<br />
axiomatic point <strong>of</strong> view, incomplete, <strong>and</strong> even worse, incompletable. Indeed, since a<br />
computer can prove only <strong>the</strong>orems based on <strong>the</strong> axioms its programmer has fed into itóit<br />
cannot, as Godel emphasized, create new axioms on its ownóit follows that in principle no<br />
computer or fully specified system <strong>of</strong> computers, even if infinite, will ever capture all <strong>the</strong><br />
truths <strong>of</strong> arithmetic (never mind <strong>the</strong> rest <strong>of</strong> ma<strong>the</strong>matics). As Godel put it, "Continued<br />
appeals to ma<strong>the</strong>matical intuition are necessary . . . for <strong>the</strong> solution <strong>of</strong> <strong>the</strong> problems <strong>of</strong><br />
finitary number <strong>the</strong>ory."<br />
The ma<strong>the</strong>matical fact <strong>of</strong> <strong>the</strong> incompleteness <strong>of</strong> formal arithmetic, moreover, is accessible<br />
not only to us, thinkers with minds <strong>and</strong> ma<strong>the</strong>matical intuitions; ironically, a computer can<br />
be programmed to prove Godel's <strong>the</strong>orems, <strong>the</strong> very <strong>the</strong>orems that establish <strong>the</strong> intrinsic<br />
limitations <strong>of</strong> computers. The truths <strong>of</strong> arithmetic, <strong>the</strong>n, cannot in principle be confined to<br />
a formal system. Here is a crucial difference between truth <strong>and</strong> pro<strong>of</strong>: a ma<strong>the</strong>matical<br />
pro<strong>of</strong>, in <strong>the</strong> sense in which we are discussing it here, is always a pro<strong>of</strong> in, <strong>and</strong> relative to,<br />
a given formal system, whereas truth, as such, is absolute. What Godel proved is that<br />
ma<strong>the</strong>matical truth is not reducible to (formal or mechanical) pro<strong>of</strong>. Syntax cannot<br />
supplant semantics. The leitmotif <strong>of</strong> <strong>the</strong> twentieth century, it turns out, st<strong>and</strong>s in need <strong>of</strong><br />
revision. Mechanical rules cannot obviate <strong>the</strong> need for meaning, <strong>and</strong> what gives us access<br />
to meaning,<br />
namely, intuition, cannot be dispensed with even in ma<strong>the</strong>matics, indeed, even in<br />
arithmetic. This was <strong>the</strong> first nail in Hilbert's c<strong>of</strong>fin.<br />
The second nail was not long in coming. Godel soon proved his second incompleteness<br />
<strong>the</strong>orem, which demonstrated, with yet fur<strong>the</strong>r irony, that if a given system <strong>of</strong> axioms for<br />
arithmetic were in fact consistent, <strong>the</strong>n it could not be proved consistent by <strong>the</strong> system<br />
itself. Put o<strong>the</strong>rwise, only an inconsistent formal system can prove its own consistency!<br />
Von Neumann, <strong>the</strong> quickest <strong>of</strong> <strong>the</strong> quick, having heard Godel announce his incompleteness<br />
results, derived, shortly <strong>the</strong>reafter, <strong>the</strong> unprovability <strong>of</strong> consistency. "I would be very<br />
much interested," he wrote Godel, "to hear your views on this. ... If you are interested, I<br />
will send you <strong>the</strong> pro<strong>of</strong> details." One can imagine his disappointment when Godel informed<br />
him that <strong>the</strong> manuscript for <strong>the</strong> second <strong>the</strong>orem was already on its way to <strong>the</strong> editors. It<br />
was Von Neumann, however, who argued, against Godel himself, that <strong>the</strong> unprovability <strong>of</strong><br />
consistency, as Godel had demonstrated it, left no wiggle room for <strong>the</strong> Hilbert program.<br />
Whereas for several years, Godel was cautious not to prejudge <strong>the</strong> question <strong>of</strong> whe<strong>the</strong>r<br />
Hilbert might discover a finitary pro<strong>of</strong> <strong>of</strong> consistency to which Godel's second <strong>the</strong>orem did<br />
not apply, Von Neumann, from <strong>the</strong> beginning, was confident that this could never happen.<br />
Assuming that one rejects Russell's controversial "axiom <strong>of</strong> reducibility," he said, "one<br />
cannot obtain a foundation for classical ma<strong>the</strong>matics via logical means." Von Neumann's<br />
striking prescience, however, concerning <strong>the</strong> full significance <strong>of</strong> what Godel had<br />
discovered may well have served only to deepen his regret that he had not been <strong>the</strong> first<br />
to make <strong>the</strong>se discoveries. Even <strong>the</strong> fact that he was one <strong>of</strong> <strong>the</strong> fa<strong>the</strong>rs <strong>of</strong> <strong>the</strong> modern