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We have... in a sense. There are number systems wherein 0 is treated rather like dx in terms of being a formal entity upon which algebraic manipulations can be performed; in such systems we can say for example that sin(0)/0 = 1 while sin(0^2)/0 = 0.

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Do you have a reference for this? I would believe you if you said "an infinitesimally small value" rather than 0. If it's true then I'm very surprised. For one, 0 = -0, so if sin(0)/0 = 1, then -1 = -sin(0)/0 = sin(0)/-0 = sin(0)/0 = 1

I'm talking about various forms of infinitesimals, yes. In a sense they're zero, but in another sense they're not.

For the reasons you mention, these models have 0 != -0. :-)

But don't these models all have a true zero, distinct from the infinitesimals, which is its own negative?

Usually, yes. But sometimes it's written as 0^\infty.

"Usually" So there is an instance where this isn't the case?

Well... notation. People come up with all sorts of ways of writing things and all sorts of weird models.

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