> The author smears the boundary between what people believe and what is logically entailed, and between mathematical techniques and the way they are applied in modelling the real world.
I think the clue here is the section mentioning Cauchy and rigor.
Without a certain flavor of rigor, "proofs" given by people, _especially in analysis_, can feel unsatisfying and can outright be incorrect, even if the thing they are trying to prove is true!
Imagine a proof of the intermediate value theorem like: well if you try to go from point A to point B you _have_ to pass through C in between eventually or else you'll never get to B.
This might be a sketch of a proof, vaguely. And it's not like the IVT is _wrong_, right? But a non-rigorous proof is not convincing. A non-rigorous proof might leave out details that would otherwise guarantee that a proof isn't left up to interpretation.
If your proof hand waves away some cases that feel trivial to you, to others that might look like a hole in your proof! Or you might think it's trivial, and actually it's not trivial.. but you haven't done it.
Anyways this is, I think, the core here. A new style of mathematics with new foundations... that haven't quite been smoothed out yet. The conclusions being reached are all kinda mostly right, but the reasons the conclusions are correct have not been actually properly set up. So skeptics can drive a truck through that contradiction.
Knowledge is about knowing the right thing for the right reasons... and in its infancy I could see a universe where a lot of mathematicians are running around using its tooling without having the right foundations for it.
We are lucky to live downstream of all this hard work. In the moment things were messier (see also calculus' initial growing pains)