Pages

Friday, April 04, 2008

Kurt Godel

He turned the lens of mathematics on itself and hit upon his famous "incompleteness theorem" — driving a stake through the heart of formalism By DOUGLAS HOFSTADTER


Source:ALFRED EISENSTAEDT/TIME LIFE PICTURES-Kurt Godel at the Institute of Advanced Study See: The Time 100-Scientists and Thinkers

Kurt Gödel (IPA: [kuɹtˈgøːdl]) (April 28, 1906 Brno (Brünn), Austria-Hungary (now Czech Republic) – January 14, 1978 Princeton, New Jersey) was an Austrian American mathematician and philosopher.

The Time 100-Scientists and Thinkers

The upshot of all this is that the cherished goal of formalization is revealed as chimerical. All formal systems — at least ones that are powerful enough to be of interest — turn out to be incomplete because they are able to express statements that say of themselves that they are unprovable. And that, in a nutshell, is what is meant when it is said that Gödel in 1931 demonstrated the "incompleteness of mathematics." It's not really math itself that is incomplete, but any formal system that attempts to capture all the truths of mathematics in its finite set of axioms and rules. To you that may not come as a shock, but to mathematicians in the 1930s, it upended their entire world view, and math has never been the same since.

Gödel's 1931 article did something else: it invented the theory of recursive functions, which today is the basis of a powerful theory of computing. Indeed, at the heart of Gödel's article lies what can be seen as an elaborate computer program for producing M.P. numbers, and this "program" is written in a formalism that strongly resembles the programming language Lisp, which wasn't invented until nearly 30 years later.


Princeton

In the late 1940s, Gödel demonstrated the existence of paradoxical solutions to Albert Einstein's field equations in general relativity. These "rotating universes" would allow time travel and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric.


Closed timelike curves

Because of the homogeneity of the spacetime and the mutual twisting of our family of timelike geodesics, it is more or less inevitable that the Gödel spacetime should have closed timelike curves (CTC's). Indeed, there are CTCs through every event in the Gödel spacetime. This causal anomaly seems to have been secretly regarded as the whole point of the model by Gödel himself, who allegedly spent the last two decades of his life searching for a proof that death could be cheated, and apparently felt that this solution provided the desired proof. This strange conviction came to light decades after his death, when his personal papers were examined by a startled astronomer.[citation needed].

A more rational interpretation of Gödel's motives is that he was striving to (and arguably succeeded in) proving that Einstein's equations of spacetime are not consistent with what we intuitively understand time to be (i.e. that it passes and the past no longer exists), much as he, conversely, succeeded with his Incompleteness Theorems in showing that intuitive mathematical concepts could not be completely described by formal mathematical systems of proof. See the book A World Without Time (ISBN 0465092942).


General Relativity

CTCs have an unnerving habit of appearing in locally unobjectionable exact solutions to the Einstein field equation of general relativity, including some of the most important solutions. These include:

* the Kerr vacuum (which models a rotating uncharged black hole)
* the van Stockum dust (which models a cylindrically symmetric configuration of dust),
* the Gödel lambdadust (which models a dust with a carefully chosen cosmological constant term).
* J. Richard Gott has proposed a mechanism for creating CTCs using cosmic strings.

Some of these examples are, like the Tipler cylinder, rather artificial, but the exterior part of the Kerr solution is thought to be in some sense generic, so it is rather unnerving to learn that its interior contains CTCs. Most physicists feel that CTCs in such solutions are artifacts.


Timelike topological feature

No closed timelike curve (CTC) on a Lorentzian manifold can be continuously deformed as a CTC to a point, because Lorentzian manifolds are locally causally well-behaved. Every CTC must pass through some topological feature which prevents it from being deformed to a point. A test particle free falling along a closed timelike geodesic transits this feature; in the test particle's frame, the feature propagates toward the test particle. This features resembles a glider in Conway's Game of Life, but in a continuous spatial automaton rather than a (discrete) cellular automaton.


Continuous spatial automaton

It is an important open question whether pseudo-photons can be created in an Einstein vacuum space-time, in the same way that a glider gun in Conway's Game of Life fires off a series of gliders. If so, it is argued that pseudo-photons can be created and destroyed only in multiples of two, as a result of energy-momentum conservation.

2 comments:

Phil Warnell said...

Hi Plato,

Nice piece about Godel and his speculations about a rotating galaxy and how it relates to the nature of time. He truly was one of the great minds of the ages; and yet although he tried never did see the world as he would have liked, in as he believed true as Plato and Descartes had. To serve perhaps as evidence of this, I leave you with a few quotes from Rebecca Goldstein's
book on Godel called [Incompleteness (The Proof and Paradox of Kurt Godel)-Atlas Books-2005-pages 259-260].

(what is contained in single quotation marks is what was said by Dr. Goldstein and in double by Dr. Wang)

“It is tempting to connect Godel’s attraction to these closed time loops with a passing remark that Hao Wang made, indicating how Godel had thought as his life as incomplete:”

“”In philosophy Godel has never arrived at what he looked for: to arrive at a new view of the world, it’s basic constituents, and the rules of their composition. Several philosophers, in particular Plato and Descartes, claim to have had certain moments in their lives an intuitive view of this kind totally different from the everyday view of the world.””

“And, again Wang made reference to a transcendental experience that Godel had awaited all his life:”

“”He also looked for (but failed to obtain) an epiphany (a revelation or sudden illumination) that would enable him to see the world in a different light. (In his conversations with me, he repeatedly said that Plato, Descartes, and Husserl all had such an experience).””

There is much more in this book that relates to your topic and our shared mutual concerns and if you haven’t already read it I would recommend it most highly. What stands out for me in this regard is what I consider to be perhaps the reason for Godel’s failure as it appears. That is he was actually caught in the limitations of his methods of reason; for though he recognized and proved that the axiom’s of mathematics although powerful could not provide the tools necessary to actual probe and then actually realize the depth to which it had. So like the axioms, Godel himself was limited by his own incompleteness.


Best,

Phil

Phil Warnell said...

Hi Plato,

Sorry, I just noticed I have made an error, for I should have said a rotating universe not galaxy. This then stand as yet another proof that we all have our limitations; one of my many being not to be able to proof read properly my own words. :-)

Editors of the world fear not, for a computer doesn't exist that has demonstrated the abilities required to replace you ;-)

Best,

Phil

P.S. The comment deleted above again serves as further testament to my claim :-<