TITLE: How did Newton invent calculus before the construction of the real numbers?
QUESTION [6 upvotes]: As far as I know, the reals were not rigorously constructed during his time (i.e via equivalence classes of Cauchy sequences, or Dedekind cuts), so how did Newton even define differentiation or integration of real-valued functions?

REPLY [2 votes]: The concepts of a real number, that arbitrary spot on a ruler between integer values was probably understood  intuitively by carpenters long before Newton or Dedekind, and the ideas of areas and volumes were also understood long before Newton invented the integral.
This question brings to mind V.I. Arnold's piece about the dangers of studying mathematics without physics:

To the question "What is 2 + 3" a French primary school pupil replied: "3 + 2, since addition is commutative". He did not know what the sum was equal to and could not even understand what he was asked about!
... as ridiculous as teaching addition of fractions to children who have never cut (at least mentally) a cake or an apple into equal parts. No wonder that the children will prefer to add a numerator to a numerator and a denominator to a denominator!"