New From Jim Holt on Infinity and More

One of my favorite past posts to write was a little tribute to David Foster Wallace for his volume on infinity, “Everything and More”:

https://math-frolic.blogspot.com/2015/02/infinity-and-angst-david-foster-wallace.html 

Jim Holt now has a new book, “When Einstein Walked With Gödel”, excerpted here with this wonderful passage on Wallace and more:

https://lithub.com/on-david-foster-wallace-georg-cantor-and-infinity/

Measuring Infinities

Fantastic article from Quanta Magazine (Kevin Hartnett) about new findings/proof of the equivalency of two variant infinities — actually findings published a year ago; am amazed it’s just now reaching the wider press (at least I’d not heard about this ’til now!):

https://www.quantamagazine.org/mathematicians-measure-infinities-find-theyre-equal-20170912/

Part of what makes the proof interesting (IF I understand matters correctly) is that it didn't require any re-statement of fundamental set theory, but only a bringing together of disparate math models that had not been linked up before. Even if you (like me) don't understand the details of the finding, just recognizing that a 50+ year problem has been resolved is very exciting. The solvers, Maryanthe Malliaris and Saharon Shelah, received the Hausdorff Medal for their work earlier this year.

…In a bit of irony, the above article got tweeted out yesterday on the very anniversary of the death of David Foster Wallace whose book on infinity, “Everything and More,” I've discussed earlier here:

https://mathtango.blogspot.com/2014/07/infinity-and-angst-david-foster-wallace.html

Paradise...?

In deference to my newest interviewee over at MathTango today, Sunday reflections from David Hilbert:

"The infinite! No other question has ever moved so profoundly the spirit of man; no other idea has so fruitfully stimulated his intellect; yet no other concept stands in greater need of clarification than that of the infinite."

"No one shall expel us from the Paradise that Cantor has created for us."

From 6th Grade to Advanced Physics...

Just a couple of links that hint at the breadth of mathematics, to start the week with:

a)  Brian Hayes, who is not a 6th-grader ;-) is off-and-running with an analysis of an arithmetic problem Fawn Nguyen previously posed to her 6th graders (...both delightful and interesting):
http://bit-player.org/2015/mrs-nguyens-prestidigitation

b)  Despite David Hilbert famously chastising the detractors of Cantor's "infinity," that "no one shall drive us from the paradise which Cantor has created for us," physicist Max Tegmark argues for finding "the true laws of physics" by finally getting rid of infinity:
http://blogs.discovermagazine.com/crux/2015/02/20/infinity-ruining-physics/#.VOs2kcZDTLX

Numerosities

For the philosophically, or foundationally, inclined, the below post argues for something called "numerosities" that give "part-whole" set relationships priority over "one-to-one" relationships, and in so doing, counter the usual Cantorian orthodoxy, which permits a partial set (say odd numbers), to be deemed equal in size to an entire set (all integers):

http://www.newappsblog.com/2014/03/counting-infinities.html

Excerpt: "The philosophical implications of the theory of numerosities for the philosophy of mathematics are far-reaching... Philosophically, the mere fact that there is a coherent, theoretically robust alternative to Cantorian orthodoxy raises all kinds of questions pertaining to our ability to ascertain what numbers ‘really’ are (that is, if there are such things indeed)."

Something for the Young, the Older, and the Conspiracy Theorist

A few items you may have already seen, but if not...:


1) Sherlock-Holmes-wannabes are still trying to figure out who "Satoshi Nakamoto," creator of Bitcoin really is. The NY Times piece below draws an unconvincing (I think) link to the former proprietor of the Silk Road website. The guessing game may be more fun than learning the true identity will ever be.

http://tinyurl.com/otavjw3

2) Another great post from MathMunch covering several matters, but what most interested me (because I'd not heard of it) is a Scrabble-like game called "Numenko" that helps young people learn/practice arithmetic:

http://mathmunch.org/2013/11/04/numenko-turning-square-and-toilet-paper/

3) And finally, my favorite recent read from the Web, the always excellent Natalie Wolchover attempts to explain (better than I've ever seen done before, though still a tough-read) two different approaches to solving the infinity "continuum" problem:

https://www.simonsfoundation.org/quanta/20131126-to-settle-infinity-question-a-new-law-of-logic/

Set aside some brain-time to take in this account of "forcing axioms" versus "V = ultimate L".

"Does Infinity Really Exist?"

Sorry for the late notice, but this upcoming live-streamed discussion of infinity (part of the World Science Festival) looks like it should be great (8pm.-9:30pm ET TONIGHT!!) if you've got no set plans for the evening:

http://www.scientificamerican.com/article.cfm?id=does-infinity-really-exist

 Moderated by Keith Devlin, with panelists Raphael Bousso, Philip Clayton, Steven Strogatz and W. Hugh Woodin. ...Like I said, oughta be great!


ADDENDUM: if you missed it, here it is:



It's fully two hours long and I found the first half-hour a bit dull/mundane (if you have any background at all with infinity), but for me it got more interesting with the entry of Hugh Woodin at around the 33-minute point and the physicist, Raphael Bousso (really enjoyed the chance to hear Woodin, who I'd not heard before and who is well-known for new work on the Continuum Hypothesis).

Infinite Trees Finitely Discussed

Keith Devlin takes on infinity via mathematical trees in this recent blog post:

http://devlinsangle.blogspot.com/2011/12/christmas-trees-from-land-of-santa.html

Wow! The Continuum Hypothesis Solved???...

...well, I doubt it, but what do I know: Apparently a major Berkeley mathematician/set theorist, Hugh Woodin, using a "radically stronger logical structure" known as "ultimate L," believes he has accomplished what no one has been able to do in well over a century, and demonstrate that Gödel's Cantor's Continuum Hypothesis is true (essentially, that there exist no infinite sets lying between the set of integers and the set of real numbers -- of course it still all hinges on the initial axiomatic system one adapts):

http://www.newscientist.com/article/mg21128231.400-ultimate-logic-to-infinity-and-beyond.html?full=true

(Coincidentally, this fascinating article is from Richard Elwes who I was just highlighting a few days back.)

If you're not interested in infinity or sets, skip this article; otherwise, dive in!

Infinity for Students and Teachers

Interested in infinity, or looking for good resources for students? This "teaching package" from plus.maths.org has lots of good links on infinity and infinite series:

http://plus.maths.org/content/teacher-package-series

Some Miscellany For Wednesday

Another look at the foundations of mathematics in upcoming book:

http://tinyurl.com/33ylyvp

If you can't get enough of Pi:

http://facts.randomhistory.com/2009/07/03_pi.html

James Tanton's newly-active Twitter feed here:

http://twitter.com/#!/jamestanton

YouTube Introduction to Cantor

YouTube Intro to Cantorian infinity for the layman:

(part 1) http://www.youtube.com/watch?v=WihXin5Oxq8
 
(part 2) http://www.youtube.com/watch?v=KhgNiqI-bt0&feature=channel

Infinity and More (or Less)....

David Foster Wallace was an amazing, prolific, creative, award-winning novelist ("Infinite Jest" his greatest popular success), who died all too early at his own hands battling depression. He also wrote some non-fiction offerings, one of which, oddly-enough, was a treatise on the history of math and the concept of infinity, for W.W.Norton's "Great Discoveries" book series.

I was looking forward to reading this 2003 take of a bright, engaging, non-mathematician writer/novelist titled "Everything And More: A Compact History of Infinity," but then came across highly critical reviews of the volume on the Web by mathematicians I much respect (the book is checkered with flaws and inaccuracies). This cooled but didn't expunge my interest in the volume. And now having read it I am amazed that this English-major/novelist, could even have tackled (...let alone in such an offbeat style) the heavy math-cerebral subject-matter surveyed in this dense 300-page book (which the author, accustomed to writing 1000-page tomes, calls a "booklet") --- so, even with mistakes-and-all, I can't help but (with a grain of salt) recommend it, as unlike any other math book you are likely to encounter; written in an informal and conversational tone about ideas that are utterly UN-informal and UN-conversational (...and the multitudinous footnotes are virtually as fascinating as the main text)! Here, a few less-negative reviews:

http://www.powells.com/review/2003_12_01.html

http://www.popmatters.com/books/reviews/e/everything-and-more.shtml

http://www.nytimes.com/2003/11/16/books/review/16PAPINET.html

Definitely not everyone's cup-o-tea, but still I think an intriguing, complex, quirky read by a brilliant, troubled mind from the humanities, covering material that has troubled a great many astute minds in the world of mathematics.