Godel's Incompleteness Theorem - Numberphile

[Xipe Totec]

That should be Gödel, but can't have umlauts in thread titles. Oh, well...

[Hokie]

1. I am a fan of Numberphile but their 1+2+3+ ... -1/12 video was nonsense

Mathologer just did a video on that one and showed why.

[Xipe Totec]

2. All well and good, but does that invalidate Godel's incompleteness theorem?

Or how is this related other than being another Numberphile video?

[Hokie]

6. No it doesn't. I watched that one by Numberphile also and thought it was well done.

Do you think Riemann's Hypothesis is one of those items that can never be proved as posited by Godel's Incompleteness Theorem?

[Xipe Totec]

9. Don't know about Riemann's Hypothesis, specifically, but if Mathematics is the language of Science

And there are truths that can not be proven with mathematics then, necessarily, there are truths beyond the reach of science as well.

Something to consider.

[Hokie]

11. Yes, that's a good point

The RH has gone unproven for over 150 years so it may be a candidate.

The intertwining of science and mathematics is fascinating. One of the greatest mysteries of all is why math works so well in science. Which one came from the other? Which one is at the basis of reality? Why did probability have to stick it's nose into physics?

Did you ever think how bizarre nuclear radioactive decay is? You can have gazillions of atoms of a radioactive material that appear to be absolutely damn identical to the outside world but you can never tell when one is going to decay and shoot out some little bit of mass and energy in the form of alpha particles, photons, neutrinos, and/or electrons. What told that atom to do that? We can never know. All we can say is that if you have a bunch of atoms a certain fraction will decay over time following a mathematical formula. To me that's mind blowing. It's also necessary for the universe to work.

[Xipe Totec]

12. Curiosity didn't kill Schrodinger's cat - random quantum fluctuations did. nt

[Hokie]

13. Or not .....

LOL

[Xipe Totec]

14. Zing!

[Xipe Totec]

15. Why the world does not exist

[Fortinbras Armstrong]

20. When I took an Intro to Philosophy class as an undergraduate

I had an argument with the instructor about Gödel's incompleteness theorem and Heisenberg's uncertainty principle. I said that they were philosophically significant because they demonstrated that some things are inherently unknowable. He disagreed with me, but I suspect that he didn't actually know anything about either one -- which is not unusual for people in his or related fields.

When I was at Harvard Divinity School, studying for my masters in history of religion, the question "what is infinity?" came up in one of my classes. I wound up giving a talk on Georg Cantor's theory of transfinite sets. I gave them Cantor's proof that the cardinality of the real numbers is greater than the cardinality of the integers. I'm afraid that most of the class did not really understand what I was talking about.

A Jesuit (of course) who did know a lot of math really confused everyone when he started a discussion with me about whether C (the cardinality of the real numbers) was equal to Aleph-1 (the transfinite set with the second-lowest cardinality). When I mentioned the phrase Zermelo–Fraenkel set theory, I literally saw the professor's eyes roll. I opined that it was probably a Gödel question.

[Xipe Totec]

21. Richard Feynman - The Uncertainty Of Knowledge

Apologies for the totally unnecessary music in the background.

[Xipe Totec]

22. I've never taken a course in pure philosophy

I've taken courses in logic, and I took a graduate course on the history and philosophy of chemistry, but have not taken a course in philosophy per-se.

But if we begin with the Socratic Paradox "The only thing I know is that I know nothing", then uncertainty and doubt are fundamental first steps towards knowledge. So anyone that pretends to know more than this, knows in fact less than nothing.

[Jim__]

23. Do you remember what your instructor's argument was?

In his book, Gödel's Theorem: An Incomplete Guide to Its Use and Abuse, Torkel Franzén may agree with your instructor.

[Fortinbras Armstrong]

24. No, I don't remember

I took this class over 45 years ago.

[Make7]

3. Gödel's Incompleteness Theorem - Numberphile

You can use html entities in post titles to do non-standard characters. For example, I used this for my reply title:

Gödel's Incompleteness Theorem - Numberphile

Although it will disappear if you preview first and then post from the preview page. Just hit the back button on your browser to post from the previous page with the html entity still showing. Also, if you edit, you'll have to add the html entities back in.

You'll have to decide if the extra effort is worth it - but it can be done.

[Xipe Totec]

4. Where do I look for the proper encodings?

Because there's lots of other accents I would dearly love to use, and I can expend the extra effort, if there's a way to do it.

[Make7]

8. This site's html specifies the character set to use as ISO-8859-1...

... so you could refer to these pages:

http://www.htmlhelp.com/reference/charset/iso192-223.html   (uppercase accents)
http://www.htmlhelp.com/reference/charset/iso224-255.html   (lowercase accents)
http://www.htmlhelp.com/reference/charset/iso160-191.html   (symbols)

Just look up the number and add the ampersand, pound sign and semicolon.

<number>;

So 214; for a uppercase umlaut.

Technically that character set only goes up to 255, for other characters I have used unicode values before - although not all browsers will display them the same but I believe most people should be able to see them as intended. You might want to test anything above 255 across a few browsers/devices to double check that they get displayed properly.

[TreasonousBastard]

5. The other day I was listening to a few mathematicians asking...

is mathematics is a thing that we have discovered, or something we invented because it works for us.

Imaginary numbers can't exist in our systems, but we do find the concept useful at times. And how do parallel lines in curved space in four dimensional space work?

[Xipe Totec]

10. Topologically speaking, we're all penta-symmetric thoroids

Starfish with delusions of grandeur.

Which is why we have five fingers and toes, five appendages (counting the head), etc.

We are ruled by numbers.

[muriel_volestrangler]

19. Nah, that doesn't really work out

We "have five appendages" because our tail (one end of our backbone) happens to have shrunk to nothingness, while our head (the other end) hasn't. For many mammals, reptiles etc. you'd say there are 6, because the tail is just as much an 'appendage' as the head. Meanwhile, our four limbs came from 4 fins, 2 on each side being sufficient for walking through mud. We have 5 fingers and toes because that happened to be the number of bones in the extremities of those fins of the ancestor from which non-fish vertebrates evolved, and that's what present-day vertebrates tend towards.

[eppur_se_muova]

7. One of the most evocative portraits ever ...

Kurt Godel by Arnold Newman (1956)

[Xipe Totec]

16. There he is, at the height of his power, while I was busy just being born. nt

[longship]

17. One could also read "Godel, Escher, Bach"

One of the best books I've ever read, by Douglas R. Hofstadter. I have read it multiple times, and given away more copies than I can remember.

It's entire title is "Gödel, Escher, Bach: an Eternel Golden Braid. A metaphorical fugue on minds and machines in the spirit of Lewis Carroll."

If one reads this, it will change your outlook on the world. It is an extraordinary book which has justifiably earned its fame, and its Pulitzer Prize. Its focus centers on Gödel's Incompleteness theorem. And travels around Bach fugues, Escher prints, and, above all, strange loops, which are central to understanding Bach, Escher, AND Gödel.

First published in 1979. My current copy is the 20th anniversary edition with a new preface by the author.

Hofstadter not only wrote the book, he typeset it as well. That means he likely knew TeX, the typesetting language invented by computer scientist Donald Knuth, whose books on computer programming are legendary. As is this book by Hofstadter.

Gödel, Escher, and Bach indeed have a common thread, an eternal golden braid. This is an astounding read.



Hofstadter's partial semantic network for GEB (by the author):


Highly recommended:
Gödel, Escher, Bach (Wiki)



"Yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation.


Thus is one of the hooks to understanding how Gödel's theorem works.

[Xipe Totec]

18. One did, in 1979

I consider it one of my formative books. I have the first edition of G.E.B on my bookshelf, next to "Methamagical Themas" and "The Mind's I".