New Bio of Gödel

 A li'l beach reading  ;)   

A new biography of Kurt Gödel is out (h/t to Natalie Wolchover for pointing it out):

https://www.amazon.com/Journey-Edge-Reason-Life-Gödel/dp/1324005440/ref=sr_1_2?dchild=1&keywords=godel&qid=1622711647&s=books&sr=1-2

Veritasium Tackles Gödel

https://www.youtube.com/watch?v=HeQX2HjkcNo

Wolchover on Gödel

Natalie Wolchover has a new piece on Gödel’s Incompleteness Theorem at Quanta Magazine:

https://www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/

As usual Natalie does a great job. I’m too lazy right now to look it up, but my favorite explanation of Gödel is probably Raymond Smullyan’s that he gives in one or more of his volumes. Rudy Rucker also does a good job for the layperson in at least one of his volumes, and Ms. Wolchover mentions the Ernest Nagel/James Newman short older volume “Gödel’s Proof” as another good source.

Lastly, I’ll re-post a quote/tribute I’ve used before from Freeman Dyson in "The Scientist As Rebel":

"Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.

"It is a curious paradox that several of the greatest and most creative spirits in science, after achieving important discoveries by following their unfettered imaginations, were in their later years obsessed with reductionist philosophy and as a result became sterile. Hilbert was a prime example of this paradox. Einstein was another…

"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral… The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Gödel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalization into unexplored territory."

Our Fragile Nation

American democracy is fragile, yet resilient. We'll soon see if Republicans turn the impeachment trial of demagogue Donald Trump into a farcical sham (as they'll no doubt attempt to do). How many Republican Senators will refuse to cave to money, power, and autocracy, and actually seriously consider the Constitution, rule-of-law, and founding principles?....

Today (because it seems timely) I am just re-posting, with minor changes, an entry from close to a year ago:

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Most readers here likely know the famous story of Kurt Gödel’s 1947 visit to an examiner’s office to apply for U.S. citizenship — it’s been briefly told many places, as I did back HERE.

Gödel, logician that he was, thought he’d found a 'logical flaw' or 'contradiction' in the U.S. Constitution that would allow a dictator to take power in the U.S., not unlike what Europe had witnessed. Friends, Oskar Morgenstern and Albert Einstein, talked Gödel out of bringing this up at his examination, believing, according to some accounts, that his worry was 'far-fetched and outlandish.'  No one knows for sure what his qualms centered upon, but the most widespread guess is that he was concerned about Article V of the Constitution allowing for amendments to the Constitution… even amendments that might weaken/eliminate various checks-and-balances and hand more authority to a despotic leader. We could theoretically amend ourselves right into a dictatorship.

The appeal of this explanation is that it reflects Gödel's well-established interest in self-reference: i.e., the Constitution could be amended, the amendments could be amended, the amendments to the amendments could be amended, etc.

While certainly possible, I’ve never been fully comfortable with that ‘guess’ of Gödel’s thought process, because Kurt probably realized what a slow, arduous, unwieldy path amending the Constitution actually is… with ample opportunity along the way to redress or put the brakes on ill-founded changes.

I’ve begun to wonder if just perhaps what Gödel had in mind was, alternatively, something far simpler, more direct, and more mathematical:

He would’ve clearly understood the math of the Electoral College (as spelled out in Amendment XII of the Constitution), and recognized that a demagogic individual could become President with a minority of the citizens' vote (not even a plurality, let alone a majority) by simply concentrating on a handful of key states. Then, with backing of a subservient Party he might run roughshod over most checks-and-balances simply with the judicious use of Executive Orders, Executive Privilege, emergency measures, martial law, judicial appointments, and the President’s function as Commander-In-Chief of the military... may or may not be a better explanation than the Article V focus, and might or might not be viewed as a 'logical flaw.'

In the oft-quoted words of Sinclair Lewis (who was contemporaneous with Gödel):

 "When fascism comes to America it will be wrapped in the flag and carrying a cross."

In short, if enough of the electorate is naive and ignorant enough to elect one, isn’t every democracy at risk of putting a despot-to-be into power, and made even easier with our Electoral College system? The German experience may be more 'normal' than we care to contemplate...

Is THAT what Gödel realized 72+ years ago?

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Re-thinking Gödel…

Pre-empting the usual musical interlude this Friday just to ruminate a bit over something....

Most readers here likely know the famous story of Kurt Gödel’s 1947 visit to an examiner’s office to apply for U.S. citizenship — it’s been briefly told many places, as I did HERE.

Gödel, logician that he was, thought he’d found a 'logical flaw' or 'contradiction' in the U.S. Constitution that would allow a dictator to take power in the U.S., not unlike what Europe had witnessed. Friends, Oskar Morgenstern and Albert Einstein, talked Gödel out of bringing this up at his examination, believing, according to some accounts, that his worry was 'far-fetched and outlandish.'  No one knows for sure what his qualms centered upon, but the most widespread guess is that he was concerned about Article V of the Constitution allowing for amendments to the Constitution… even amendments that might weaken/eliminate various checks-and-balances and hand more authority to a despotic leader. We could theoretically amend ourselves right into a dictatorship.
The appeal of this explanation is that it reflects Gödel's well-established interest in self-reference: i.e., the Constitution could be amended, the amendments could be amended, the amendments to the amendments could be amended, etc.

While certainly possible, I’ve never been fully comfortable with that ‘guess’ of Gödel’s thought process, because Kurt probably realized what a slow, arduous path amending the Constitution actually is… with ample opportunity along the way to redress or put the brakes on ill-founded changes.

I’ve begun to wonder if just perhaps what Gödel had in mind was, alternatively, something far simpler, more direct, and more mathematical:

He would’ve clearly understood the math of the Electoral College (as spelled out in Amendment XII of the Constitution), and recognized that a demagogic individual could become President with a minority of the citizens' vote (not even a plurality, let alone a majority) by simply concentrating on a handful of key states. Then, with backing of a subservient Party he might run roughshod over most checks-and-balances simply with the judicious use of Executive Orders, Executive Privilege, emergency measures, martial law,  and the President’s function as Commander-In-Chief of the military... may or may not be a better explanation than the Article V focus, and might or might not be viewed as a 'logical flaw.'
In the oft-quoted words of Sinclair Lewis (who was contemporaneous with Gödel):

"When fascism comes to America it will be wrapped in the flag and carrying a cross."

In short, if enough of the electorate is naive and ignorant enough to elect one, isn’t every democracy at risk of putting a despot-to-be into power (and made even easier with our Electoral College system)?

Is THAT what Gödel realized 72 years ago?

Or heck, maybe I'm just makin’ crazy talk? ;)

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[sidenote: Though it’s not a planned ongoing feature, I will have another mathematician “profile” of sorts up here this coming Sunday, and a couple of suggestions for others to profile have come in so there might(?) be others in the future — if you wish to suggest someone feel free to email or DM me.]

Some Bits Crossing My Mind This Month

A miscellany today...:

1)  FIRST, in case you've been living under a rock... on the planet Zorka... in Galaxy 134-18B this last week and don't know, TOMORROW (Monday) Michael Atiyah is giving a 45-min. talk entitled simply, "The Riemann Hypothesis" at the Heidelberg Laureate Forum, claiming "a simple proof using a radically new approach." I have no idea how serious of a "proof" this is (other than Atiyah being a serious mathematician, but still hard to take this at face-value, given how many radically new, simple approaches have already been tried). Either way, math cyberspace should be abuzz tomorrow with commentary following the presentation:

I believe the talk will be live-streamed and recorded at the HLF YouTube channel here:

https://www.youtube.com/user/LaureateForum

Also, a couple of the Aperiodical bloggers will be in attendance and reporting on the meeting (I assume they'll check in after the dust settles, but maybe they'll do some live-blogging or tweeting  as well? -- surely there will be some live-tweeting from #HLF18).

==> ADDENDUM 9pm. 9/23... the proof, by contradiction, has now been posted here (h/t to @sigfpe on Twitter):

https://www.dropbox.com/s/pydoj0a8hguebc6/2018-The_Riemann_Hypothesis.pdf?dl=0

ADDENDUM II 9/24:  one of the live-tweeted threads from Atiyah's talk (now over) is here:
https://twitter.com/mpoessel/status/1044131977950109696

...needless to say, a lot of skepticism being expressed across the Web by those who understand the math/logic; no doubt there will be a lot more commentary today, and even if negative, much food-for-thought may still emerge from this.

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2)  Speaking (loosely) of proofs... logician George Boolos would’ve been 78 this month… had he not died at the relatively young age of 55 in 1996. For any relative newbies, one of my favorite mathy pages on the Web is his famous, delightful single page explaining Gödel’s second incompleteness theorem “in words of one syllable” (worth reading for fun at least once every year!):

https://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

It can probably even make a nice introduction to Gödel for younger folks.

———————————

3)  Curses, curses, curses to Jordan Ellenberg who has gotten me regularly reading Martin Shkreli’s prison-composed blog, ever since Jordan cited it in a tweet.  I first wrote about it back here:

https://math-frolic.blogspot.com/2018/08/calling-all-number-theorists.html

And in his 9/6/18 entry, after seeking help with some math he was working on, Shkreli ended by writing:

“Thank you to all the professors, postdocs and other math professionals who have reached out to help me. It has been great to communicate with you all. Bear with me as I order my thoughts and respond in the limited way I can.”

I don’t know if this is bluster, bluff, or actuality, but if it is for real, I’d sure be curious to hear about what substantive math any “math professionals” have taken up with Martin, if you’d care to share? Ought to be some sort of interesting backstory there.

[...Also, Martin regularly recommends Bio-Pharm stocks to buy or avoid (or short), and even though I don't dabble in bio-pharm stocks myself I'd be curious if anyone else has found his judgments useful/profitable.]

———————————

4)  For those interested in cognition, science writer John Horgan has a new Web-accessible volume out on mind-body problems. I’ve enjoyed John’s writing in the past, but also tire a bit of this topic that seems forever shrouded in sound and fury, without much ever resolved. As John says, the book “offers my subjective takes on my subjects’ subjective takes on subjectivity.” So I wasn’t expecting too much from his latest, but in fact enjoyed it immensely, partly because of the portraits it paints of specific diverse, fascinating thinkers; their foibles and makeup, in addition to their academic or cerebral selves, while delving into their thoughts on mind/body issues. You can read the whole volume here: 

https://mindbodyproblems.com

…or you can download it from the Web for a small price.

There are probably many Douglas Hofstadter fans out there, so as one sample chapter, I recommend Chapter Two which is with Dr. Hofstadter (p.s… one small side-note that I learned here, and didn’t even realize before, is that David Chalmers did his PhD. under Hofstadter):

https://mindbodyproblems.com/chapter-two/

Speaking of books, Scott Alexander (just a bit behind the times) offers a long review of Nassim Taleb’s “The Black Swan” here (followed by 250+ comments):

http://slatestarcodex.com/2018/09/19/book-review-the-black-swan/

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5)  For those with the chops to follow it, Steve Strogatz recently passed along this history of the Langlands Program:

http://www.ams.org/journals/bull/2018-55-04/S0273-0979-2018-01609-1/S0273-0979-2018-01609-1.pdf

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6)  And I'll close out with this gem that surfaced on my Twitter feed yesterday:
https://twitter.com/jayvanbavel/status/1042838461173116928

Elusive Truth...

Sunday reflection:

“Gödel didn’t believe that truth would elude us. He proved that it would. He didn’t invent a myth to conform to his prejudice of the world — at least not when it came to mathematics. He discovered his theorem as surely as if it was a rock he had dug up from the ground. He could pass it around the table and it would be as real as that rock. If anyone cared to, they could dig it up where he buried it and find it just the same. Look for it and you’ll find it where he said it is, just off center from where you’re staring. There are faint stars in the night sky that you can see, but only if you look to the side of where they shine. They burn too weakly or are too far away to be seen directly, even if you stare. But you can see them out of the corner of your eye because the cells on the periphery of your retina are more sensitive to light. Maybe truth is just like that. You can see it, but only out of the corner of your eye.” 

— Janna Levin (from “A Madman Dreams of Turing Machines”)

Gödel's Loophole

I don't have a lot of potpourri picks for you this week (over at MathTango), so I'll add this long, older pdf that seems timely, on the well-known story of Kurt Gödel and a possible loophole in Article V (perhaps) of the U.S. Constitution (a surprisingly interesting read, coming from a law review journal!):
http://tinyurl.com/zs3tvb3

Just Sayin’…

Judge Philip Forman:  “Now, Mr. Gödel, where do you come from?” 

Kurt Gödel:  “Where I come from? Austria.”

Judge:  “What kind of government did you have in Austria?” 

Gödel:  “It was a republic, but the constitution was such that it finally was changed into a dictatorship.” 

Judge:  “Oh! This is very bad. This could not happen in this country.” 

Gödel:  “Oh, yes, I can prove it.” 

With the intensity of the political conventions passed I thought I could drag myself away from politics… but, well, not quite yet!:

Kurt Gödel was no slouch of a thinker or logician, and many or most of you will know the legendary story of his trip to attain U.S. citizenship in 1947. 

The above conversation snippet is part of the purported discussion between Kurt Gödel and the presiding examiner, Judge Philip Forman, when Gödel went (with his friends Albert Einstein and Oskar Morgenstern as character witnesses) to a hearing applying for U.S. citizenship. Using pure reasoning, Gödel was certain he had found a flaw (a self-contradiction) in the U.S. Constitution that would permit a dictator to attain power in the country (in fact, according to some versions of the story I’ve seen, he believed it almost inevitable, given enough time). And, he thought he could prove it!

Einstein and Morgenstern knew Gödel was courting disaster (in applying for citizenship) if he insisted on pushing his view that the country was bound for possible dictatorship, so they helped maneuver Gödel nervously through the mundane procedure, and all ended well.

The story has been told widely, though never with complete clarity, nor with details as to what Gödel’s specific argument with the Constitution was. Many believe it had to do with Article V which spelled out the right to amend the Constitution… in Gödelian recursive thinking this would imply also the right, at some point, to amend the Constitution to say it may no longer be amended, following a despot gaining power. There are, however, likely several other spots in the Constitution that could harbor the seeds of eventual dictatorship. 

For a further long legalistic discussion of Gödel’s story legend see pdf link here:

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2010183

Anyway, I find it interesting that now, almost 70 years since mathematician Gödel expressed his concerns to Einstein and Morgenstern, there is suddenly more talk of fascism and dictatorship in America than probably ever before in history.  As a recent tweet I saw on Twitter said, "When Fascism comes to America, it won't release its tax returns" ;-) 
One can almost imagine Kurt Gödel, somewhere in the Great Beyond, nodding knowingly...

Introduction to Incompleteness

"Incompleteness is one of the most beautiful and profound proofs that I’ve ever seen. If you’re at all interested in mathematics, it’s something that’s worth taking the effort to understand."  -- Mark Chu-Carroll 

Mark Chu-Carroll (of "Good Math, Bad Math") is in the process of re-posting his own splendid discussion/explanation of Gödelian Incompleteness this week. If it's a subject that interests you, or you've always wanted a detailed introduction, his first three four posts (with more to come) are here:

http://www.goodmath.org/blog/2014/08/25/godel-reposts/

http://www.goodmath.org/blog/2014/08/26/godel-numbering/

http://www.goodmath.org/blog/2014/08/27/godel-part-3-arithmetic-and-logic/

[just added] http://www.goodmath.org/blog/2014/08/28/gdel-part-3-meta-logic-with-arithmetic/

Sunday Thoughts…

From Freeman Dyson in  "The Scientist As Rebel":

"Gödel's theorem shows conclusively that in pure mathematics reductionism does not work. To decide whether a mathematical statement is true, it is not sufficient to reduce the statement to marks on paper and to study the behavior of the marks. Except in trivial cases, you can decide the truth of a statement only by studying its meaning and its context in the larger world of mathematical ideas.
"It is a curious paradox that several of the greatest and most creative spirits in science, after achieving important discoveries by following their unfettered imaginations, were in their later years obsessed with reductionist philosophy and as a result became sterile. Hilbert was a prime example of this paradox. Einstein was another…

"Science in its everyday practice is much closer to art than to philosophy. When I look at Gödel's proof of his undecidability theorem, I do not see a philosophical argument. The proof is a soaring piece of architecture, as unique and as lovely as Chartres Cathedral… The proof is a great work of art. It is a construction, not a reduction. It destroyed Hilbert's dream of reducing all mathematics to a few equations, and replaced it with a greater dream of mathematics as an endlessly growing realm of ideas. Gödel proved that in mathematics the whole is always greater than the sum of the parts. Every formalization of mathematics raises questions that reach beyond the limits of the formalization into unexplored territory."

"Genius Recognizing Genius"

83 Years ago yesterday...:

"…the steps leading up to Gödel's startling conclusions are both logically tricky and intricately intertwined…
"An indicator of the degree to which Gödel's results were unexpected can be found in the reaction to his original announcement of the theorem at a philosophy-of-science symposium in Königsberg, Germany, on September 7, 1930. Ironically, Königsberg happened to be Hilbert's hometown, which perhaps partially accounts for the lukewarm reception given to Gödel's presentation of his results. In fact, the transcript of the discussions at the meeting gives no indication whatsoever of Gödel's remarks, and there is no mention of Gödel at all in an article published later summarizing the papers given at the meeting! So like many belief-shattering ideas, Gödel's appears to have been so unexpected and revolutionary that even the professionals didn't at first understand what he had accomplished.  But one participant who did see immediately the implications of the work was John von Neumann, who cornered Gödel after his talk and pressed him for more details -- a case of genius recognizing genius, I suppose."

          -- from John L. Casti's "Searching For Certainty"

and the rest, as they say, is math-logic-philosophy history....

Happy Birthday To You, Happy Axiomatic Birthday To You...

(via WikimediaCommons)

 A Happy Birthday today to Kurt Gödel (...wherever you are)!

In his honor I'll link once again to the logician George Boolos' clever explication, using only monoysllabic words(!), of Gödel's Second Incompleteness Theorem... always makes me grin:

http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

ADDENDUM: for those with a deep interest in Gödel the "Gödel's Lost Letter..." blog has just put up a wonderful ranging post also in honor of the birthday today:

http://rjlipton.wordpress.com/2013/04/28/happy-birthday-kurt-gdel/

Boltzmann, Cantor, Gödel, Turing

From the BBC "Dangerous Knowledge" series:






If you've not seen these wonderful videos on 4 greats, try to set aside enough time at some point to enjoy them (each ~45 mins.).

Mathematical Truth...

A great, older interview with Rebecca Goldstein on Gödel, Platonism, and everything in-between, from 2005:

http://edge.org/conversation/godel-and-the-nature-of-mathematical-truth

an excerpt:

 "Gödel mistrusted our ability to communicate. Natural language, he thought, was imprecise, and we usually don't understand each other. Gödel wanted to prove a mathematical theorem that would have all the precision of mathematics—the only language with any claims to precision—but with the sweep of philosophy. He wanted a mathematical theorem that would speak to the issues of meta-mathematics. And two extraordinary things happened. One is that he actually did produce such a theorem. The other is that it was interpreted by the jazzier parts of the intellectual culture as saying, philosophically exactly the opposite of what he had been intending to say with it. Gödel had intended to show that our knowledge of mathematics exceeds our formal proofs. He hadn't meant to subvert the notion that we have objective mathematical knowledge or claim that there is no mathematical proof—quite the contrary. He believed that we do have access to an independent mathematical reality. Our formal systems are incomplete because there's more to mathematical reality than can be contained in any of our formal systems. More precisely, what he showed is that all of our formal systems strong enough for arithmetic are either inconsistent or incomplete. Now an inconsistent system is completely worthless since inconsistent systems allow you to derive contradictions. And once you have a contradiction then you can prove anything at all."

Gödel Incompleteness 101

(Kurt Gödel via Wikimedia Commons)

From a prior RadioLab show, with Steven Strogatz, this podcast segment recounts Gödel 'Incompleteness' for us layfolk (starts with a minute+ of advertising):

Deja Vu Gödel

I've linked to this before… and will again… 'cuz it's one of my favorite citations. A phenomenally short, succinct, inspired, mind-twisting, 1994 monosyllabic explanation of Gödel's Second Incompleteness Theorem from George Boolos. Gotta luv it!:

http://tinyurl.com/cgexvsy

or, available here as pdf:

http://www2.kenyon.edu/Depts/Math/Milnikel/boolos-godel.pdf

Gödel Simplified... sort of

I got a big kick out of this 1994 1-page (pdf) explanation of Gödel's Second Incompleteness Theorem from George Boolos, all in single-syllable words (literally)... pretty impressive!:

http://bit.ly/bWrBqz

Gödel In a Nutshell

A short, descriptive bio of Kurt Gödel here:

http://drvitelli.typepad.com/providentia/2011/07/deconstructing-godel.html

"Picking Holes In Mathematics"

If you're interested in the underpinnings of mathematics, an interesting article from plus.maths.org here, entitled "Picking Holes In Mathematics":

http://plus.maths.org/content/picking-holes-mathematics

Early on it starts off in this vein:

"The logician Kurt Gödel proved in the 1930s that if you set out proper rules for mathematics, excluding leaps of faith or intuition as admissible moves, you lose the ability to decide whether certain statements are true or false. And this isn't because you chose the wrong rules: for any set of rules, as long as they're strong enough to make sense of whole number arithmetic, there will be statements you can't prove or refute.

"This is rather shocking and you may wonder why Gödel's result hasn't wiped out mathematics once and for all. The answer is that, initially at least, the unprovable statements logicians came up with were quite artificial and didn't touch on ordinary everyday mathematics."

...and then it goes on to recount some of the work of Harvey Friedman of Ohio State University in the arena of Gödel incompleteness.