Division by 0, this time; and how scaling might explain it
I have a book about math. And this book seems to be saying that the idea of a limit, in calculus, "revolves around the meaning of 0/0". It goes on to use the following demonstration that 0/0 is "ambiguous, meaningless, and undefined":
n * m == k can be written as m == k / n (just bring n over from the left to the right).
For example, 3 * 2 == 6 can be written as 2 == 6 / 3.
Now, what if n is 0? Then, you have to say:
1) 0 * m == k can be written as 2) m == k / 0.
Here's the problem with that. 0 * m is 0. And that's true no matter what m is. So, using 1), 0 == k no matter what m is. So, using 2), m == 0 / 0 no matter what m is. Which makes no sense.
And that senselessness supports the argument that 0/0 is ambiguous, meaningless, and undefined. The explanation in the book is a lot less clear than the above, but that's what he's trying to say. But the argument looks like nothing more than a notational trick to me. It looks like the act of a person with blind faith in notation moving terms from one side of an equality to the other without thinking about the meaning nor significance of what they're doing. To inject some meaning into what's being done above, I thought (again) about scaling. And that gave me something a bit more concrete and practical with which to reason about dividing by 0.
Let's think about scaling again for a moment and then use that as a lens through which to re-examine the argument above.
Typically, when you scale by some value S, you can get back to where you were by scaling again by 1 / S or, as I prefer to write since I dislike division, S-1. Because you can get it back, the original size of the scalee must be, in a sense, preserved in the output of the scaling operation. It can't be present in S, nor in S-1, because that's constant for every scalee, so the only place it can be preserved is in the output of the first scaling. Somehow it makes me think of a hologram which, if broken into tiny pieces, each of its pieces preserves the entire image. I say "typically", because S == 0 is a counter-example. If you scale by 0 then the output is 0. And scaling again does not get you back to where you were. It can only get you to where you already are, which is 0. So scaling by 0 throws away the information of the original size. There's nothing you can scale 0 by to get back to the original size. As I said in the last post, 0 can't be scaled. So there's something special about 0, as well as 1.
But scaling explains why 0/0 makes no sense. First, 0/0 means you want to scale 0 by the reciprocal of 0. 0 is the one thing that you can't get a reciprocal of. The reciprocal graph abhors 0. So you can't scale by that thing, whatever it means. Second, you're clearly trying to scale 0 by something, here, and 0 can't be scaled. You're trying to scale a-thing-that-can't-be-scaled by a-thing-that-can't-be-calculated (which makes me think of Oscar Wilde describing fox-hunting as "the unspeakable in pursuit of the inedible"). In general, any attempt to scale 0 fails and 0 stays at 0. But here we can't even be sure of that because 0-1 has no meaning.
So now back to the original argument. n * m == k can be written as m == k / n. This just means that you can scale m by n to get k and then you can get back to m by scaling again by n-1 (the reciprocal of n). The only time that process fails is when n == 0 because the output, k, contains no value. It doesn't encode the original value of m (as it usually would: keeping it safe and ready to be unlocked at any time by scaling again by n-1).
So, I think the puzzle of division by zero is possibly a red-herring. Any graphics programmer who's made the mistake of scaling a cube by 0 and then trying to scale it back can see that the real issue here is actually the flip-side of division-by-zero, which is scaling-by-zero. It's lossy. You lose all your info by doing that. You throw everything down the drain. And it's meaningless to try to reason about scaling back.
The book has more to say about zero (although it seems to wilfully confuse the represented with the representation in order to make its points about ambiguity). For example, it says, "zero apples evokes the idea of apples only to deny that there are any. This is a little strange." It then asks whether 0 oranges is the same as 0 apples, but the book doesn't seem to offer any opinion on the answer from what I can tell.
I have an opinion on the answer: no, they're not the same. I also don't believe that fully-transparent red (that is, invisible red) is the same color as fully-transparent yellow (that is, invisible yellow) even though you might look at both of them and think they look identical (that is, invisible). Watch this video (from about 1:28) if you're curious for an explanation.
-Steve