I find these discussions on objectivity quite informative, and whilst I am still taking on board points raised by both you and Disinfo Agent I would like to clarify for myself your important view on the separation of the observer and the observed.

I think what you are saying is that the ultimate reality of the world and our place in it is an inseparable whole. What science does is to introduce the notion of separating the observer from the observed in order to arrive at objective conclusions. But this separation should not be thought of as a real separation of observer and observed, it does not expose that part of nature which may be imagined as existing independently of our involvement, it is rather a means of providing a consistent repeatable map of nature that works for anyone who may choose to use that map. But that map, in the overall scheme of things has our fingerprints all over it (to use one of your phrases). It still seems to me that at the quantum level, our finger prints become much more pronounced - the spin of a particle clearly depends on how we set up the detector, but I suppose you could say that in the classical world the attributes of a bullet having a trajectory are fundamentally linked to our collective perceptions of such concepts as time and space. But I take your point that fundamentally, at the quantum and classical level, in order to do science, we have to stand back and apply an objective measurement methodology and accept that that is the best we can ever do regardless of how much we are entwined with the experiment at a very fundamental level.

If I have got this right, then it goes some way to clarifying for me this whole business of objectivity and observations. We cannot separate the observer from the observed in a real sense, every experiment has our finger prints over it, but, importantly, we can make the process objective - and this process is science. But put like this, it becomes clear that science is not in the business of discovering nature as she really is, it is in the business of discovering in an objective way nature and us as an entwined entity.

I'm still pondering over objectivity and mathematics, nothing I have said adds to that discussion, but I think I am a little clearer about what objectivity means within the context of representing the physical world.