[Haskell-cafe] Re: 0/0 > 1 == False
Jonathan Cast
jonathanccast at fastmail.fm
Fri Jan 11 22:10:20 EST 2008
On 11 Jan 2008, at 10:12 AM, Achim Schneider wrote:
> David Roundy <droundy at darcs.net> wrote:
>
>> Prelude> let x=1e-300/1e300
>> Prelude> x
>> 0.0
>> Prelude> x/x
>> NaN
>>
>> The "true" answer here is that x/x == 1.0 (not 0 or +Infinity), but
>> there's no way for the computer to know this, so it's NaN.
Didn't catch this the first time around, but: only to a physicist.
(I mean no disrespect to the author of darcs, but nevertheless the
point stands). Back in the real world, 0 / 0 may be defined
arbitrarily, or left undefined. (Defining it breaks the wonderful
property that, if lim (xn) = x, lim (yn) = y, and x/y = z, then lim
(xn / yn) = z. This is considered a Bad Thing by real
mathematicians). In fact, in integration theory 0 * inf = 0 for
certain 'multiplications', which gives the lie to 0 / 0.
>>
> Weeeeeeeelllllllll...... math philosophy, Ok.
>
> You can't divide something in a way that uses no slices. You just
> don't
> cut, if you cut zero times. Which is what you do when you divide by
> one, mind you, not when you divide by zero. Division by [1..0] equals
> multiplication by [1..].
Right. (Although 0 * inf is defined by fiat, as noted above).
> You can't get to the end of either spectrum,
> just axiomatically dodge around the singularities to axiomatically
> connect the loose ends.
`Axiomatically' --- you mean by re-defining standard notation like *
and / to mean what you need in this case. I think this is a
different thing than setting up ZFC so everyone agrees on what a
`set' is from henceforth.
> There is no true answer here, the question is wrong.
Exactly.
jcc
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