[theoh]

As thanatropism said, "coordinate-free" is not synonymous with (Einsteinian) relativity. It's a general mathematical approach, and it doesn't have any philosophical consequences that I'm aware of. Just because spacetime is a curved manifold doesn't mean it can't be dealt with using coordinates in a higher space in which it is embedded.

[neutralid]

Sometimes, symmetry reductions become easier using a coordinate-free approach. This means that you could better determine invariances in the system without having a bias towards a particular coordinate system. Philosophically, finding commonality in systems is powerful and any bias (coordinates) distorts those commonalities making them harder to perceive. From the article, I inferred that finding coordinate-free approaches in social sciences could be fruitful.

[theoh]

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.

[neutralid]

The focus is on the subtle impact of using some framework for analysis rather than the gross impact of individual bias (obviously that's an issue every scientist should be and probably is aware of).

For example, statistical causal models are widely used in the social sciences. After searching within the field, I looked for a problem associated with explanation ambiguity due to how the models were framed (or could be equivalently framed leading to alternate explanations) and found this paper:

MacCallum, R. C., Wegener, D. T., Uchino, B. N., & Fabrigar, L. R. (1993). The problem of equivalent models in applications of covariance structure analysis. Psychological Bulletin, 114(1), 185-199. doi:10.1037/0033-2909.114.1.185

I'm not sure if they could extend their approach using manifold theory and an equivalence class for the set of models, but I wouldn't be surprised if someone did something similar. It could then be an example of a coordinate-free approach to data analysis in psychology.