Intuitions ...

See: Essential Metaphysics?

There's no reason why the intuitions which support first order speech should come out as 'true' when articulated in higher order conversations - in fact, this expectation can be very misleading. There is no harm in mathematicians being platonists, but if they want to explore meta-mathematical issues which turn on exactly this intuition, they will get into trouble, because it will turn out to be false (or even unintelligible).

(The reason it will be false is that it is a metaphysical or ontological issue, and these change their character - especially whether they should be taken 'literally' or 'seriously' - in a way which depends on the level of enquiry.)

The trouble is that the way things seem to 'go wrong' here is very disturbing, because some of the intuitions in question appear to underpin the very capacities which can then be turned on them to show their incoherence. Also, first level performance can be damaged by too much higher level enquiry. If you're trying to solve differential equations, you don't want to be distracted by ontological uncertainties about their subject matter. Worse, the internal heuristic models and images we conjure with often seem to be essential to solving the problem.

This is another version of the phenomenological mistake (see 'Thoughts on thoughts').