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The intuitionists argue that mathematics precedes logic, whereas Hilbert and his followers (their position being Platonism today) argue as you do (mathematics has its roots in logic).

Both branches of mathematics disagree on basic logical principles (for a Platonist "A or Not(A)" is universally true, but for an Intuitionist it is provably false in some instances). This leads to simple properties such as trichotomy on the reals (given any number, it is <0, =0, >0) failing for intuitionism but being valued for the Platonist.

Godel's incompleteness essentially tells us we can never know which position is "the right one", as no system can prove it's own consistency (i.e no system can ensure itself will never lead to a false result).

Both are acknowledged as consistent systems with respect to one another within academic journals. It is very much a matter of philosophy as to which one is accepted as true.