What is the use of mathematics in biology?

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I can think of two uses:

1. Statistical summary of complex phenomena - so they may be comprehended 'at a glance' - or patterns seen. A prime example of this is the tabulation of measurements.

It's hard to argue against the usefulness of this kind of summary use of maths in biology - certainly I've done it a lot; but equally it is a long way from being the essence of biology - and it certainly doesn't 'tell you' what is true and what isn't.

(For example, statistical 'tests of significance' have nothing (i.e. nothing) to do with determining biological causality - in the sense that they are orthogonal to understanding causality.). 

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2. Modelling - when biological entities are given mathematical identities, and real life processes are selectively summarized in mathematical processes - and the resultant construct is used to simulate reality.

It's a case of if this and that and the other, in such and such amounts, and if nothing else matters - then the following will follow.

In modelling, pretty much everything depends on the 'ifs' - and it is always hard to know when nothing else significant is going-on in reality so that the model really does capture the essence; but in sum, modelling is - or ought to be - only a small part of biology; embedded (as it were) in the larger subject.

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To put it another way - when maths is applied to biology, reality is transformed into abstraction - quantities and processes - in order that reality can be studied and manipulated in various ways.

Assuming no calculation errors, the validity of modelling therefore hinges entirely on whether the transformation of reality into an abstract mathematical entity is valid.

We must always evaluate whether any particular model - an abstract mathematical entity - matches-up with biological reality in its essential attributes: and evaluating this match-up is something which the model itself (obviously!) cannot do.

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It is noteworthy that the greatest users of mathematics in biology, who are among the greatest biologists, were generally immersed in biological reality in one way or another, or had spent a significant chunk of their lives in this fashion; so that their mathematical representations were in a feedback-relationship with biological reality.

This was the value of large scope sciences and practical subjects as a training for science - Biology and Medicine, for instance - replete with a lot of field work, clinical work and the like, which provided an open-ended and immersive (personality-changing) contact with the realities of the subject.

These early years of immersive experience have the potential to discipline the emotions in-line with the realities of the subject - so that people may develop a feeling for the subject - so that their 'gut feelings' may become a (generally) valid guide to reality.

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This broad and immersive experiential education leading to an emotional identification with the-science-as-a-whole, is probably (or so it seems ) the only antidote (albeit partial) against scientists becoming captured by their methodology - and coming to believe that the tool in which they personally have expertise is the only thing that matters; believing that their technique captures the essence of the whole of biological reality.

And this belief rapidly hardens into delusion - because once you believe that your own technique captures all that is important in biology, then you will use that technique to evaluate everything in biology and reject anything which disagrees with this evaluations, and reject anything which is not amenable to this evaluation.

Modern science is full of such people - indeed there are few (very few) people in modern science who are not of this type, and modern science is run by deluded methodologists.

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But (insofar as this is how they function) such people aren't scientists - no matter how powerful or prestigious, no matter how clever or ingenious - they are technicians posing as scientists.

They are merely a specific instance of the general phenomenon that when you are a hammer, everything looks like a nail.

A real biologist is not a hammer, but a person who wields a hammer, as necessary - plus many other tools, as appropriate.

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