Section 3: SYSTEMATIC REPRESENTATION OF ALL SYNTHETIC PRINCIPLES OF PURE UNDERSTANDING (p. 194)

• The Pure understanding is the source of:
  • All Principles.
  • Rules in respect of that which happens.
  • Principles according to which everything that can be presented to us as an object must conform to rules.
• Mathematics is made up of pure a priori principles that we may not ascribe to the pure understanding which is the faculty of concepts.
  
• 1 & 2, which correspond to categories of Quantity and Quality, are entitled mathematical principles.
  • Because they allow of intuitive certainty of evidential force and a priori application to appearances.
• 3 & 4 are entitled dynamical principles because they allow of only discursive certainty.
• Mathematical and dynamical are so named for their application not their content - they are all principles of pure understanding.

Description of the Principles in the Table:

• 1) Axioms of intuition (p. 197)
  • Principle: All intuitions are extensive magnitudes.
  • Proof:
    1. All appearances are extensive magnitudes.
    2. Consciousness of the synthetic unity of the manifold is the concept of magnitude.
       THEREFORE: Appearances, as intuitions in time and space, must be represented through the same synthesis whereby time and space are determined.
  • A magnitude is extensive when the representation of the parts makes possible, and therefore necessarily precedes, the representation of the whole.
  • There are no axioms of magnitude but there are some synthetic propositions.
  • Propositions of numerical relation are synthetic (i.e. 7+5=12) but are only formulas.
  • This transcendental principle of mathematics makes pure mathematics applicable to objects of experience.

• 2) Anticipations of Perception (p. 201)
  • Principle: In appearances, the real, that is, an object of sensation, has intensive magnitude, that is, a degree.
  • Proof:
    1. Perception is empirical consciousness (of sensation).
    2. Appearances are not pure intuitions like time and space. In addition to intuition they contain the real of sensation (the matter of some object) as subjective representation.
    3. From empirical consciousness to pure consciousness a graduated transition is possible.
       THEREFORE: there is also possible a synthesis in the process of generating the magnitude of a sensation.
       THEREFORE: since the sensation is not itself an objective representation, and since neither the intuition of space nor time has met with it, its magnitude in not extensive, but intensive.
  • Anticipation: all knowledge by means of which I am enabled to know and determine a priori what belongs to empirical knowledge.
  • What corresponds in empirical intuition to sensation is reality.
  • A sensation occurs in an instant and is not extensive through time.
  • The real, which corresponds to sensation in general, represents the concept which includes being.
  • Of magnitudes in general, we can know a priori only a single quality, that of continuity.
    • In all quality (the real in appearance) we can know a priori only their intensive quantity, i.e. they have a degree. All else is left up to experience.

• 3) Analogies of Experience (p. 208)
  • Principle: Experience is possible only through the representation of a necessary connection of perceptions.
  • Proof:
    1. Experience is an empirical knowledge.
    2. It is a synthesis of perceptions, not contained in perception but itself containing in one consciousness the synthetic unity of the manifold of perceptions.
    3. Since time itself cannot be perceived, the determination of the existence of objects in time can take place only through their relation in time in general. (And therefore only through the concepts that connect them a priori.)
       THEREFORE: since these always carry a necessity with them, experience is only possible through a representation of necessary connection of perceptions.
  • The three modes of time are duration, succession, and coexistence.
  • The general principles of the three analogies rest on the necessary unity of apperception at every instant of time.
  • These principles are not concerned with appearances, only with existence and relation in respect to existence.
    • Existence can never be known a priori.
    • Existence can not be constructed (like mathematical principles) so these principles will be only regulative.
  • These analogies are valid for empirical, not transcendental, employment of understanding.
  • In the principle we make use of the category but in its application to appearances, we use the schema.

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