Metaphysics of the coming age | 236 comments (236 topical, 0 editorial, 0 hidden)

Re: Metaphysics of the coming age (none / 0)

But then, Gödel's theorem is just applying the presocratic paradox of "this statement is false" to formal logic. I'm not sure how it makes formal logic so unstable...

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Wed Jan 2nd, 2008 at 02:32:59 PM EST
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Re: Metaphysics of the coming age (none / 0)

It doesn't make it unstable, it just proves that truth functions are incomputable.

We have met the enemy, and he is us — Pogo

by Migeru (migeru at eurotrib dot com) on Wed Jan 2nd, 2008 at 03:11:55 PM EST
[ Parent ]

Re: Metaphysics of the coming age (none / 0)

It's possible that first learning about the undecidability aspects of computer theory made the Gödel theorem lose its surprise factor...

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Wed Jan 2nd, 2008 at 03:28:09 PM EST
[ Parent ]

Re: Metaphysics of the coming age (none / 0)

Gödel proved the incompleteness of Arithmetic. Specifically, he demonstrated that in any axiomatic system capable of Arithmetic there are statements that can neither be proved or disproved within the axioms.

To fully understand what that meant and means requires an understanding of what Logic and Mathematics were 'on about' from around 1870 to l933. To (greatly) simplify, Frege, Hilbert, Russell, Whitehead, & All That Crew were attempting to construct an intellectual tool that would always, when correctly used, derive a True and Valid answer.

Gödel's Proof = You Can't.

She believed in nothing; only her skepticism kept her from being an atheist. -- Jean-Paul Sartre

by ATinNM on Wed Jan 2nd, 2008 at 10:56:09 PM EST
[ Parent ]

Re: Metaphysics of the coming age (none / 0)

Yes, and it failed on "this statement is false". That's the basic point of Gödel's proof. I don't know how they expected to attribute a truth value to that statement, those formal logicians ; but that you can't attribute a truth value to it sounds even a bit reconforting... Logic doesn't get out of the realm of human experience about truth values.

Un roi sans divertissement est un homme plein de misères

by linca (antonin POINT lucas AROBASE gmail.com) on Thu Jan 3rd, 2008 at 04:13:57 AM EST
[ Parent ]

Re: Metaphysics of the coming age (none / 0)

Many antinomies (paradoxes,) such as that one - a variation on wossname's 'All Cretans are liars ...' - have an oscillating first order Truth Value with the TV changing depending on the step: On True Then False Then True & etc XOR On False Then True Then False & etc. Since they are infinitely oscillating there is no resolution possible.

By moving to the Second Order, on the other hand, the True/False OR False/True oscillation is, sometimes, capable of resolution.

But that, very often, takes you outside the original axiomatic system.

She believed in nothing; only her skepticism kept her from being an atheist. -- Jean-Paul Sartre

by ATinNM on Thu Jan 3rd, 2008 at 05:27:32 AM EST
[ Parent ]

It ALSO failed on (none / 1)

"Arithmetic is consistent"

That was the sailing-over-the-edge moment.

Now it may be that arithmetic really is consistent, but that is another truth function that is "uncomputable."

The Fates are kind.

by Gaianne on Thu Jan 3rd, 2008 at 04:48:01 PM EST
[ Parent ]

Metaphysics of the coming age | 236 comments (236 topical, 0 editorial, 0 hidden)

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