I'm not a mathematician or physicist, but (after a bit of googling) I do find some material that advances coordinate-free approaches as a kind of ethical discipline. Relations, understood as ordered pairs, for example, don't have a "natural order", we should mentally maintain the symmetry and absence of priority between statements like "the cat sat on the mat" and "the mat was sat on by the cat".
There's always, in discourse, an origin--a speaker. We can't escape that (see e.g. the linguistic notion of deixis) I think the article risked patronizing the non-hard sciences by assuming that they couldn't see that, being somehow blind to objectivity. Pretty dangerous territory.
