The transcendental aesthetic depends on an understanding of things in themselves.
Since all of the objects in space and time are deductive, we can deduce that, so far as regards the pure employment of our concepts, the transcendental objects in space and time would thereby be made to contradict our sense perceptions. The manifold may not contradict itself, but it is still possible that it may be in contradiction with our concepts. It remains a mystery why the paralogisms can not take account of the things in themselves; certainly, our faculties can never, as a whole, furnish a true and demonstrated science, because, like space, they have lying before them analytic principles. I assert, in natural theology, that our ideas can be treated like our synthetic judgements; still, the phenomena prove the validity of the objects in space and time. Because of our necessary ignorance of the conditions, we can deduce that our ideas constitute the whole content for the objects in space and time.
Certainly, it remains a mystery why our faculties manufacture the whole content for the things in themselves.
Um, yeah. So, like, these faculties, like, your synthetic judgements of your analytic principles -- narrowly ("broadly") speaking -- are, like, right there, man.
So, the thing is, things in themselves are not in the indefinite sense of "self" (or "not-self") self-sensing, or to put it another way, the sense of the sense of self is not the sense of self, but rather the sense of a sense, or as you might say an anti-sense. This is all very sensible, but the "self" (or "self" to the naïve) has a quality of "selfhood" which implies the non-being ("being") of the "self", if you catch my drift. Right? The concept of the sense of self-sensing conceptual selves, or "selves", is itself a self-sensing concept, or, conceptually speaking, a self-conceptualizing sense of conceptual "selfness" (or "selfness", broadly ("narrowly") speaking).
From this we can see that a syncretic understanding of the being or non-being of any given "being" is ("is") not in and of itself a self, but rather a "self", or non-self, the being being not "being" but rather being a "being" in the "sense" that its faculties ("senses") are either synthetic in themselves or in themselves "synthetic", or in a Heideggerian sense a manifold "nothing" in the sense of the thing that the thing "is not" (which is to say, "is").
I hope I've brought some clarity to the matter.
In the Transcendental Aesthetic of his Critique of Pure Reason, Kant argues that space and time are subjective conditions of human sensibility. This he opposes to the Newtonian, absolutist theory, that space and time exist objectively and independently of objects; and to the Leibnizian relational theory, that space and time are objective relations among objects. Kant also argues that it is an error to apply the concepts of space and time to things in themselves. According to the Aesthetic, not only are space and time subjective conditions of experience; they are solely subjective conditions of experience, applicable only to appearances and not at all to things in themselves. Kant succeeds in demonstrating that, considered as representations, space and time are necessary conditions of our perception of outer and inner objects. However, we will show that he fails in establishing that space and time are actually prior to perception, and in showing that space and time can exist only as representation. In fact, we show that the Kant's arguments do not preclude the possibility of an objectively relational space-time, ŕ la Leibniz.
Kant's argument, at least in the B edition of the Critique, has a simple structure. The short introduction (§1, B33--36) defines important terms used in the Aesthetic (and elsewhere in the Critique). This is followed by an discussion of space. This treatment is divided into three parts. In the first, the ``Metaphysical exposition of [the] concept [of space]'' (B37--40), Kant argues from first principles that space must be a priori and must be an intuition. In the Transcendental exposition of space (B40--42), he attempts to demonstrate that the synthetic apriority of geometry depends on space being an a priori intuition. Finally, the section of conclusions (B42--45) demonstrates that, if space is indeed an a priori intuition as has been (supposedly) proven, then it must be the sensible form of outer experience; it can apply only to appearances and not to things in themselves; and it is empirically real but transcendentally ideal.
The treatment of time parallels that of space. Like the exposition of space, there is a metaphysical exposition (B46--48), arguing from first principles that time is an a priori intuition; and a transcendental exposition (B48--B49), which argues that time must be an a priori intuition in order for us to make use of the synthetic a priori concept of succession. There is also a section of conclusions (B49-53), which demonstrates that, because time is an a priori intuition, it is the sensible form of all inner intuition and hence, mediately, the sensible form of all experience.
Following the treatment of time are the Elucidation (B53--58) and the ``General remarks on the transcendental aesthetic'' (B59--72). These sections make claims not integral to the matter of space and time as the forms of sensibility. We will thus not treat those sections here.
We now consider the first two arguments in the metaphysical exposition of space (B38--39). These arguments attempt to prove the apriority of our representation of space. The first argument claims that, in order for sensations to be related to things outside us, those sensations must already have space as their ground. That is, we could not conceive the existence of objects other than ourselves without a prior representation of space underlying them. Thus, claims Kant, space precedes experience, making it a priori.
It is not entirely clear here why space necessarily precedes the objects we place in it. In addition, Kant does not support his claim that sensations must be represented in space before we can distinguish ideas within ourselves from objects without. Thus one alternative to the apriority of space is that we are somehow able to, without a representation of space, differentiate between objects which are within us and objects which are without us. Then we place those external objects in an objective space which is determined by objective relational properties of the objects. Then, space as a representation would not be a priori, but rather empirical in nature. This is, in fact, one possible interpretation of the Leibnizian relationist position. Kant offers other arguments against this possibility later on, but we shall see that those arguments are not flawless themselves.
The second argument claims that, since we can represent space without objects but not objects without space, space is necessary to our representation of objects, and thus a priori. Unfortunately, Kant does not define ``represent'' here. What does it mean to ``represent'' an object? If it is taken to refer to conceptualisation, surely we can have a concept of an object without placing the object in space: for example, I can have a concept of God, but, since God is not an appearance, I cannot place God within space. Thus it is more likely that ``represent'' here means ``intuit''. That is, I cannot intuit or perceive an object without putting it somewhere in space.
While it seems clear that we cannot sense, and hence cannot intuit, external objects without space, it is not clear that we can intuit space without objects. A relationist would argue that, since space is nothing but relations among objects, it is meaningless without objects. Thus space is not necessarily prior to objects, but may be intuited simultaneously with them.
The third and fourth arguments in the metaphysical exposition of space (B39--40) argue that space is an intuition. The first of these arguments, number 3, proceeds from the singularity of space. Since we perceive individual spaces as parts of the single all-encompassing Space, spaces are defined by their limitations within singular Space. Thus, claims Kant, an a priori intuition grounds all concepts of space.
The argument here is not clear at all. Apparently, Kant wishes to claim that, being singular, space must be an intuition (since concepts are always general in nature). Since any concept of an individual space must be contained within the intuition of singular Space, that intuition must precede the concepts, and hence be a priori. The argument against necessary apriority given above holds here as well; we can imagine our representation of space being simultaneous with our representation of the object; then, since the concept of space is not a priori, there is no reason for the intuition that precedes it to be so, either. Note that we are not here arguing against our representation of space being an intuition: however, if it is not a priori, we are not obliged to accept that no real thing, neither concept nor intuition, can underlie this representation.
The fourth argument begins with the statement that space is represented as an infinite given magnitude. Kant claims that, while concepts can contain infinite numbers of representations under them (e.g., there are an infinite number of possible intuitions corresponding to my concept of ``dog''), concepts cannot contain such infinities within themselves. Since an infinite number of spaces are contained in Space, and thought through it, space cannot be a concept, so it (being a representation) must be an intuition.
This argument seems to be based on an unsupported assumption, namely that concepts cannot be infinite. Kant makes no attempt to prove or even support this claim in the Aesthetic. Neither is there an explanation of why intuitions can be infinite. One objection to Kant's argument is that, when we represent space, we do not necessarily cognize the infinity of possible spaces contained within. Kant is here unclear as to what it means for a representation to ``contain an infinite set of representations within itself'', so it is difficult to argue for or against Kant's claim here. It may well be that Kant is equivocating between ``within X'' as meaning ``thought immediately through X'' and as meaning ``conceptually derivable from X''.
In the transcendental exposition, Kant argues from the synthetic apriority of geometry that space must be an a priori intuition. In this case, it is not even necessary to consider the validity of the argument: it is unsound. If geometry is considered as a pure, formal mathematics, its proofs consist of logical deductions from given axioms and are hence analytic. If it is considered as a science of space, there is another problem. Modern physics, especially general relativity, has satisfiably demonstrated that spacetime is indeed non-Euclidean. Thus the ``common-sense'' and (Kant would say) a priori axioms of Euclidean geometry do not apply to actual space on astronomical scales. Geometry as a true and valid study of space, therefore, must have an empirical basis if it is to be accurate, and is hence a posteriori. Since in neither case is geometry synthetic a priori, the entire argument of the transcendental exposition fails.
The first and second conclusions that Kant draws in this section (B42) are that space does not represent a property of things in themselves, and that it is not a relation among objects. This is because space, being a priori, is intuited prior to the intuition of the existence of the objects it comprehends or that it relates. However, we have argued above that Kant does not sufficiently demonstrate the apriority of space. In fact, we have shown that the relationist position is indeed tenable despite Kant's arguments to the contrary.
Kant's next conclusion (B42) is that space is ``the subjective condition of sensibility, under which alone outer intuition is possible for us''. This may well be the case for space qua representation. Though Kant does not go so far as to prove (rather than simply state) that, without space, we can not distinguish the us from the not-us, it is the case that our intuition of outer appearances is intimately bound to our representation of space. However, if there is indeed a relation of space outside of its rôle as representation (which we have argued above to be a possibility), that relation is not merely a subjective condition: it is an objective reality. Certainly, Kant has not proved that space is ``nothing other than merely the form of all appearances of outer sense'' (emphasis added).
Another conclusion made in this section (B43) is that space applies to appearances only and not things in themselves. Space as representation indeed applies only to appearances; a representation cannot refer to a thing in itself. However, space as an objective relation, if indeed it is, may well apply to things in themselves. We could never prove this, since metaphysics can meaningfully deal only with appearances, but Kant has not proved that space cannot apply to things in themselves. Thus, while the empirical reality of space is still most readily affirmed, the transcendental ideality is only necessary if we consider space as a human representation.
In general, the most important conclusions on space depend on the existence of space solely as a mental representation. Once we have shown that an external, objective relation modelled by this representation is possible, Kant's conclusions become much more narrow. They then apply only to space as a representation---and it is tautologous to state that a representation is transcendentally ideal, or that is is subjective.
Since the metaphysical arguments presented concerning time are parallel to those concerning space, similar reasoning may be used to show that Kant's reasoning is faulty. We thus consider Kant's transcendental arguments on time. The first, curiously, appears as the third argument in the metaphysical exposition; the second is in the transcendental exposition.
The third argument of the metaphysical exposition (B47) is that we know apodictically that different times are not simultaneous; thus, the reasoning goes, non-simultaneity of time is an a priori rule under which alone experience is possible. The problem here is that Kant does not define ``simultaneous''. We must therefore take the usual definition, which is ``occurring together in time''. Then the statement that ``different times are not simultaneous'' is mere tautology. The fact that this rule is a priori is no surprise: the rule is analytic! Kant has thus proved nothing about the apriority of time itself.
Even if we grant Kant the benefit of the doubt here, by taking a different, unspecified, definition for ``simultaneous'', we encounter another problem. That is, Einsteinian special relativity has shown that ``simultaneity'' of two events depends on the frame of reference from which the events are being observed. Hence, we cannot really call time ``simultaneous'': events A and B can be simultaneous in frame X, and B and C in frame Y, without A and C being simultaneous in either frame of reference.
The other transcendental argument is presented in the ``Transcendental exposition of the concept of time'' (B48--49). This argument demonstrates that time must be an a priori intuition. No concept, it is claimed, could explain an object having two contradictory predicates (one at time A, one at time B); thus time qua representation must be an intuition. The transcendental exposition also claims that, since the theory of motion is synthetic a priori, the representation of time that underlies it and allows for succession must be a priori. However, Kant does not explain in the Aesthetic why, precisely, the theory of motion should be a priori. We may make the same claims here as we did for geometry: our common-sense ``a priori'' notions of motion do not necessarily correspond to the full complexities of reality, so the theory of motion is not necessarily a priori. Thus, the representation of time may well be empirical in the same was as space. Also, much as for space, it is only demonstrated here that our representation of time is an intuition; if this intuition is not a priori, then it could well be founded on some external thing which is not an intuition.
The first conclusion Kant draws (B49) is that time cannot exist independently, for then ``it would be actual without an actual object''. This is, unfortunately, the whole of the argument. It appears that here time is being treated as a representation (hence it has an object); however, it is obvious that a representation cannot exist alone. Kant fails to prove that an objective concept of time (whether absolute or relational) is impossible; his arguments fail for the same reasons as the arguments concerning space. Likewise, Kant's claim that ``time cannot attach to things as an objective determination'', because it must precede the objects, depends on an apriority which Kant has not satisfactorily proved.
Kant also labels time as an inner (rather than outer) intuition (B49--50). This is apparently because time, not having shape or position, cannot exist in space (hence ``outside''). Rather, we give objects of outer intuition positions in time mediately, through the (inner) representations we have of those outer objects. Thus Kant places all appearance subjectively within time. As mentioned above, he has not satisfactorily shown that time does not exist independently of our intuition. Thus time is not necessarily purely subjective; the argument against Kant here is much the same as that for space.
Though we have attacked nearly every proposition made in the expositions of space and time, much of Kant's philosophy remains. Space and time are still, when considered as representations, necessary for perception as we know it. However, the representations of space and time are not necessarily prior to representations of the objects they describe. Finally, Kant is not at all successful in showing that space and time are solely representations, and that they do not apply to things in themselves. The most that can be said is that our representations apply only to appearances, and that a non-subjective relation of spacetime might not apply to things in themselves.
The Transcendental Aesthetic is the first real meat n' bones section to the Critique of Pure Reason, in which Kant demonstrates that space and time are conditions we impose on the outside world, independent of any particular experience. This section is called the transcendental aesthetic because it inquires into aesthetics — as the study of experience, not beauty — from a critical viewpoint by abstracting away the particulars of day-to-day experience and looking only at what is left.
Aesthetics up until this point in philosophy had developed similarly in many different contexts. Even in Parmenides we see a tendency to think of experience as misleading, or at least not always reliable. The major difference in pre-Kantian systems of aesthetics is whether or not there are things out there to be experienced. Empiricists followed after Aristotle and claimed that there were, whereas idealists followed after Plato and weren't as sure.
Kant doesn't choose either way, yet. He writes, "Objects are given to us by means of sensibility, and it alone yields us intuitions; they are thought through the understanding, and from the understanding arise as concepts." (A19/B33) This sets up the bare terminology that he needs in order to speak of experience. The way he uses the word "object" is vague enough to encompass both the empiricists' notion of a real thing in a real world and the idealists' notion of deceptive senses. Whatever the cause, the senses receive a lot of data about how the object looks, where it is, and so forth; this sensory data is called an intuition. Understanding is Kant's name for the cognitive faculty that logically categorizes sensory data into concepts.
I always keep a notebook handy when I read Kant so I can draw pictures:
Object (Apperance) → Intuition → Concept (Knowledge)
So, we proceed to reflect on all our past experience and focus only on the universal properties of experience. After some soul-searching, Kant provides us with two: space and time. This section of the Critique is pretty straight-forward (for being part of the Critique), but I've read here and elsewhere that some guy named General Relativity and his weapon of mass destruction, non-Euclidean geometry, somehow refute these two sections of the Critique.
Well, in a way, they're right. Minkowski (of Gundam Wing fame) once wrote, "From henceforth, space by itself, and time by itself, have vanished into the merest shadows and only a kind of blend of the two exists in its own right." On the other hand, the two sections in the Critique are so similar as to be treated simultaneously. Kant is lucky he didn't claim space and not time, or time and not space — then he'd be in a pickle. But as things stand, Kant treats space and time equally.
With that out of the way, Kant gives four arguments for the apriority of spacetime:
The relationship "is different from", from which the concept of spacetime can be derived, is inherent to all experiences. If we did not know this relationship, all locations and times would coincide like some bad doomsday plot on Doctor Who.
Further, it is impossibly hard to imagine a possible experience that does not take place in spacetime. No, a vacuum is just an empty space. Even if I were able to cast a spell to stop time, time would still appear to move for those inside and outside the area of effect.
Spacetime is indefinite. There's a famous science television show that shows a bunch of schoolkids fencing off a square meter area, and then it zooms out an order of magnitude at a time until the entire Milky Way is contained in the area. Well, there's no limits to how far we can imagine to zoom out; even if the universe were finite, we could still write down bigger and bigger numbers and attach them to various units of space and time. However, any particular experience is definite, contained within a specific space for a specific duration of time.
Finally, spacetime is not an empirical concept because, while empirical concepts (e.g., "tree") can apply to infinitely many different experiences (e.g., me seeing a tree from 1m away; 10m away; while sitting in its branches; etc.), they cannot contain an infinite within itself. For example, the concept of an infinitely big tree, can never be a concept derived from a particular experience, because human experience is finite — if nothing else, it is restricted to one lifetime. In the same way, spacetime is not an empirical concept because no matter how big or long a specific extent of spacetime we perceive, we can always imagine one bigger and longer.
Ha, ha. I get one free innuendo per philosophy writeup, so shut up.
(The above neatly interprets and summarizes § 2 and § 4 of the Transcendental Aesthetic in the B version of the Critique of Pure Reason, for the folks reading at home.)
So, updating our picture:
Space and Time | Object (Apperance) → Intuition → Concept (Knowledge)
The discovery of non-Euclidean geometry shell-shocked mathematicians, physicists, and philosophers, because it demonstrated a now trivial fact about model theory: a single language can have more than one model. But it should not shock Kant:
"Time and space are, therefore, two sources of knowledge, from which bodies of a priori synthetic knowledge can be derived. (Pure mathematics is a brilliant example of such knowledge, especially as it regards space and its relations.) (A39-40/B55), emphasis mine
"Time and space are, therefore, two sources of knowledge, from which bodies of a priori synthetic knowledge can be derived. (Pure mathematics is a brilliant example of such knowledge, especially as it regards space and its relations.)
(A39-40/B55), emphasis mine
According to Kant, then, when Euclid set down the postulates of geometry, he was acting on his natural intuitions about space and time. Certainly, however, other types of geometry could be created by placing or removing emphasis on particular aspects of our natural concept of space — specifically what happens to parallel lines that are extended indefinitely. But these alternative geometries all have their foundation in the natural concept of space and the imagination of possible experiences. Knowing about them doesn't change how we perceive the world; in fact, it's only because we perceive the world that we know about them at all.
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