Part One of The Main Transcendental Problem. How Is Pure Mathematics Possible?
§ 6. Mathematics consists of synthetic a priori knowledge. How was it possible for human reason to produce such a priori knowledge? If we understand the origins of mathematics, we might know the basis of all knowledge that is not derived from experience.
§ 7. All mathematical knowledge consists of concepts that are derived from intuitions. These intuitions, however, are not based on experience.
§ 8. How is it possible to intuit anything a priori? How can the intuition of the object occur before the experience of the object?
§ 9. My intuition of an object can occur before I experience an object if my intuition contains only the mere form of sensory experience
§ 10. We can intuit things a priori only through the mere form of sensuous intuition. In so doing, we can only know objects as they appear to us, not as they are in themselves, apart from our sensations. Mathematics is not an analysis of concepts. Mathematical concepts are constructed from a synthesis of intuitions. Geometry is based on the pure intuition of space. The arithmetical concept of number is constructed from the successive addition of units in time. Pure mechanics uses time to construct motion. Space and time are pure a priori intuitions. They are the mere forms of our sensations and exist in us prior to all of our intuitions of objects. Space and time are a priori knowledge of a sensed object as it appears to an observer.
§ 11. The problem of a priori intuition is solved. The pure a priori intuition of space and time is the basis of empirical a posteriori intuition. Synthetic a priori mathematical knowledge refers to empirically sensed objects. A priori intuition relates to the mere form of sensibility; it makes the appearance of objects possible. The a priori form of a phenomenal object is space and time. The a posteriori matter of a phenomenal object is sensation, which is not affected by pure, a priori intuition. The subjective a priori pure forms of sensation, namely space and time, are the basis of mathematics and of all of the objective a posteriori phenomena to which mathematics refers.
§ 12. The concept of pure, a priori intuition can be illustrated by geometrical congruence, the three–dimensionality of space, and the boundlessness of infinity. These cannot be shown or inferred from concepts. They can only be known through pure intuition. Pure mathematics is possible because we intuit space and time as the mere form of phenomena.
§ 13. The difference between similar things which are not congruent cannot be made intelligible by understanding and thinking about any concept. They can only be made intelligible by being intuited or perceived. For example, the difference of chirality is of this nature. So, also, is the difference seen in mirror images. Right hands and ears are similar to left hands and ears. They are not, however, congruent. These objects are not things as they are apart from their appearance. They are known only through sensuous intuition. The form of external sensible intuition is space. Time is the form of internal sense. Time and space are mere forms of our sense intuition and are not qualities of things in themselves apart from our sensuous intuition.
Remark I. Pure mathematics, including pure geometry, has objective reality when it refers to objects of sense. Pure mathematical propositions are not creations of imagination. They are necessarily valid of space and all of its phenomenal objects because a priori mathematical space is the foundational form of all a posteriori external appearance.
Remark II. Berkeleian Idealism denies the existence of things in themselves. The Critique of Pure Reason, however, asserts that it is uncertain whether or not external objects are given, and we can only know their existence as a mere appearance. Unlike Locke's claim, space is also known as a mere appearance, not as a thing existing in itself.
Remark III. Sensuous knowledge represents things only in the way that they affect our senses. Appearances, not things as they exist in themselves, are known through the senses. Space, time, and all appearances in general are mere modes of representation. Space and time are ideal, subjective, and exist a priori in our all of our representations. They apply to all of the objects of the sensible world because these objects exist as mere appearances. Such objects are not dreams or illusions, though. The difference between truth and dreaming or illusion depends on the connection of representations according to rules of true experience. A false judgment can be made if we take a subjective representation as being objective. All the propositions of geometry are true of space and all of the objects that are in space. Therefore, they are true of all possible experience. If space is considered to be the mere form of sensibility, the propositions of geometry can be known a priori concerning all objects of external intuition.
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