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)

I agree that there is a lot of vague language around the practice of mathematics as a social and philosophical construct ('analysts' vs 'synthetics') but I'm not sure how that indicates the author does not understand what truth is. My understanding of the history of mathematics and science is that these areas of knowledge were much more intertwined with philosophy and religion than they are considered to be today.

So Newton saw no issue with working on the calculus at the same time as being an alchemist and a non-trinitarian. Understanding the world was often a religious activity - by understanding Nature, you understood God's creation - and in Naples it seems that understanding analysis was tied to certain political and nationalist ideas.

What do you mean? I searched the page for "are", it doesn't appear much at all, I'm ruling that one out. So do you mean for instance this statement - ?

  "This zealous quest for universal problem-solving algorithms is precisely what made the synthetics uneasy."

"The author smears the boundary between what people believe and what is logically entailed"

This is not the fault of the author. This is a fairly accurate description of the societal situation back then, and the article is more about societal impacts of math than math itself. Revolutionary, and later Napoleonic France had very high regard for science, to the degree that Napoleon took a sizeable contingent of scientists (including then-top mathematicians like Gaspard Monge) with him to Egypt in 1799. The same France also conquered half of the continent and upended traditional relations everywhere.

This caused some political reaction in the, well, more reactionary parts of the world, especially given that the foundations of modern mathematics were yet incomplete. Many important algebraic and analytic theorems were only discovered/proven in the 19th century proper. Therefore, there was a certain tendency to RETVRN to the golden age of geometry, which also for historical reasons didn't involve any French people (and that was politically expedient).

If I had to compare this situation to whatever is happening now, it would be politicization of biology/medicine after Covid. Another similarity is that many scientists were completely existentially dependent on their kings, which didn't give them a lot of independence, especially in bigger countries, where you could not simply move to a competing jurisdiction 20 miles away.

If your sovereign is somewhat educated (which, at that time, was already quite normal; these aren't illiterate chieftains of the Carolingian era) and hates subversive French (mathematical or otherwise) innovations with passion, you won't be dabbling with them openly.

The Naples state at that time was around 5 million people. You had the landowners (I imagine) looking around at the 'enclosures' of common land in Britain and other parts of Europe and thinking about rents. You had the engineers and Jacobins thinking about new roads and canals and all. The ones who lost out appear to have been the peasants as they lost the feudal protections and access to common lands. And so it goes.

> Using another set of metaphors: Analysts followed the fast flights of their feverish and uncontrolled imagination, while synthetics kept their feet on the ground. Their procedures were slow but safe.