Dispositions and Perception
Let’s start with something easy, before we get to more difficult questions. Namely: dispositions cannot be seen. You can gaze at salt for as long as you like and you will not be able to see its disposition to dissolve in water. It has that disposition, and it has it at the time you are gazing at it, but the disposition will not reveal itself to your senses no matter how much you scrutinize the salt. Nor will using a microscope help. The only way to find out if salt is soluble in water is to place it in water and observe the outcome. Solubility is not a manifest property of salt, unlike its whiteness or salty taste—the senses can’t resonate to it. Why is that? Because solubility is a property that concerns the future and possible situations: salt will dissolve in water and it would if it were now placed in water—and we can’t see what will happen or would happen in certain possible circumstances. These facts are absent from your field of vision as you stare at the salt, involving other times and circumstances. If salt had only dispositional properties, it would be imperceptible; but it has various manifest properties as well as dispositional properties—as we are inclined to put it. Dispositions imperceptibly coexist with other non-dispositional perceptible properties. It is, indeed, in the very nature of dispositions not to be perceptible, since they bring in the temporally remote and merely possible. The same goes for powers, capacities, tendencies, aptitudes, and propensities. The merely potential is not a possible object of perception.
Puzzlement begins when we consider color. It has been widely held that colors are dispositions to produce sense experiences, but dispositions are not perceptible and colors are. You can no more see a disposition to produce color experiences than you can see solubility, and for the same reasons—yet you see objects as colored. Doesn’t that prove that colors aren’t dispositions? But isn’t the dispositional view of color plausible? The key to resolving this puzzle is to distinguish the perceptible color property itself from the disposition upon which it depends (supervenes)—the latter being invisible.  We can call these two properties “perceptual color” and “objective color” (the terms are not ideal), thus distinguishing two things that might be called color: color as we perceive it, and color as it exists in objects. The color property we perceive is not a disposition, and hence is perceptible; but the color property that objects have is a disposition, and hence is not perceptible. There are two things that are called “red”! Dispositional color doesn’t enter perceptual content, while perceptual color does (by definition). The latter is a kind of primitive monadic property deployed by the mind in perceiving the world, while the former is a complex relational property that cannot feature as a perceptual constituent. We could accordingly say that color terms are systematically ambiguous. The important point is that the dispositional theory cannot work as a theory of what we see when we see color—the quality that we see is not a disposition. Of course, the two properties are closely connected, but they are not identical, on pain of rendering color imperceptible.
Now consider shape and other so-called primary qualities. Suppose we are attracted by the view that such qualities are best analyzed as dispositions (or powers).  That view has its appeal because of the intimate connection between shape (say) and dispositions—what kind of holes things can fit into, et cetera. There is little daylight between such a property and the dispositions or powers that come along with it, so an identity thesis seems indicated. What is it to be round, say, but to be capable of rolling along a flat plane, fitting into a round hole, and so on? But now there is a question about perception: how can such properties be perceived if they are dispositions? Shouldn’t everything be invisible if the world consists au fond of dispositions? But the world is not invisible, so doesn’t that refute the dispositional theory? Taking our cue from color, we can reply that this doesn’t follow once we distinguish perceptual shape from objective shape: there is the shape we see and the shape things are—the latter being dispositional, but not the former. We spread perceptual shape on the world as a primitive “categorical” property, but objective shape consists in an array of dispositions to act in certain ways: we perceive what we spread, but we don’t perceive objective shape. Objectively, shape is dispositional; subjectively, it is categorical. There are actually two sorts of shape property; “square” is systematically ambiguous.
In the case of primary qualities we can say that objective shape properties come from the world (and are dispositional) while subjective shape properties come from the mind and are imposed on the world (and are not dispositional). Thus we can accept the dispositional view of primary qualities while agreeing that dispositions cannot be perceived, without rendering the world invisible—by postulating two levels of shape property. The shapes we see are not the shapes things objectively have; in fact we don’t see objective shape at all (except in some derivative sense).  It isn’t that primary qualities are secondary qualities, since the dispositions that constitute them are not dispositions to produce shape experiences; but they are alike in that perceived shape is imposed on the world like perceived color. And it is fortunate that this duality of properties exists or else the world would be invisible to us, what with dispositions being invisible.
It might be thought that the dispositional theory of shape (etc.) is not plausible because all dispositions need a “categorical basis” in virtue of which they hold. But this doctrine is not as pellucid as it may seem: for how cannon-dispositional properties give rise to dispositions? By what magic is this accomplished? (Compare mind and body.) How can dispositions obtain “in virtue of” a categorical basis if that basis has no modal nature—if it concerns merely the present and actual? This isn’t to say that macro dispositions don’t have a basis in micro properties—as solubility has a basis in the properties of molecules—but this basis consists in turn of dispositionsof molecules not merely “structural properties”, as if mere geometry could give rise to power. The molecules are disposed to move and disperse in certain ways when salt is placed in water: that is the “categorical basis” of solubility—except that it isn’t “categorical”. Maybe there is some ultimate basis for all dispositions in nature that is not just another disposition, but the usual so-called categorical bases are really tacitly dispositional (they are about behavior not mere form or substance). It is true that perceived shape, like perceived color, is not dispositional, but that doesn’t stop objective shape from being dispositional; and the two are closely connected. It is as if perceived shape gives rise to dispositions but really the underlying objective shape properties are the things that operate as dispositions. Put simply: the world is the totality of invisible dispositions, but the mind imposes non-dispositional qualities on the world that render it perceptible. The mind supplies what is necessary in order to make the world perceptually accessible to us; if it did not, the world would be as invisible as solubility is.
The position we have arrived at is distinctly Kantian. There is an objective reality of imperceptible dispositions (the noumenal world) and superimposed on that a layer of sensible qualities whose origin is the mind itself (the phenomenal world). The reason for the divide is the mismatch between dispositions and perception. The world consists of dispositional facts that cannot be experienced (metaphysics), but we gain perceptual contact with things by imposing from our own resources a layer of facts that are not dispositional (epistemology). There are thus two worlds tightly joined together. Think of it like this: when the physical world came into existence it consisted entirely of invisible dispositions; when conscious beings came along that needed to know about the world they overcame the problem of invisibility by inventing a whole new range of properties that they used to mediate between their minds and reality. They penetrated the cloak of invisibility by painting the world in colors they could recognize; or rather, they dealt with the inevitable invisibility of the world by substituting properties they could see. Perceptual systems didn’t make the dispositions visible (that would be impossible); instead they created another world connected to the first world that suited their receptive apparatus better. These beings then imagined that the world they created was the world that exists objectively, instead of being a proxy for it; but the world never became truly visible, because dispositions in their nature cannot be seen. We imposed our a priori geometry on the world along with our color space (and other sensory qualities), but we never gained perceptual access to the underlying realities. To repeat, it is not possible to see what would happen if (or what is happening in close possible worlds), but that is what a disposition is. We have the illusion that we experience real shape and color, because we impose non-dispositional qualities that we do perceive; but really objective reality is perceptually closed off to us.
Compare phenomenalism: for a table in the other room to be square is for certain counterfactual conditionals to be true, to the effect that I would have an experience of a square table if I had experiences of moving into that room. But such conditional facts cannot be perceived, so they cannot be what I see when I look at the table. So is the table invisible according to phenomenalism? Yes, if that is all there is to say: I cannot see a square table if that fact consists in having a disposition to appear in a certain way to perceivers, since I cannot see any such disposition. Likewise, if we decide on metaphysical grounds that reality is best understood as consisting entirely of dispositions (powers, potentials), then we face the question of how it can be perceived—and we are led to the kind of two-worlds Kantian view I have sketched. 
Ever since Plato we have been schooled to accept a realm of universals that serve to constitute reality as we experience it. The idea of a basic duality of universals is not part of that tradition. But our current reflections suggest that universals divide into two quite distinct groups: those that reduce to dispositions and hence are not perceptible, and those that are perceptible but don’t reduce to dispositions (I leave aside other categories such as the abstract and the moral). The latter have their source in the mind (“universals of the mind”), while the former exist in objective reality (“universals of the world”). The mind constructs a world by exploiting the first kind of universal, while the other world exists by dint of its own devices and is entirely dispositional. The two sorts of universal run in parallel while never merging and each belonging to a different layer of reality.
 Actually the view sketched is closer to Schopenhauer than to Kant, since it posits an active world of powers (“Will”) as constituting objective reality; what Schopenhauer would call “Ideas” corresponds to the level I am calling perceptual properties. Other variants of the general bipartite structure are conceivable: the noumenal as constituted by natural laws (also imperceptible) or supernatural spirits (ditto).