Existence exists.

[frank harley]

Godel ultimately concluded that no axiomatic system can be formally demonstrated, meaning deductively proven. That is, no axiomatic system is complete. Systems can be axiomatized still. Even if incomplete, that doesn't mean axiomatization is impossible, it just means you need a method besides a formal deduction to demonstrate the validity of a system.

 

Carnap is different, to the extent he still made claims about reality. All Godel did was talk about proving and validating statements, and on his own terms, there is no problem. To Carnap, reality is a problem for verification.

 

Plasmatic, that book was published in 1931, so maybe it still has Carnap's view that he later rejected as untenable. I checked out the book from my school's philosophy department's library. Anyway, the verification method he talks about is a deductive endeavor, because he doesn't think empirical statements have meaning until they are given a formal description. With a formal description, then you could proceed to verify - but observation itself is not sufficient, the senses are irrelevant for verification. Keep in mind I wrote that all empirical statements are indeterminate under Carnap's view, but that isn't to say no verification is possible with indeterminacy. Because of indeterminacy though, logical syntax is required to say anything meaningful about an empirical statement. I shouldn't have said "empirical fact".

Well, yes, Godel wrote that all mathematical axioms need to be supported outside of its own context. In other words, contrary to Russell and Whitehead (Principia), math isn't a 'complete' system.

 

The conceptual link here is the hope that all knowledge in general might be axiomitized like math is; moreover, non-axiomed knowledge is nonsense. Godel simply cut the tree down by destroying the roots.

No knowledge is sufficintly complete.

 

No knowledge is sufficiently complete to be an axiom. language, form Wittgenstein onwards, reinforces this issue, ostensibly up to Derrida...

[Eiuol]

Well, yes, Godel wrote that all mathematical axioms need to be supported outside of its own context. In other words, contrary to Russell and Whitehead (Principia), math isn't a 'complete' system.

 

Rand didn't make a claim like this. She never implied that knowledge is complete in the sense Russel or Godel meant it, where completeness is the ability to deductively prove an entire system including its axioms. I don't see your point. Having axioms doesn't mean claiming anything about completeness, and axioms don't require completeness to be true. If you mean that Rand created an axiomatically derived philosophy, Rand emphatically said she didn't do that, she advocated induction, which is one valid way to demonstrate truth not precluded by anything Godel concluded.

 

All Godel did is explain the futility of Russel's and Whitehead's project - at least not them exactly, but related things. Don't extend its conclusions farther than it was ever meant to apply. Please at least relate the discussion to things people have said rather than misrepresenting at least 3 different thinkers based on what they allegedly demonstrated according to you. For one, Wittgenstein literally failed to comprehend Godel's theorem according to Godel himself. So there is no apparent valid argument regarding existence as an axiom, even as a devil's advocate.

[Plasmatic]

Leaving aside whether or not the claims made of certain philosophers are accurate....

No knowledge is sufficintly complete.

No knowledge is sufficiently complete to be an axiom. language, form Wittgenstein onwards, reinforces this issue, ostensibly up to Derrida...

The assumption that all knowledge is linguistic or conceptual is here being imported to somehow be a foil against foundationalism. First of all Godel's theorom applies to the language of symbolic logic. Second Oist conception of axioms aren't the same as other philosophers. Third, not all knowledge is conceptual.

[Harrison Danneskjold]

On what basis can non-axiomatized knowledge be considered insufficient?

An axiomatized basis?

Gödel's theorem absolutely proved that the infallibility of a deductive system can never be extended across all human knowledge. Far from crippling our epistemological foundations (as some believe) this precisely demonstrates the problem with Descartes' epistemology, and the necessity of an alternative- such as Isaac Newton's and, incidentally, Ayn Rand's.

Gödel's theorem demonstrates precisely why induction is absolutely necessary.

[frank harley]

Rand didn't make a claim like this. She never implied that knowledge is complete in the sense Russel or Godel meant it, where completeness is the ability to deductively prove an entire system including its axioms. I don't see your point. Having axioms doesn't mean claiming anything about completeness, and axioms don't require completeness to be true. If you mean that Rand created an axiomatically derived philosophy, Rand emphatically said she didn't do that, she advocated induction, which is one valid way to demonstrate truth not precluded by anything Godel concluded.

 

All Godel did is explain the futility of Russel's and Whitehead's project - at least not them exactly, but related things. Don't extend its conclusions farther than it was ever meant to apply. Please at least relate the discussion to things people have said rather than misrepresenting at least 3 different thinkers based on what they allegedly demonstrated according to you. For one, Wittgenstein literally failed to comprehend Godel's theorem according to Godel himself. So there is no apparent valid argument regarding existence as an axiom, even as a devil's advocate.

Yes, axioms would require either completeness (True in all cases) or boundary conditions. Other wise thety're jusyt convenient starting points vulnerable to attack.

 

The intent of R/W's project was to establish a system of math that was axiomatically complete, as did Bourbaki in France, btw. That none was found is, IMHO, no big deal, as I'm a second-principle naturalist anyway, mathematically speaking.

 

But back then the insistence was: complete math = complete formal logic= linguistic completeness= complete philosophical system. Obviously, the break vcame withthe study of the ambiguity itself; said A really doean't mean 'A'.

 

W disagreeed with the applicability of Godel because his Investigations revealed that math was useless as a model for thought (other than that off math!).

[frank harley]

On what basis can non-axiomatized knowledge be considered insufficient?

An axiomatized basis?

Gödel's theorem absolutely proved that the infallibility of a deductive system can never be extended across all human knowledge. Far from crippling our epistemological foundations (as some believe) this precisely demonstrates the problem with Descartes' epistemology, and the necessity of an alternative- such as Isaac Newton's and, incidentally, Ayn Rand's.

Gödel's theorem demonstrates precisely why induction is absolutely necessary.

Yes, i agree that a good application of godel to deduction would be that every triangle of knowledge hangs by its own fulcrum.

 

No one ever said that induction wasn't 'necessary'. Rather, form Hume onwards, that induction itself cannot explain how science works.

 

To this end, perhaps Quine's ontology is a better description: accepting  simple fact can alter an 'ontology'. or offer a different world view.

[frank harley]

Leaving aside whether or not the claims made of certain philosophers are accurate....

The assumption that all knowledge is linguistic or conceptual is here being imported to somehow be a foil against foundationalism. First of all Godel's theorom applies to the language of symbolic logic. Second Oist conception of axioms aren't the same as other philosophers. Third, not all knowledge is conceptual.

Yes, the linguistic turn says that all knnowledge is essentially a narative told by a language.

This does, indeed, contradict Aristotle, who believed thast we have direct access to what is (essen).

 

Russell called this the 'bundle'--definitions simply collect all that can be said of what's being defined. The real turnaround  came with Kripke na d his 'causal reference'. More than one Objectivist commentator has noted a similarity with Rand....