CPR: The System of Principles: Axioms of Intuition, Anticipations of Perception

The task at hand is [B188]

to exhibit as systematically linked the judgements that the understanding … actually brings about a priori [which is what happens in the context of the schematism].

I take this to mean that the Schematism provides the basic framework within which we can go on to investigate in more detail precisely in what way each of the categories are linked to experience in judgement. In other words, how the categories are translated into principles of judgement.

Recall that the Analytic of Principles, or transcendental doctrine of the power of judgement, is made up of:

• The sensible condition under which the categories can be used: Schematism
• The synthetic judgements that are made on the basis of the categories, under the condition above: System of Principles

The Supreme Principle of All Analytic Judgements

This is the domain of general logic, and the supreme principle is the law of non-contradiction. And there isn’t much more to say about analytic judgements here.

The Supreme Principle of All Synthetic Judgements

In making a synthetic judgement one joins two concepts, neither of which contains the other. There must be a third thing that makes this possible. It is inner sense, and therefore time, because this is the thing that contains all presentations. But there is more to it. There are three things that the possibility of synthetic judgements depends on (is based on):

• Inner sense / time
• Imagination
• The unity of apperception

This because presentations are all in time owing to a synthesis of the imagination, and have a synthetic unity owing to the unity of apperception.

Now, synthetic propositions must refer to actual or possible experience. Experience is of objects, thus synthetic propositions referring to experience are objective.

But experience rests on the synthetic unity of appearances, otherwise it would be merely a “rhapsody of perceptions”.

Hence experience is based on a priori principles, because there must be principles as to the form of this unity.

These principles are universal rules of unity in the synthesis of appearances

The supreme principle of synthetic judgements is this:

Every object is subject to the conditions necessary for synthetic unity of the manifold of intuition in a possible experience.

Hence synthetic a priori judgements are possible if we say that the conditions for possibility of experience as such are the same conditions as those for the possibility of objects of experience. Thus synthetic a priori judgements have objective validity.

Systematic Presentation of the Synthetic Principles of Pure Understanding

The synthetic principles break down according to the categories. Like this:

Note: I didn’t cover the difference between mathematical and dynamical when it came up in the metaphysical deduction. [B110]

Mathematical Principles

Kant describes the principles that concern intuition as mathematical. Why is this? Because they concern intuition in general, which is exactly what mathematics is about—intuition as such, or the form of intuition, i.e., space for geometry. Dynamical principles, on the other hand, are concerned with actuality, with, as Kant says, existence, and the relations between existing things.

Both kinds principles, he says, are “capable of complete certainty” [B201], but with intuitions this certainty is immediate. You can just see it, owing to the faculty of synthesizing appearances in space. A good example is the following non-rigorous yet strikingly right and certain proof of Phythagoras’ Theorem. The central square that drops down from the hypotenuse, side c of the right-angle triangle indicated, is obviously—once you get it—exactly twice the area of each of the squares on sides a and b.

Although we should be wary of misusing Kant’s technical terms, it might be not quite a coincidence that such visual mathematical proofs are often called “intuitive”.

Mathematical principles justify the application of mathematics to appearances, and deal with appearances in regard to their mere possibility. Another way to put this is that mathematical principles deal with the structure or form of experience, and dynamical principles deal with things (at least in terms of their existence as such).

Breaking things down further, the mathematical principles are those concerned with extensive and intensive magnitude, i.e., the axioms of intuition and anticipations of perception, respectively.

Axioms of Intuition

[The Transcendental Aesthetic is relevant to this, so re-read it]

The principle of these axioms:

All intuitions are extensive magnitudes.

This is the principle that serves as the basis for geometry—for the judgements of geometry. Kant calls the principle the “transcendental principle of the mathematics of appearances”. [B206]

Basically, the axioms of intuition are the axioms of geometry.

Anticipations of Perception

The principle of these anticipations:

In all appearances the real that is an object of sensation has intensive magnitude, i.e., a degree

This is a principle that governs all sensation. Intensive magnitude is the degree or intensity of sensations. Examples are density (or _specific gravity), luminescence, etc. Kant [B215] uses the former, and says that while the substance is extends equally in cases of different densities, what changes is intensive magnitude.

But Kant is not doing physics here; he is demonstrating the possibility of making judgements in these terms. In other words, he is demonstrating that degree or intensity is a fundamental dimension of the way that we judge in matters of measurement in physical science, one that is separate from extensive magnitude.

As he says…

Now to this presupposition, for which they could find no support in experience, and which is therefore purely metaphysical, I oppose a transcendental proof, which does not indeed explain the difference in the filling of spaces, but completely destroys the supposed necessity of the above presupposition, that the difference is only to be explained on the assumption of empty space. My proof has the merit at least of freeing the understanding, so that it is at liberty to think this difference in some other manner, should it be found that some other hypothesis is required for the explanation of the natural appearances. [B215]

Note: The Anticipations are principles of the category of Quality, but it turns out that they are based on a kind of quantity: intensive magnitude.

The Difference Between Extensive and Intensive Magnitudes

Is that extensive magnitude is synthesized successively, by adding parts one by one to form a whole, whereas intensive magnitude is perceived instantaneously.

Why “Anticipations”?

And without anything to sense there is sensation registering zero, such that we can “anticipate” sensation of just the same kind only more intense. That is to say, we can a priori know the “degreeness”—the property of having an intensive magnitude—because of the knowledge we have of the way our sensibilities are affected. But how can Kant say it is a priori? Because the real is a category necessarily coupled with the synthesis in empirical consciousness. Empirical consciousness, through empirical, is founded on a transcendental synthesis, namely the synthesis of apprehension, which Kant calls “the transcendental ground of the possibility of all knowledge in general” [Deduction A102], therefore what is known in empirical consciousness—such as that sensations have degrees—can be called a priori, especially in the absence of those sensations. This knowledge is an “anticipation” of forthcoming perceptions, or of future variations in the intensity of sensation (of whichever kind).

Kant explains it at the very end of the section [B217]:

The quality of sensation (e.g., colors, taste, etc.) is always merely empirical and cannot at all be presented a priori. But the real—as opposed to negation, = 0—that corresponds to sensation as such presents only something whose concept itself contains a being, and sugnifies nothing but the synthesis in an empirical consciousness as such. For empirical consciousness can in inner sense be raised from 0 to any higher degree, so that the same extensive magnitude of intuition (e.g., an illuminated surface) arouses as great a sensation as does an aggregate of much else (that is less illuminated) taken together. We can, therefore, abstract entirely from the extensive magnitude of appearance, and can yet present in mere sensation in one moment a synthesis of uniform ascent from 0 to the given empirical consciousness. Hence although all sensations, as such, are given only a posteriori, their property of having a degree can be cognized a priori.