tim@isrnix.UUCP (06/28/83)
the idea that one could have a strictly delimited set of laws to be
operational for all eternity as Tom Craver suggests utilizes a model that
is no longer valid for its own field. That model is logic or a complete
logical system similar to the one developed by Russell and Whitehead in
"Principia Mathematica". Wittgenstein made some very interesting attacks
on this model but the final blow was Godel's theorem-his proof that no
logical system could be self-contained and totally consistent. If such
is true for LOGICAL systems how can it possibly be true for POLITICAL
systems or moral systems? I would suggest that Tom Craver get his head
out of Ayn Rand (seemingly his only reading source) and read Michael
Polanyi's "Personal Knowledge" or Hofstadter's "Godel,Escher, Bach"
for an explication of Godel's theorem and possible implications.
Morality (which is what the best politics should be) is not something
which can be frozen at one point in time but must develop just as dynamically
as science, technology, and all other human endeavors. But it is typical
for Conservatives to try to freeze morality at an anachronistic level which
fails to match current conditions. At one time it was "moral" for humans
to reproduce as often as possible for example if we wished to propagate
the species. But now when overpopulations threatens the worldwide ecological
balance and thousands of other animals and plants it is "moral" to try
to stabilize or, if possible, reduce the human population.
Tim Sevener
decvax!pur-ee!iuvax!isrnix!timlvc@cbscd5.UUCP (06/30/83)
The news item posted by Tim Sevener concerning 'limited laws for all time' contains much misunderstanding over Godel's proof. This news item will deal only with this and not discuss such laws or any justification for gov't. Godel's proof states that under a "certain set of circumstances" there exists a problem which cannot be proven true using statements of the same "form" as the problem. A few examples will illustrate what I mean. 1 ) The problem x * x + 1 = 0 can be solved only by using complex numbers yet it is stated without using complex numbers. 2 ) The problem x * x - 2 = 0 can be solved only by introducing irrational numbers, yet they are not in the problem. Ultimately what Tim is saying is that the facts of reality are insufficient to explain reality. This is simply absurd. He states "If such is true for LOGICAL systems how can it possibly be true for POLITICAL systems or moral systems?" The answer quite simply is that Godel's theorem does not apply. If the premises are not met the conclusion does not follow. I would suggest that Tim Sevener get his head out of Hofstadter (a mathematical lightweight) and read "Introduction to Metamathetics" by S. C. Kleene. It is an excellent text, and quite carefully proves Godel's proof. Larry Cipriani cbscd5!lvc
tim@isrnix.UUCP (07/01/83)
In response to Larry Cipriani's assertion that I am claiming that
"the facts of reality are insufficient to prove the facts of reality"
I can only say one thing-since when did Tom Craver's attempt to set up
strictly delimited set of laws for government ever incorporate all the
facts of reality? Since when has it incorporated any facts at all for
that matter? Are you claiming a mathematical system that will embrace
everything we will or could ever know about reality? That would be
quite interesting! My point is indeed relevant to claims for some
eternally valid set of laws for humanity-the idea of complex numbers
was developed historically as one way to deal with a tricky mathematical
problem. It was not a part of the original foundation of that problem.
At one time it was thought that Euclidean geometry was all there was-
this is it folks, we got all the laws that could ever be for geometry
right here! Of course that has later been agreed not to be the case,as
non-Euclidean geometry has been incorporated into Einstein's theories.
Most people think of mathematics as something indubitable-you're either
right or not. By pointing out that even in the realm of abstractions
(Kant's synthetic propositions?)from reality where everything would
seem to be cut and dried it really isn't. If mathematics then develops
and cannot be put into some box that will fit it for all time, once
again I ask, how can the morality of human beings and their relations?
anyhow it probably IS true that "the facts of reality are insufficient
to prove the facts of reality"!
Tim Sevener
decvax!pur-ee!iuvax!isrnix!timee461@rochester.UUCP (07/04/83)
Whoops .. I hope you are not reading this for the second time - when I submitted it for the first time, the first few lines got lost. I'm not sure if I managed to cancel this first posting... Now the full story: The followup posted by Larry Cipriani, where he tries to explain Godel's theorem to Tim Sevener, introduces only more misunderstanding over the subject. This news item will deal only with this. The subject is more suitable for net.math newsgroup, but Godel's theorem received the crippling treatment here, in net.politics... First, a quotation from Larry's article: > Godel's proof states that under a "certain set of circumstances" > there exists a problem which cannot be proven true using statements of > the same "form" as the problem. > A few examples will illustrate what I mean. > 1 ) The problem x * x + 1 = 0 can be solved only by using complex > numbers yet it is stated without using complex numbers. > 2 ) The problem x * x - 2 = 0 can be solved only by introducing > irrational numbers, yet they are not in the problem. The theorem, as it appears in the original Godel's work, is preceded by some 30 pages of definitions necessary to make the precise statement. The attempt to compress these to: "certain set of circumstances" and "form" appears to fail completely. Let me state the main result in an acceptably (I think) simplified version: Godel says that within each system of axioms that is consistent and powerful enough to express the arithmetic of integers there are THEOREMS that can not be proved WITHIN the system. (Theorem means here: a statement that is true). Note the difference: "theorems", NOT "problems"! Also, note that the proof is required to use only the fundamental axioms of the system and theorems derivable from these axioms (this is the meaning of: "within the system"). When Larry's examples are restated in the form of theorems, they are PROVABLY FALSE WITHIN THE RESPECTIVE SYSTEMS, hence they bear no relevance whatsoever to Godel's theorem. For example the following statement: "there exists a rational number x such that x*x - 2 = 0" can be disproved within the arithmetic of rationals. Note again: NOT "a number x", but "a rational number". Stay within the system! One of the most famous examples of assertions that appear to be true but nobody was able to prove them yet, is Cardano's theorem (a**n + b**n = c**n has no integer solutions in a,b,c for integer n>2). Another example: for any integer n there are two primes a,b such that a + b = 2*n. However, Larry is perfectly correct in that Godel's theorem was misused by Tim Sevener. Tim, unless you are discussing a moral or political system that contains the arithmetic of integers (wow! this would be something!) Godel's theorem does not apply. Krzysztof Kozminski
myers@uwvax.UUCP (07/15/83)
Yes, Godel's theorem applies only to systems dealing with relationships between integers. The interesting thing about integers, though, is that symbols in ANY formal system can be represented as integers! Hence, any relationship expressable withing a formal system is representable as a relationship between integers. That is the amazing thing about Godel's theorem. What the theorem says about formal systems dealing with non-negative integers is applicable to ALL formal systems. If ya don't believe me, just read Chapter 1 of "Theory of Computation" by Brainerd and (Madison's own) Landweber. It's clear enough. Enjoy, Jeff Myers@uwvax