Hume's critical philosophy rendered the whole enterprise of traditional metaphysics dubious. If there is possibly no metaphysical reality, or even if there is, we cannot meaningfully talk about it, then metaphysicians should close shop for want of a subject. To be sure, this in itself is no big deal, but if metaphysics is a basic human need--and it arguably is, for it is no trifling question in anyone's life whether there is a God or not, or whether they have an immortal soul or not, and whether these things can be known or not, etc.--then Hume's conclusions entail that all of us, qua humans, have to get out of this business.
Therefore, for the first time in history, it became a legitimate question whether metaphysics as a science is possible at all--a question Kant set out to answer in its full generality.
Now, since any scientific theory is a set of known judgments, this question in its fullest generality is equivalent to the question whether known metaphysical judgments are possible. And since such judgments are to provide us with knowledge about some insensible, purely intelligible reality, they have to be known a priori (not on the basis of experience), and they have to be synthetic (i.e., they have to be about reality, and not only about the structure of our complex concepts analyzable in analytic judgments). So the general formal question turns out to be whether synthetic a priori judgments are possible, and if so, how?
The answer of "Hume's Fork" to this question was a simple 'no'. (As Hume argues, if the concept [idea] of the predicate is not contained in the concept [idea] of the subject, then, on the basis of his principle of free recombination of simple ideas, the proposition has to be about a matter of fact, whence it has to be contingent, and thus it cannot be known a priori. So, the synthetic a priori for Hume [and for many after him as well] would be an oxymoron.) But Kant reprehends Hume for not recognizing that we in fact do have synthetic a priori propositions [saps -- henceforth], the clearest examples of which, according to Kant, we can find in math. [n. 272-273.] For, according to him, it is impossible to account for the necessary truth of mathematical propositions purely in terms of the analysis of its concepts. [n. 269.] As he argues, the quantitative concept of 'shortest distance', for example, is by no means contained in the qualitative concept of 'straight line', however you analyze it. But then 'A straight line is the shortest distance between two points' is synthetic. However, if it is, how can we know it a priori?
Kant answers that the only possible way is that this synthesis [of straightness and shortest distance] somehow precedes any actual experience. To explain this precedence, he provides us with a careful analysis of experience in general. Any experience consists of intuitions, i.e., sensory representations of singular objects in their individuality. [As opposed to the intellectual representations of singular objects that represent them in a universal manner, which are the concepts of the understanding -- more about these later.] But any individual sensory representation (whether a color, a shape, a sound, a smell, etc.), precisely because it is individual, has to be determined in space and time. Indeed, space and time are nothing but those determinations of such sensory contents [the empirical intuitions, as Kant calls them], which determine their individuality [say, this round patch of green here may differ from that one, only because this one is here and that one is there -- make them overlap, and they become one individual]. Therefore, space and time, as the necessary, formal determinations of all empirical intuitions in their individuality precede all experience. [Of course, they precede them not in time, but logically, as a necessary precondition precedes anything of which it is a precondition. But space and time are precisely such necessary preconditions of all experience, because no experience can be had without its being determined in space and time.] So, in this sense, space and time are the forms of all intuitions, and as such they precede all experience as its necessary precondition. But then any judgment that concerns only these forms in themselves, disregarding their empirical contents, can be known a priori, "before", that is, not depending on, any actual experience. However, judgments of geometry are precisely such judgments about space, and judgments of arithmetic are precisely such judgments about time [provided we conceive of numbers as the result of the successive addition of units in the temporal process of counting]. Therefore, mathematical propositions, despite the fact that they are synthetic, can be known a priori on the basis of our having the pure intuitions of space and time.
There are two important consequences of this explanation from the point of view of the general enterprise.
1. In this explanation we have the model of a general method of establishing the possibility of saps. The method [which Kant refers to as a 'transcendental deduction'] consists in the discovery of those necessary preconditions of all possible experience which alone make it possible for objects to appear to us as they do in experience. [nn. 284-285.] Synthetic judgments, then, which express some general features of these preconditions will always be knowable a priori.
2. Saps can characterize all possible objects of experience, not as they are in themselves, but only as they appear to us. [n. 285.] Indeed, the only way we can know anything about objects of experience a priori is that the way they appear to us is constituted by our cognitive faculties. This does not turn these objects into mere illusions, yet it necessitates the distinction between an object as it appears to us [phenomenon] and an object in itself (noumenon, Ding an sich). [nn. 289-293]
Since human experience does not consist only of sensory intuitions, but also of objects insofar as they fall under our empirical concepts in our understanding, after examining the constitution of experience by the forms of sensibility (space and time), Kant goes on to examine in an analogous manner the constitution of experience by the forms of the operations of the understanding, which he calls the categories. In the constitution of human experience, categories are to empirical concepts as the forms of intuition are to empirical intuitions; therefore, they can also serve as the basis of forming synthetic a priori judgments, making pure natural science possible.
Empirical concepts are universal representations of classes of individual intuitions (say, the concept 'centaur' is the universal representation of all individual centaurs that can appear in possible experience; that they actually don't appear in our experience indicates that there are no centaurs; or perhaps there are, we have just not found them yet). Pure concepts, on the other hand, are concepts applicable to empirical concepts. For example, the concept of existence is not properly applicable to individual centaurs (for how could you possibly deny the existence of something which is actually a centaur?), but it is applicable to the concept: to say that there are no centaurs is to claim that there are no individual objects of possible experience that would fall under this concept; the empirical concept of centaurs is actually empty.
In a more rigorous reconstruction, along the lines of the ideas of Gottlob Frege, we can draw this distinction in the following manner. The concept of centaurs is like a function, the domain of which is the set of all possible objects of experience (intuitions), and the range of which is the set of truth-values: true or false.
Centaur(x) = T, if x is a centaur; otherwise Centaur(x) = F
The concept of existence is a concept operating on such concepts; that is to say, its domain is the set of all functions like the one above, and its range is again the set of truth-values. Thus, if we define the extension of the above concept as the set of individuals for which the concept yields the value T, that is, Extension(Centaur(x)) := {x: Centaur(x) = T}, then the concept of existence will operate on the concept of centaurs in accordance with the following rule:
($x)(Centaur(x)) = T, if Extension(Centaur(x)) is not the empty set; otherwise ($x)(Centaur(x)) = F
In general, if C is any empirical concept (that is, C(x) = T if x is a C, otherwise C(x) = F), and its extension is Extension(C (x)) := {x: C(x) = T}, then the concept of existence can be defined as the following function:
($x)(C(x)) = T, if Extension(C(x)) is not the empty set, otherwise ($x)(C(x)) = T
In any case, the point of all this is that the concept of existence, as a category, is a logical function that operates on empirical concepts (which in their turn are functions that operate on intuitions). It is the recognition of this two-tier structure of concepts that serves as the basis for Kant’s subsequent investigations.
As a result of his transcendental deductions, Kant provides us with what he regards as an exhaustive list of all such categories, on the basis of the forms of judgment they constitute, and the list of saps they generate as the principles of pure natural science (pp. 46-47).
Without going into the details of the system (which is contained in the Critique of Pure Reason), it is worth our while to see how this analysis works in Kant’s solution of Hume’s problem.
The concept of cause, being a category, operates on empirical concepts, just like the concept of existence as analyzed above. The difference, however, is that the concept of cause is a two-argument function, that is, it takes two empirical concepts as its arguments. Accordingly, if E1 and E2 are two empirical concepts, we may provide an analogous analysis as follows:
(Cx,y)(E1(x),E2(y)) = T, if whenever an x which is E1 occurs, then, on account of the occurrence of x, a y which is E2 occurs, in all possible situations, otherwise (Cx,y)(E1(x),E2(y)) = F
Of course, Kant does not even hint at such technical details. Nevertheless, what he does say can be captured quite clearly by the general idea of this reconstruction. For on the basis of this reconstruction it is easy to see that the concept of cause is indeed a pure concept added to two empirical concepts in a judgment of experience, as opposed to a judgment of perception, which contains only empirical concepts (check the relevant entries in the Index!). And since the concept of cause is defined for all possible empirical concepts, it is also clear that for any such empirical concept there has to be some empirical concept to which it is connected by the non-empirical concept of cause, and this is precisely what is expressed by the synthetic a priori principle of causality, namely, that every object of possible experience has to have some cause, and that that type of cause will always cause the same type of effect in any possible situation. [Formally: for every E2, there is some E1, such that (Cx,y)(E1(x),E2(y)); the point is that even if we may not know for any E2 which E1 it is that satisfies this formula, nevertheless, we can know a priori that there is some such E1.] But then, this solves Hume’s problem. For in this way we can know a priori that every event must have some cause and that similar causes will always have similar effects (that is, if any x that is E1 occurs at any time, then a y which is an E2 will also occur at that time, whether in the past, present or future), for this is the only way in which an object of possible experience can appear to us, as falling under the concepts of the understanding.
But then, again, just as in the case of the explanation of the possibility of mathematics, besides providing the solution of Hume’s problem concerning phenomenal objects, this analysis also generates another problem concerning what these objects are. For, again, since objects of possible experience, as they can only appear to us, are constituted by the categories, for any event and any object appears to us as part of a universal causal order, and the categories do not belong to objects in themselves but to our understanding, the objects of experience again have to be distinguished from objects in themselves. It is only about these objects of experience, the phenomenal objects, that we can have synthetic a priori knowledge, but about objects in themselves we can have no such knowledge at all. On the other hand, since we know that objects of experience are constituted appearances, and appearances have to be the appearances of something, we know that these appearances are not empty illusions; we know that they are of objects in themselves, we just must not confuse the two. [See nn. 360-361]
Another “limiting” consequence of this solution is that since the categories are second-order concepts of our first-order, empirical concepts, they can only apply to objects of possible experience. Therefore, it is impossible to apply them to anything beyond the limits of possible experience. Indeed, since the totality of all experiences, the observable world, is not one of the experiences, we cannot use these concepts even to prove conclusions concerning this totality. So, for example, since the concept of cause is interpreted on the set of all possible experiences, but not for this set, we cannot meaningfully talk about the cause of this totality [n. 316; 332], whence we certainly cannot prove the existence of God as the cause of the world.
But then this has the immediate further consequence that metaphysics in the old, “dogmatic” way is impossible, according to Kant. For that metaphysics promised to prove the existence of God, the immortality of the soul, etc. in general, a number of theses about objects beyond possible experience, by means of concepts of pure understanding, such as cause, substance, existence, etc. (See Kant’s table of the categories.)
On the other hand, precisely because the pure concepts of understanding are interpreted on this totality, another cognitive faculty, reason, necessarily reflects on, and forms its own cognitive acts, the ideas, in connection with this totality. [nn. 327-329] It is reason and its ideas that are responsible for the human desire to reflect on the totality of the world of experiences and what lies beyond it, that is, for the human desire to reach metaphysical knowledge. As we have seen, in a certain way, in which it has been attempted to be satisfied by dogmatic metaphysics, this desire is doomed to frustration.
For what dogmatic metaphysics attempts is impossible: to apply the concepts of pure understanding beyond the sphere of their applicability, to apply them to the totality of the world of experiences and what lies beyond, instead of applying them within this world, where they can only have their proper use, the constitution of phenomena as coming under the necessary laws of nature. But reason’s desire to apply its ideas necessarily gives rise to the “dialectical illusion” of metaphysics, an apparently endless series of arguments and counter-arguments concerning what Kant says are demonstrably undecidable questions.
Thus, reflection on the totality of subjective experiences generates the idea of the soul as the permanent subject of all these experiences. But establishing that the soul is the permanent subject of all these experiences cannot guarantee that it will be the permanent substance of our egos beyond the totality of these experiences, so proofs of the immortality of the soul are doomed to failure. (n. 335.) Of course, the opposite cannot be proved either, since that would require proving that the dissolution of the body is the end of all subjective experiences, and nothing can prove that, except that you wait and you’ll see; oh, well, or you won’t.
Likewise, the totality of all objective experiences cannot be proven to be infinite or finite, either extensively or intensively; that is to say, the infinity or the finitude of the world in space and time, and the infinite divisibility of matter are all undecidable, for these all would involve attributing properties to something beyond all experiences, and thus neither positions can be proven to be true (in fact, both are meaningless). [nn. 340-342]
On the other hand, trying to prove that everything in the world is determined by antecedent causes, and hence there is no free will, or that some agents are free and thus not everything is determined is a vain attempt, for in fact these two positions are compatible. For of course everything in the world of all possible experiences is determined by causal laws, because it is only in this way that any objects of experience can appear to us. But the underlying subject of all subjective experiences, which we grasp by the idea of the human soul, is not one of the objects in this world of all possible experiences, so no wonder it does not come under the causal laws of this world, and thus the only way we can conceive of the soul is by thinking of ourselves as free, autonomous agents, acting on, but not determined by, the world of experiences. [nn. 343-347]
Again, in the same way, the positions that there is some necessary cause of the whole world, and that there is nothing necessary in itself, but everything is contingent on something else are in fact compatible, so they do not decide the question of the existence of God one way or another. For of course within the world of experiences, everything must have a cause, but that does not mean that the world as a whole must have a cause. Yet, this does not exclude the possibility that there is something which is related to the world as something which is a cause within the world is related to its effect. [nn. 347-349]
In fact, these two last antinomies show not only that dogmatic metaphysics is impossible, but also how a critical metaphysics is still possible. For both of them show that the metaphysical ideas (not concepts—those belong to understanding) of God and the soul are not constitutive concepts of the world of experiences, but regulative ideas marking out the limits of the world of objective and subjective experiences. Indeed, these ideas are formed by means of analogies, relating to these two totalities. For the idea of the soul is formed as the underlying subject of all subjective experiences (consciousness), by the analogy that just as the appearances of objects of the world are related to their underlying, permanent subject, their substance, so is the totality of the phenomena of consciousness related to something else, the soul. Again, just as objects of experience in the world are related to other things as their causes, so is the world as a whole related to something else, that is, to God.
In this way, if we realize that we can form the ideas of pure reason by means of these analogies, we can also see that there is also a positive, non-illusory function of these ideas. For being related to the totalities of all experiences, they serve to mark out the limits of the concepts of understanding, and also serve to help us see the whole world of experiences “in perspective”. So they do not have a constitutive function within experience, as the concepts of understanding do, but they have a regulative function in positioning ourselves in relation to the world. As such, the ideas of God and the soul, although they cannot be the constitutive elements of a dogmatic metaphysics, they do have a regulative role in the critical metaphysics of the foundation of morals. We have no knowledge of an immortal soul or of an omnipotent God, but the only way we can conceive of ourselves as moral agents is to look upon ourselves as if we were the free, autonomous creatures of a loving, omnipotent God, to whom we owe moral duty. [nn. 359; 363] That is to say, we have to conceive of ourselves as citizens of two worlds: the autonomous moral agents of the world of noumena, acting in the causally determined physical world of phenomena.