We know that calculus does not take real values and takes variables like infinity and approaching zero. So can we conclude that even calculus cannot define reality and is different from the "real" value?
closed as unclear what you're asking by Keelan♦, James Kingsbery, jeroenk, Nick R, Swami Vishwananda Sep 27 '15 at 4:31
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All physical measurements in the real world are approximate. To the extent that calculus is used to model physical phenomena, it is an approximation to reality [if one believes in an underlying reality at all ... a separate question.]
However, within the scope of mathematics, calculus is exact. That's because calculus is based on a logically rigorous theory of the real numbers, which are based on a logically rigorous account of set theory.
That is, we can start from the Zermelo-Fraenkel axioms of set theory, published in the early 1920s, and we can then develop the theory of the real numbers and calculus from those axioms. In that respect, calculus is an exact theory of some imaginary or fictional entities; since (as far as we know) there are no real numbers in the physical universe.
So, you are entirely correct that calculus is only an approximation to the real world, whatever that may be. But within the domain of pure mathematics, calculus is exact. That's because the real numbers and sets in general are only abstract mathematical entities. They do not necessarily have any analog in the physical world.
A clarification of when you say 'calculus does not take real values and takes variables like infinity and approaching zero'. This is not correct, and might be leading to confusion.
The central notion in calculus, which underlies all of of the other notions is that of a limit, i.e. the limit of a function at a particular point. For this, we usually write something like, as x→a, f(x)→b, with important instances being f(x)→0 and f(x)→∞. I expect that this is where you get the idea that calculus 'takes variables like infinity and approaching zero'.
But these notions are precisesly defined, and the functions for which they are defined are precise, not approximate.
In particular, the idea that a function tends to infinity at a point means that, although the function itself does not take a value at that point, it takes values arbitrarily close to that point, and we can make the value arbitrarily large by getting sufficiently close to that point. This is spelled out rigorously by what is called the ε-δ definition. (Rather than explain it here, it's probably best to look at the Wikipedia article.)
Calculus can take real values - it results in finite difference calculus. Starting with the gradient formula you just leave the 'infinitesimals' and their powers in when you're doing the algebra. On whether conventional smooth calculus (and by implication physics) defines reality or not: not exactly, because there is no perfect circle, or perfect anything. But we do have good approximations, simplifying assumptions, finite element analysis and fuzzy logic to model reality.
When you talk about "real," you start from the implication that some thing can be distinguished as "real" as opposed to "modeled." That distinction is more important than the mentioning of calculus.
To use a Tarskiian truth: Calculus only defines reality if and only if Calculus defines reality. It would be completely valid to state "Calculus defines a reality, which is the reality defined by calculus."
I think the deeper question is whether the reality "you" live in is defined by calculus, and by FAR the hardest part of that question is defining what "you" are. From there, the rest is easy. However, that step has been a source of disagreement for philosophers since the invention of language.