Yes, I take an extreme Platonist position. We do not "construct" the primes - we discover them. Or did the number "29" not exist until the first human counted past 28?
Well then, speaking as an Extreme Platonist can you tell if the real line well-ordered? Or equivalently, are there infinite sets so big that one cannot tell if one is larger than another? Or equvalently, does there exist an unmeasurable set of real numbers (outer measure 1 and inner measure 0 for example.)
These are serious questions for the Platonist position.