The non-visualizibility of concepts is integral to arguments for the immanteriality of the intellect. (It is also a major victory over crude empiricism. Sometimes small gains are enough.) All that is needed to show that the mind is more than an empirical viewing box, as Descartes did with his chiliagon illustration, is elicit acts of conception which exceed the powers of empirical perception. What we know is more than what we can see, with our eyeballs and with our “mind’s eye.” The point is not that our conceptions exist outside our minds --after all they are our conceptions-- but that they exist in our minds in an intellectual, as opposed to perceptual, way. Do we deny chiliagons are possibly materially real? (If I make one tonight out of a sheet of construction paper, dear reader, I’ll mail it to you.) Do we deny they are conceptually real even when not materialized? They exist apart from our tactile and neural constructions of them.
If things like “literally” and “chiliagons” existed only in our minds (i.e., as phantasms), we would never come up with the idea, since we can’t visualize them. Hence, they do exist in a way that exceeds, or so to speak awaits, our grasp of them. Such abstract things have esse intelligibile et naturale (i.e., an intelligible existence as their own nature), but can also exist in a different mode by esse intentione (i.e., with an ‘intentional’ existence).
Now, suppose someone objects, "This demonstrate that we can conceive of something immaterial, but doesn’t prove that this immaterial whatever-it-is has real existence outside our minds. I mean, I can visualize world peace, but it doesn’t do a whole lot to stop the folks in Darfur."
I would immediately challenge this claim about Darfur. You cannot visualize world peace, since, who can really say in clear detail what would have to change, who would have to die, etc., for it to come about? What you can do is conceive of “world peace” and associate mental images with that concept. Can we visualize “humanity”? No, but we can conceive of it. Can you visualize your self, viz., your enduring identity at every spatiotemporal point of its existence? No, but you can grasp such a concept. Can you visualize God? No, but etc. etc.
People say their life flashed before their eyes at a moment of near death–which for all I know may really happen visually, but I doubt it–yet that kind of claim only makes sense to others by conception, not visualization. Even though we cannot “see” our entire life, such a thing exists, and in a way that does run through but does not stay within our minds.