Immanuel Kant, easily the most influential modern philosopher, used his proof of synthetic a priori judgments to form the foundation of three areas of science: mathematics, natural science, and metaphysics. With transcendental idealism, one can form a priori methodologies for each of these areas to make them true pure sciences. The innovation of these synthetic a priori judgments refutes the claims of skeptics such as Hume that to have true knowledge of metaphysics is impossible, and eliminates the need for dogmatism. With Kant’s philosophy, not only can we reason about the nature of the universe and the relationship between the real and the perceivable worlds, but these metaphysical judgments also constitute a science.

Pure mathematics is the first science Kant attempts to prove is possible. When we think about how we perform a mathematical operation, such as 645 * 32, we realize that this type of mathematical concept is not true by definition, but requires reason and analysis of experience, and thus they must be synthetic concepts. However, mathematical principles such as x+0=x are also necessarily true and therefore a priori truths. One of the first hurdles Kant must overcome, then, is how math can be deduced a priori, without any previous knowledge or experience. The answer to this dilemma is that mathematics is based on principles that are gained through pure intuition instead of empiricism. Whereas empiricism is the a posteriori awareness of external objects via sense perception, pure intuition is the a priori visualization of pure forms in one’s mind. This pure intuition does not require experience in order to function. How, then, can we imagine something we have never seen? The answer is that intuition does not represent things as they are in the real world, but only the form of sensibility of real-world objects. Thus mathematics is possible through the intuition, the structuring of sensibility.

Now that we know pure mathematics exists, we can attempt to elucidate how it is possible. Through our awareness, we perceive an object. This object is then structured in time and space by our faculty of sensibility; through experience these perceptions are formed into an understanding of the object. Thus time and space are not a result of experience; they are prior to experience. They are simply how the faculty of sensibility works to structure perceptions. The bases of empirical intuitions of the real world objects that are necessary for mathematics are pure intuitions of space and time. For example, our sensibility imposes the intuition of space on the objects around us. Then, by making synthetic judgments on that space, we form the concepts of geometry. These principles of geometry are necessarily true, because they can never be proved false by experience. Thus pure geometry must exist and must consist of a priori synthetic truths. Just as geometry is based on synthetic judgments on space, arithmetic is based on synthetic judgments of time. In order for one number to be “after” another, this concept of time must exist, and through the sequentiality of numbers, one can deduce operations and truths concerning these operations. Hence, as geometry and arithmetic use synthetic judgments on space and time, all areas of mathematics similarly impose these intuitions on the objects around us to form the concepts of mathematics. For similar reasons, pure natural science, or physics, must also be possible. We know that pure science exists because there are universal laws, such as “substance is permanent” and “every event is determined by a cause according to constant laws” (Kolak, 652). These laws must not be a posteriori, because experience can only teach us what exists and how it exists, but not that it must exist. Neither are they a priori, for we must make our deductions from observations. However, the conformity of experience to constant laws must be an a priori understanding.

This understanding is the result of both judgments of experience, which are always valid and are based on a priori concepts of the understanding, and judgments of perception, which are subjectively valid and are based on simple observation. When a judgment of perception, or an intuition, is subsumed under this a priori pure understanding, it becomes a judgment of experience that is then objectively true and universal. This pure understanding consists of twelve concepts of understanding, called categories, which are a priori concepts derived from logical judgments. Thus, through our awareness we have perceptions. Then our sensibility, using the concepts of pure understanding, structures these perceptions into experiences which we use to form science. This process is called the schematism of pure understanding, where schemata are notions of objects categorized and structured in time. The categories can only subsume schemata, and not awarenesses.

This system of natural science solves Hume’s problem of cause and effect. Instead of universal concepts being derived from experience using the innate human tendency to classify according to cause and effect, experience is derived from perceptions as they are subsumed under the pure universal concepts, such as cause and effect. We still do not know, however, if these principles describe the actual world, or only our perceived world. However, because our experience is derived from pure universal concepts, which are only valid when applied to perceptions, the principles of science belong to the phenomenal world of appearances. Thus Kant has shown that through the pure concepts of understanding, we can derive principles of natural science concerning the phenomenal world.

Because science and mathematics seem more concrete and objective than metaphysics, it is easier to accept the process through which they are possible. While math is possible through its inherent nature, and physics through experience and pure understanding, metaphysics is based on pure rational concepts, which has no basis in experience. Thus metaphysical truths cannot be deduced through observation, but only by reasoning. Just as the categories of understanding allow one to deduce universal truths from experience, so do necessary concepts of the faculty of reason allow one to deduce universal truths from the understanding. These two classes of synthetic a priori judgments are entirely separate: the one based on experience, and the other never able to be proved or disproved through experience.

The origin of ideas, or pure rational concepts, must be in the faculty of reason, even as the origin of the categories is in the understanding. The transcendental ideas -- psychological, cosmological, and theological -- are founded on the three syllogisms of reason: categorical, hypothetical, and disjunctive. These three ideas are the basis of all possible metaphysical statements, or all the claims of pure reason. Because these ideas belong to reason, and metaphysics is simply the application of these ideas to concepts of reason, our faculty of reason must be able to examine its own processes and deduce true facts about them, which is metaphysics. But why does metaphysics exist? Our minds are not comfortable with simply observing the sensuous world and its connections through universal laws; it requires some knowledge of things in themselves to be content. The three concepts of reason provide the mind with concepts beyond the sensual world - the soul, the real world, and God - and with these assumptions (not cognitions) reason is more satisfied.

Metaphysics may seem to be impossible because often two people may both reason correctly, and yet both arrive at contradictory conclusions to metaphysical questions. This transcendental illusion is the result of applying the understanding and sensibility beyond their limits. Although the objective rules may be the same in each case, the subjective idea of causal connection can lead to different deductions. This dialectic of pure reason can only be avoided by limiting our conclusions to the transcendental world. Therefore, metaphysics as it applies to the transcendental world must exist through the application of pure rational concepts to understood objects.

Not only must metaphysics exist, but it also must exist as a science. To be a science, metaphysics must be more than mere theory; it must contain certain principles of the three transcendental ideas. Unlike other sciences, metaphysics is complete. Because its concepts are held within reason instead of observation, no new events or discoveries will increase its scope. Kant claims that before his system, metaphysics was not a science, for each new philosopher simply proposed new theories instead of proving new concepts a priori. Metaphysics also often used probability and common sense, neither of which is valid. A basis of probability means that the principle is not necessarily true, and can only be shown to be true by induction. Kant has proven this method to be unreliable. Common sense is a subjective basis, and therefore not universally valid or necessarily true. Because we cannot visualize concepts of metaphysics a priori without experience, and whenever we deduce a metaphysical claim we must have a logical justification, metaphysical claims constitute a science like mathematics or physics.

The possibility of synthetic a priori judgments is the basis for each of Kant’s answers to how mathematics, natural science, and metaphysics are possible. Experience combined with each science’s a priori concepts allows one to make conclusions in each area. Mathematics has the a priori concepts of time and space that allow the understanding to combine observations of shape and quantity into the science of math. Natural science has the twelve categories that subsume experience in order to form laws of nature. Metaphysics has the three transcendental ideas through which reason structures understanding into general metaphysical principles, the justification of which require that metaphysics be a science and not mere speculation. These three sciences, with their common thread of synthetic a priori ideas, allow us to refute skepticism and to make true statements about the phenomenal world.

Kolak, Daniel. The Mayfield Anthology of Western Philosophy. Mountain View: Mayfield

Publishing Company, 1998.