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Transcendental: Transcendental philosophy tries to determine the conditions for the possibility of knowledge prior to and isolated from all experience [This is NOT transcendent , or, that which surpasses the limits of experience].
Aesthetic: Normally, we would define this as the study of art and beauty, but, this definition was coined in the mid 1700 ’s by Alexander
Baumgarten from the Greek root from which it was derived, aesthesis , perception.
So, when he gets to his third critique, Kant’s aesthetic means the study of art and beauty, here, in the first critique, aesthetic means, more properly, sensible perception.
TRANSCENDENTAL AESTHETIC: The science or conditions of all principles of a priori sensibility.
– i.e., the conditions for the rules that are independent of experience that govern sensible experience.
These principles will be space and time.

Knowledge is from two fundamental sources of the mind:
(1) the capacity for receiving representations (the receptivity for impressions, i.e. intuitions, which permit sensibility the immediacy of relation through which the object is given)
(2) the power of knowing an object through these representations
(understanding as the spontaneity in the production of concepts through which the object is thought).
Thus: Intuitions (from sensibility) and Concepts (from understanding) constitute the elements of all our knowledge
—we CANNOT have knowledge without their union , their relation, their synthesis.
BUT the synthesis goes even further: sensibility and understanding have a priori structures through which they perceive and conceive.

Understanding gives an a priori structure to sensibility, through which sensibility will perceive objects and give them as intuitions to understanding to by synthesized through further a priori structures to yield concepts.
In other words: the objects of experience and knowledge and the conditions of objectivity both have their origin in the knowing subject. (Objectivity is ultimately subjective.) There is a real, independent, objective realm of spatio-temporal objects that get presented to us in sensation. We are capable of a priori , real, objective knowledge of this realm. We can have true knowledge (i.e., make true judgments) because the mind produces the objects and the conditions of their existence.

INTUITION: that by which knowledge has an immediate (non-mediated) relation to objects. Takes place when the object is given to us. Given when mind is affected.
SENSIBILITY: Capacity or receptivity for receiving representations through the mode by which we are affected by objects.
The capacity is like an openness to objects being given.
UNDERSTANDING: What thinks through intuition and yields concepts.
Unifies the synthesized sensory manifold.
SENSATION: effect of an object on the faculty of representation.
EMPIRICAL: the intuition in relation to an object through sensation (the thought of the thing that we perceive).
APPEARANCE: undetermined object of an empirical intuition.
We never actually encounter appearances (in themselves) because they are always determined by space and time and the categories. What we do encounter is the phenomenon —the product of sense and thought.

MATTER: (Content) that in the appearance that corresponds to sensation. Must be a posteriori .
FORM: that which determined the manifold of appearance that it allows of being ordered in certain relations. Must be a priori in mind (it can be considered apart from all sensations).
PURE: Representations without any sensation.
PURE FORM: Must be a priori .
PURE INTUITION: Pure form of sensible intuitions.
A priori . (For example, take away all representations of a body that understanding thinks of it, and you have pure intuition still of extension and figure —it persists even if you have no actual object of sensation or sensibility there, akin to Descartes’ wax analogy).
TRANSCENDENTAL AESTHETIC: The science of all principles of a priori sensibility. i.e., the science of the conditions for the rules that are independent of experience that govern sensible experience.

Definitions:
OUTER SENSE: the sense of otherness that cannot be separated from us; a property of the mind, that which represents objects as outside us and in space; by this
“distance” from us, we can determine the shape and magnitude of objects and the relations between objects.
INNER SENSE: the means by which the mind intuits itself, not as object, but as self ( i.e.
, the self is the subject of intuition and is not an object presented); its form is in time.
Time cannot be outwardly intuited just as space cannot be inwardly intuited. So, what, then are space and time? In order to answer this, Kant gives a brief exposition (first of space) in four points:

1) Space is not an empirical concept derived from outer sense; space must be presupposed.
2) Space is a necessary a priori representation that underlies all outer sense. It is the very condition for the possibility of there being appearances at all.
3) Space is not discursive; i.e.
, it is not a general concept of relations ( à la
Leibniz ), but is a pure intention. It is essentially one (we do not think of a patchwork or conglomeration of discreet spaces; space is one: any spaces depends upon the idea of a delineation of this one space).
4) Space is represented as an infinite given manifold. [Space is intuition, not an object given (represented) through intuition, but is intuited itself. An infinite given manifold, as shown in Transcendental Dialectic, cannot be a possible experience, thus, space is not empirical.] We think infinite possibility in space; we cannot do this with any other concept; thus, space is not a concept but is an a priori intuition.

Q1) So, space is a pure intuition, but geometry, that which studies things in space, is both a priori and synthetic. How?
A1)
Space’s origin is in intuition.
Q2) How can there be an outer intuition prior to the perception of an object and by which the concept of the object can be determined a priori ?
A2) “Manifestly, not otherwise than in so far as the intuition has its seat in the subject only, as the formal character of the subject, in virtue of which, in being affected by objects, it obtains immediate representation , that is, intuition , of them; and only in so far, therefore, as it is merely the form of outer sense in general” (B41).
i.e.
: the objects of experience and knowledge and the conditions of objectivity both have their origin in the knowing subject (objectivity is ultimately subjective). Through sensibility, the a priori intuition of space permits the immediacy of the subjective relation between subject / object.

1) Space does not represent any property of things in themselves or their relations to other things. Nothing about these things can be intuited prior to the things themselves.
2) Space is nothing but the form of all appearances of outer sense (it is the subjective condition of sensibility under which outer intuition is possible).
As form, space can be prior to all perception, thus, exists in the mind a priori and (as pure intuition) permits the determination of the relation of things.
3) Space is necessarily a subjective condition for knowledge; it is not a thing that could be known outside of a subject knowing it.
4) The exposition on space establishes the reality (objective validity) of space, i.e.
, what can be presented to us outwardly as object — AND — the ideality of space, i.e., what can be considered about objects in themselves through reason (w/o sensibility) (B44/A28). Thus, space has both empirical reality and transcendental ideality.

R EALISM : posits that things exist independent of I; things have objective, independent reality.
I DEALISM : posits that things do not exist independent of the mind; things have subjective, dependent reality.
A T RANSCENDENTAL R EALISM : would hold that space and time are things-inthemselves; they are real, objective, and independent of the mind.
A T RANSCENDENTAL I DEALISM : (like Kant’s) would hold that space and time are the a priori forms of sensibility and objects therein of human cognition are appearances that originate in the human mind.
BUT … Kant supplements his TRANSCENDENTAL IDEALISM with an EMPIRICAL
REALISM : permitting appearances in space and time have both objective reality and the possibility of being directly perceived.

But …
(1) there is
besides space that that can be a subjective representation of an outer thing that is at once both objective and
(B44/A28).
(2) this examination is transcendental; all appearances in space are not things-in-themselves, they are
only, representations of our sensibility alone, while they do point beyond themselves to real things, we
them as representations of our sensibility whose form is space. We cannot get to the thing-in-itself through representations.