Actually, there's exactly one algebraic structure that has a well defined division by 0: the set that contains only a single element. As long as your fine with being able to do no useful operations whatsoever, then yes you can have division by zero. Otherwise, saying that dividing by zero has a value is sloppy terminology. If you're fine with using terminology knowing that it's wrong, that's up to you. Just don't expect to get whatever argument you're making published in a peer reviewed mathematical journal.

Actually, there's exactly one algebraic structure that has a well defined division by 0: the set that contains only a single element.

No. This might be true only if you refuse to drop certain axioms about uniqueness, but obviously you can't expect to insist on field axioms when you're not, you know, working with a field. You can, for instance, create a perfectly reasonable set of operations that includes division by zero on a real projective line. Note that this doesn't make a distinction between positive and negative infinity; otherwise division by zero works exactly the way one would expect intuitively.

As long as your fine with being able to do no useful operations whatsoever, then yes you can have division by zero. Otherwise, saying that dividing by zero has a value is sloppy terminology. If you're fine with using terminology knowing that it's wrong, that's up to you. Just don't expect to get whatever argument you're making published in a peer reviewed mathematical journal.[/quote]

I'm trying to see this the way you do, but I can't. Is the example I gave above satisfactory to you, or are you convinced that rigorously defining division by zero is antithetical to "serious" mathematics?

You're right, I was making an implicit assumption about uniqueness.

No. This might be true only if you refuse to drop certain axioms about uniqueness, but obviously you can't expect to insist on field axioms when you're not, you know, working with a field. You can, for instance, create a perfectly reasonable set of operations that includes division by zero on a real projective line. Note that this doesn't make a distinction between positive and negative infinity; otherwise division by zero works exactly the way one would expect intuitively.

I have a question on your link that I'm confused about that's not perfectly related to the OP, but somewhat related.

How do they justify A*infinity=infinity and infinity+infinity=undefined? Why is infinity*infinity=/=2*infinity=infinity? I can understand it for all their other operations except this one.

[quote name='cowsarenotevil' timestamp='1325203962' post='4897983']
No. This might be true only if you refuse to drop certain axioms about uniqueness, but obviously you can't expect to insist on field axioms when you're not, you know, working with a field. You can, for instance, create a perfectly reasonable set of operations that includes division by zero on a real projective line. Note that this doesn't make a distinction between positive and negative infinity; otherwise division by zero works exactly the way one would expect intuitively.

I have a question on your link that I'm confused about that's not perfectly related to the OP, but somewhat related.

How do they justify A*infinity=infinity and infinity+infinity=undefined? Why is infinity*infinity=/=2*infinity=infinity? I can understand it for all their other operations except this one.
[/quote]

I'm not absolutely certain, but I'm pretty sure it's roughly as follows (excuse my imprecision): infinity on the real projective line is generally analogous to the "infinity" that results from limits of real numbers, but when taking limits the result can be either positive or negative infinity (consider the limit of 1/x as x goes to zero from the left vs. the right). In the real projective line, though, there is no distinction between positive and negative infinity. This means that (infinity + infinity) is actually undefined for the same reason as (infinity - infinity); that is, they're exactly the same thing. This isn't a problem for (infinity * infinity) though, since no matter what the result will not be finite.

Regardless of what mathematicians bicker about, I think the IEEE is unnecessarily confusing.

'Not a number' is really pretty much the least sensible answer to the operation 0 / 0. If we define division as the inverse of multiplication, we see that any number satisfies the predicate 0*x=0. 'not a specific number', or less verbose but semantically identical, 'a number' is the actually correct answer. Which is the formal negation of what the IEEE would have us believe.

'Not a number' would infact be the most satisfying answer to the operation nonzero / zero. There is no such number that satisfies the predicate 1 / x = 0, so 'not a number' very well applies here. 'infinity' is not a number but a concept, so to introduce it into the result set of a supposedly numeric type like a float is kindof improper.

I'm not absolutely certain, but I'm pretty sure it's roughly as follows (excuse my imprecision): infinity on the real projective line is generally analogous to the "infinity" that results from limits of real numbers, but when taking limits the result can be either positive or negative infinity (consider the limit of 1/x as x goes to zero from the left vs. the right). In the real projective line, though, there is no distinction between positive and negative infinity. This means that (infinity + infinity) is actually undefined for the same reason as (infinity - infinity); that is, they're exactly the same thing. This isn't a problem for (infinity * infinity) though, since no matter what the result will not be finite.

I think if I'm grasping this correctly it's more like
X=infinity
Y=infinity
X+Y=undefined
Where I was thinking of it like:
X=infinity
X+X=2X
2*infinity=infinity

The former being more accurate as the two infinities can be thought of as separate values as far as this property is concerned.

It's just like defining taking numbers to the zero power or saying that the number of real numbers is greater than the number of integers: the concepts are "arbitrarily" defined and so not debatable per se, but somehow Cantor's theory of infinite sets was quite controversial, mainly because the usefulness of defining operations in some particular way is not always so apparent.

Right. Defining to the power zero as one is something I think nobody takes exception to.

The analogy with Cantor's theory is slightly off though; the rules of exponentiation dont tell us what to the power zero should be, so we pick the continuous continuation for convenience. Cantors diagonal argument relies not on the introduction of another axiom to constrain previously undefined notions; it relies on the rejection of a previously embraced axiom to render an overconstrained argument consistent. That is, it is rightly controversial. Or bullshit, one might say, if one rejects this choice of axioms.

[quote name='cowsarenotevil' timestamp='1325199551' post='4897959']
It's just like defining taking numbers to the zero power or saying that the number of real numbers is greater than the number of integers: the concepts are "arbitrarily" defined and so not debatable per se, but somehow Cantor's theory of infinite sets was quite controversial, mainly because the usefulness of defining operations in some particular way is not always so apparent.

Right. Defining to the power zero as one is something I think nobody takes exception to.

The analogy with Cantor's theory is slightly off though; the rules of exponentiation dont tell us what to the power zero should be, so we pick the continuous continuation for convenience.[/quote]

I actually meant 0^0 which is not well-defined by limiting behavior, even though it's often convenient to treat it as 1.

Cantors diagonal argument relies not on the introduction of another axiom to constrain previously undefined notions; it relies on the rejection of a previously embraced axiom to render an overconstrained argument consistent. That is, it is rightly controversial. Or bullshit, one might say, if one rejects this choice of axioms.[/quote]

I'm not quite sure I understand. I'm not convinced that the notion that infinite sets are equal in size if and only if there is a bijection between them was widely accepted beforehand.

In any case the only point I wanted to make is that there are different (actually useful) contexts that call for different axioms which define things like a/0, 0^0 or infinity in different ways, as well as sometimes not at all, and that in any of these cases it's disingenuous to say "this is obviously this" or "this doesn't make sense so it can only be undefined" to the exclusion of all other sets of axioms.

[quote name='cowsarenotevil' timestamp='1325208918' post='4898007']
I'm not absolutely certain, but I'm pretty sure it's roughly as follows (excuse my imprecision): infinity on the real projective line is generally analogous to the "infinity" that results from limits of real numbers, but when taking limits the result can be either positive or negative infinity (consider the limit of 1/x as x goes to zero from the left vs. the right). In the real projective line, though, there is no distinction between positive and negative infinity. This means that (infinity + infinity) is actually undefined for the same reason as (infinity - infinity); that is, they're exactly the same thing. This isn't a problem for (infinity * infinity) though, since no matter what the result will not be finite.

I think if I'm grasping this correctly it's more like
X=infinity
Y=infinity
X+Y=undefined
Where I was thinking of it like:
X=infinity
X+X=2X
2*infinity=infinity

The former being more accurate as the two infinities can be thought of as separate values as far as this property is concerned.
[/quote]

You're right, but I don't think that's really any different from what I said. If you sum two limits that tend toward positive infinity, the result will be positive infinity. On the other hand, if you sum two limits, one of which tends toward positive infinity, and one of which tends toward negative infinity, you can get positive infinity, negative infinity, or even something that isn't infinite at all. Thus the sum of positive and negative infinity is undefined without this information.

The possibility of them being "separate values" only arises precisely because "negative infinity" and "positive infinity" are always assigned the same one value under the real projective line, so by basing the behavior of operations involving infinity on analogous operations on limits that tend toward infinity, you have to treat (infinity + infinity) as any of the cases of summing limits that tend toward infinity, including the ones that are not infinite at all.

Likewise, (2 * some limit that tends toward infinity) will always still be infinite.