Space (Transcendental Aesthetic, B37-B46)

Continental Philosophy / critique of pure reason February 14, 2024

In theTranscendental Aesthetic Kant develops a thesis that might be stated as follows:

  1. Our sensory experience is a precondition for any knowledge of the world.

Despite its foundational importance, this thesis might appear unilluminating, given that Descartes, Locke, Hume, and others made a similar claim.1 But in the chapter on Space in the Critique, we are introduced to a new way of considering space that transforms our understanding of the foundations of empirical reasoning. In Kant’s hands the thesis above could be read as transformed into the following expression:

  1. The transcendental concept of space is the precondition for any understanding of our experience whatsoever.

In what follows we will unpack what this new perspective amounts to, utilizing the textual resources given us in the chapter.

Contents:

  1. Toward a Transcendental Perspective
  2. A Priori [Synthetic]
  3. Synthetic [A Priori]
  4. What does Kant’s mean by the term “Transcendental”?
  5. Notes
  6. References

Toward a Transcendental Perspective

At the beginning of Section 3 (B40) titled “On the Transcendental Exposition of the Concept of Space,” Kant gives us a working definition of what he means by the term “transcendental.” Kant describes the transcendental in different ways, but in B40 he highlights the idea that if a concept is a transcendental one, it should serve as an originative source of knowledge. In his own words, it may then be considered to be

a principle from which the possibility of other a priori synthetic knowledge can be understood.”

B40

As will be seen, the concept of space fits this description perfectly. No derivative concept, such as measurement, or volume, or area, is as fundamental to our experience of the world as space itself. For Kant, who considered space transcendentally, this meant that it was the origin of a priori ideas about our experience in a way that was truly originative with respect to our understanding of the physical world. In a way unlike other authors before him, space became a form of forms, a primary concept.

Descartes’ deductive starting point was the cogito; Locke’s was our sense experience; and for writers such as Spinoza and Leibniz, their conception of God became the primary deductive principle of their systems. But for Kant, the true foundation of our understanding2 is the concept of space. Inasmuch as it originates the possibility of knowledge within a certain field of application, it establishes a domain within which our understanding may apply itself. That domain is the field of our sensible intuition.

This description resonates with an important point made in an earlier essay about the concept of innate ideas as developed in Locke’s discussion about them:3 innate ideas ought to be associated with highly general concepts so as to be fundamental to certain cognitive operations and not merely derivative.

Their level of generality applicability ought to be such that other kinds of knowledge follow from them inasmuch as, by their presence in the mind, they make such knowledge possible. Past examples include such a priori concepts as Plato‘s Forms, which, in the Meno, are held to make a knowledge of geometry possible. If the ideas we take to be innate were not originative in this sense, the door would be left open for the supposition that we might be able to get our first principles from our experience rather than some innate resource. Thus, innate ideas ought to provide a basis for our capacity to recognize propositions such as 2+2=4, 2+3=5, 2+4=6, etc., as true, rather than the mathematical propositions themselves.

A Priori [Synthetic]

For Kant there is no question of whether geometry is an example of a priori reasoning. Kant’s question is “How this is possible?” Because no a priori knowledge (strictly universal and necessarily true) of our experience is possible, experience cannot be the source of such knowledge. Therefore, by process of elimination, it must originate from within us. Their a priori aspect is rooted in the absolutely a priori character of space itself. Kant writes,

this intuition [of space] must be a priori, that is it must be found in us prior to any perception of an object, and must therefore be pure, not empirical intuition. For geometrical propositions are one and all apodeictic, that is, are bound up with the consciousness of their necessity; for instance, that space has only three dimensions.”

B41

Here he takes our everyday experience of the three dimensionality of objects in our outer intuition to be a universal characteristic of the way we perceive them.

Space is not merely one form among others; rather, it is an innate “form of forms” with respect to our sensibility:

the intuition [of space] has its seat in the subject only, as the formal character of the subject, in virtue of which, in being affected by objects, it obtains immediate representation, that is, intuition, of them; and only insofar, therefore, as it is merely the form of outer sense in general.”

B41

When he turns to consider the a priori origin of our idea of space within us, he does so by abstracting from all sensibility and all elements of our understanding (such as being substance or divisible). Kant tells us in the opening section of the Transcendental Aesthetic that all appearances have two components: (a) their matter, which “corresponds to sensation” and (b) their form, which “so determines the manifold of appearance that it allows of being ordered in a certain relations,” (i.e., so that form is a spatial organizing principle both for objects themselves and in relation to other objects).5 When all elements of sensibility or empirical form are abstracted away, all that is left to the mind to consider is a “pure” intuition of space as indefinite extension and figure.6

Space itself is clearly a presupposition of Geometry, inasmuch as Geometry attempts to deliver to us a knowledge of the forms space may take. Points, lines, and circles are all intelligible as determinations of space. But what exactly is the connection between our pure intuition of space (as lacking any object of the senses) and the pure intelligibility of geometrical propositions? The answer might be given by first considering the “purity” of our pure intuition of space. The “purity” of such an intuition indicates its greater, more immediate intelligibility. Kant writes,

reason is the faculty which supplies the principles of a priori knowledge. Pure reason is, therefore, that which contains the principles whereby we know anything absolutely a priori.”

Introduction, Section VII (B24)

The term “pure” flags that whatever is being considered is contemplated as something absolutely ideal, hence, as purely intelligible apart from any sensible qualities. The move to treat space as a pure intuition is a move toward treating it with respect to its pure intelligibility. Consider this with respect to the following proposition

  • All sensible appearances have the property of spatiality.

Having discovered space as a pure form (of sensibility) he concludes that space is the precondition for all representations of sensible forms. When he turns to consider sensible forms in their highest generality, the abstract forms of geometry, such forms are in the same originative position as space with respect to empirical objects taken directly from sensation.

It was said above that in the introductory section of the Transcendental Aesthetic Kant tells us that extension and figure remain to form our concept of space once anything sensible has been abstracted from it. It might be noticed that extension and figure are precisely the remaining “pure” forms required for our inner representation of space geometry. As pure forms, when applied as such to the construction of figures in space so as to form representations of finite spatial forms, they operate within a space of intelligibility that is removed from our immediate experience. Consider the following proposition as an example,

  • Whenever three lines are brought together to form a triangle, they enclose a space.

The truth of this proposition is both strictly universal and necessary. The intelligibility of Geometrical figures is derivative, but only with respect to pure forms of spatiality that determine the form that any sensible object must take.4

Synthetic [A Priori]

What remains yet to consider is the synthetic aspect of such propositions. In what way, for example, does a proposition from geometry involve our experience? Or again, if such propositions can be construed so as to involve our experience, how then can that form of experience lead us to a priori, necessary truth? This set of questions recalls again the background against which Kant was making his argument about the synthetic a priori. For the authors who followed Descartes in both the rationalist and empiricist traditions, the facts of experience could lead us to abstract ideas that could subsequently, as ideas, be the basis for necessary truths. But the facts of experience themselves could never be used as the basis for inferring strictly universal truths. The question then arises: how did Kant manage to merge synthetic with a priori truth?

The answer is one that requires some explanation of the experiential side of our capacity to inferentially connect non-analytic concepts. That is to say: how is the mind able to perceive a necessary relationship between any two concepts that are not related purely by the meaning of the terms involved? This question gains added importance when it is reflected that it asks us to get beyond language itself as providing the only possible structural principles for our experience. Let’s begin with an example and proceed to analyze it.

  1. Let’s start with a synthetic proposition taken from geometry, the claim that from a succession of points a line can be created.
  2. This proposition should be considered synthetic because from the mere definition, or even more broadly, the meaning of the term “point” there is no way to affirm its potential use in the construction of lines (although it might seem very obvious). If we were to suppose someone very new to geometry they would not be in a position to know this in advance if they had only been given the definition of a point.
  3. The leap from the notion of a point to the construction of a line can, however, be made by connecting a succession of points in time, in our inner or outer experience as the case may be.
  4. It may be reflected that the proposition that a succession of points can be used to generate a line is, moreover, universally true in the strict sense–no counterexample is even conceivable.
  5. Since the construction of the line occurs in experience (or in our imagined experience) and cannot be discovered strictly on the basis of the meaning of the term “point,” we have an example of a synthetic a priori truth.

Some readers might prefer a different example. One of Kant’s examples is the relationship between a triangle and the fact of its interior angles being equal to 180 degrees. Again, the important thing to keep in mind is whether the fact that its interior angles are equal to 180 degrees is something that can be known prior to any experience strictly on the basis of the meaning of the term “triangle.” In this case, our prior knowledge might depend upon the definition we have been given to start with. But suppose for the sake of argument that someone (again, someone very new to the field of geometry) was merely given “a three-sided figure” as a definition of a line. the question then becomes: can the leap be made from “three-sided” to “having angles equal to 180 degrees.” It would seem not: the notions of “180 degrees” or “interior angles” would not occur to a very recent beginner who is not already familiar with the terms and doesn’t already find them in the notion of three sidedness. The proof (for example, constructing a square and dividing it in half) would require going beyond the mere meaning of the term “triangle.” Furthermore, once the connection between being a triangle and having interior angles equal to 180 degrees has been made, it is impossible to see how it could be otherwise.

Thus, we have examples of synthetic a priori truths. They are true in virtue of the way our mind structures our experience, rather than the other way around. We, in effect, discover the truths of geometry as derivative consequences of the fact that we have space in us a priori as an originative concept in the sense discussed above.

What does Kant’s mean by the term “Transcendental”?

(The following is a discussion of the term “transcendental” as is appeared in a earlier version of the same article)

Next we come to the question of what is meant by the term “transcendental.” Consider what Kant writes at the beginning of Section 3 of the TA in relation to the remarks above about geometry:

“I understand by a transcendental exposition the explanation of a concept, as a principle from which the possibility of other a priori synthetic knowledge can be understood. For this purpose it is required (1) that such knowledge does really flow from the given concept, (2) that this knowledge is only possible only the assumption of a given mode of explaining the concept.”[B 40]

What Kant is after here is some “concept” (e.g. space) that stands outside, in a certain sense, the discursive processes of a particular branch of knowledge and that grounds whatever a priori claims to knowledge it uses as the basis for further knowledge. For example, one might reason (as Kant does) that any two sides of a triangle must be greater than any third side. This can be taken to be a synthetic a priori truth of geometry. It expands our knowledge and reveals a necessary and universal truth, but not by the simple analysis of concepts. But its a priori truth is based upon a prior “transcendental” truth, the necessity of space as a determining factor of the way objects represent themselves to us. It is not a truth based upon an assumed definition that determines the basis for further deductions as is the case with Euclid’s axioms, but upon the way things must be for us, inasmuch as spatiality must be taken to be a precondition for the representation of an object in space. Although we might attempt to define it, as a precondition for any representation of an object, it stands outside the interplay of experience and conceptual refinement and so is a priori in a special sense. In this sense, “space” transcends the study of geometry.

As a confirmation of this reading, consider what Kant says in the sentences that immediately follow the quotation above:

“Geometry is a science which determines the properties of space synthetically, and yet a priori. What, then, must be our representation of space, in order that such knowledge of it may be possible? It must in its origin be intuition; for from a mere concept no propositions can be obtained which go beyond the concept–as happens in geometry….Further, this intuition must be a priori, that is, it must be found in us prior to any perception of an object, and must therefore be pure, not empirical, intuition. For geometrical propositions are one and all apodeictic, that is, are bound up with the consciousness of their necessity; for instance, that space has only three dimensions. ”[B 40-41]

The final clause gives the most explicit sense of what Kant is after: our way of representing objects spatially is necessitated by the way in which we represent them to ourselves. That space (as opposed to time, an object of inner intuition for Kant) has only three dimensions is necessitated by the way in which we represent objects spatially, i.e., three dimensionally. This truth of our experience is not a concept developed out of our experience, so Kant argues, but a precondition for sensory experience itself and so can act as the basis for universal and necessary claims about the nature of experience in general. Furthermore, its transcendental aspect lies in its origin, our intuition of it as a basic reality, which precedes any attempt to define space itself axiomatically. Because of its immediate, nondiscursive intuitability, it has a claim to precede the science of geometry, which begins with axioms and definitions.

Notes

  1. In Meditation VI, where Descartes discusses the existence of material things, he writes of their clarity and liveliness as a way of distinguishing them from the ideas upon which we reflect on the way to showing that global skepticism of their reality is unjustified. Furthermore, he writes “I saw that the ideas that I formed were, for the most part, made up of elements of sensory ideas. This convinced me that I had nothing at all in my intellect that I had not previously had in sensation.” (A&T 74). Locke does something very similar in the first chapter of Book II of his Essay Concerning Human Understanding, as does Hume, who contrasted the “force and vivacity” of sensory impressions with that of our ideas about them in his Treatise on Human Nature. ↩︎
  2. God does appear later in the Critique as the first principle of reason, but not as an object of our understanding. Our Understanding can apply many forms to our thoughts about things, but requires the material of sensation in order to go beyond the merely imaginary. ↩︎
  3. Please see a discussion of Locke’s treatment of innate ideas in a prior article that discusses the topic: Kant’s Introduction to the Critique » Structure and Flux ↩︎
  4. See B34 for Kant’s forms of appearance. ↩︎

References

Tags:

You may also like...

Leave a Reply