Posted on 02/15/2005 2:39:04 PM PST by snarks_when_bored
Ping
"Later" ping
Perhaps you'll be more specific about what it is you're calling 'nonsense'?
Completeness is a vital, crucial, essential theoretical attribute that is necessary in the design of algorithms. Without proof of its existence, algorithms are rendered useless.
Goedel's ideas surrounding his proof of the incompleteness of formal reasoning systems are as important if not more important than any philosopher of any age including Einstein.
His philosophy is not anti-science but is rather a view on its limitations inside a particular theory. Theories can be expanded to be more general but they may never capture everything.
Completeness then defines what a theory can ascertain and ensures that it ascertains all that is specified in its scope. To say a theory is complete is to say at once that it is reliable and valid within its ***scope***.
I would think that Goedel's theorem would keep scientists humble, as mentioned in post #6.
Rebecca Goldstein is the author of four novels, including THE MIND-BODY PROBLEM, and a collection of short stories, STRANGE ATTRACTORS. Her work has won numerous prizes, including two Whiting Awards. In 1996, she was named a MacArthur Foundation Fellow. She holds a Ph.D. in philosophy from Princeton University, where her work was concentrated in the philosophy of science and was supported by a National Science Foundation fellowship. She resides in Cambridge, Massachusetts.
I'm reasonably certain that you are unfamiliar with Godel's Proof.
There is nothing left-wing or nihilistic in Godel's lucid, brief proof for the incompleteness of any non-contradictory formal system.
It was part of Symbolic Logic II when I was an undergrad.
Nothing communist about Godel.
I can't believe that I have to post this.
|
js1138's Law: The first to post a negative comment on a science thread hasn't read the article.
Depressing, innit? ;)
Ping
Math threads never get very many pings. They're worth posting from time to time, but you can't expect a whole lot of action.
One function it is impossible to calculate is the interaction (or interference, if you will) upon the observed by the observer. Since you can't predict the latter wouldn't you end up with a model that is unstable over time? Or is it merely a function with infinite solutions (or do they only seem infinite?).Either way, it seem incomplete to me.
No problem.
...Godel's lucid, brief proof for the incompleteness of any non-contradictory formal system...
Two (friendly) comments. First, Gödel's proof was stunning, but not brief (nor was it lucid in the ordinary meaning of that word). Second, the formal system has to be strong enough to express natural number arithmetic with multiplication; systems weaker than thatsay, systems that include only the operation of additioncan be proved to be both complete and consistent.
Here's a useful link to an article by R.B. Braithwaite which provides some background and details on Gödel's Incompleteness Theorem:
"On Formally Undecidable Propositions Of Principia Mathematica And Related Systems"
Russel countered with his Theory of Types, but it was thoroughly unsatisfactory and you could tell his heart wasn't in it. That idea essentially ruled out self-reference; that is, metastatements of the order of "this statement is false" were illegal. But in fact there turned out to be no reliable way of deciding where the boundary between statement and metastatement really lay; in fact, there isn't one, as Russell concluded himself.
But that does not invalidate logical systems per se, as the postmodernists fervently believe, it simply dictates that they have boundaries. It does not state that there is no truth or logic, or that all points of view are therefore equally valid, quite the opposite, in fact. It simply states that formal logical systems have limitations. Most adults should be able to live with that.
What Godel showed was twofold: first, that within any formal logical system of sufficient power, a statement could be made that was true but undemonstrable (incompleteness) and second, that a statement could be formulated following that system's rules that could be shown both true and false (incoherence).
Two (friendly) comments. First, it's important to say "undemonstrable within the system" rather than simply "undemonstrable". Second, it's not quite accurate to say that "a statement could be formulated following that system's rules that could be shown both true and false (incoherence)"if that were true, the system would be inconsistent, which nobody desires. The aim of the constructor of a formal logical system is to rule out the possibility of contradiction, but at the same time to insure the possibility of proving the greatest number of true propositions. What Gödel showed was that if a system has the expressive resources needed to formulate the axioms of natural number arithmetic with multiplication, then it is impossible to prove within the system every true proposition without incurring the penalty of inconsistency, i.e., the ability to 'prove' a contradiction (such as, 1=2). Since no formal system constructor wants an inconsistent system, (s)he's forced to give up the hope of being able to prove every true proposition within that system.
In short, in a 'strong enough' system, the penalty for completeness is inconsistency, and the penalty for consistency is incompleteness.
Of course, by expanding the system to include new axioms, propositions which were previously known to be true but unprovable, become provable. But, again, in the expanded system, new propositions can be formulated which can be known to be true, but turn out to be unprovable within the expanded system. And so on ad infinitum.
Disclaimer: Opinions posted on Free Republic are those of the individual posters and do not necessarily represent the opinion of Free Republic or its management. All materials posted herein are protected by copyright law and the exemption for fair use of copyrighted works.