Personally, I like to split the difference: the physical continuum definitely exists, to whatever extent any physical thing exists, but the real number line (and indeed the completed inductive set of integers) may just be a human-constructed fiction. The physical continuum is not necessarily identical to the real continuum; the latter is just a very useful model that lets us do human things like calculus.

(And the discrepancy might not be in the physical continuum being simpler than the mathematical reals, as some here postulate, but rather in the continuum being far stranger than the reals, in ways we may never observe nor comprehend.)