This may be eerily similar to this, since the concepts of $0$ and of $\infty$ go hand in hand, but $0$ is more natural and is considered a number. And I am probably missing something very obvious, but here is the question.
With regard to all of the trouble that $0$ causes (singularity in otherwise nice function), which manifest in the physical world (center of black hole in general relativity for example), wouldn't it be simpler to also regard $0$ as a concept, similar to the concept of $\infty$, while having the smallest possible number, 1 unit, be something that is agreed upon among mathematicians, some $\epsilon$. Of course, if we want to refer to the current $0$, we simply say nothing.
This not only avoids the concept of $\infty$ in the computational cases, but allows for the use of the concept of $\infty$ when making an analytical argument. Furthermore, one usual concern, which is differentiation and integration (Calculus), can be thought of as finding the approximation of "two points" toward $0$, while the actual computation is of those "two points" 1 unit away from one another.
It seems to me that mathematics construct an "infinite/continuous expansion" of the "finite/discrete world" that everything lives in for the sake of simplifying analysis and computation. But with the rise in computing power, wouldn't it be appropriate to reconstruct the number system and other important concepts from continuous/infinite toward discrete/finite? I understand that there is the field of discrete mathematics, but it feels very lacking (apology for my ignorance). For example, theorems on discrete functions are far and few between, e.g. we have not figured out analytically the "simple" logistic chaos in the discrete cases.
I apologize if this question seems silly, but it bothers me since I have been thinking for a long time that if we use a discrete system to reformulate the mathematics of relativity and quantum, then we would not have any trouble unifying these grand theories.