On Fri, Jun 24, 2016 at 02:40:15AM +0200, Timon Gehr via Digitalmars-d wrote:
> On 24.06.2016 01:58, H. S. Teoh via Digitalmars-d wrote:
> > On Fri, Jun 24, 2016 at 01:33:46AM +0200, Timon Gehr via Digitalmars-d
> > wrote:
> > > On 24.06.2016 00:53, H. S. Teoh via Digitalmars-d wrote:
> > > > > > Because 0^^0 = 1, and 1 is representable.
> > > > > >
> > > > > > E.g. n^^m counts the number of functions from an m-set to an
> > > > > > n-set, and there is exactly one function from {} to {}.
> > > > This argument only works for discrete sets.
> > >
> > > No, it works for any cardinals n and m.
> >
> > It doesn't. What is the meaning of m-set and n-set when m and n are
> > real numbers? How many elements are in a pi-set? How many
> > functions are there from a pi-set to an e-set? Your "number of
> > functions" argument only works if m and n are discrete numbers.
> > ...
>
> Most real numbers are not (identified with) cardinals, so I don't see
> how that contradicts what I said, or how what you are saying is the
> same thing as what you previously stated.
You said n^^m counts the *number* of functions from an m-set to an
n-set. How do you define "number of functions" when m and n are
non-integer?
[...]
> I don't care about any of the limits. ^^ has an essential
> discontinuity at (0,0), so the limits don't need to have a bearing on
> how you define the value, but if they do, consider that there is only
> one direction in which the limit is 0, and an uncountably infinite
> number of directions in which the limit is 1.
Are you sure about this? I'm pretty sure you can define f(x)^g(x) where
f(0)=g(0)=0, such that the limit as x->0 can be made to approach any
number you wish.
> Have a look at this plot: http://www.wolframalpha.com/input/?i=x%5E-y
> Can you even spot the discontinuity? (I can't.)
That's because your graph is of the function x^(-y), which is a
completely different beast from the function x^y. If you look at the
graph of the latter, you can see how the manifold curves around x=0 in
such a way that the curvature becomes extreme around (0,0), a sign of an
inherent discontinuity.
[...]
> Why do you think that is? Again consider my example where a+b is actually
> a+b unless a=b=2, in which case it is 5.
Your example has no bearing on this discussion at all. + has no
discontinuity around 2, so I don't understand what this obsession with
2+2=5 has to do with x^y at (0,0), which *does* have a discontinuity.
Besides, it's very clear from basic arithmetic what 2+2 means, whereas
the same can't be said for 0^^0.
[...]
> > just like 0! is chosen to equal 1 because it makes the definition of
> > factorial "nicer". But it's still an arbitrary choice.
> > ...
>
> n! = Γ(n+1).
> 0! = Γ(1) = 1. Consider it arbitrary if you wish.
The gamma function is only one of many possible interpolations of the
integer factorial to real numbers. It is the one we like to use because
it has certain nice properties, but it's far from being the only
possible choice. Again, an arbitrary choice made on aesthetic grounds
rather than something inherently dictated by the concept of factorial.
> (The offset by 1 here is IMHO a real example of an unfortunate and
> arbitrary choice of notation, but I hope that does not take away from
> my real point.)
This "unfortunate and arbitrary" choice was precisely due to the same
idea of aesthetically simplifying the integral that defines the
function. Look what it bought us: an inconvenient off-by-1 argument that
has to be accounted for everywhere the function is used as an
interpolation of factorial.
Defining 0^0=1 in like manner makes certain definitions and use cases
"nicer", but not as nice in other cases. Why not just face the fact that
it's an essential discontinuity that is best left undefined?
> Anyway, 2+2=4 because this makes the definition of + "nicer". It is
> not an arbitrary choice. There is /a reason/ why it is "nicer".
This is a totally backwards argument. 2+2=4 because that's how counting
in the real world works, not because it happens to correspond to some
abstract sense of aesthetics. Integer arithmetic came before abstract
algebra, not the other way round. Ancient people knew that 2+2=4 long
before the concept of associativity, commutativity, function continuity,
etc., were even dreamed of. All of the latter arose as a *result* of
2+2=4 (and other integer arithmetic properties) via generalizations of
how arithmetic worked. People did not learn how to count because some
philosopher first invented abstract algebra and then decided to define +
as an associative commutative operation continuous in both arguments.
Whereas 0^0 does not reflect real-world counting of any sort, but is a
concept that came about as a generalization of repeated multiplication.
The two (0^0 and 2+2) are not on the same footing at all.
T
--
Let's eat some disquits while we format the biskettes.
