December 31, 2014

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 Nature of 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 by Nature to 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 Nature had in mind for its journal as well. But since Nature bills itself as “An international journal, published weekly, with original, groundbreaking research spanning all of the scientific disciplines [emphasis mine]” Mumford assumed the readers of Nature would 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 Nature were 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:

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.

Posted at December 31, 2014 6:36 AM UTC

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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.

Posted by: Roger Witte on December 31, 2014 5:02 PM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

A group of scientists and engineers in the UK (and some years ago) applied for a research grant that was to a call for interdisciplinary projects. The group included biologists, mathematicians, software engineers, mathematical biologists, and even us category theorist at Bangor. We got through the first selection process with flying colours, but were sunk in the second round when a referee with a jaundiced view of such projects said that they never work! None of the other interdiscilinary projects was funded.

Recent comments on the REF in the UK, also show that universities are reluctant to have such projects as with them it is difficult to attribute success to one small key group.

In the project we were involved with there was very little difficulty of communication of the ideas relating to networks and the use of categorical language to handle them. Rather there was a lot of interest and that was 15 years ago.

Posted by: Tim Porter on January 1, 2015 10:22 AM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

It is sad that most public math (even for a scientific audience) has to be dumbed down to a high school level, and the current high school level is fairly dumb with mostly rote learning and little in terms of explanatory frameworks, other than piece meal.

A friend once asked me to explain a math comment I made and insisted that she, as a college graduate with a successful career using math in business, could understand it if I only explained it correctly.

I forget what my comment was. We had a long digression down to some crux that there are various notions of “equality” (possibly from strict up to homotopy). I think I got her to understand that “equals” may not mean “equals” with this example: in some senses $3 \times 5$ is the same as $5 \times 3$ and in other senses they are different. Two closets with those dimensions can hold the same volume of stuff but would differ for the task of finding clothing hanging on a rod.

And I had thought that “explain it so a MBA can understand it” was a joke.

Posted by: RodMcGuire on January 1, 2015 1:37 AM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

The notion of equality in maths is tricky, and is interpreted in Homotopy Type Theory as a process, and using groupoid concepts. Thus $= \;$ does not really mean “the same as”, and this can be confusing for people. Again $f=g$ for functions on the natural numbers should mean that $f$ and $g$ take all the same values; but a computer cannot check all values. Maths is full of such of abuses of language which have been found convenient, but can also lead to difficulties, and confusion for beginners, since they are not explained at all.

Posted by: Ronnie Brown on May 20, 2015 4:12 PM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

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.

I have the sneaky feeling that there is an underacknowledged symmetry here, and that a specialist in any subject could argue that their specialization is perceived as too cut off from the world. “Nobody thinks they need to learn a bunch of dead kings,” the history teacher complains, to which English Lit replies, “Everybody says they’ll never need to know about sonnets when they grow up!” By symmetry, we should drop each class and teach its material “as needed” in the others.

As a practical matter, it seems easier to me to reorganize the mathematics curriculum and retrain math teachers to bring in more and better examples from other subjects, than to reorganize everything and retrain all the teachers, including those who (like so many adults) say they “just never got math.”

I went to a somewhat offbeat private school for my fourth- through sixth-grade years. We had occasional projects where the whole class would be divided into groups of a few students apiece and would work on a project which drew upon all the subjects we studied. I suspect that was beneficial, although the way American schools are set up, it only seems feasible for the earlier grades. There’s an organizational phase transition between elementary school, where the same class goes through each subject together, and middle school, where you go from one teacher to the next for different subjects.

Posted by: Blake Stacey on January 1, 2015 7:02 PM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

Ensuring that teachers are adequately trained in their own subjects is, like making sure textbooks are both available and decent, one of those infrastructure problems which are too dreary to attract attention. They’re not Visionary, Iconoclastic Education Reform! But they matter—and we’re not doing a good job with them.

We have surveyed how well prepared in terms of disciplinary course work teachers at various levels felt for teaching various mathematics topics in what is a fairly representative sample of 60 districts. In general, we would summarize the findings by stating that many teachers felt ill prepared to teach mathematics topics that are in state standards and in the new Common Core State Standards for mathematics. Why did these teachers feel so ill prepared?

There is perhaps a simple answer for the elementary and middle school teachers: They felt ill prepared because if we examine the coursework they studied during their teacher preparation, they were ill prepared.

Posted by: Blake Stacey on January 1, 2015 7:21 PM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

If they feel ill-prepared, maybe that’s a good thing, given the Dunning-Kruger effect.

Posted by: John Baez on January 2, 2015 8:13 AM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

I am a grad student in geometry and I found the explanation hard to follow - at least the things I did not already know. Maybe I’m too dense, but it might also be that this is too technical indeed.

I often find that when I try to explain things and I try to skip over the details in order to make an explanation shorter, people (even specialists) understand me more poorly than if I make the explanation longer and more complete. I think is kind of happened here as well.

Posted by: Kosta on February 11, 2015 3:52 PM | Permalink | Reply to this

Re: Can One Explain Schemes to a Biologist?

I agree that it was too technical, and used unnecessary jargon. For example, “the symmetry group of the algebraic numbers” would be okay if you told the biologists what “algebraic numbers” are, which they forgot to do… but adding “known as its Galois group $Gal(\overline{\mathbb{Q}}/\mathbb{Q})$” is completely unhelpful, merely intimidating. Readers will either know that jargon and thus not need to be told about it, or not know it and not gain anything from seeing it… given that this is an obituary, not a textbook.

Posted by: John Baez on February 11, 2015 7:33 PM | Permalink | Reply to this