Hot take: the author sneaks in a premise that synthetic mathematics is per se "reactionary", but this is itself pure reactionary copium for not getting it: https://ncatlab.org/nlab/show/synthetic+mathematics https://en.wikipedia.org/wiki/Synthetic_mathematics . There's nothing wrong with wishing to pursue a "coordinate-free" approach to any mathematical field: the old geometers were quite right about this.
> The Neapolitans did not reject modern analysis simply because they considered it French.
And yet after reading the article, it sounds like that is exactly what happened. They took some minor philosophical dispute in math and blew it up for cultural reasons to stick it to the invader. It doesn't sound like it ever really was about the math for most people in that context.
War often pushes people to the limit
I really want to read an essay on this topic by someone I'm more confident actually understands what math is. Or truth, for that matter. 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. They persist in phrasing their statements about how people conceptualize math in terms of "is" and "are", which I tend to assume is a stylistic choice to speak in the perspective of their subjects, but they're so sloppy about perception and truth and "reason" in the rest of the piece that I can't be sure.