What do mathematicians mean when they say that the calculus that Newton and Leibniz invented was not rigorous?

Question

Was it that Newton and Leibniz didn’t really know what they were doing? I presume that they had to create proofs for how their integrals and derivatives worked.

Were their calculations incorrect because they weren’t rigorous?

In: 3

Answer 1

**TL;DR** It’s not that Newton was “wrong” *per se*, but that the playground in which he was “right” was too small and we needed new rules before we could expand the playground.

For the assumptions under which they were working, their stuff works just fine. And, in their time, there was no real reason to search for additional rigor because these new tools dropped and there was a lot of work that could be done without issue. It took about 200 years, and many great mathematicians using this “unrigorous” version of calculus, before the exceptions and pathologies built up to the point of needing better foundations for it.

For instance, there was just a general assumption that if a function is continuous then – outside of isolated points – it should also be differentiable. For the most part this is true for things like orbits of planets and trajectories of ballistics. But it’s just not a true statement thanks to the [Weierstrass Function](https://en.wikipedia.org/wiki/Weierstrass_function) which is continuous everywhere but differentiable nowhere.

But, eventually, we began to use Calculus in more sophisticated settings and the pathologies were too much for this broader use. This is when limits really began to make a difference because they are able to rigorously formulate a lot of the more wishy-washy parts of Newtonian/Leibnizian Calculus. This allows us to articulate more precisely what we can do and when we can do it. They’re also insanely practical for approximations.

As far as we know, limit calculus is pretty self consistent but it isn’t the end of the story. We have much more broadly expanded the context in which we want to do “Calculus stuff” to places where classical formulations of limits don’t make sense. We want to use that intuition, but it would be unrigorous to apply those rules. So we create even more nuanced formulations of stuff like continuity so that we can do Calculus in places where it shouldn’t make sense. This is not unlike expanding from simple Newton-esque functions to more general differentiable functions that we need limits for.