[Haskell-cafe] Re: 0/0 > 1 == False
Achim Schneider
barsoap at web.de
Sat Jan 12 07:06:31 EST 2008
Kalman Noel <kalman.noel at bluebottle.com> wrote:
> Achim Schneider wrote:
> > Actually, lim( 0 ) * lim( inf ) isn't anything but equals one, and
> > the anything is defined to one (or, rather, is _one_ anything) to be
> > able to use the abstraction. It's a bit like the difference between
> > eight pens and a box of pens. If someone knows how to properly
> > formalise n = 1, please speak up.
>
> Sorry if I still don't follow at all. Here is how I understand (i. e.
> have learnt) lim notation, with n ∈ N, a_n ∈ R. (Excuse my poor
> terminology, I have to translate this in my mind from German maths
> language ;-). My point of posting this is that I don't see how to
> accommodate the lim notation as I know it with your term. The limit of
> infinity? What is the limit of infinity, and why should I
> multiplicate it with 0? Why should I get 1?
>
n * n = 1 where
lim lim
n -> 0 n -> oo
You don't get 1, you start off with it. If you want to find the area
of a function, you slice 1^2 into infinitely many parts and then look
how much every single slice differs from lim( 0 ) * 1, all that
lim( inf ) many times. When you've finished counting pebble, you know
how to scale this 1^2 to match it with your "normal" value of 1.
n = 12
n = 1 * n
now, 1 is twelve. QED: The wrath of algebra. "One" as a pure concept is
a very strange beast, as it can mean anything.
Like, if you take something and try to understand it by dividing it
successively into infinitely many parts, the meaning of each part will
approach zero, as you don't change the thing you're analysing but the
nature of your lenses.
If you ever want to watch a zen master frowning or despair, tell him
exactly that.
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