Diagonalization and Other Mathematical Wonders

It’s only a slight exaggeration to say I’m a mathematician because of Cantor’s diagonalization arguments (both the proof that the rationals are countable and the proof that the reals aren’t). I was already enjoying my intro to proofs class when we got to it, but it was the first theorem in the class (or in math generally) that truly astonished me.

In a highly scientific survey of all the mathematicians who live in my house about mind-blowing mathematics, diagonalization made a strong showing. Image: Jochen Burghardt, via Wikimedia Commons.

Mike Lawler recently wrote a post about math that made him go whoa! that links to a recent Reddit thread on the same subject. Lawler’s list includes the residue theorem and Galois theory along with Polya’s theory of counting, a topic I was unfamiliar with. Unsurprisingly, I am not alone in being astonished by the diagonalization argument, but people love a lot of other mathematics as well.

If you’re feeling a little blah after a long semester and months of dwindling daylight (Southern Hemisphere-dwellers, just imagine you’re reading this in six months), a trip through that Reddit thread might cheer you up. It’s got some of the greatest hits of undergraduate math and a good dose of heavier research math to boot. (Incidentally, maybe I need to bite the bullet and read the proof of the Riemann-Roch theorem in Hartshorne, which comes highly recommended by at least one Redditor.)

Reading through other people’s top mathematical moments naturally led me to reflect on my favorite bits of mathematics. I was 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.

For me, teaching complex analysis this semester was refreshing. Revisiting the basic material and really feeling like I understood how all the pieces fit together was gratifying. I have never appreciated the Cauchy integral formula more. Another recent time I really felt like I had a new appreciation for a theorem came last semester, when I was teaching undergraduate geometry and topology. The Gauss-Bonnet theorem seemed to tumble from the classification of surfaces in a way that felt so much more natural to me than it had when I first learned it.

Finally, I can’t reminisce about mind-blowing math without pointing to my favorite piece of mathematics communication of the year, and SMBC comic about Kempner series. (Kevin Knudson’s Forbes post on the topic is fun, too, although it doesn’t use the word “balls” as many times.)

Image: Zach Weinersmith, SMBC Comics.

When was the last time math made you say “What the balls?!”

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4 Responses to Diagonalization and Other Mathematical Wonders

1. Sue VanHattum says:

I agree: What the f-*!

I have no idea how one would prove that taking the ones with 9s out would make it finite. Can someone point me to a proof? (Or maybe I’ll play around with it some…)

• evelynjlamb says:

I’m glad I’m not the only mathematician who just learned about that recently! The SMBC comic is the first I remember hearing about it, and I was legitimately flabbergasted. You’ll enjoy playing with it for yourself, but Kevin Knudson’s post (http://www.forbes.com/sites/kevinknudson/2015/11/20/harmonic-convergence-adding-up-reciprocals/) has an explanation. Harvey Mudd’s page about it is good too: https://www.math.hmc.edu/funfacts/ffiles/20005.3.shtml. It has some suggestions for incorporating it into a class if you wanted to use it there. Enjoy!

• Mike Lawler says:

Thanks for the mention in the article, Evelyn.

We played around with the series in the SMBC comic (thanks to you tweeting about it, of course!). We didn’t go all the way to showing that the new series converges, but working with the new series did prove to be a neat counting exercise for kids:

https://mikesmathpage.wordpress.com/2015/06/30/counting-and-a-fun-harmonic-series-fact/

2. Barbara says:

The only thing better than the original Riemann Roch (which you can read anywhere, it’s always beautiful) is Grothendieck (Hirzebruch) Riemann Roch. I suggest reading the version for projective morphisms of smooth varieties in Fulton, or if you’re in a hurry google-image the original note of Grothendieck: short, in German, and with drawings of devils on it :).