What do you mean by "randomly constructed statement" (I request example).
Boolean formula (e.g. 3-SAT) with a random number of clauses, say around one googol (a very small example if the number of clauses is truly a random number).
Then I would agree that intuitions don't provide evidence for such statements, although it is a little bit confusing to say that there is no correlation between intuitions and truth for this class of statements since people don't have any intuitions pertaining to members of this class of statements.
To further nitpick, one googol is small as compared to what exactly?
(I'd find it way more natural for "randomly constructed statement" to be selected from statements people actually construct and have intuitive opinions about, given the context of the discussion.)
Closely related to some of Luke's recent discussions about philosophy, philosopher Paul Thagard has recently called for changes to the way we do philosophy:
In the same article, Thagard also lists eleven areas where modern philosophy goes awry. For example:
Source: Philosopher, Paul Thagard