A more conservative approach is to point out that some of our (mathematical) models are useful: they let us predict what happens when we perform an experiment, while others are not useful in that sense (though they can still can help you get through a grad school, write a grant application, gather a few karma points when posted here, or maybe even found a new church, if your model is untestable but convincing enough).

If you are a natural scientist, it's the "useful" models that should matter to you, while if you are a philosopher, there is little distinction. Your musings are clearly of the latter kind.