But the objects of mathematical thought are not matters of fact. Mathematical truth is not factual truth. I am inclined to say that the mathematician has a clear and coherent understanding of the objects with which mathematics deals, or lacks such understanding. The judgments of the mathematician are either true or false, correct or incorrect, clear or unclear. They are never more or less probable.
Mathematical problems are either solvable with certitude, or they are undecidable.
There are two dimensions in which the philosopher does not have metaphysical or moral knowledge, but rather philosophical understanding. One is the consideration of ideas as intelligible objects. That understanding may be clear or unclear, adequate or inadequate, but it is never true or false. In addition to the understanding of intelligible objects, philosopher may have clear and adequate understanding of various intellectual disciplines, such as the philosophy of mathematics, the philosophy of science, and the philosophy of history. This distinction between knowledge and understanding conforms to Aristotle's analysis of the intellectual virtues.