Mathematics is all about abstraction and logic: numbers are just one
of the things which make sense to study mathematically. Numbers, to
continue with this example, *abstract* the idea of counting things.

Abstraction is a kind of *deliberate, creative ignorance*, where we
choose a few things to pay attention to, what we may call *salient features*,
and then *ignore everything else*. In the case of numbers, there
are actually two different functions served by numbers: *order*
and *quantity*. In the case of *quantity*, we are asking 'how many are there'?
For example three apples: it does not matter which apple we count first,
provided we count each apple once. And no matter how we count them, provided
we count each one once, we will end up with denoting the last one
'apple number 3'. The other function, *order*, we make use of the idea
that, if we start counting at 1, then the number 1 comes first, followed
by 2, followed by 3, and so on.

Now the thing with abstraction is that we all do it, all of the time.
Abstraction is not a mathematical technique, but a *fundamental mental technique*.
Abstraction underlies everything you think. What mathematics has done is
to take that and become deliberately aware of the process of abstraction.
In everyday scenarios, if we talk about 'driving a car', we are dealing
with an *abstract concept* of a car. We are *ignoring* all the details
of a particular car being driven. If we talk about *electric cars*, then
we are selecting, from all *cars*, those whose drivetrain is electric.

The problem arises because most abstraction we do in our daily thinking
we do without insight into the abstraction process we are utilising.
Sometimes the things we are ignoring matter, and often we will be unaware
that those ignored features do matter until something goes wrong. Sometimes
by the time that something goes wrong, it will be too late to do anything
about it. And that is the danger of abstraction: the danger of ignoring
something that wasn't safe to ignore.

Thus it is good to be aware of the process of abstraction, and it is
imperative that we *choose our abstractions wisely*. In mental health,
what worries me is that it is all too easy to take a diagnostic label
such as 'mania', and, in seeking a 'treatment for mania', essentially
ignoring all other aspects of a particular instance of mania. What
led to it? What role did anxiety play? Was the person taking drugs of
some sort? and so on. If any of these details matter, then treating
mania as a single homogenous thing, where all that matters is whether
or not a person is experiencing an episode of mania, is an easy
trap to fall into. If, just as knowing *what type of fire you are dealing with*,
it is important to know *what type of mania you are dealing with*, then
in trialling 'treatments for mania', it is necessary to subdivide
'mania' into many subtypes. And then the spectre of brain complexity
arises: into how many subtypes do we need to divide 'mania' until it
makes sense to seek remedies for a particular subtype of mania.
I say more about this worry [elsewhere on this wiki](/psychiatry/criticism).