From: rhinoceros (rhinoceros@freemail.gr)
Date: Thu Jun 19 2003 - 17:12:36 MDT
[Rafael Anschau]
I noticed that the word Fact is widely used to turn a statement into an axiom, but as a Godel pointed out, every axiom is grounded on another non-provable axiom...
[rhinoceros]
"Fact" has several meanings. It can be an event which actually occured or it can be an account of one such event. I guess we are talking about the second case. The judgement that a statement is a fact can be objective (this rock falls on the ground), subjective (her tits are nice) or even inter-subjective (Jesus kicks ass). The common thing is that the speaker does not see any reason to justify these "facts" any more.
Apparently, most of us will see through that and question these facts until we reach at something objective or something satisfactory to our own "facts". Hopefully, if we are in a big enough, diverse enough, and happily prickly company with some minimal social skills, we'll arrive at something close to an objective fact (still testable, falsifiable etc). Can we then call it an axiom?
Yes and no. First, it can't be just any kind of fact. Axioms are supposed to be few, or else they would not be useful. "This stone fell" is not good enough. "Stones fall" is a little bit better. That said, if we define an axiom as an irreducible statement which does not need more sustification and is at the end of a chain of reasoning, then we can say we have an axiom. Here is an interesting entry about "regress" and "coherence" in reasoning:
http://www.wikipedia.org/wiki/The_regress_argument_in_epistemology
However, this is not what Goedel was talking about. The axioms with which he was concerned were the axioms used for building a logical or mathematical formal system. Those axioms, like the other ones, were irreducible and did not need to be justified, but there were two differences:
(a) They did not need to represent any "fact".
(b) They were chosen for building a formal system which should be free of contradictions and hopefully complete (a failure, according to Goedel's proof).
This seems to imply that (a) Goedel had nothing to say about facts, and (b) incompleteness is something which we will face only when we try to impose on the real world an axiomatic formal system with an ennumerable set of possible statements (as Goedel did for the formal systems).
[Rafael Anschau]
Could it be that the meta-axiom we all agree to hold as true, IN SPITE of Godel's Theorem is the computer ? This is, when we talk about facts we're saying that a variable has such value, or that a computation has yielded a specific result ? I define a fact as computational analysis of some system, and that works for me. Any comments ?
[thinoceros]
I didn't quite get that... Aren't computers dumb? What kind of computational analysis? It is up to us to specify it... What variables will the computer look for? It is us who will have to decide about its input and provide it with the suitable means of "perception". And what kind of analysis is it to perform on this input?
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