[Haskell] Typing in haskell and mathematics
Benjamin Franksen
benjamin.franksen at bessy.de
Tue Feb 1 06:32:25 EST 2005
On Tuesday 01 February 2005 02:41, Jeremy Gibbons wrote:
> <benjamin.franksen at bessy.de> wrote:
> > BTW, 'sigma sin' is not a function.
>
> I'm missing something here. I don't have an integral symbol to hand,
> which is what I meant by the "sigma",
I understood it that way. But let us use '\int' or '\integral' so we
won't get confusion with the symbol for discrete sums (although the
latter are only a special variant of general integration).
> so perhaps I was unclear. I'd
> say the integral of the sine function is itself a binary function,
> taking lower and upper bounds as arguments.
If you interpret it this way, you are right, of course. The problem is
that such an interpretation is bound to functions on real numbers and
even there it cannot be easily generalized. In Integration Theory, you
usually integrate over an arbitrary measureable set, not only
intervals. In general, such sets may not even be representable as a
subset of \R^n (= finite-dimensional euclidian space). They need not
even have a topology.
Thus, in most of the more abstract mathematical fields, 'integral f'
means the value of the integral over the whole domain (if it exists &
is finite). Integration over a subset is indicated by a subscript
denoting the subset.
Ok, if you want to nitpick, you could say then that '\integral f' is a
function from the set of measureable subsets of the domain of f to the
codomain of f.
Ben
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