In answer to your questions below, it seems to me that my work has philosophical significance in several ways. First, it shows that the reach of Quine’s (and perhaps Putnam’s) indispensability argument is extremely limited (for whatever that’s worth). Secondly, I believe it shows that one can’t sustain the view from Galileo to Tegmark that mathematics (and the continuum in particular) is somehow embedded in nature. Relatedly, it does not sustain the view that the success of analysis in natural science must be due to the independent reality of the real number system.
My results tell us nothing new about physics. And indeed, they do not tell us that physics is somehow conservative over PA. In fact it can’t because if Michael Beeson is right, quantum mechanics is inconsistent with general relativity; see his article, “Constructivity, computability, and the continuum”, in G. Sica (ed.) Essays on the Foundations of Mathematics and Logic, Volume 2 (2005), pp. 23-25. It just tells us that the mathematics used in the different parts of physics is conservative over PA.
Finally, to be “quite happy with ZFC” is not the same as saying that there is a good philosophical justification for it.