The Incomplete Gödel

American Scientist Online ^ | September-October, 2005, issue | Gregory H. Moore

[snarks_when_bored]

Are you familiar with Gödel's slingshot?

No, I wasn't. But I've now googled a bit about it. I see that it has something to do with modal logic. I'll have to do some reading later. In the meantime,

I don't necessarily believe that it's possible for modal logic to be given an effective implementation.

(rimshot)

[Blind Eye Jones]

>
The Great Frankenstein Einstein Debate at Princeton

Little known fact: the Frankenstein Einstein debate at Princeton in the late 30s. Frankenstein was coming off a few hit horror films with Dracula and bought his way into Princeton. The high point was a math debate at the student union hall where Einstein said, "pi r2" and Frankenstein rebutted with, "No! Pie are round!" It caused a big stir amongst the faculty and they chased Frankenstein back to the dorm with pitchforks and torches.

[RightWhale]

Goldstein's philosophy classes might be interesting, but like many who teach philosophy she is no Hegel [that is a hard act to follow]. What it comes down to is that the student has to develop his own understanding. The teacher can only point the way, or several ways or possible points of departure, and maybe one of them will be a sufficient spark. If Chomsky can do it, there is hope that all of us can do it.

[Blind Eye Jones]

Forgive me for that picture S_W_B, Halloween is around the corner and I couldn't resist. All The Best!

[snarks_when_bored]

Hey, folks, I've pinged you because you've evinced an interest in Kurt Gödel and his work on past threads. FYI, the current (April, 2006) issue of the Notices of the American Mathematical Society is devoted in its entirety to Gödel. Each article is a PDF file and must be downloaded individually.

Here's the link:

Kurt Gödel special edition of the Notices of American Mathematical Society

Enjoy!

[AmishDude]

I was at a talk with a half-dozen prominent Hungarians when Erdos died.

[Lancey Howard]

Thanks for the link.

[snarks_when_bored]

Of course, I should've mentioned in my ping in post #45 that it's the Gödel Centenary. He was born on April 28, 1906 (and died on January 14, 1978).

[BykrBayb]

Thanks for that link. Looks like they have a good collection of articles there. It should keep me busy for a while.

[snarks_when_bored]

Thanks for that link. Looks like they have a good collection of articles there. It should keep me busy for a while.

Me, too, BB! And you're welcome...

[ThePythonicCow]

Thanks!

[D-fendr]

thanks snarky.. great stuff.

[D-fendr]

Gonna take another shot at this:

I was only responding to D-fendr's citation of the passage, pointing out that the hypothesis, i.e., that the human mind is capable of grasping all mathematical truths, doesn't appear at all likely

How about a human mind capable of coming up with the Incompleteness Theorem?

What I'm pointing to is the ability to see the greater context of, transcend, the program and see a meta-program, and on up...

[GOPJ]

Thanks for the ping.

[mathluv]

Thanks for the ping. Bookmark for later.

[longshadow]

Hence, to make the argument that Gödel wanted to make, the argument that Yourgrau summarizes, one must suppose it possible that mathematical truth in its entirety might somehow be graspable,...

Which violates Gödel's Incompleteness Theorems, as there no way to know if a Gödel statment is true or not without supplemental axioms, which creates a new set of unproveably true Gödel statements.... and so on, as you pointed out.

In effect, the argument being made is a case of Begging the Question, because it requires you to assume that which contradicts Gödel's Theorems (that we can can somehow prove true a Gödel statement with out creating more Gödel statements) order to arrive a conclusion (that we can grasp Mathematical truth in its entirety) that violates Gödel's Theorems.

[longshadow]

Nevertheless, there are large parts of mathematics that are catagorical. Not only are all theorems provable, there is a proof scheme for these theorems. Euclidean (and consequently, all the non-Euclidean) geometry is one such. First order logic is another as is Pressburger arithemetic.

The other thing to keep in mind is that the limitation imposed by Gödel's Incompleteness theorems has little if any practical effect on most Mathematicians. Since the only sort of theorem used by Mathematicians to prove other theorems are theorems which have already been proven, Gödel's Theorems never come into play. IOW, that which we can prove, we can prove.

It's only the meta-Mathematicians, the Russells, the Whiteheads, et al., whose lofty goals were trashed forever by Gödel's results. Life for mortal Mathematicians goes on as before, they hardly even feel a bump when the pass over Gödel's Theorems.

[IslandJeff]

Thanks!

[Recursive Prophet]

Just found this link today and signed on. As a long time Gödel fan I’m pleased to see so much interest and thanks to Snarks for all the links. I came here looking for more facts on Kurt’s life. I highly recommend “A Madman Dreams of Turing Machines” also for insight into his personality. I read it only recently, after starting my own story on a website dedicated to the ABC show LOST. My goal is to introduce the magic of numbers to an audience not really very familiar with them. It’s an experiment.

You don’t need to ever have see LOST to follow this thread. Those interested in Turing, Gödel, and Riemann might find it amusing. Remember its a fiction oriented toward those with little background in math but who enjoy puzzles. Any suggestions on ways to improve it would be most welcome.
http://www.4815162342.com/forum/viewtopic.php?t=31267

I look forward to participating in this forum when I’ve had time to read more. Hard to believe I didn’t come across this site before.

[Billthedrill]

A Godel BTT. He and Einstein were walking buddies at the Institute for Advanced Studies. The story I heard is that no one else would walk with them because no one else had the faintest idea what they were talking about.

I'm guessing it wasn't "Hey, yo, how 'bout them Packers, baby?"...