On the one hand, the human mind must base all of its knowledge on sensation, but on the other hand, we distinguish sense knowledge from intellectual knowledge. To be more precise, we would have to distinguish even among sense knowledge, we can distinguish the exterior senses- like touch and sight and hearing- from the interior ones, like imagination. When we speak of all our knowledge being based on sensation, we mean it takes its origin in the exterior senses.
But though our knowledge arises from exterior senses, our judgments about the things we know are not always judgments about things given in exterior sensation. This is obviously true in the case of fictional characters- our opinions about Hamlet or Don Quixote are not judgments about actual persons given in sensation, which is why we don’t call the police when Hamlet kills Leartes. But this is no less true of rigorous scientific entities- like mathematicals. Even if every orbit and fragment of projectile motion failed to be a perfect ellipse, it would not change the definition of the ellipse- or any of the proofs that follow from the definition. Even though we take our knowledge of mathematicals from sensation, no mathematical entity is perfectly given in sensible things. In St. Thomas’s terms, mathematical things are taken from sensation, but terminate in the imagination. Said in our terms, although our knowledge of mathematical things comes from our exterior sense experience (because we could never know them without sight or touch), our judgment about mathematical things is not a judgment about something in exterior sense experience, but about something in imagination. We give a similar account of fictional characters, groups and classes of individuals, hallucinated beings, etc. In fact, the ability of something to be in imagination that is not in sense is a major cause of error and even psychosis. There is no error in dreaming, but there is in believing that the things we dream are given in exterior sensation. A similar account can be given of some psychiatric disorders, or errors.
This distinction between things terminating in exterior sense and those terminating in imagination provides us with a light to see the even more important distinction between those things that terminate in sensation, and those that terminate in intellect. Our judgments about things given in divine science are given neither in exterior sense nor in imagination, even though our knowledge of these things takes its origin in things we know from sensation. Our inability to sense those mathematicals we imagine is an analogy for our inability to imagine those things we judge to exist by intellect. One cannot see or touch any of Euclid’s circles, but he can imagine them; and in a similar way one cannot imagine his own soul or agent intellect, or universals, or the angels, and yet they are known. The kind of knowledge we can have is radically imperfect- for notice that even though we imagine mathematicals, there would be no impediment to our actually sensing them: we could touch Euclid’s triangles if such things existed. Nevertheless, we could never sense universals, even if they existed outside of our minds- neither will we ever be able to look at the angelic nature even though it does subsist outside of our mind. The knowledge we have of the things that terminate in intellect alone is taken by separationfrom the things in sensation. This separation happens either by a direct negation of the things given in sensation- which gives us concepts like immaterial, or by way of showing the loftiness of things above sensation. If we visualize our knowledge as a line that goes from A to B, with A being exterior sensation and B being intellect, we can imagine metaphysics as either about the “not-A”, or the “above A”, but we can never have a perfectly clear, distinct third concept of “B alone” (if one posited imagination [X]on the line between A and B, he could, on the contrary, speak of an “X alone”, which is why one does not need to define mathematicals by either negation or excellence.)