Following Evelyn’s last post about the new Breakthrough Prize in Mathematics, I will now discuss the opposite of wild mathematical success.

Depending on how excited you are about public speaking, the moments before giving a talk at a math conference may be full of anticipation or anxiety. So what happens when the speaker says something incorrect? At best, it’s embarrassing – like messing up in the middle of a recital in front of other musicians who know that music. What if the speaker presents an argument that is somehow fundamentally flawed? We talk a lot about how to handle students’ mistakes and how valuable they are – see Evelyn’s February blog post. But what about colleagues’ mistakes? How can we take it ins stride when *we* make mistakes in front of peers?

That’s one topic of the entertaining short story “*The Penultimate Conjecture*” by the late celebrated writer Leonard Michaels. One of a series of seven stories featuring fictional mathematician named Nachman, this story is read by Rebecca Curtis at the New Yorker’s monthly podcast posted on the first of this month. In just 45 minutes listeners can soak up some literature with a mathematical flavor and a dark sense of humor. According to Alex Kasman, who maintains the site Mathematical Fiction, this is the most mathematical of all of Michaels’ stories about Nachman, who, after attending fellow mathematician Lindquist’s much-anticipated presentation of the proof of the long-outstanding Penultimate Conjecture, realizes that the presenter’s proof is flawed. Should he say something to Lindquist? It’s not clear what he will do, and we share his uneasiness at being the messenger of bad news. Kasman files all the Nachman stories under the “anti-social mathematician” banner, but unlike the clichés in other stories about mathematicians, this one seems more true-to-life to me. Like the New Yorker’s fiction editor who chats with Ms. Curtis at the end of the reading, I am interested to know what the Ultimate Conjecture might be, and I tend to agree with this blogger that it is probably meant to be that of Nachman. However, one thing that isn’t discussed during the New Yorker podcast is the remaining possibility that Lindquist’s work will be fruitful in other ways besides what he aimed. My thoughts turn to the Math Overflow post concerning the Most Interesting Mistakes in Mathematics , in which many fascinating examples of mistakes made by preeminent mathematicians led to innovations. The most recent example mentioned concerns the Perko Pair, a pair of knots once thought to be non-isomorphic (due to a theorem that was later disproven), and later shown to be isomorphic (by a lawyer named Perko).

Most of us lack preeminence, and for the pessimists among us, the quality of famous mathematicians’ mistakes might just be a reason to save the contents of every Field’s medalist’s wastebasket. But for the optimists among us, it’s also encouraging to think that exercising ones intellect is bound to be fruitful even if it’s not in the manner intended. An interesting side note – in an effort to find a discussion about how mistakes during talks should be handled, I found pretty much nothing, which reinforces my belief that mathematicians are a very polite (or perhaps just confrontation-averse) bunch. I did, however, find a post on Math Overflow about how to correct mistakes in published work. Of course, step one is always to email the author. I also ran into this recent nice post by Orr Shalit at his blog Noncommutative Analysis, in which Shalit discusses how he handled the discovery of a mistake in a 16-year-old paper. What are your thoughts on mistakes in our field? On mathematical literature?

Voevodsky has a radical response to mistakes in Mathematics. Have a look at these slides: http://www.math.ias.edu/~vladimir/Site3/Univalent_Foundations_files/2014_IAS.pdf.

Get HoTT, it’s cool.

Thanks for this great link! I see how mistakes in mathematics are motivation for the use of a different foundation from ZFC so that computers might be able to truly assist in checking mathematical proofs. Also, not to give it away, but for our readers who may be wondering, HoTT is a book on Homotopy Type Theory 🙂

For some reason, I couldn’t get the link to work to Voevodsky’s slides the first time, so interested readers who shared this difficulty should first go to

http://www.math.ias.edu/~vladimir/ , click on Univalent Foundations, and look for lecture #10.

Michael Giudici & Sarah Hart identify somebody’s mistake thus: “However, the proof forgets that GS can contain elements whose square lies in S”. They delicately blame the proof not the prover.