Well here it is, my second installment of notes on the Critique of Pure Reason. This post walks through the first part of the “Transcendental Doctrine of Elements,” “The Transcendental Aesthetic.” Once again, I am using the Guyer-Wood translation. Enjoy and please comment!
The Transcendental Aesthetic <A>
This part of the Critique contains Kant’s famous arguments on space and time being the conditions by which synthetic a priori judgments are possible. He starts off with some definitions before jumping into the arguments. First, Intuitions are relations between cognitions and objects immediately given to the knower either through sensory experience or in the imagination. In his words, “objects are…given to us by means of sensibility (i.e. the way we are affected by objects), and it alone affords us intuitions; but they are thought through the understanding, and from it arises concepts.” (A19) So we are given objects through sensibility and we form intuitions about those objects which then can further develop into concepts through use of reasoning.
But what kind of objects are we given? Well there are objects of sensation (i.e. empirical objects) that we encounter in the world from which we form empirical intuitions, and if those empirical intuitions are of undetermined objects then they are appearances. If we are given an object sans empirical datum, then it is a pure intuition. If an intuition is empirical, then the appearance which corresponds to the sensation is its matter; if an intuition is pure, then the “manifold of appearances” (A20) to be intuited Kant calls the form of appearance. Thus, the book on my desk is an object given to me through the senses and corresponds to the empirical intuition in my mind of the book; the idea of books as a class of thing is given to me purely mentally and corresponds to the pure intuition of “books” in my mind.
The purpose behind Kant’s exposition into the nature of space is to find out if space meets the conditions necessary to ground the synthetic a priori judgments found in fields such as mathematics. He makes five small arguments supporting his thesis which I shall formalize:
- A posteriori intuitions are represented as being outside of me, different from each other, and in different locations.
- If (1), then there already is a representation of space acting as a ground for them.
- There already is a representation of space acting as a ground for a posteriori intuitions. (1,2)
- If (3), then space cannot be an empirical concept drawn from outer experiences.
- Space cannot be an empirical concept drawn from outer experiences. (3,4)
- It is impossible to represent to oneself the absence of space (not to be confused with representing to oneself space devoid of all objects).
- If (1), then space is a necessary representation a priori.
- Space is a necessary representation a priori. (1,2)
- Geometry operates with apodictic certainty which comes from being grounded in a priori necessity.
- If (1), then Geometry cannot be represented a posteriori (for a posteriori representations lack the sort of necessity required for apodictic certainty).
- Geometry cannot be represented a posteriori. (1,2)
- Space can only be represented as single and all-encompassing. To speak of “spaces” within space is only to speak parts of that all-encompassing whole.
- If (1), then space is a pure intuition (not discursive or a general concept of relations of things in general).
- Space is a pure intuition (1,2)
- If (3), then Geometry is derived a priori with apodictic certainty.
- Geometry is derived a priori with apodictic certainty. (3,4)
- A general concept of space can determine nothing in respect to magnitude.
- If (1), then space is represented as a given infinite magnitude.
- Space is represented as a given infinite magnitude. (1,2)
There are seven conclusions worth noting that can be drawn from the arguments above which show the progression of Kant’s argument:
i. There already is a representation of space acting as a ground for a posteriori intuitions.
ii. Space cannot be an empirical concept drawn from outer experiences.
iii. Space is a necessary representation a priori.
iv. Space is a pure intuition
v. Geometry cannot be represented a posteriori.
vi. Geometry is derived a priori with apodictic certainty.
vii. Space is represented as a given infinite magnitude.
From (i) and (ii) we can see that Kant does not theorize space as any kind of empirical entity that we could point to and say “ah, space.” Space serves as the grounding for all empirical observations, but is not itself an empirical observation. That leads us to (iii), for if a representation cannot be known a posteriori, then the only option left is a priori. Not only is space represented a priori, but it is a pure intuition (iv) meaning the representation is void of any empirical content and satisfies the sufficient condition for synthetic a priori judgments.
(v) and (vi) hold, for Kant, the driving idea behind these arguments; geometry provides the best example of why space must be represented a priori. There is a necessity and universality to geometry (and mathematics in general) that Kant simply does not see as possible without being able to ground geometry in a priori intuitions and concepts. If the “representation of space were a concept acquired a posteriori which was drawn out of general outer experience, the first principles of mathematical determination would be nothing but perceptions.” (A24) And if that were true, then geometry would only have empirical certainty which is never universal and necessary in a strict sense.
The [E] argument has a couple things going on in it which I think are important. First there is the infinite magnitude with which space is represented to us. Space is not confined to any certain magnitude within some larger non-space; it is an all-encompassing whole which we cannot subtract from itself. Second, the fact that the general concept of space cannot determine anything with respect to the magnitude of space also suggests that there are perhaps other things that the general concept of space cannot represent. And if that is true, and if space is a pure intuition, then this points to how space can act as the grounding for synthetic a priori judgments.
Conclusions from these Concepts
Kant first points out that “space represents no property at all of any things in themselves nor of any relation of them to each other.” (A26) Space is not some object that can be counted amongst all the other objects in the universe, it is the framework by which all those other objects have a place to exist. This means that space is the “form of all appearances of outer sense, i.e., the subjective condition of sensibility, under which alone outer intuition is possible for us.” (A26) Now, since the receptivity of the subject (i.e. human) necessarily precedes the intuitions of objects that might be given them, it follows that the pure form of space must exist prior to all other outer intuitions a priori.
Following this line means we can only talk about space and the external objects within it from our own (i.e. human) standpoint. As Kant puts it, “if we depart from the subjective condition under which alone we can acquire outer intuition, namely that through which we may be affected by objects, then the representation of space signifies nothing at all.” (A26) “The constant form of (our) receptivity, which we call sensibility, is a necessary condition of all the relations within which objects can be intuited as outside us, and, if one abstracts from these objects, it is a pure intuition, which bears the name space.” (A27)
It is our pure intention of space that makes possible all outer objects of the world being represented to us, and in the case of mathematics (as one example) of how synthetic a priori judgments are possible. These intuitions are unique to the human knowers, and in that sense, anything we might think of as objective has to be defined in such a way as to coincide with transcendental ideality. Also unlike the idealism of Berkeley, Kant’s ontology contains two distinct realms: the realm of appearances where we humans can possess knowledge of and interact with objects in the world, and the realm of the things-in-themselves, which is completely unknowable and unbroachable; more on this later.
Kant’s arguments on time closely follow those on space with few exceptions:
- A posteriori intuitions of time are represented as simultaneous or successive.
- If (1), then there already is a representation of time acting as a ground for them.
- There already is a representation of time acting as a ground for a posteriori intuitions. (1,2)
- If (3), then time cannot be an empirical concept drawn from outer experiences.
- Time cannot be an empirical concept drawn from outer experiences. (4,3)
- It is impossible to represent to oneself the absence of time (not to be confused with representing to oneself time devoid of all appearances).
- If (1), then time is a necessary representation a priori.
- Time is a necessary representation a priori. (1,2)
- The axioms of time operate with apodictic certainty which comes from being grounded in a priori necessity.
- If (1), then the axioms of time cannot be represented a posteriori (for a posteriori representations lack the sort of necessity required for apodictic certainty).
- The axioms of time cannot be represented a posteriori. (1,2)
- Time can only be represented as single and all-encompassing. To speak of “times” within time is only to speak parts of that all-encompassing whole.
- If (1), then time is a pure intuition (not discursive or a general concept of relations of things in general).
- Time is a pure intuition (1,2)
- If (3), then further propositions about time cannot be derived from the general concept of time, making them synthetic.
- Further propositions about time cannot be derived from the general concept of time, making them synthetic. (3,4)
- Time is represented as an infinite magnitude.
- If (1), then every determinate magnitude of time is only possible by limiting the single time that grounds it.
- Every determinate magnitude of time is only possible by limiting the single time that grounds it. (1,2)
- If (3), then the entire representation (i.e. the infinitude of time) cannot be given though concepts; the immediate intuition must be the ground.
- The entire representation (of time) cannot be given through concepts; the immediate intuition must be the ground. (3,4)
Here there are eight conclusions we can draw from these arguments to help elucidate Kant’s position:
I. There already is a representation of time acting as a ground for a posteriori intuitions.
II. Time cannot be an empirical concept drawn from outer experiences.
III. Time is a necessary representation a priori.
IV. The axioms of time cannot be represented a posteriori.
V. Time is a pure intuition
VI. Further propositions about time cannot be derived from the general concept of time, making them synthetic.
VII. Every determinate magnitude of time is only possible by limiting the single time that grounds it.
VIII. The entire representation (of time) cannot be given through concepts; the immediate intuition must be the ground.
For the most part these arguments parrot those above on space with some minor exceptions. Looking first at [H], with the arguments on space Kant seemed to referring to something quite different than space itself, namely geometry; while the parallel argument on time talks about the “axioms of time” that seem to be referring to rules or laws governing our understanding of time. However, here one could ask “what is geometry?” I submit that it is nothing more than the rules or laws governing our understanding of space. These arguments are actually doing the exact same work with respect to their subject matter; it is just the terminology that might trip someone up.
The [I] argument on time seems to go a bit further than its counterpart did on space; specifically, the move that makes any propositions beyond the general concept of time synthetic. Now I did allude above to the mention of “general concept” in [E] on space, suggesting that Kant is making reference to further propositions on space as being synthetic, and here we see it explicitly stated. Both pure intuitions of time and space are the limitless forms by which all of our knowledge is contained and whenever anything is added to it (whether it is done a priori like geometry or a posteriori like a bouncing ball), it is done so synthetically.
The [J] argument on time does not really give us anything we have not yet seen, but it does pack more in than its counterpart. Rather than concentrate on the “general concept” of time as he does in the [E] argument (and as he does in the [I] argument), Kant’s focus here is on time as an infinite realm that, stripped to its pure form, cannot be made into a concept. The point here is that time (as well as space) is given to us as an infinitely large field and the only way we can move past pure intuition into the realm of concepts is to limit it in some way.
Conclusions from the Concepts
There are a couple of differences between space and time that Kant points out. First is that while with space we were concerned with “outer sense” (that is, we comprehend everything that is outside of us), with time we are concerned with “inner sense” (everything we intuit in our minds). This means that not only is every representation of the mind in time, but also every a posteriori representation that is given to the mind as well. As Kant tells us, “Space, as the pure form of all outer intuitions, is limited as an a priori condition merely to outer intuitions. But since, on the contrary, all representations, whether or not they have outer things as their object, nevertheless as determinations of the mind themselves belong to the inner state, while this inner state belongs under the formal conditions of inner intuition, and thus of time.” (A34) In other words, since every representation of space also is given to the mind, they are subjected to the pure inner intuition of time.
Kant goes on to make the same points with time as he did with space: It takes a human subject intuiting the pure form of time in order to have time as we understand it. Time as a thing-in-itself is a nonsensical notion. Time (and space) only has objective validity insofar as they are appearances to the human subject. In Kant’s words, “Our assertions accordingly teach the empirical reality of time, i.e., objective validity in regard to all objects that may ever be given to our senses. And since our intuition is always sensible, no object can ever be given to us in experience that would not belong under the condition of time. But, on the contrary, we dispute all claim of time to absolute reality, namely where it would attach to things absolutely as a condition or property even without regard to the form of our sensible intuition. Such properties, which pertain to things in themselves, can never be given to us through the senses.” (A36)
Kant considers an objection that alterations are real in our representations, and since time is required for alterations to be possible, then time must be something real. This however misses what Kant is saying. Time as the pure intuition of the inner sense can still do all the same work as time that “real.” Since the only representations we notice change in are the appearances given to us through the sense, and not the things-in-themselves, an absolute or objectively real (not in the transcendental sense) time is not necessary. The two-part system created by Kant’s argumentation provides a very easy sidestep to objections of this nature.
There is a great quote at (A39) in which Kant is explicit about synthetic propositions: “Time and space are accordingly two sources of cognition, from which different synthetic cognitions can be drawn a priori, of which especially pure mathematics in regard to the cognitions of space and its relations provides a splendid example. Both taken together are, namely, the pure forms of all sensible intuition, and thereby make possible synthetic a priori propositions.” And about appearances and things-in-themselves: “But these a priori sources of cognition determine their own boundaries by that very fact, namely that they apply to objects only so far as they are considered as appearances, but do not present things in themselves.”
Lastly in this section, Kant tells us that space and time are the only two elements of the transcendental aesthetic, for every other concept belonging to the sensibility which unites both elements presupposes something empirical. This would exclude concepts such as motion, alteration, mathematics, physics, as well as others.
Here I am skipping over the General Remarks on the Transcendental Aesthetic and pressing onward to the second <B> version.
The Transcendental Aesthetic <B>
Note: I am only concerning myself with covering the changes that occur in the second edition. All material covered in the first edition will be skipped over for brevity.
Kant divided the “Transcendental Doctrine of Elements” from the beginning of the “Transcendental Aesthetic” to the end of the “Transcendental Deduction of the Pure Concepts of the Understanding” into twenty seven sections.
In the first paragraph after elucidating the difference between inner and outer sense Kant defines exposition as “the distinct (even if not complete) representation of that which belongs to a concept; but the exposition is metaphysical when it contains that which exhibits the concept as given a priori.
The [E] argument above has been rewritten thusly:
- Space is represented as an infinite given magnitude.
- Concepts are representations that are contained within infinite sets of possible representations.
- If (1) and (2), then space cannot be a concept.
- Space cannot be a concept. (1,2,3)
- If (4), then space is an a priori intuition.
- Space is an a priori intuition (4,5)
Above I formalize the argument so that (1) is the conclusion. The three sentences in that argument (the <A> version) go as follows:
- Space is represented as a given infinite magnitude.
- A general concept of space can determine nothing in respect to magnitude.
- If there were not boundlessness in the progress of intuition, no concept of relations could bring with it a principle of their infinity.
Looking at this argument in light of the <B> version I now have a better idea of what Kant was up to in the <A> version, but since, in my opinion, the [E] argument above is still faithful to what Kant is saying, I am going to leave it as is and move on. The point here is that space is not a concept but rather an intuition, and the argument revolves around an infinite set of concepts not itself being able to be a concept. Modern set theory shows that Kant assumed that notion incorrectly, but as this is more a note taking session than a refutation party, I shall move forward.
The [C] argument from above has been deleted in favor of §3 “Transcendental Exposition of the Concept of Space.” By transcendental exposition Kant means “the explanation of a concept as a principle from which insight into the possibility of other synthetic a priori cognitions can be gained.” (B 40) He goes on to remind us that pure intuitions cannot be concepts, “for from a mere concept no propositions can be drawn that go beyond the concept, which…happens in geometry (Introduction V),” and “this intuition must be encountered in us a priori…for geometrical propositions are all apodictic…(and) such propositions cannot be empirical (Introduction II).” (B41) Keeping in mind the necessary conditions for geometry to be represented by synthetic a priori propositions, Kant tells us that the only way this outer intuition (i.e. geometry) can inhabit the mind in such a way as to precede all other objects is by having its seat in the subject (i.e. the human subject). Thus, the pure form of outer intuition (i.e. space) is located within the subject and geometry is synthetically cognized a priori from it.
In §5 “Transcendental Exposition of the Concept of Time” Kant adds some remarks to the [H] argument from above. He tells us that alteration and motion are only “possible through and in the representation of time – that if this representation were not a priori (inner) intuition, then no concept…could make comprehensible the possibility of an alteration.” (B48) Just as with space, the pure intuition of time cannot be a concept because it would involve all other concepts being tethered to a concept that has infinite magnitude, which for Kant is not possible.
Here (again) I am skipping over the General Remarks on the Transcendental Aesthetic. While there are some interesting remarks in there, for the purposes of this note taking session I simply want to concern myself with elucidating the arguments themselves, and not concern myself with the broader implications.
Back to the Preface/Introduction Forward to Introduction to Transcendental Logic