How does the argument from geometry work?

Kant’s argument from geometry is situated in the Transcendental Aesthetic under the Transcendental Exposition of the Concept of Space. Kant, having established that our concept of space is a pure intuition in the Metaphysical Exposition of the Concept of Space, aims to answer the question, how is our synthetic a priori knowledge, specifically our knowledge of mathematical geometry, possible? The question can be further articulated as, what is the enabling condition for us to obtain our knowledge of geometry? How do we come to formulate our knowledge of geometry given that our concept of the space is a priori and intuitive? How is our knowledge of geometry generated from our concept of space?

Kant addresses these questions in the Transcendental Exposition of the Concept of Space, aiming to prove that our concept of space as pure intuition is the only possibility for us to obtain knowledge of geometry. However, in that account, our concept of space is merely a necessary condition for our knowledge of geometry. We also need a demonstration of our knowledge of  geometry is indeed constructed via our intuition of space. In the Discipline of Pure Reason in Dogmatic Use in the Doctrine of Method, Kant supplements this argument with an analysis of  mathematical methods. While the first account provides an analytical reasoning of the necessary condition for the possibility of geometrical knowledge, the second account offers a synthetic demonstration of mathematical methods for the construction of geometrical knowledge. I present both accounts mentioned above to show that Kant’s answer to the question, how is synthetic a priori knowledge derived from our concept of space, is a detailed account of space being both a  pure intuition and the pure form of sensibility that orders our perceptions of geometrical and outer bodies.

At the beginning of the Transcendental Exposition, Kant states,

“I understand by transcendental exposition the explanation of a concept as a principle from which insight into the possibility of other synthetic a priori cognitions can be gained. For this aim it is required 1. That such cognitions actually flow from the given concept, and 2. That these cognitions are only possible under the presupposition of a given way of explaining this concept.”1

In other words, to demonstrate that our concept of space gives rise to the possibility of our  synthetic a priori knowledge of geometry, Kant needs to prove that 1. our knowledge of geometry is based on our concept of space and that our concept of space being pure and intuitive is the necessary condition for our knowledge of geometry; 2. our knowledge of geometry is only possible if our concept of space is the pure form of sensibility that orders our sensations and  enables us to cognize outer objects.

Kant’s proof of the first claim is straightforward. He begins with the assumption that our  knowledge of geometry is synthetic a priori, and he analytically derives that our concept of  space has to be a priori and an intuition. That is, our concept of space as a pure intuition is the sole possibility for us to obtain such synthetic a priori knowledge of geometry. Kant defines geometry as “a science that determines the properties of space synthetically and yet a priori.”2 Here, Kant first observes that geometry is a study of the properties of space and therefore our  concept of space applies in our derivation of geometrical knowledge. He adopts the common  understanding of geometry from Wolff that geometry concerns about spatial magnitude and also “bodily objects about their length, breadth, and width.” Second, Kant argues that given  geometrical knowledge is synthetic,3 it must be drawn from intuitions, for concepts do not  provide knowledge that “go beyond the concept.”4 Third, since geometrical knowledge is apodictic and carries necessity, it must be derived from a priori, non-empirical knowledge, or otherwise our knowledge of geometry would be dependent upon experiences. Concluding the  results above, we have come to establish that we must have the concept of space as a priori and  an intuition for the possibility of our synthetic a priori geometrical knowledge. That is, the only  possible explanation for our having knowledge of geometry is that we have the concept of space as pure intuition.

However, the above account only indicates that our concept of space is the necessary condition for our knowledge of geometry. Does our knowledge of geometry actually “flow from” our  concept of space? How is our knowledge of geometry come from our a priori intuitive concept  of space? Kant provides his answer through his analysis of mathematical methods. He states, “mathematical cognition [is] from the construction of concepts.”5 He explains that to construct a  concept “means to exhibit a priori the intuition corresponding to it.”6 What he means by this is that geometrical reasoning of any concept must presume our cognition of a concrete object. In  other words, Kant believes that to cognize a geometrical concept, we must be able to mentally represent the object first. This can be best explained in the example of proving all triangles have  inner angle sum of 180 degrees. In the proof of this proposition, the geometer first mentally  constructs a triangle. He then proceeds by extending the lines of the three sides of the triangle and employing the relation between parallel lines and the corresponding angles to demonstrate  that three angles together can be presented on a straight line. The geometer consequently concludes that the inner angle sum of the triangle is equal to 180 degrees. During the process of  this reasoning, no empirical data or method is used. Instead, the process is wholly guided by  intuition in three senses. First, the geometer’s mental construction of the triangle is necessarily  embedded in the space, which is already shown to be a pure intuition. Second, the propositions  that the geometer uses, is “a chain of inferences that is always guided by intuition.”7 Finally, the  geometer utilizes singular entities, such as parameters of the angles, to conduct his argument. Our intuition of space plays an essential role in this geometrical proof. Similarly, other geometrical proofs are also derived from our concept of space as pure intuition and our intuitive inferences of spatial relations.

The two explanations above – space as pure intuition being the necessary condition for the possibility of our knowledge of geometry, and our geometrical knowledge is established via our intuition of space – have proved Kant’s first claim, that is, our knowledge of geometry flows  from our concept of space. Now, we proceed to examine Kant’s second claim, that our concept of space is the pure form of sensibility that a priori orders our cognitions of outer bodies.

Kant presents the following question in the Transcendental Exposition: “now how can an outer  intuition inhabit the mind that precedes the objects themselves, and in which the concept of the  latter can be determined a priori?”8 Here, the paradox can be formulated as, given that our  knowledge of geometry relies on our intuition of space in concreto, but also that our cognition of  geometrical objects are in turn determined by our knowledge of geometry, how is this possible? Kant believes that there is only one possibility for this to happen, that is, space is a subjective,  pure form of sensibility that orders our sensations and structures our cognitions of outer bodies. Space is the “form of outer sense in general” that acquires us immediate representation, i.e. intuition of outer bodies. This argument proves Kant’s second claim, that our knowledge of  geometry is only possible if our concept of space is the pure form of sensibility that orders our sensations and enables us to cognize outer objects.

Space insofar has been established as: 1. a pure intuition, and 2. the pure form of sensibility. As a pure intuition, it provides the foundation for our knowledge of geometry. As the pure form of sensibility, it orders our knowledge of geometry. These two roles of space give rise to our synthetic a priori knowledge of geometry, as well as outer spatial objects. Our geometrical knowledge is possible exactly because the two roles space plays in our cognition.

Kant’s purpose of presenting the argument from geometry in the Transcendental Exposition has stirred debates among commentators. Some argue that the argument from geometry aims to prove that space is a pure intuition while others contend that it supplements Kant’s transcendental idealism by demonstrating space as a pure form of sensibility through the account of geometrical knowledge. The argument of geometry, in my explanation above, provides a bridge between the claim that space is a pure intuition and the claim that space is the pure form of sensibility. It does so by demonstrating the role of space in our knowledge of geometry and  exhibiting the applicability of geometrical knowledge in our cognitions of outer bodies. Therefore, the argument of geometry offers a strong account for Kant’s transcendental idealism.

Footnote:
1 B40.
2 B41.
3 Geometrical knowledge is synthetic because merely thinking about geometrical concepts alone cannot  give us knowledge provided by geometrical propositions such as “the distance of a straight line between two points is the shortest.” Geometrical propositions cannot be derived alone from concepts. (B16)
4 B41.
5 B741.
6 Ibid.
7 B745.
8 B41.