[Haskell-cafe] Re: 0/0 > 1 == False

[Jonathan Cast]

On 12 Jan 2008, at 4:06 AM, Achim Schneider wrote:

> 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

wtf?

>
>
>
> You don't get 1, you start off with it.

If you start off with anything, you can often end up with it as  
well.  Enlightenment comes when you realize that sometimes you don't,  
and you acquire the ability to change your mind.

> If you want to find the area
> of a function,

Under, not of.

> you slice 1^2 into infinitely many parts and then look
> how much every single slice differs from lim( 0 ) * 1

Beg pardon?  lim(0) = 0.  lim(0) * 1 = 0.

> , all that
> lim( inf ) many times.

You can't do this.  lim(inf) = inf is an extended real number, and is  
completely unrelated to aleph_0 = the least upper bound of N.  inf is  
not a cardinality and cannot be a number of 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.

`Algebra'

> "One" as a pure concept is
> a very strange beast, as it can mean anything.

To you.  Mathematicians assign a different meaning to the concept.

>
jcc