[Haskell] Typing in haskell and mathematics
benjamin.franksen at bessy.de
Mon Jan 31 07:59:58 EST 2005
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.
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