Cantor's Attic

 Just can't get enough of Georg Cantor?...want to explore his work/ideas further?... Is that what's buggin' you! In a tweet yesterday, Richard Elwes pointed out this site doing just that:

http://cantorsattic.info/Cantor%27s_Attic

Cantor Weirdness

Fantastic treatment of the fractal Cantor Set and the “Devil’s Staircase” (Cantor function) from PBS’s “Infinite Series." Is it any wonder Cantor was driven to a sanatorium!:

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."

A Couple From Today

Couple of posts that appeared too late to make it into my MathTango weekly wrap-up:

a)  If The Cantor Set doesn't already blow your mind enough, don't despair, 'cuz Evelyn Lamb is here to blow your mind with the "fat Cantor Set":
http://blogs.scientificamerican.com/roots-of-unity/2015/04/30/a-few-of-my-favorite-spaces-fat-cantor-sets/

b)  and in some more deep stuff, Michael Harris has an ongoing discussion of homotopy theory:
https://mathematicswithoutapologies.wordpress.com/2015/05/01/pantheism-and-homotopy-theory-part-3/

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)."

Cantor's Paradise…

Keith Devlin's latest "Devlin's Angle" blogpost was inspired (in part) by his moderation of the World Science Festival panel (shown in my May 31 post):

http://devlinsangle.blogspot.com/2013/06/will-cantors-paradise-ever-be-of.html

He muses about whether the "study of infinity – in particular the hierarchy of larger infinities that Cantor bequeathed to us – would ever have any practical applications." (...and he thinks likely, not).

Part of what makes Cantor's work so eternally interesting is the huge divergence of opinion about it from his very own time. As Devlin writes, "Reactions to Cantor’s revolutionary new ideas ranged from outraged condemnation to fulsome praise." And these completely opposed viewpoints came from individuals equally-well-established and respected in the field. One of the paradoxical aspects of infinity is the equal ease with which it may be discussed in either direction: i.e., the infinitely large or the infinitesimally small. In any event, it was David Hilbert who eventually coined the term "Cantor's Paradise" for the new Cantorian thought that did slowly take hold.

Even so, still today, "infinity" can be such a difficult concept to grasp, that Cantor's ideas remain a frequent target of attacks by "crackpots" of the sort Mark Chu-Carroll often hears from:

http://scientopia.org/blogs/goodmath/category/bad-math/cantor-crankery/


In other news... the 99th Carnival of Mathematics is now up at Wild About Math blog for your delectation:

http://wildaboutmath.com/2013/06/02/carnival-of-mathematics-99/

Sunday Afternoon Reading

just passing along a few things from the last week:

1) A nice little introduction to Cantor, for any who need it, from Curious Wavefunction over at Scientific American:

http://tinyurl.com/a4u5n8r

2) Another bit of interesting Ramanujan biography here:

http://tinyurl.com/b44jgyk

3) For educators especially, Dan Meyer brings up a discussion of 'pattern matching' as it may apply to Khan Academy (this relates back to the subject of Benny's Rules which I've written about previously):

http://tinyurl.com/agoyxyn

(be sure to read the comments as well)

4) and lastly, for sheer entertainment (more chemistry than math, but really general scientific literacy), this Wikipedia page came across my Twitter feed, originally I think from Alexander Bogomolny, on the "dihydrogen monoxide hoax":

http://en.wikipedia.org/wiki/Dihydrogen_monoxide_hoax 

(be sure to read the "Public Efforts..." section... and weep)

accompanying photo here:

https://twitter.com/CutTheKnotMath/status/295265488156622848/photo/1

 

The Maddening Set

 I can never read (or write) about the 'Cantor Set' without thinking to myself, "Well, of course Cantor was driven to madness…!" (an infinite number of points having 1-1 or 0 measure might do that):

http://tinyurl.com/44szvot

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.).

Good Math, Not-So-Good Math

Mark Chu-Carroll over at 'Good Math Bad Math' has never suffered math cranks very well (...and he gets his share of them):

http://scientopia.org/blogs/goodmath/2011/11/05/yet-another-cantor-crank/

For more crankish entertainment you can visit here:

http://www.crank.net/maths.html

Rumbling In Cantor's Paradise

 "No one will drive us from the paradise which Cantor created for us." -- David Hilbert

I wasn't aware there was very much serious controversy over Cantor's proof that the real number set is uncountable (versus the set of integers which is countable), but RJ Lipton is aware of the naysayers out there and takes a stab at reaching them here:

http://tinyurl.com/3mo7mt4

(Not sure Lipton will win over any doubters with his argument, but for most, Cantor probably doesn't even require a defense; at any rate some interesting comments below Lipton's post.)

Kickin' Back With Cantor

...Whenever I have a day off (like today), and just want to relax a bit with some pizza and imported beer, I enjoy kicking back and... like yourself no doubt... watching some rollicking video on Georg Cantor ;-) :

Cantor Set

Georg Cantor derived some of the most mind-blowing insights of any mathematician; indeed many of his contemporaries found his ideas so wackalooney (excuse the technical language ;-)) that they simply dismissed his notions out-of-hand. But today, while his ideas are no less mind-boggling, they are widely accepted.
One of the most fascinating discoveries, though probably less famous than many of his conclusions regarding infinity, was the "Cantor Set." It is commonly depicted by taking a unit length and then deleting the middle third, to leave two bookend pieces, which in turn have their middle third removed, and on and on and on, infinitely. Many of the properties of this simple, final Cantor Set are quite remarkable...

From the mathacademy.com site (see: http://www.mathacademy.com/pr/prime/articles/cantset/ ):

"What remains after infinitely many steps is a remarkable subset of the real numbers called the Cantor set, or “Cantor’s Dust.”
At first glance one may reasonably wonder if there is anything left. After all, the lengths of the intervals we removed all add up to 1, exactly the length of the segment we started with:

 

Yet, remarkably, we can show that there are just as many “points” remaining as there were before we began! This startling fact is only one of the many surprising properties exhibited by the Cantor set."

The Cantor Set also exhibits self-similarity or fractal-like properties throughout.

...and another site here to explore the Cantor Set step-by-step:

http://personal.bgsu.edu/~carother/cantor/Cantor1.html

One often wonders, in the case of Cantor (and certain other mathematicians), did some sort of madness drive him to many of his remarkable, penetrating insights... or, did his unique insights drive him to madness???
Occasionally (and luckily, only occasionally) it seems as if there is a fine line between succeeding at the highest levels of math... and insanity. Chapter 4 (entitled "Mathematics as an Addiction") of Reuben Hersh's recent "Loving and Hating Mathematics" interestingly addresses the link (if any) between math and mental illness. (He concludes "No," but does add: "Still, there is something different about mathematicians compared to, say chemists or geologists or even English professors. It is possible to be 'crazy' -- that is conspicuously eccentric, very odd, even antisocial -- and still hold a job as a math professor.")
...It all reminds me a bit of Steven Wright's oft-cited observation that "There's a fine line between fishing... and standing on the shore looking like an idiot." ;-)

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