[Haskell] Typing in haskell and mathematics

[Benjamin Franksen]

On Monday 31 January 2005 04:24, Jeremy Gibbons wrote:
> Despite being a fan of generic programming, I have my doubts about
> this kind of automatic lifting. It works fine in "ordinary
> mathematics", because there is no fear of confusion - one hardly ever
> deals with functions as entities in their own right. 

May I please beg to differ? When I studied math, things were quite 
different, at least. I remember whole branches of mathematics 
completely dedicated to dealing with "functions as entities in their 
own right". One notable example is Functional Analysis, of which I 
happen to know a little. And, as far as I remember, we used notation 
which reflected this, i.e. nobody wrote 'f(x)' when actually they meant 
just 'f', which is the same as '\x -> f x', which in math is usually 
written 'x |-> f(x)'.

> (Witness "sigma 
> sin(x) dx", involving a term sin(x) and a dummy variable x, rather
> than the more logical "sigma sin", involving the function.)

The notations for 'integral' and 'differential quotient' stem from a 
time when dealing with functions as entities in their own right was 
indeed not yet a common concept in mathematics, i.e. earlier than 1900.

BTW, 'sigma sin' is not a function.

Ben