I was having a fun conversation about the Numberphile 1 + 2 + 3 + . . . . = -1/12 video on twitter yesterday, but the limitations of twitter were making the conversation difficult. So I decided to spend a little time today gathering my thoughts on the video, and perhaps more to the point, the video’s potential impact on the public perception of math. It is proving to be more difficult than I expected to pin down my own thoughts, but I’ve spent all the time I can, so here goes nothing!
**Update** – February 10th, 2014 – Ed Frenkel has an amazing 10 minute video interview about his reaction to the Numberphile video. Wonderful. I put this interview at the top of my post here because he says much of what I wanted to but far more eloquently than I could have:
** End update **
It almost feels as though there must be some sort of uncertainty principle for popular math presentations -> rigor * popularity = constant. The more rigor, the less popularity, and the more popularity, the less the rigor will have to be.
Who knows if that uncertainty principle is true or not, but even if it is true, I’m not sure it would bother me all that much. It would mean that occasionally there will be popular presentations of interesting math that fall short on rigor. Maybe the difficulty that I’m having gathering all of my thoughts is simply because the Numberphile video falling short on rigor doesn’t bother me all that much. I do not believe that people who are out to tell stories about interesting or unusual math are out intentionally trying to deceive anyone. If the cost of bringing math to the masses is an occasional angry lecture from the internet about things like analytic continuation, then so be it.
Essentially everything that could possibly be written about the Numberphile video is covered in this wonderful blog post from Aperiodical:
http://aperiodical.com/2014/01/an-infinite-series-of-blog-posts-which-sums-to-minus-a-twelfth/#more-11423
For the purest of the pure math people, the two Terry Tao pieces tell the story. Evelyn Lamb’s post is probably the best place (by far) to start if you don’t live and breathe math everyday. If the video made you cringe, then Cathy O’Neil’s post will probably make you stand up and cheer.
Rather than rehashing all of the math, I want to focus more on why I liked the video despite the flaws, and also talk through my reaction to two other popular math videos for comparison.
Let’s start with a fun one – Vi Hart’s “Wind and Mr. Ug” :
I wanted to start with this one because it is one of the most brilliant, if not the most brilliant, presentation of math for the masses that I have ever seen. The first time I saw it I thought she should win a genius grant. The only difficulty that I have with this video is that I don’t really know what you should do with people (kids mostly) after watching it. Sure, you can make a few Moebius strips – and I’ve done that every time I’ve watched this video with kids – but that seems to be about it. It is a beautiful video, but it floats away like a balloon after it ends. Truthfully, I’m not at all sure how I’d use it in a class if I was still teaching. The one and only thing I’ve seen that used this video in a clever way is hardly math-related at all. It is still fun though, so check out this great parody video that tells the story from Mr. Ug’s perspective:
Next up is a video I don’t particularly like – Khan Academy’s video about the Golden Ratio and the moon:
For me, the example in this video is so contrived that I have a hard time seeing how students would find it interesting. Unlike the two videos above that I love (and watched all the way through before posting them to this article), I struggle to make it even 30 seconds in to this one. There’s so much beautiful math behind the golden ratio, why in the world you’d want to spend time on example like this is beyond me. While I am probably a bigger fan of Khan Academy than most folks, and happily use their exercises with my kids from time to time, I worry that videos like this make math seem lifeless. These contrived ideas make my blood boil, and so before I get too far down the road of a full-blown rant, let me stop and simply say that I wrote about it here:
https://mikesmathpage.wordpress.com/2013/12/12/the-golden-ratio-jumping-the-shark/
Finally, the Numberphile video.
As I write my post here, the video has been viewed just over 1.3 million times. Wow.
Yes there are mathematical flaws, and yes some articles have point out more flaws than I had noticed myself the first time I saw it, but I liked it and still like it despite the criticism. I watched it with my kids who screamed at the screen that the equation was wrong (especially my 7 year old!). I asked them about it again today at breakfast. My 7 year old just repeated that it was all wrong. My 10 year old said that it was just a bunch of physics nonsense (ha!). At least their emotional reactions stuck with them.
If I was still teaching college or high school kids I would definitely watch the video with them. Spending a day or two with students talking through some of the ideas in this video would, I think, be absolutely amazing. There are so many different, and so many interesting paths to go down to help understand this crazy looking sum.
There are a couple of reasons that It didn’t really bother me that they were cutting corners on the math details. First, many of those loose ends / errors are very interesting things to talk about in their own right and would be great things to talk through with students after watching the video. Many of the blog articles explain in varying amounts of detail, how to correct these errors. Second, I’m sure that a lot of physicists think about the world exactly the way that these guys in the video do, and I believe that they were trying to explain the rather odd result here as best they could without getting too caught up in ideas of analytic continuation or unusual interpretations of infinite sums.
A lot of the criticism reminds me of the criticism I heard directed at a new theory from Seiberg and Witten back in the 90s. They had found a completely new way of thinking about some ideas in physics that had amazing applications in pure math. Some mathematicians were worried about the lack of rigor. Others ran with the theory and some big problems that had been unsolved for 100’s of years fell like dominoes. I remember attending a few standing room only lectures by Cliff Taubes at Harvard. I’ll never forget the excitement and buzz in the math world then, and I’ll also never forget the healthy skepticism either – mathematicians crave rigor.
Physicists aren’t nearly as worried about rigor, though. For example, here’s a write up on the Numberphile video from a physics blog:
http://physicsbuzz.physicscentral.com/2014/01/does-1234-112.html?m=1
Two of the items emphasized in this blog are (1) the infinite sum result is well known going back to Euler and Ramanujan, and (2) it gives the right answers in Quantum Electro Dynamics (QED) so as strange as the bizarre some of the series seems, there must be something there. I’m sure there are some physicists who would want more rigor, but my guess is that those two points of emphasis would carry the day most of the time in the physics community.
Away from the mathematical rigor, one other very important point of focus was mentioned by Patrick Honner in our twitter conversation yesterday:
I have a couple of reactions. First, if you want to talk about fun and interesting math, I do not think you can ever completely eliminate a this problem. If you are worried that people won’t get the math, you can report the proof of Fermat’s Last Theorem, or note that Harald Helfgott proved the weak Goldbach conjecture, but you can never give any details. With regard to the specific topic of this video, covering analytic continuation or non-standard techniques used in infinite sums would alleviate the problem only because no one would watch the video. Both of the topics are too technical for inclusion in a 10 minute presentation to a general audience.
Second, push back against ideas in science isn’t new or unusual and I hope that a large part of the message to the public about math is that it is ok to ask questions and ok to not understand. Sometimes the pushback can be terribly unhealthy – see the story of Georg Cantor presented in “The Mystery of Aleph” for example. But it can be healthy, too. “The Infinity Puzzle” gives an absolutely fascinating (and a little controversial) description of how ideas in quantum physics have evolved over time. Gary Kasparov’s series “My Great Predecessors” provides an astonishing presentation of the evolution ideas in chess. Of the 1.3 million people who have viewed the Numberphile video, there are probably at least 1 million who thought “I don’t get this.” There were many resources that came to light for people looking to understand it, which I think is great. I really hope that these resources found they way to anyone whose reaction to the video was similar to the reaction Mr. Honner is worried about.
Third, I do share Mr. Honner’s concern in general, and worry that a lot of popular math presentations end up being pitched in ways that are not always the best. One that I saw recently really struck me because the author has done, and I’m sure will continue to do, more for math in the US than just about anyone. If you look at some of the promotional material for Jordan Ellenberg’s new book “How not to be Wrong” you’ll find that “Ellenberg learned algebra at the age of 8 and got a perfect score on his Math SATs as a 12 year old.”
http://www.penguincatalogue.co.uk/hi/press/title.html?catalogueId=257&imprintId=1166&titleId=19456
Of all the accomplishments he’s had, and all of the amazing things he’s done, I wouldn’t have even dreamed about putting his age 12 SAT score near the top. I worry about a subtle message to kids that reinforces the idea that math is simply about innate ability – you are either a math person or you aren’t, and you might know by age 12 either way. I worry a lot about the reaction of someone who thinks “I don’t get math” to this message.
Finally, if you are looking to find new and interesting and certainly non-controversial ways to pitch fun math, look no further than . . . . Numberphile! Check out their interview with Ed Frenkel:
Frenkel’s quote at the end really struck me: “What if I told you there is this beautiful world out there and you don’t even have to travel anywhere to find it. It’s right at your fingertips . . . . this is the coolest stuff in the world.”
At this point there are many people out there working in various different ways to communicate fun and interesting math to the public. While those people might have differing views about Numberphile’s infinite series video, I’m sure that group broadly agrees with Frenkel’s position. His note at the end inspired me to do a series about 4 dimensional spheres with my kids (see my blog post right before this one), and hopefully other people who agree with Frenkel will continue to put new and fun math resources out for public consumption. Can’t wait for the next public math controversy 🙂
(two ways, actually!) and also how to find that the volume of a sphere was 
That talk with my kids had been inspired by a paper that Stephen Strogatz had posted earlier in the week that discussed the volume of a sphere in any dimension. Pretty neat stuff for sure, but I stopped with just the circle and sphere with my kids.
, I was wondering if either of the kids would be able to recognize the number. Turns out that my oldest son actually did guess the right number after I asked him a few questions. My youngest son thought it was pretty neat that we found a value for the sum, and also really cool that the volume of the 4d sphere with radius 1 was 
though that might require another week or two of pondering on my end!
in their formulas. That result reminded me of another famous problem whose solution also has powers of 
and 
Mike's Math Page 