The estrangement he observes aren't that surprising, in either direction. Many universities have both Math and Applied Math departments. Why have both unless the mathematicians in the Math department don't want to work on applications? I have spoken with people who say if you're working on an application, "it's not really math."
In biology, there is almost certainly a self-selection effect in which the field attracts people who want to study science but are not comfortable with math, or just people who have a particular interest in plants or animals, which is uncorrelated with math skills.
I suspect there is a self-selection effect in the other direction too. I was always good at math, but I never wanted to major in it or go to grad school in it. I got a PhD in AI and machine learning, which was quite mathematical enough, and yet I can't recall ever interacting with anyone from the math department. As far as I knew, they wanted to do "pure math" and weren't interested in applications. So the people who want to do practical things select them selves into other majors like physics, engineering, and computer science.
> Many universities have both Math and Applied Math departments. Why have both unless the mathematicians in the Math department don't want to work on applications?
"Applied Mathematics" as a field is not literally "mathematics applied to something"; it's a fuzzy group of related topics (things like numerical analysis, PDEs, or computational linear algebra) that's grown large and culturally distinct enough to have its own department, much like theoretical CS or statistics. There are plenty of "applied" mathematicians who don't work on applications, and some "pure" mathematicians who do.
Applied Mathematics certainly has the intention of using Math to solve problems from other fields though. I studied Robotics, and shared several classes with people from the Applied Mathematics course;
* Transportation Modelling (using math to model transport on roads, rail, and shipping, and solve optimisation problems)
* Computer Vision (using math to recognise patterns in images)
* Biomechanics (using math to model the movement and formation of all things biology; we did everything from a sperm cell locomotion to an Achilles tendon spring strength)
There were others, but all of them had a very practical purpose, and most of the people I spent time with on the Applied Mathematics course were actively pursuing a career in engineering of some sort, while the Mathematics course was made up of either Pure Math people looking to go into academia, or people destined for finance.
Each field has its own culture and its own way of thinking. People are often socialized to a particular culture in the university or even earlier, and it sticks. You can learn other fields on your own, but it's hard to adopt the culture without socialization.
I did theoretical computer science in the university, leaning towards more applied stuff by the end of my PhD. I'm still a computer scientist at heart. I can follow some topics in research mathematics, but I don't think like a mathematician and I'm not interested in the same things. I work in bioinformatics these days, but I often zone out when people start talking about the stuff that goes in the results section of a paper. I'm not a bioinformatician, and I'm not interested in the same things. I've seen a similar culture gap between bioinformatics and "proper" biology, but I don't have first-hand experience with that.
There are currently 251131639 sequenced proteins in UniProt[^1], so, that's a very lower bound on the number of things a modern biologist has to amuse themselves with. Many still consider biology as the study of each individual biological organism, system, or protein. But since there are so many of those, I argue that biology must become a science of methods of understanding, and not a science of bare understanding. It's the difference between a company that produces mining machinery and a company that sends miners with pick and shovel underground. And that transformation is going to require for biologists to become system scientists and engineers, steeped to the brim in math, biochemistry and computer sciences.
Why is there any science at all above pure maths? Why isn't physics, chemistry, geology etc etc just maths?
That's because choosing the right level of abstraction is really important for making practical progress.
For example penicillin was discovered and used to save millions of lives without any rigorous mathematical understanding of how the drug interacts with it's target.
I'm not saying maths isn't incredibly useful and increasingly important in the study of biology, I'm just saying that approaches that don't need maths ( beyond simple counting et al ) are also very important as well - biology is so complex, it's too easy to get bogged down in the detail.
Also I do wonder sometimes whether mathematicians don't actually understand some of the maths they work on - they can follow the mathematical logic but can't "see it". ie then find their way through the logic maze by following a logical thread in the darkness - better than stumbling around randomly - but it doesn't mean you understand the maze - and because they don't understand it beyond the 'following the logical thread' they can't communicate it to others.
Perhaps the latter is unfair - I'm not a mathematician - I'd be interested to hear other views on that.
Also I do wonder sometimes whether mathematicians don't actually understand some of the maths they work on
We don’t. One of the first steps to mathematical maturity is learning to let go of the need to understand, the need to visualize. Much of mathematics is a formal affair of making arguments to satisfy necessary and sufficient conditions. Trying to understand infinite-dimensional spaces or highly abstract sets and objects is too much, and unnecessary.
”Young man, in mathematics you don't understand things. You just get used to them.”
> Much of mathematics is a formal affair of making arguments to satisfy necessary and sufficient conditions.
Some areas have a formalist culture like this (descriptive set theory seems to, for instance, though that might just be what it looks like from the outside), but it's far from universal. At the other end of the spectrum I find it hard to imagine anyone getting far with algebraic geometry without building intuition. And then of course in mathematical physics the intuition-frontier is always decades ahead of the formal one.
I would disagree. Mathematicians certainly don't need to visualize everything but "intuition" is a commonly used phrase which is a notion of understanding.
Although math merely requires proving some statement, often having an intuition / understanding of how concepts interact with each other helps figure out which things are likely to be true.
I never felt I properly understood a proof unless I understood it both intuitively and formally. The formalism is to make sure your intuitions are water-tight. But there are proofs you can accept are formally correct without intuitively understanding them - I would accept the truth of such proof but not feel like I understood them.
My point was that there is only so much we can understand. Let me give you a concrete example, one which I have chosen to be easy to understand--the irony!-- : You are a biologist and are given the task of reverting skin senescence in a billionaire client of yours. I'm choosing this example because senescence is a very individual process, with different biological pumps[^1] stopping at different points in time and for different reasons. You can choose to understand how the processes worked together to produce the present system state and skin condition. But that's not your task, your task is to revert it. Understanding seems like a logical first step, but along the way, you (always) discover that these processes involve tens of thousands of interactions between an order of magnitude more of metabolites working at different stages and compartments, and that you can't keep a general intuition of them in your mind[^5], other than the very basic "sh*t breaks". But that's okay. You can always put all of it in a database. Then you only need to remember where the database is, and the dozens of different simulations that are interacting with that database. You will also need to understand the organization of the database, and what the simulations are doing, but there are way less of those and they follow human-made ontologies, sometimes they even come with documentation. If you play with those toys correctly, you will come with an individual intervention for your billionaire that you know will be sound, even if you don't have a comprehensive chain of reasoning of why expressing your customer's variation of 3D9S[^2] 7.5% less will help make his skin better[^3].
In any case, this is an area where there is some vigorous debate[^4] right now.
This is somewhat similar to how we don't understand the precise effect of a weight amount trillions in a LLM, but we can still architect, build and profit from the LLM.
[^1] That's a name I use for clusters of connected pathways, but the distinction is arbitrary and in this case the clusters were created by a graph clustering algorithm.
[^3] If you are thinking that I should have made this example about cancer: the most frequent cause of cancer is cellular senescence. I couldn't muster the cynicism of making an example about the symptom instead of the cause. But most of my colleagues in search of public funding will. Go figure.
[^5] Or, worse, you risk holding to the wrong intuition or understanding. Because we tend to misunderstand complex things much more easily than simple things, you know.
Isn't skin ageing much simpler - the structural proteins in the extracellular matrix like fibronectin get damaged by sunlight over time - crosslinks are formed and the skin loses elasticity?
ie it's not a complex cellular biology thing - just a wear and tear thing, for components that weren't designed to be replaced - ( like adult teeth for example ).
So there might not be an existing biological process you can hijack or reverse - so understanding existing biology might not help you at all.
As to your main point about the complexity of the system. Bottom line biology has evolved to maintain stable patterns - if it was always on a knife edge you'd be dead - so while there might be lots of moving parts the control surface and the state machine has to be much smaller - with the controls being rather forgiving.
As an analogy - you don't need to be a mechanic to be able to drive a car - you can abstract the cars complex mechanics to some very simple high level characteristics - and you can pile those abstractions ( if they don't leak ) on top of each other - so there is a carburettor - you don't need to fully understand how the internals work to understand it's role in the car, but you don't need to know about a carburettor to be able to press the accelerator.
> Isn't skin ageing much simpler - the structural proteins in the extracellular matrix like fibronectin get damaged by sunlight over time - crosslinks are formed and the skin loses elasticity?
If that were all there was to it, sunburns you get as a kid will make your skin look permanently older. Barring very severe burns, that doesn't happen. There is however a slower rate of replacement of all sort of proteins and structures as you age. About why it slows down, a biologist will say "the cause it's not well understood". They should instead say "the many causes are not well understood," which is kind of my point.
In this case, there definitely isn't "a" single process to hijack or reverse. The idea of a magical drug is, well, ludicrous. Using your simile, it's like trying to use a car to solve a town's transportation problem. But if increase your complexity budget quite a bit, there are all sort of interventions that will get you where you want.
There is an element to this, but at the same time things tend to get incredibly messy at the level of whole complex organisms. (Btw., that also holds for mining/geology: you can do a lot of complex measurements from the surface, drill some holes and still be wrong about what you encounter underground).
While Pachter enumerates differences in culture and breakdowns in collaboration, I feel the root cause is individual social attitude. The two cultures differ because their self-selected members differ in personality. What drew me to study math was the department's attitude of anarchy and irreverence. Status didn't matter, funding didn't matter, appearance didn't matter, prerequisites didn't matter, you just needed two people and two pieces of chalk and a blackboard and an afternoon. By contrast biologists I've worked with have been acutely attuned to hierarchy and funding strategy and marketing and credit attribution - social maneuvering that would fall as flat in math departments[1] as "third cohomology group" falls flat on Nature's reviewers.
[1] in my limited experience of two math departments
Status matters. Politics are nasty. Every subfield has its own culture, its own royalty. Better funded professors get more and higher status students. Bigotry is common, and so are "quirky personalities" -- and due to the tolerance of weirdos, bigotry is assumed to not exist. Mathematicians are not without their people problems. Just like every other slice of humanity, they lie to themselves.
Sure a segment of the population aims for department head, but another segment is happy just being left alone to do their thing and pay via teaching. It was this less ambitious more self-motivated group who attracted me.
While I agree that it's very questionable that Nature would invite someone to write an obituary (or what have you) and then reject it, I fail to see what's so controversial about the text being overly technical. I work in Physics, and "[...] locus of solutions of sets of polynomial equations by combining the algebraic properties of the rings of polynomials with the geometric properties of this locus, known as a variety" is still an incredibly tough sentence to parse on the first pass. I cannot imagine how it would read for someone who is either unfamiliar (or only passingly familiar) with, for example, the concept of a ring; "algebraic properties of a ring of polynomials"? This just seems like a case of https://xkcd.com/2501/ , with a hint of arrogance in thinking everyone working in STEM must be as comfortable with abstract concepts of mathematics as mathematicians are.
this is unfortunately, author keep talk about this is useful in phylogeny, but biologists who work on phylogeny is not popular in the group of biologists who frequently publishing articles on Nature.
> ... what mathematicians can deliver to genomics that is special and unique, is the ability to not only generalize, but to do so “correctly”.
I think this isn't really special or unique to mathematics. Certainly it's something that some mathematicians work hard to be good at, but many great mathematicians never play this game. Look at like Terry Tao, the man is undoubtedly one of the (if not the) greatest living mathematician, but IMO his best work tends to be these crazy mind-bending proofs or developments within specific areas of math. He's not a Grothendieck or a Hilbert who reorganizes concepts in elucidating ways or creates powerful generalizations. This isn't a knock on Tao, it's just pointing out that research fields are broad and require different skillsets. In terms of hard science it's IMO kind of the difference between a brilliant theorist and a brilliant experimentalist.
Taking that comparison one step further, biology also has its theoreticians and its experimentalists. Being a skilled theoretician, understanding how to organize abstract concepts to the right level of generality, is definitely something that math can help you improve at, but it in no way is limited to mathematics. For example, Stephen Jay Gould was IMO brilliant at operating abstractly, but he had no formal mathematical training I'm aware of. Critical thought belongs to every field, even ones outside of research science (ex. Law, Philosophy).
> But wouldn’t it be better if mathematicians proved they are serious about biology and biologists truly experimented with mathematics?
For the reasons above, this isn't clear to me. Does a first-year Ecology PhD really need to think critically about Hilbert spaces? They might find it to be a fun exercise, and I could see how they could get benefits from it, but they could get similar benefits from like any advanced philosophy course, IMO. I'm all for collaboration when it benefits both fields, but collaboration for collaboration's sake seems like a time sink without an obvious impact.
caveat: this is all said 10 years after the post was written, I do think the cultural divide the author talks about has closed somewhat since writing, so maybe this arrangement is now just more palatable to me.
This article was written in 2014 and I do not know if the specifics are still true. However, I think mathematicians and biologists work in two different levels of abstraction.
One is a lawful good with occasional venture into chaotic good, only to reform the chaos. The other is a true neutral with lot of expeditions into chaotic evil just for fun.
Well, it's 2024, I'm a biologist, I read the thing about schemes, and I still don't understand what they are or how they would help me understand biology any better.
I'm an applied mathematician (Ph.D. in math but never a professor) and I did not want to try to understand the description of schemes in the obituary. I thought that the obituary was too technical for me. On the other hand, if I thought that schemes were useful for my work, then I would dig into them and try to understand them.
For a ten year old article, it holds up remarkably well. Grant writing and academic research are typically divergent skill sets. Since research grants (the vast majority from DHHS, NSF, and DoD) have become a major source of revenue for universities, the most valuable faculty to a university are those who acquire grants, leading universities to hire faculty who will likely acquire many grants. These faculty select for similar students, and the cycle continues. Because pure mathematics research is intrinsically resource-minimal, the same paradigm doesn't work over there.
In a possible reflection of that reality, I have a strong feeling that, on average, university biology departments are housed in much newer and nicer buildings than university mathematics departments.
These kinds of articles are not helpful. "Why isn't A talking to B? Isn't that so sad?".
People are busy (on both sides). If Mathematicians want to want to get Biologists' attention, they should do something like Deepmind's AlphaFold - tackle a long-standing extremely difficult problem in biology using mathematical approaches.
In biology, there is almost certainly a self-selection effect in which the field attracts people who want to study science but are not comfortable with math, or just people who have a particular interest in plants or animals, which is uncorrelated with math skills.
I suspect there is a self-selection effect in the other direction too. I was always good at math, but I never wanted to major in it or go to grad school in it. I got a PhD in AI and machine learning, which was quite mathematical enough, and yet I can't recall ever interacting with anyone from the math department. As far as I knew, they wanted to do "pure math" and weren't interested in applications. So the people who want to do practical things select them selves into other majors like physics, engineering, and computer science.