Fausty said:
Err, I don't even know how to respond to your "explanation" of the incompleteness theory. I wonder if you aren't talking about an entirely different proof than the one the rest of refer to as "Godel's theory." The one I've studied has nothing to do with "deduction" nor "induction."
From wikipedia: There are two distinct senses of the word "undecidable" in contemporary use. The first of these is the sense used in relation to Gödel's theorems, that of a statement being neither provable nor refutable in a specified
deductive system (my emphasis). (entry on the incompleteness theorems)
Godel was a logician (and mathematician, and philosopher) and his incompleteness theorem applies to logical systems which are inevitably deductive. If you don't see the relevance of deduction to Godel's theorem's, I think it suggests a confusion on your part, not mine.
Of course Godel's incompleteness theorems have little to do with induction. That was the whole point of my post. But questions of whether or not every event (or effect) has a cause are problems of induction. Likewise with questions about constitution. So my point was that in appealing to Godel's theorems as showing anything about these inductive problems, you were misapplying his theorems.
As to "truth," ummm. . . where did that loaded word manage to sneak in?
It's widely accepted that the incompleteness theorem's show that there are true propositions that are unprovable by any given system. That's why they are "incomplete". I don't see what your problem is here.
Also note Godel's own discussion of his first incompleteness theorem (from Logical Journey, 1996):
I represented real numbers by predicates in number theory… and found that I had to use the concept of truth (for number theory) to verify the axioms of analysis. By an enumeration of symbols, sentences and proofs within the given system, I quickly discovered that the concept of arithmetic truth cannot be defined in arithmetic. If it were possible to define truth in the system itself, we would have something like the liar paradox, showing the system to be inconsistent… Note that this argument can be formalized to show the existence of undecidable propositions without giving any individual instances. (If there were no undecidable propositions, all (and only) true propositions would be provable within the system. But then we would have a contradiction.)… In contrast to truth, provability in a given formal system is an explicit combinatorial property of certain sentences of the system, which is formally specifiable by suitable elementary means…
Admittedly, I come at Godel's work from a mathematical perspective - he was a mathematician, after all. And I see the flowering of his work in large measure in the field of computer science, and in fundamental axioms of computational intractability. It also underlies Shannon's information theoretic framework, and everything flowing therefrom.
If, somehow, it has also inspired non-mathematical philosophers in their own leaps of rhetorical efflorescence. . . well, ok I guess. Lots of things can provide creative impetus for non-scientific cogitation.
Well, Godel was just as much a logician as a mathematician and even wrote several papers of philosophy. He's not a man easily categorized. But, Mathematical and logical proofs do not show anything about causality, or about the constitution of physical objects. You seemed to suggest that Godel's theorem's do show something about these subjects - so I think if anyone is making "leaps of rhetorical efflorescence" it's the scientists, not the philosophy guy who also has studied formal logic.
Does Godel "say anything" about causality, as such? Yeah, he does - alot. Not directly, but it is all first-order implication of his fundamental proofs. In fact that's sort of the foundational element of Godelian computational approaches. It enabled Turing to model the "universal calculator" (which we now refer to as a "universal Turing engine") and helped him to avoid dead-ends in seeking definitive links in causality relationships within abstracted algorithmic computation. It's all "part of" Godel, though perhaps less sexy when one isn't marinated in the practical implications flowing therefrom.
So, there's nothing in the proofs that directly discusses causality. Yet, the proofs directly make conclusions about deductive systems. And you are claiming that 1) the proofs "say a lot" about causality and 2) have nothing to do with deduction. Let me ask you directly: have you studied the proofs first hand? It's hard to believe that you have when you are unclear about how deduction is related to them.
Anyway, I suppose if you see science as an attempt to create some grand deductive system of the universe then Godel's incompleteness theorems have shown that they will inevitably leave something out. But, long before Godel there were problems with this view of practical science anyway.
Godel a Platonist? Well, I'll say only that Godel was a freaky genius. I think I've seen glimpses of the world he was trying to describe and explore, in his work. But, I know very few mathematicians with sufficient genius to make claims of fully understanding what he was up to - and I suspect more than a few of those are just flat-out wrong. So if you've now managed to boil down Godel to one word - Platonist - from outside of any understanding of his mathematical milieu itself, I can only say. . . wow. Perhaps a tad reductionist?
It's pretty obvious that by saying "Godel was a kind of Platonist" I wasn't speaking reductively. "Godel was a kind of Platonist" is obviously not an identity statement like "Water is H2O", rather 'Platonist' in this sentence is an adjective that means, "someone who thinks Platonism is true". And, taken in context, the Platonism referred to is obviously Platonism about mathematical entities. You can be a Platonist without Platonism accounting for everything that you are.
Anyway, some quotes from Godel:
I am under the impression that after sufficient clarification of the concepts in question it will be possible to conduct these discussions with mathematical rigour and that the result will then be…that the Platonistic view is the only one tenable. (Gödel Gibbs Lecture Published 1995, p. 322).
and listed under "What I Believe" in his published notes dated around 1960:
Materialism is false.
Concepts have an objective existence
See also his lecture: "Is Mathematics a Syntax of Language?" for a defense of Platonism against the view described by the title.
Anyway, your complaint about my claim that "Godel was a kind of Platonist" seems entirely unjustified. Obviously I was not claiming that everything about Godel can be explained by Platonism. Rather I was making a claim as to his defended views about the objective, non-physical nature of mathematical entities (and concepts). This view is properly described as a kind of Platonism and Godel uses the term this way himself. Likewise, your complaints about my use of the word "truth" and "deductive" in describing Godel's theorems seems unjustified. All of this is now much more well supported than anything you have said about Godel in this thread.