It depends what you mean by 'entity'. Zero is a theoretical construct, and in that sense, exists. Algebraically, the real numbers are a "group" under addition, and zero represents the identity element of that group (ie.: x + 0 = x). All groups have identities. Of course, the ideas of abstract algebra arose AFTER zero was adopted in most number systems, but the motivation is the same: you need an answer for the question "what is (1 - 1)?". So I guess there's no really good answer to your question, or rather the question is somewhat awkwardly posed. If you're asking whether you can hold zero in your hand, no you can't. But the existence of zero is part of our number systems (or at least some of them), and so you can't 'prove' the existence within the same system of logic. It would be like performing brain surgery on yourself! As for your comment about advanced calculus, I again am not sure what you mean. The domain of many continuous real functions includes zero (f(x) = 2x, for instance). If you're referring to the idea of the "limit of a function as it approaches zero", then you'd have to consider the idea of the "limit of a function as it approaches one" (or any number for that matter).

Zero has been called a "placeholder" so that our number system works. For example when you write "10.02" you mean one "ten" + no "ones" + no "tenths" + two "hundredths" in decimal notation (base 10). You must have something in the ones place representing an "empty set," or null value for the notation to be consistent and meaningful. I do not know what you mean by a "true entity." Is a set with no members an entity? (Remember set theory?) It is certainly a valid numerical construct. The reason that differential calculus uses the limit as delta-x or 'dx' (for example) "approaches zero" is that this difference is on the bottom of a fraction: e.g. the expression "dy/dx" as dx approaches zero is used to find the instantaneous slope of a function of y with respect to x. We can not actually use dx=0 because division by zero is not meaningful - you can not divide something into zero pieces or into pieces of zero size. We can, however approach zero and as dx and dy both get infinitesimally small, you approach the actual "instantaneous slope" of a function.

This is more of a philosophical question than a math question. Still, it's the philosophy of mathematics, which is important at the highest levels of the discipline. "Zero" is as true an entity in every way as "one" is. If you think carefully about it, all numbers are abstractions that don't exist anywhere in reality. To see this distinction, consider the following: can you show me "one"? In fact, you cannot. You can show me one *of something*, a toothpick, say, but I could protest and say, "That's not 'one.' That's just a toothpick. I want to see 'one!'" When we discuss mathematics, it's usually in relation to a particular problem under study. We understand that when we add one and one, we're adding one toothpick and one toothpick to arrive at an answer of two toothpicks. The larger abstraction of "one anything" plus "one anything" equalling "two anythings" is only a concept easily grasped by our minds because we've worked so many specific examples. However, without instances backing those abstractions, the statement 1+1=2 becomes somewhat meaningless. If I rush up to you and say, "I have one. It would be great if you could lend me one because I need two. Can you lend me one?" You're likely to respond: "One what?" The abstraction, in and of itself, is meaningless without application to a particular context. Clearly, though, such abstractions do have value. Once we realize we're dealing with an abstraction, the value is that we can apply the logic without having to figure it out again and again. I don't have to rederive 1+1 because I'm adding watermelons instead of toothpicks, and that's the power of abstraction in a nutshell. However, consider this: if the abstraction itself could never, ever be grounded, would it be of any use at all? In other words, if we invited a number on the numberline called "foo", but it is not a number of watermelons or toothpicks or anything we can actually have in reality, what is foo? Would it even exist? (Now replace "foo" in the last statement with "infinity" and you'll see what I'm getting at. It's not a perfect analogy because you have some sense of where infinity is on a number line--that being: ---> that way--whereas foo is completely unspecified.) Plato said yes...in fact, Plato's philosophy is grounded in the believe that these abstractions were the *only* things that really exist (and by "really" he meant on a higher plane that we are incapable of perceiving directly). For him, the abstractions themselves were the "real" things ("one plus one") and the instantiations of these abstractions ("one watermelon plus one watermelon") are mere shadows of the abstractions themselves. He called these abstractions "forms," and philosophers today know this theory as "Platonic forms" after him. Descartes' famous quote, "I think, therefore I am," is along the same lines of thought--the abstractions such as "zero" and "one" are real, but we cannot perceive them directly. The universe, according to Descartes, and all the matter in it is the only limited way in which our minds can perceive these abstractions, so in forming those perceptions, our thoughts quite literally create the universe and everything in it, including ourselves. Pretty weighty stuff, but it makes the point that zero is just as real as any other number in the number line or abstract mathematical concept. So we know zero is "as real as one," but that leaves us with the question: Ok then, how real is one? The answer: your mileage may vary. :-)