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.
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.