Because the intuitions on which those opinions are based, actually are beliefs: They arise from your experience with subdividing things. The fifth axiom of Euclid's geometry may be a more helpful example here; it is the one which states that given a line and a point not on it, exactly one line may be drawn through the point which never meets the first one. By denying this axiom we get Riemannian or hyperbolic geometry. Now, of the three versions of the axiom, which one does it feel like you believe? I rather strongly suspect it is the Euclidean one; there's a reason it took two thousand years to formalise the other kinds of geometry. Yet they all describe the behaviour of actual straight lines in different circumstances.
(nods) That's an excellent answer. Thank you.
Very brief recap: The logical positivists said "All truths are experimentally testable". Their critics responded: "If that's true, how did you experimentally test it? And if it's not true, who cares?" Which is a fair criticism. Logical positivism pretty much collapsed as a philosophical position. But it seems to me that a very slight rephrasing might have saved it: "All _beliefs_ are experimentally testable". For if the critic makes the same adjustment, asking "Is that a belief, and if so -" you can interrupt him and say, "No, that's not a belief, that's a definition of what it means to say 'I believe X'."
A definition is not true or false, it is useful or not useful. Why is this definition useful? Because it allows us to distinguish between two classes of declarative statements; the ones that are actual beliefs, and the ones that have the grammatical form of beliefs but are empty of meaningful belief-content.
It seems to me, then, that both the positivists and their critics fell into the trap of confusing 'belief' and 'truth', and that carefully making this distinction might have saved positivism from considerable undeserved mockery.