Kant’s main project in the Prolegomena to Any Future Metaphysics is to find the source of metaphysical cognitions in order to understand metaphysics and to determine whether metaphysics can offer us any knowledge. In dealing with cognitions, Kant must make distinctions between judgments of form and content to precisely characterize cognitions of metaphysics.

As such, distinctions are necessary in order to properly understand Kantian metaphysics, I will highlight some of the differences in kinds of knowledge. I will then offer my own analysis of Kant’s conception of mathematical knowledge and its relation to space.

The first distinction Kant makes is between forms of a priori and a posteriori knowledge. Although a priori literally means “before experience,” it is more precisely defined as knowledge, which is not derived from experience. Examples of a priori are found throughout the field of mathematics, such as 2+2=4. You cannot prove that 2+2=4 through experience, but it is nevertheless logically true.

A posteriori knowledge, on the other hand, is knowledge that is derived from experience. Empirical science deals mainly with a posteriori knowledge. Knowing the Earth is round, for example, can only be known through experience and therefore will be categorized under a posteriori knowledge.

Kant also makes a distinction in content of different kinds of knowledge, the first of which is analytic and the second synthetic. Analytic knowledge is merely explicative, as Kant puts it, or rather, its truth depends wholly on its predicate.

Kant gives the example, “Gold is a yellow metal’; this is analytically true because the concept of yellow metal is necessarily contained in the predicate, the concept of gold. Indeed, the only method to determine whether a priori knowledge is true or not is by the law of contradiction, that is, if it were logically impossible.

The statement, for instance, “All bachelors are married” contradicts the concept of a bachelor, which is defined as an unmarried man; because no one can be married and unmarried at the same time, the statement is thus false. Moreover, analytical truths are a priori, because analysis of a concept requires no reference to experience. If knowledge is not analytic, it must be synthetic. Synthetic knowledge is ampliative, that is, it augments knowledge of a given predicate that could have begotten through mere analysis of the predicate. Copleston gives the example, “All members of tribe X are short.”

Because it is impossible to analyze membership of tribe X and necessarily elicit the concept of shortness, the concept of shortness is being added to the concept of membership and is therefore synthetic. In fact, all a posteriori knowledge, such as Copleston’s proposition, must be synthetic, because the other kind of knowledge, analytical knowledge, is not grounded in experience and there must be some knowledge that captures experience.

The type of knowledge Kant associates with metaphysics is synthetic a priori knowledge. Kant believes synthetic a priori knowledge connects the predicate and the subject not by mere analysis but by the necessary and universal conditions for human understanding.

Mathematics is one form of synthetic a priori knowledge. Arithmetical judgments are synthetic, as the famous example goes, because under examination of “7+5=12,” 12 is not obtained through mere analysis of “7,” “+,” and “5.” Rather I depend on the help of intuition outside of the aforementioned concepts to arrive at this equation. Essentially, I am augmenting the concept of “7+5” by attaching to it that it is equivalent to “12,” and thus arithmetical judgments must be synthetic.

Geometry, too, is the synthesis of at least two concepts to induce geometrical principles. While Kant admits there are some geometric concepts that are analytic, they only act as identical propositions; on their own, they could never construct a geometrical principle, like Pythagoras’ Theorem. It is important to note that by geometry Kant is referring only to Euclidean geometry.

What ensures mathematics apodeictical truth is its transcendental deduction from the pure intuitions of space and time, which are necessarily and universally required to understand the world. After all, our pure intuition of space motivates the creation of geometry and arithmetic is compelled by adding successive units of time.

Without its inherent connection to space and time, mathematics would never be understood. Indeed, anything that the mind does not put in context of space and time is outside of the realm of human understanding. Kantian metaphysics is grounded on the idea that the mind must order the world in a certain way to understand it. As such, the ordered understanding of the world is made possible by synthetic a priori knowledge.

In Kant’s time there was only one kind of geometry, Euclidean; therefore, when Kant implied that geometry was deduced from pure intuitions of space, his audience understood only one form of geometrical representation of space, Euclidean space. However, after Kant’s death, mathematicians discovered new forms of geometry.

Indeed, using the same logical architecture, except for one principle, which Euclid himself was never able to prove, mathematicians found that there were, in fact, infinite many non-Euclidean geometries. Where one space was possible before, infinite spaces are possible now. The important question is which of these spaces we understand when we view the world.

No a priori knowledge can differentiate between equally feasible and logical geometries, we must experience space to determine its geometry. Whether our geometrical understanding of the world is universal is also at stake, because, if there are many different logically possible geometries, why must we understand the same one, when there is no tool to differentiate between these geometries?

Moreover, if there were one geometry that was universally taken to be space, why would it be this geometry and not any of the others? It would have to be completely arbitrary. Therefore, our geometrical conceptualization of space may be neither a priori nor universal. Thus the geometrical representation of space is not necessarily synthetic a priori.

Kantian metaphysics is grounded in necessary and universal synthetic a priori knowledge, including geometry. If geometry were not universally conceived in a certain, there will necessarily be different understanding of space, that is, not in the form of different pure intuitions, rather apply different ordering principles to the world to understand it.

Thus, Kantian metaphysics may not be entirely transcendental. If this holds the least bit of credibility, Kant’s transcendental idealism, which is entirely dependent on the universality of synthetic a priori concepts, collapses.