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.