A Proof to Knock Your Socks Off!
A recent YouTube video (January 9, 2014) by Numberphile has created a stir! You know it is unusual when it results in the reprint of an article in your local newspaper, which usually does not recognize mathematics as a topic in existence.
After watching the proof, the first thing I wanted to do was get my hands on a copy of Polchinski's String Theory text. Sure enough...the book exists, the relationship exists, and it is even proven before it is used:
First, watch the video of this astounding proof. If you had a Deja Vu feeling, it may be because I also suggested this video a month ago.
Still there? And is your mind with you? Almost 2 million people have watched this video, many reacting with puzzling comments.
You have just watched a "valid" proof by two physicists of an obviously "wrong" statement, yet its "truth" is used in quantum theory. How can the sum of the positive integeres 1 + 2 + 3 + 4 + .... = -1/12, not infinity as you would expect.
The follow-up newspaper article "In the End, It All Adds Up to – 1/12" (February 4, 2014) by Dennis Overbye, science writer for The New York Times, is also intriguing.
You can find more reactions and comments on the website SPLOID. Admittedly, the "proof" bothers me, but I am not as adamant as many other viewers, who appeal to other types of infinities, Cesaro sums, Heisenberg problem types, and the math God name Euler! One responder Carlos Osuna adds : "Regardless, this kinds of exercises are fun and prove that you can bend theory as much as you want, but reality will stuck you in the head like a hammer."
After some more thoughts, I can rest comfortably that some mathematics must still work right. For example, if 1 + 2 + 3 + 4 + .... = -1/12, then by the distributive property, -1 + -2 + -3 + -4 + .... = -(-1/12) = 1/12, which means that STILL ..... + -4 + -3 + -2 + -1 + 0 + 1 + 2 + 3 + 4 + .... = 1/12 + -1/12 = 0, as it has always been! (Note: before anyone jumps on me, this is somewhat tongue-in-cheek mathematics.)
Nonetheless, I suggest that you rewatch the video, share it, and then wander around for a day or two trying to ease your mind and views about mathematics and the idea of proof.