[Haskell-cafe] mathematical notation and functional programming

[Henning Thielemann]

Over the past years I became more and more aware that common mathematical
notation is full of inaccuracies, abuses and stupidity. I wonder if
mathematical notation is subject of a mathematical branch and whether
there are papers about this topic, e.g. how one can improve common
mathematical notation with the knowledge of functional languages.

Things I'm unhappy about are for instance

f(x) \in L(\R)
   where f \in L(\R) is meant

F(x) = \int f(x) \dif x
   where x shouldn't be visible outside the integral

O(n)
   which should be O(\n -> n) (a remark by Simon Thompson in
                               The Craft of Functional Programming)

a < b < c
   which is a short-cut of a < b \land b < c

f(.)
   which means \x -> f x or just f


All of these examples expose a common misunderstanding of functions, so I
assume that the pioneers of functional programming also must have worried
about common mathematical notation.