Whoooa!

Taking off from an earlier post by Mike Lawler on mathy things that make us go "whoa!," including Cantor's diagonalization proof,  Evelyn Lamb posts about some of her own "mathematical wonders" (with several good further links):

http://blogs.ams.org/blogonmathblogs/2015/12/21/mind-blowing-math-reminiscence/

Lamb writes at one point that as "a late mathematical bloomer"... "Not a lot of math really blew my mind in college because my attitude at the time tended towards the utilitarian. Diagonalization notwithstanding, I didn’t often appreciate the beauty of what I was learning or even know that I should be surprised by it. As time passes, I gain more and more respect for many ideas in math, even ones I’ve been familiar with for years."

Somehow, I find that a fascinating confession, since I imagine (maybe incorrectly?) most professional mathematicians arriving at their destination specifically because of an early captivation with the wonders/beauty of math and (in Wigner's terms) its "unreasonable effectiveness," versus duller, mere utilitarian application. But the detour-ridden roads to our final destinations are often long and winding, and mathematics, with its many possible footpaths, side-tracks, byways, may be no different than any other.

Anyway, there are too many 'whoa'-inducing ideas in math to pick a favorite, but I will link once again to one of my own mind-blowing faves, the Cantor Set:
http://platonicrealms.com/encyclopedia/Cantor-set

Honestly, it's not hard for me to imagine how Cantor was driven from sanity, given the matters he persistently tackled and wrestled with. If you stare at the sun you risk going blind, and if you stare at the heart of mathematics, as Cantor did, perhaps there are risks as well.

Rebecca Goldstein wrote a couple of decades back, "Mathematics and music are God's languages. When you speak them...you're speaking directly to God." 

I like that metaphor; whether it be God, Creation, the center of the Universe, or some other essence-of-being, when you "speak" mathematics or music (or, I would add certain forms of prayer/meditation), you reach a place, outside the narrow human realm, unattainable by any other means. WHOOOA indeed!