[OT] ...

[Timon Gehr via Digitalmars-d]

On 24.06.2016 08:49, H. S. Teoh via Digitalmars-d wrote:
> ...How do you define "number of functions" when m and n are
> non-integer?
> ...

I don't. But even when n is an arbitrary real number, I still want empty 
products to be 1.

> ...
>
>> 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.

It's a mirror image.

> If you look at the
> graph of the latter,

Then the point (0,0) is hidden.

> 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.
> ...

Well, there's no question that the discontinuity is there. It's just 
that, if you want to expose the discontinuity by taking limits along 
some path not intersecting the y axis, you need to choose it in a 
somewhat clever way in order not to end up with the value 1. Of course, 
this is not really a strong argument for assigning any specific value at 
(0,0).

>
> [...]
>> 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. ...

It's another example of a notational convention. I was trying to figure 
out if/why you consider some well-motivated conventions more arbitrary 
than others.

> Besides, it's very clear from basic arithmetic what 2+2 means, whereas
> the same can't be said for 0^^0.
> ...

That's where we disagree. Both expressions arise naturally in e.g. 
combinatorics.

> ...
>
>> (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. ...

n! = ∫dx [0≤x]xⁿ·e⁻ˣ.
Γ(t) = ∫dx [0≤x]xᵗ⁻¹·e⁻ˣ.

One of those is simpler, and it is not Γ.

> ...
> 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?
> ...

Because you don't actually care about continuity properties of x^y as a 
function in (x,y) in the cases when you encounter 0^0 in practice.

I agree that you sometimes don't want to consider (0,0). If you want to 
study x^y as a continuous function, just say something to the effect of: 
"Consider the continuous map ℝ⁺×ℝ ∋ (x,y) ↦ xʸ ∈ ℝ". This excludes (0,0) 
from consideration in the way you want (it also excludes the case that x 
is a negative integer and y is an integer, for example).

There is no good reason to complicate e.g. polynomial and power series 
notations by default. Either x or y often (or even, usually) does not 
vary continuously (and if y varies, x is usually not 0).

>
>> 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,

Yes, and 0^0 = 1 because that is how counting in the real world works.

> ...
>
> Whereas 0^0 does not reflect real-world counting of any sort,

Yes it does. It counts the number of empty sequences of nothing.

> but is a
> concept that came about as a generalization of repeated multiplication.

That's one way to think about it, and then you would expect it to 
actually generalize repeated multiplication, would you not?


BTW: I found this funny:

http://www.wolframalpha.com/input/?i=product+x+for+i%3D1+to+n

http://www.wolframalpha.com/input/?i=product+0*i+from+i%3D1+to+0
  ('*i' needed to pass their parser for some reason.)

http://www.wolframalpha.com/input/?i=0%5E0


By treating 0^0 consistently as 1, you never run into this kind of 
problem. Doesn't this demonstrate that they are doing it wrong? How 
would you design the notation?