[DrScientist]

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.

[chongli]

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.”

— John von Neumann

[nyssos]

> 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.

[viknesh]

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.

[jvvw]

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.

[dsign]

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.

[DrScientist]

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.

[dsign]

> 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.