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?
 A: In Leibnitz' case (who co-invented) by (quite wrongly) assuming that the Greek atomos (indivisible) really was that: a smallest indivisible part he called "Monaden", and which also explains that his integral sign is just a stylized "S" for sum as he really believed you could sum those parts up. His philosophy was generally quite wonky (as Voltaire would agree). It just turned out to work as it should (usually), but books even a century later by great mathematicians like Euler had errors as the infinitesimal concepts were not fully understood (eg, why it matters to sometimes have uniform continuity, not only continuity).
In newton's case, I remember this less except that his "fluxiones" were motivated by physics and again turned out to do the right thing, even though a foundation wasn't present yet...which is arguably part of his (and their) genius.
A: I have read that Newton imagined time as flowing in very short, discrete ticks, and treated the derivative as linear interpolation between the ticks.  This let him consider rates of change that varied with time and position.  But to Newton, nailing down the precise definition of the "derivative" was not so important as having a symbolic framework to quantify his laws of motion and produce empirical predictions, much as to modern physicists, nailing down things like Feynman integrals are not so important as having a symbolic framework to generate predictions for particle collisions.
Generally speaking, before the first half of the twentieth century, mathematics was not conducted as rigorously as it is today.  The inventors of calculus did not need to give super-rigorous proofs, because as you point out, the concepts necessary for a rigorous foundation of calculus did not yet exist -- in fact, calculus motivated their invention (or discovery)!  
Perhaps more pertinently, the clean, neat construction of calculus seen in many real analysis classes (construct reals, define limits with $\epsilon$-$\delta$s, proceed with definition-theorem-proof) is made in hindsight.  Its actual historical development would have proceeded in jumps and starts, often (from the perspective of the real analysis student) backwards and sideways.  We don't see that in class because real analysis is centuries-old theory, digested and regurgitated by so many generations of mathematicians that the standard presentation is now streamlined and clean.
A: 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!"

A: You don't need a formal definition of the real numbers to understand the real numbers.  Decimal fractions, with which Newton was quite familiar, had been invented a hundred years before, and provide a sufficiently rich and uniform model for a good understanding of the real numbers.  Add to decimal fractions the  notion, of which Newton was well aware, that $0.abc9999\ldots = 0.ab\{c+1\}000\ldots$, and what you have is exactly the real numbers.
So I think the premise of your question is wrong. Mathematician rarely invent some formalization out of thin air, and then study its properties; rather, they have an object which has certain properties, which they formalize in order to examine more closely.  The real numbers were invented and understood long before they were formalized by Cantor and Dedekind. Had Cantor or Dedekind's formalizations failed to capture the properties of real numbers as they were already understood, they would have thrown away those formalizations and started over. (On review, I see that Andres Caicedo said this in a comment.)
Similarly the notion of a continuous function long predates its formal definition by Weierstrass. The early notions don't always exactly match the modern formal notion, but they match very closely, because the modern formal notation is intended to formalize the earlier intuitive notion.
A: 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).
