The two cultures of mathematics and biology (2014)

> ... 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.

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.

[^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.

> 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.