To have knowledge consists in possessing the truth. False knowledge is impossible. One cannot say "I know" and add "but what I know is false or incorrect."
There are two modes of truth -- theoretical and practical. Theoretical truth is descriptive truth. We have such truth when our judgments conform to reality -- a reality that is independent of our minds. We have practical truth when our judgments about what should be sought or what should be done to conform to right desire.
The dichotomy of true and false is exhaustive only when we suspend judgment and entertain a proposition without making judgments. Our judgments may be probable or improbable. They have certitude only when they are beyond the shadow of doubt. Otherwise they are highly probable when they are beyond reasonable doubt; they have a lower degree of probability when, at a given time they are supported by reasons and evidence that merely tip the scales in their favor.
The major divisions in the field of knowledge are (1) common-sense knowledge, (2) the knowledge obtained by scientific and historical research, and (3) the knowledge of reality that is philosophical -- knowledge that is metaphysical or moral, knowledge obtained without research and investigation, by means of rational reflection on common sense and common experience.
Like philosophical knowledge, mathematics results from armchair thinking. 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. We should never say that philosophers know such objects as truth, good and evil, and beauty, but rather that they understand these intelligible objects -- these objects of thought.
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. What we have called knowledge he treats as science, distinguishing it from understanding and wisdom, which is the understanding of first principles and final ends.