How did Newton invent calculus before the construction of the real numbers? 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?

Way earlier, the Greeks invented a surprising amount of mathematics. Archimedes knew a fair amount of calculus, and the Greeks proved by Euclidean geometry that $\sqrt{2}$ and other surds were irrational (thus annoying Pythagorus greatly).
And I can do a moderate amount of (often valid) math without knowing why the the co-finite subtopology of the reals has a hetero-cocomorphic normal centralizer (if that actually means something, I apologize).
