### Can One Explain Schemes to a Biologist?

#### Posted by John Baez

Tonight I read in Lior Pachter’s blog:

I’m a (50%) professor of mathematics and (50%) professor of molecular & cell biology at UC Berkeley. There have been plenty of days when I have spent the working hours with biologists and then gone off at night with some mathematicians. I mean that literally. I have had, of course, intimate friends among both biologists and mathematicians. I think it is through living among these groups and much more, I think, through moving regularly from one to the other and back again that I have become occupied with the problem that I’ve christened to myself as the ‘two cultures’. For constantly I feel that I am moving among two groups — comparable in intelligence, identical in race, not grossly different in social origin, earning about the same incomes, who have almost ceased to communicate at all, who in intellectual, moral and psychological climate have so little in common that instead of crossing the campus from Evans Hall to the Li Ka Shing building, I may as well have crossed an ocean.

I try not to become preoccupied with the two cultures problem, but this holiday season I have not been able to escape it. First there was a blog post by David Mumford, a professor emeritus of applied mathematics at Brown University, published on December 14th. For those readers of the blog who do not follow mathematics, it is relevant to what I am about to write that David Mumford won the Fields Medal in 1974 for his work in algebraic geometry, and afterwards launched another successful career as an applied mathematician, building on Ulf Grenader’s Pattern Theory and making significant contributions to vision research. A lot of his work is connected to neuroscience and therefore biology. Among his many awards are the MacArthur Fellowship, the Shaw Prize, the Wolf Prize and the National Medal of Science. David Mumford is not Joe Schmo.

It therefore came as a surprise to me to read his post titled “Can one explain schemes to biologists?” in which he describes the rejection by the journal

Natureof an obituary he was asked to write. Now I have to say that I have heard of obituaries being retracted, but never of an obituary being rejected. The Mumford rejection is all the more disturbing because it happened after he was invited byNatureto write the obituary in the first place!The obituary Mumford was asked to write was for Alexander Grothendieck, a leading and towering figure in 20th century.

Continuing to quote Pachter:

My colleague Edward Frenkel published a brief non-technical obituary about Grothendieck in the New York Times, and perhaps that is what

Naturehad in mind for its journal as well. But sinceNaturebills itself as “An international journal, published weekly, with original, groundbreaking research spanning all of thescientific disciplines[emphasis mine]” Mumford assumed the readers ofNaturewould be interested not only in where Grothendieck was born and died, but in what he actually accomplished in his life, and why he is admired for his mathematics. Here is the beginning excerpt of Mumford’s blog post explaining why he and John Tate (his coauthor for the post) needed to talk about the concept of a scheme in their post:John Tate and I were asked by Nature magazine to write an obituary for Alexander Grothendieck. Now he is a hero of mine, the person that I met most deserving of the adjective “genius”. I got to know him when he visited Harvard and John, Shurik (as he was known) and I ran a seminar on “Existence theorems”. His devotion to math, his disdain for formality and convention, his openness and what John and others call his naiveté struck a chord with me.

So John and I agreed and wrote the obituary below. Since the readership of

Naturewere more or less entirely made up of non-mathematicians, it seemed as though our challenge was to try to make some key parts of Grothendieck’s work accessible to such an audience. Obviously the very definition of a scheme is central to nearly all his work, and we also wanted to say something genuine about categories and cohomology.What they came up with is a short but well-written obituary that is the best I have read about Grothendieck. It is non-technical yet accurate and meaningfully describes, at a high level, what he is revered for and why. Here it is (copied verbatim from David Mumford’s blog)…

Well, at this point I’ll turn you over to that blog article:

- David Mumford, Can one explain schemes to a biologist?

But the rest of Pachter’s article is interesting too. I’ll quote just a bit more:

What biologists should appreciate, what was on offer in Mumford’s obituary, and what mathematicians can deliver to genomics that is special and unique, is the ability to not only generalize, but to do so “correctly”. The mathematician Raoul Bott once reminisced that “Grothendieck was extraordinary as he could play with concepts, and also was prepared to work very hard to make arguments almost tautological.” In other words, what made Grothendieck special was not that he generalized concepts in algebraic geometry to make them more abstract, but that he was able to do so in the right way. What made his insights seemingly tautological at the end of the day, was that he had the “right” way of viewing things and the “right” abstractions in mind. That is what mathematicians can contribute most of all to genomics. Of course sometimes theorems are important, or specific mathematical techniques solve problems and mathematicians are to thank for that. Phylogenetic invariants are important for phylogenetics which in turn is important for comparative genomics which in turn is important for functional genomics which in turn is important for medicine. But it is the the abstract thinking that I think matters most. In other words, I agree with Charles Darwin that mathematicians are endowed with an extra sense… I am not sure exactly what he meant, but it is clear to me that it is the sense that allows for understanding the difference between the “right” way and the “wrong” way to think about something.

Since I’m trying to think about biology these days, this is encouraging. But the story of the huge culture divide is not. It has a less tragic ending than you might think from what you’ve read here so far: a revised version of Mumford and Tate’s obituary was ultimately accepted by *Nature*.

However, ironically, the accepted version will not be freely available, thanks to *Nature*’s repressive policies — while the rejected version appears on Mumford’s blog, so you can read that.

Mumford concludes:

The whole thing is a compromise and I don’t want to say Nature is foolish or stupid not to allow more math. The real problem is that such a huge and painful gap has opened up between mathematicians and the rest of the world. I think that Middle and High School math curricula are one large cause of this. If math was introduced as connected to the rest of the world instead of being an isolated exercise, if it was shown to connect to money, to measuring the real world, to physics, chemistry and biology, to optimizing decisions and to writing computer code, fewer students would be turned off. In fact, why not drop separate High School math classes and teach the math as needed in science, civics and business classes? If you think about it, I think you’ll agree that this is not such a crazy idea.

## Re: Can One Explain Schemes to a Biologist?

I just read your post and the two referenced posts. I think that part of the problem is that the focus is on academe. I don’t work in mathematics or in biology, but in transportation planning and engineering. However my experience is that teams in industry are always multidisciplinary.