>>You mean an "additional" reason?
>
>I must confess to being somewhat of a Platonist. From that
>viewpoint, I meant _the real reason_. :-)

No doubt. I just would add an IMHO.

>>... and any level Calculus book, of course.
>
>No. For example, just look in their sections on power
>series: x0 = 1, even when x = 0. (Perhaps you can
>find some texts which don't do that, but that is done in all
>the texts I've taught from.)

It was a while that I last taught Calculus (or, for that matter, anything else). If the memory serves, they did universally use the shorthand Sakxk for the infinite sum a0 + a1x + ..., but I have no recollection of any instance where the proper point has been made or the matter of x0 = 1 at all taken up.

>
>>>Therefore, I also briefly address the issue of 0^0 below.
>>>
>>>we can always think of as being present in a product,
>>>whether there are any other factors or not.
>>
>>This is a delicate point. We are not exactly talking of the
>>integers.
>
>I'm not certain that I understand your point. Are you
>perhaps wishing to distinguish between the integer 0 and the
>real number 0 when used as exponents?

No, I just meant to point out that the terminology here became somewhat lax. With integers, there is no question about a factor being present in or absent from a product. It's not the case with real x's. You admit this in the next paragraph. Consider what you said previously:

"So let's think about x0 in general. How many factors of x
does it have? Well, none, of course; the exponent tells us that it
has 0 factors of x."

In the context of real x, asking "how many factors" is different from having this question for integer x. Suggesting that along with x comes also 1/x is not altogether compelling because:

there could be more factors, like x·y2·1/(xy)·...
this brings up the simple conventional argument that xa - a = 1.

>> Would you call x a factor in
>>
>>x×1/x = 1.
>
>Yes, the expression on the left does have a factor of x.
>Indeed, it has factors of x in both numerator and
>denominator. (Furthermore, I'll note that those factors of x
>can legitimately be "cancelled", giving 1 as the result,
>except when x = 0.)
>
>>(I understand what you wanted to say and accept that. The
>>above is to underscore a possible ambiguity in your
>>argument.)
>
>What is that possible ambiguity?

As above: too many factors may conceal a mutual cancellation of x and 1/x or just to bring up the basic xa - a = 1. Ambiguity may not be the right word. But "circular reasoning" may.

>>>x0 is the empty product, and must
>>>therefore be 1, regardless of x.
>>
>>The argument is nice, but by the same token, 0x =
>>0 regardless of x.
>
>I don't know what you mean by "the same token".
>0x is not an empty product unless x happens to be
>0. We should have

"Going to the extreme": having an empty product, admittedly a nice concept, is analogous to taking 0 to a "no power". For any (positive) power it's 0. Hence, it is also 0 for a no-power.

>0x = 0 if x > 0, 1 if x = 0, and (assuming we
>want to stay in the real number system) undefined if x < 0.
>
>>>Since I've already presented an argument presumably showing
>>>that 00 must be 1, perhaps I should also now, in
>>>fairness, present reasons for leaving 00
>>>undefined. But I can't! I am aware of not a single
>>>compelling mathematical argument for leaving 00
>>>undefined.
>>
>>I do not believe that.
>
>Well, it's true. Note the important word "compelling".
>
>>To me, the ambiguity created by
>>"0x = 0 for all x" is sufficiently compelling to
>>have second thoughts.
>
>I'd rather that you had second thoughts about 0x
>= 0 for negative x! ;-)

Always have them. I even dream twice a night. Monogamy is especially hard on me.

>>>So then you're wondering: Why is 00 often left
>>>undefined, at least in lower level texts?
>>
>>With a view to Calculus and for the same reason that 0/0 is
>>left undefined: there are ways and ways to approach 0.
>
>Approach? There is no "approaching" going on, at least not
>in what I've been talking about. I have been talking about
>00, which is an _arithmetic_ expression. The base
>and the exponent are _constant_. (I cannot stress that point
>too strongly.) But I now see what is perhaps causing your
>consternation, Alex.

No, no. It's much rather an amusement. I simply do not believe that there is always one best way or vene one _real_ explanation. As a platonist, I believe in a beauty of an argument on its own merits. I absolutely enjoyed your "empty sum" and "empty product" concepts. It's just I can envisage that for somebody this arumentation will be outlandish, let alone un _real_.

>When we consider limits in Calculus, we have certain forms
>which are called indeterminate ... we are not literally
>talking about 00.

We are not; somebody is.

>One should not confuse the
>arithmetic expression and the limit form!

They will, they will ... You'll yet be surprised.

>>>Unfortuantely, to
>>>explain that gaffe would take us far beyond what's
>>>appropriate for a middle school audience.
>>
>>This is why ... Is it not? I can't be 100% sure, but perhaps
>>the sources chose to avoid giving an explanation that "would
>>take the audience far beyond what's appropriate for a middle
>>school audience."
>
>Yes, I suspect that confusion between the arithmetic
>expression and the limit form could well contribute to the
>problem.

>>But to say it's a mistake to have Calculus in mind, is a misteak.
>
>Of course, it's no mistake to have Calculus in mind. The
>mistake is to think that Calculus would keep us from
>defining 00 as it should be!

Again, I see your point and enjoy your argument but I can't accept the value judgement concealed behind "... as it should be." Look how far we (the humanity, I mean) got (developing mathematics) without this point having been settled.

>>(See B. Cipra's or E. J. Barbeau's books.)
>
>I'd be happy to look at any particular passages you'd care
>to mention, if you still think they're relevant.

What's relevant here is an inordinate amount of ways the kids get confused in. The "empty sum" and "empty product", however natural they are to your (or mine) mind's organization are bound to confuse some. I can't substantiate my opinion in any way but I think that they may be enjoyable more than helpful. In other words, they provide an additional angle to what one has already accepted or they bring more confusion to one who is struggling anyway.