Two Mysteries About Mathematician Alexander Grothendieck

Environmental & Science Education, STEM, History of Science, Maths 

Ed Hessler

I'm not enough of a mathematician or historian of mathematics to say that mathematics can be divided into two periods:before Grothendieck and after Gronthendieck. I take heart in making the distinction in this comment made by M. I. T. mathematician Michael Artin on his contributions to mathematics from Rivka Galchen's fascinating essay in The New Yorker (May 16, 2022) on Grothendieck's life. "'Well, everything changed in the field. He came, and it was like night and day. It was a revolution."'  

I must add that Artin worked with Grothendieck during his career. The Artin quote is preceded by support for his statement by logician and philosopher of mathematics at Case Western, Colin McLarty.

So much in Galchen's reporting captured my attention and made me think. As usual I have picked only a very few items to highlight and urge that you read it. The reporting is a beautifully told story, based on facts - Grothendieck's mathematics personal - it is a beautifully told story.

Galchen begins with an anecdote about Grothendieck learning the definition of a circle - a foreshadowing of his mathematics. "He rewrote definitions, even of things as basic as a point, his reframings uncovered connections between seemingly unrelated realms of math."  While a child in an interment camp, Grothendieck criticized "his textbooks as lacking 'serious' definitions of length, area, and volume." 

In addition to definitions,Grothendieck "'named things,'" which may not sound like much at first but "according to mathematician Ravi Vakil 'there's a lot of power in naming..'" Galchen makes this observation. "In the forbiddingly complex world of math, sometimes something as simple as new language leads you to discoveries." And Vakil provides an example. "'It's like when Newton defined weight and mass. They had not been distinguished before. And suddenly you could understand what was previously muddled.'" *

It was his abrupt leaving of mathematics and what followed - two mysteries - in which Galchen places Grothensieck's  mathematical life. One observation reminded of an idea that is important in teaching and learning mathematics: soak time. When stuck consider and  re-consider the problem, even set it aside, looking for an opening. This is not at all easy to learn do. We want answers. Now. When one is  not forthcoming we give up with declarations such as I'm not good at maths. Neither are my Mom and Dad. Q.E.D. "Grothendieck spoke of problem solving as akin to opening a hard nut. You could open it with sharp tools and a hammer, but that was not his way.. He said that it was better to put the nut in liquid, to led it soak, even to walk away from it, until eventually it opened."

In a recent New York Times (April 6 2022) article by Jenny Anderson, republished in the Minneapolis StarTribune (April 1, 2022) discusses "the learning pit" which is "a metaphor (for) one of several common educational strategies that lean into the idea that struggle is something to be embraced." The idea is to learn to get comfortable "with being a little uncomfortable." Of course the problems should be age appropriate and tools need to be provided to help students" climb out of the pit. The twin ideas of struggle and learning your way out are "vital to learning (and) well established."  The aim is a constructive mindset. The article has several references if you are interested in pursuing this educational intervention. Here is the link to the story in the NYT. It may be available if you haven't exceeded your limited free access. I can't find it in the StarTribune.

Grothendieck had a very sharp side that infected all his relationships, including his family. He allowed himself to ride his motorbike to work but his wife, Mireille, was not allowed to drive a car."  The family shopping was done on foot. Mathematician Barry Mazur and his wife once spent tine with the Grothendiecks near Paris. Upon arriving at the Mazurs for dinner, "'(Gronthendieck) came in and saw the spread and said with a big smile This is wonderful! And then he turned to Mireille and said in a harsh voice, See how easy it is to make a vegetarian meal!" "Mazur added, 'Of course, it was Mireille who had the burden and responsibility of taking care of all those people." Grothendeick also vilified mathematicians who solved a problem that "didn't use the foundational systems Gronthendieck had established." Galchen relates an interesting, perhaps apochraphyl,  story about Pythagoras who also demanded a cult-like following and the result for one who didn't.

American mathematician, Leila Schneps, who upon reading a Grothendieck ms. recommended to her by Pierre Lochak who is now her partner said she didn't think she "would be drawn to Grothendieck's work."  She was and told Galchen "'One idea in there is that we have been writing math in a way that is all wrong.'" "Grothendieck argues that mathematicians hide all of the discovery process, and make it appear smooth and deductive.'" "He said that, because of this, the creative side of math is totally misunderstand. He said it should be written in a different way, that shows all the thinking, all the wrong turns---that he wanted to write it in a way that emphasized the creative process."

She and her husband set out to find Grothendieck and evenutually found him, a "thin bearded man, buying vegetables in the market." What followed was "a tremendous, demanding, tumultuous friendship." Shortly before his death, "Grothendieck shed or burned most of  his meagre possessions;" one he kept was "a painting of his father in the internment camp."

I'd never heard of Alexander Gothendieck and finished this brief history of a life that had such deep implications for a field, quite an accomplishment, wishing it had been longer. I hope you like it as well.

* Research on misconceptions in science has demonstrated that K-12 students have difficulty with a surprising number of scientific concepts and that it takes time and effort to learn new scientific ways of thinking about them as well as in using them correctly. Interestingly, Project 2061's Benchmarks for Scientific Literacy (1993),  made this observation about the distinction mentioned several that are especially difficult. "Heat energy itself is a surprisingly difficult idea for students, who thoroughly confound it with the idea of temperature. A great deal of work is required for students to make the distinction successfully, and the heat/temperature distinction may join mass/weight, speed/acceleration, and power/energy distinctions as topics that, for purposes of literacy, are not worth the extraordinary time required to learn them." ((p. 81).

I'm not current with recent research on any of the above but I thought this was worth mentioning.