Musings on a Mathematician’s Duties

Come with me to take a dive into the abc conjecture and the ethical duties of mathematicians! Image courtesy of HaPe_Gera via Flickr CC.

As I mentioned in my last post, I wish a genie would grant me thorough understanding of the proof Shinichi Mochizuki proposed for the abc conjecture. Much of this wish is motivated by a desire for the divisive debate to be settled concerning the more than 500 page text. Last year, it was announced that Go Yamashita had written a summary of Mochizuki’s proposed proof, but that summary was still 294 pages and didn’t settle debates about Mochizuki’s potential proof.

In July, Ivan Fesenko, who has organized conferences on the inter-universal Teichmüller(IUT) theory that underlies Mochizuki’s proposed proof, released a document titled “Remarks on Aspects of Modern Pioneering Mathematical Research,” which heavily focuses on Mochizuki’s IUT theory and the abc conjecture.

Here are some highlights from that document.

• “There is a fundamental aspect of speaking in tongues in the IUT, this aspect reflects the nature of the theory and the work on it,” Fesenko wrote.
• “There is no obligation for the author of a breakthrough theory to write at the level of a well polished textbook…All younger experts in IUT have not found the study of the theory more difficult than their study of other theories,” he added. (All?)
• “The ethical responsibility of mathematicians includes a certain duty, never precisely stated in any formal way, but of course felt by and known to serious researchers: to dedicate an appropriate amount of time to study each new groundbreaking theory or proof in one’s general area,” Fesenko wrote.

After reading that text, it is unclear to me how Fesenko defines “appropriate amount of time” or what constitutes a “groundbreaking theory or proof in one’s general area.” It also left me with questions about where certain duties lie. For example, is it the responsibility of someone who seeks to understand a new theory to bridge gaps between their understanding and the available explanations? How much of that responsibility also lies with the researchers behind the theory?

John Baez discussed Fesenko’s text in his post “The Duties of a Mathematician” on the n-Category Café group blog. Peter Woit also wrote about “abc News” (and Fesenko’s remarks) on his “Not Even Wrong” blog.

Woit begins by sharing a claim he had heard but was waiting to discuss:

“The last couple months I’ve heard reports from several people claiming that arithmetic geometers Peter Scholze and Jakob Stix had identified a serious problem with Mochizuki’s claimed proof of the abc conjecture,” he writes, adding that he was waiting to publicly discuss the matter until the three had time to meet and “sort out the mathematics” and “have something to say publicly.” However, Fesenko’s text mentions the matter between Mochizuki, Scholze and Stix in a footnote, which Woit discussed in his blog post. Two of the sentences which most resonate with me from Woit’s post are “Fesenko’s document has a lot of unpleasant things to say about those who have written anything at all skeptical concerning Mochizuki’s claimed proof, mostly without naming names” and, concerning Peter Scholze, who was recently awarded a Fields Medal “That his questions about Mochizuki’s proof were ‘shallow’ seems highly unlikely, to me at least.”

I think Baez’s first reaction to Fesenko’s remarks on “the ethical responsibility of mathematicians” is highly relatable:

“My first reaction to this claimed duty was quite personal: namely that I couldn’t possibly meet it. My research is too thinly spread over too many fields to ‘study each new groundbreaking theory or proof’ in my general area. While Fesenko says that ‘truly groundbreaking theories are rare, and this duty is not too cumbersome,’ I feel the opposite…More importantly, while it is undeniably good to know what’s going on, that doesn’t make it a duty.” I believe mathematicians should be free to study what they’re interested in.”

Many of the comments written on those two blog posts were also illuminating. For example, Simon Willerton commented (on “The Duties of a Mathematician” post) “The AMS has some ethical guidelines [http://www.ams.org/about-us/governance/policy-statements/sec-ethics] which includes the following under the responsibilities of mathematicians with regard to mathematical research and its presentation.

• To endeavor to be knowledgeable in their field, especially about work related to their research;

Do you feel that’s different to what Fesenko is asking for? The ethical guidelines perhaps give the AMS’s answer to your question about the duties of a mathematician.”

What do you think about the duties of a mathematician and related ethics, or steps that should be taken to settle the debate about Mochizuki’s proof for the abc conjecture? Please share your ideas in the comments or reach out to me on Twitter @writesRCrowell!