At the superficial or architectonic level, Kant has already introduced the Categories and derived them from the Table of Functions of Unity in Judgment, but that, as we have seen, is an inadequate justification of the Categories. Their appearance in the text is thoroughly ad hoc, and even if we accept Kant's claim that they are the rules for synthesis lying ready in the mind, we have no idea how they are supposed to function. What Kant needs is an argument that derives the Categories from the account of synthesis. Logically speaking, they are posterior to, not anterior to, that account.
What might such a derivation look like? Well, synthesis, Kant tells us, is essentially reproduction of a manifold of perceptions in imagination according to a rule. In the act of performing this reproduction, Kant says, the mind affects itself, and by so doing sets the spatially organized manifold in a temporal sequence. What are the most elementary ways in which such a temporally organized reproduction can take place?
There are three possibilities. First of all, two perceptions can be separated out from the entire diversity of perceptions and reproduced in a temporal order, with the precise temporal order not otherwise determined. In other words, which perception is to be reproduced first and which second is specified as undetermined. [Note: There is going to turn out to be a difference between saying that the perceptions are reproduced in an unspecified order and saying that the rule for their reproduction specifies that they may be reproduced in any order. This sounds like gobbledygook, but trust me, it isn't.]
Second, two perceptions can be separated out from the entire diversity of perceptions and reproduced in a specified temporal order -- this one first, then that one. Now, a very subtle but crucial point. The rule specifies which perception is to be identified as the first and which as the second, even if they are not called up in imagination in that order! Think for example of the rule which specifies the relative rank ordering of privates and corporals in the army. The order determined by the rule [from top down, as is appropriate in the army] is "corporals first, privates second." I could just as well have said "privates second, corporals first." Merely mentioning privates first does not alter the order specified by the rule.
Finally, two perceptions can be separated out from the entire diversity of perceptions and reproduced according to a rule that specifies that first the one perception is to be reproduced, then the other, then the other, followed by the first, so that the order [a la poetic rhyme schemes] is ABBA.
These are the three most fundamental rule-types by which a set of perceptions may be reproduced in imagination. [I say "rule types" for a reason that will become apparent shortly.] if you now take a look at the three Categories of Relation in the Table of Categories, you will find that they are "Of Inherence and Subsistence, Of Causality and Dependence, and Of Community." [The last is also glossed by Kant as "reciprocity between agent and patient."] To put it informally in the language of Kant's day, the first is the Category of Substance, the second is the Category of Cause and Effect, and the third is the Category of Action and Reaction [as in the case of two bodies that mutually exert a gravitational pull on one another.].
The Categories of Relation, Kant tells us in the Analogies, exactly correspond to the three different basic ways of reproducing perceptions in imagination according to a rule, as described above. How so? At this point, Kant helps us along with another of his all-too-rare examples -- the house and the boat.
Consider our perceptual experience of a house as compared with our experience of a boat sailing down a river. In both cases, our experience consists of a sequence of perceptions. In the case of the house, we may, for example, be walking around it, seeing first the front, then one side, then the back, and then the other side. In the case of the boat, we see it first upstream, then opposite us as we stand on the shore, then downstream. But there is this fundamental difference: the house is an object; the movement of the boat is an event. The four sides of the house [along with the foundation, the roof, and the inside] co-exist in time. The concept in accordance with which we are reproducing our perceptions in imagination so as to constitute them a unity specifies that the order in which they are to be reproduced is undetermined. Thus I can equally well walk around the house in the other direction, or start with the back, look then at the front, and only then look at the two sides, and so forth. In the case of the movement of the boat downstream, however, the order in which the perceptions are reproduced is determined by the concept -- first upstream, then midstream, then downstream.
I will leave it to you to work through the unpacking of the concept, from Newtonian physics, of equal and opposite gravitational attraction in the interaction of two masses in space.
OK. Now for a number of complications that must be introduced to capture what Kant is actually saying. I have already noted, in the example of the army rankings, that what matters is not the sequence in which the perceptions occur in the process of reproduction, but the rule-governed order that is specified for them. When I am recalling my experience of seeing a boat sail down a river, I can perfectly well think first about how it looked downstream, then how it looked upstream, and finally how it looked midstream. But as I recall it, I recall it as ordered upstream, midstream, downstream. Furthermore, although Kant does not say so, and even seems to suggest the contrary, it is perfectly physically possible that my first perception of the boat should be downstream, and my last perception be upstream. How so? Well, my upstream perception might be an auditory perception of the boat's whistle and my downstream perception might be a visual perception of the side of the boat facing me. Given the difference in the speed with which light and sound travel, it does not take much effort to work out how such a sequence might occur. Think of watching the halftime show at a football game from the last row in the upper stands. Even though you have every reason to believe that the band is marching in time to the music it is playing, you are far enough away so that their steps seem to lag behind their beat. Once again, the difference between the speed of sound and the speed of light.
Second complication, and this one really is complicated. My perceptions of the house and the boat, in addition to being perceptions of a physical object or an event, are also my mental contents. That is to say, they are perceptions in the empirical consciousness of Robert Paul Wolff. In fact, all perceptions have what I call in my book a double nature. They are all both contents of consciousness [bound up in the unity of consciousness with which we started this argument] and also representations of something other than themselves. My consciousness has a subjective temporal order -- first I am conscious of this perception, then of that one, then of the other one. And this subjective temporal order, which is the order of the empirical object that I call my mind, may very well be quite different from the order in which I reproduce them when forming the experience of an object external to my mind.
Scientific investigation tells me that my visual perceptions, as contents of subjective consciousness, are empirically caused in space and time by interactions between physical objects and my retina and optic nerve by the intermediation of photons reflected from the surface of the objects and striking my retina. So the elaborate rule for the reproduction of my manifold of perceptions in a rule governed temporal order must incorporate into itself this complicated situation. The Appearance, that Phenomenal Object that is Robert Paul Wolff, is just one of many objects in space and time obeying the universal causal laws discovered by Newton and other researchers. Once the original diversity of sense contents has had imposed on it a spatial order, and then in the process of reproduction in [my] imagination has had a rule-governed temporal order as well imposed on it, the result must be the concept of a world order of causally determined objects of which my own empirical consciousness is a part.
Right about now, someone out there is going to be asking, "What is all this about Robert Paul Wolff's empirical consciousness? I'm not Robert Paul Wolff. Don't I count for anything in this story?" That is the right question to ask, and as we shall see in not too long, it is a question that poses the most profound problems for Kant's ethical theory. It may occur to those of you with a secret guilty attraction to nineteenth century German Idealism that maybe the problems implicit in this question are what gave rise to the splitting off of Geist or the World Soul from the individual consciousness. To put the same point another way, what we have here, in all its messy difficulty, is what early twentieth century analytic philosophers called "the problem of other minds."
Next complication. What exactly is a physical object in space and time? The account just given makes it sound as though it is some sort of Tinker Toy or Lego construction out of sense contents, my mind then being another construction out of the same bag of parts. Not a bit of it. Kant's view in this, the most unabashedly Phenomenalist strain of his thinking, is that objects are structures of judgments, not structures of perceptions. And my mind also is a structure of judgments. Hence, as Kant signaled way back at the beginning of this exposition, empirical minds and empirical objects are on a par with one another. They are all mere phenomena, appearances in space and time.