The two cultures of mathematics and biology (2014)
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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.
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.
"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.
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.
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.
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.”
— John von Neumann
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.
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.
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.
[^2] https://www.rcsb.org/structure/3D9S
[^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.
[^4] https://direct.mit.edu/posc/article-abstract/31/5/594/115643...
[^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.
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.
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.
[1] in my limited experience of two math departments
Allow me to doubt that mathematics departments are immune to status competition.
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.
The two cultures of mathematics and biology - https://news.ycombinator.com/item?id=8819811 - Dec 2014 (69 comments)
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.
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.
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.
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.