1. Space is not an empirical concept, abstracted from outer experiences, for space is presupposed in referring sensations to something external, and external experience is only possible through the presentation of space.
2. Space is a necessary presentation a priori, which underlies all external perceptions; for we cannot imagine that there should be no space, although we can imagine that there should be nothing in space.
3. Space is not a discursive or general concept of the relations of things in general, for there is only one space, of which what we call “spaces” are parts, not instances.
4. Space is presented as an infinite given magnitude, which holds within itself all the parts of space; this relation is different from that of a concept to its instances, and therefore space is not a concept but an Anschauung.
The transcendental argument concerning space is derived from
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geometry. Kant holds that Euclidean geometry is known a priori, although it is synthetic, i.e., not deducible from logic alone. Geometrical proofs, he considers, depend upon the figures; we can see, for instance, that, given two intersecting straight lines at right angles to each other, only one straight line at right angles to both can be drawn through their point of intersection. This knowledge, he thinks, is not derived from experience. But the only way in which my intuition can anticipate what will be found in the object is if it contains only the form of my sensibility, antedating in my subjectivity all the actual impressions. The objects of sense must obey geometry, because geometry is concerned with our ways of perceiving, and therefore we cannot perceive otherwise. This explains why geometry, though synthetic, is a priori and apodeictic.
The arguments with regard to time are essentially the same, except that arithmetic replaces geometry with the contention that counting takes time.
Let us now examine these arguments one by one.
The first of the metaphysical arguments concerning space says: “Space is not an empirical concept abstracted from external experiences. For in order that certain sensations may be referred to something outside me [i.e., to something in a different position in space from that in which I find myself], and further in order that I may be able to perceive them as outside and beside each other, and thus as not merely different, but in different places, the presentation of space must already give the foundation [zum Grunde liegen].” Therefore external experience is only possible through the presentation of space.
The phrase “outside me [i.e., in a different place from that in which I find myself]” is a difficult one. As a thing-in-itself, I am not anywhere, and nothing is spatially outside me; it is only my body as a phenomenon that can be meant. Thus all that is really involved is what comes in the second part of the sentence, namely that I perceive different objects as in different places. The image which arises in one’s mind is that of a cloak-room attendant who hangs different coats on different pegs; the pegs must already exist, but the attendant’s subjectivity arranges the coats.
There is here, as throughout Kant’s theory of the subjectivity of space and time, a difficulty which he seems to have never felt. What induces me to arrange objects of perception as I do rather than other-
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wise? Why, for instance, do I always see people’s eyes above their mouths and not below them? According to Kant, the eyes and the mouth exist as things in themselves, and cause my separate percepts, but nothing in them corresponds to the spatial arrangement that exists in my perception. Contrast with this the physical theory of colours. We do not suppose that in matter there are colours in the sense in which our percepts have colours, but we do think that different colours correspond to different wave-lengths. Since waves, however, involve space and time, there cannot, for Kant, be waves in the causes of our percepts. If, on the other hand, the space and time of our percepts have counterparts in the world of matter, as physics assumes, then geometry is applicable to these counterparts, and Kant’s arguments fail. Kant holds that the mind orders the raw material of sensation, but never thinks it necessary to say why it orders it as it does and not otherwise.
In regard to time this difficulty is even greater, because of the intrusion of causality. I perceive the lightning before I perceive the thunder; a thing-in-itself A caused my perception of lightning, and another thing-in-itself B caused my perception of thunder, but A was not earlier than B, since time exists only in the relations of percepts. Why, then, do the two timeless things A and B produce effects at different times? This must be wholly arbitrary if Kant is right, and there must be no relation betwen A and B corresponding to the fact that the percept caused by A is earlier than that caused by B.
The second metaphysical argument maintains that it is possible to imagine nothing in space, but impossible to imagine no space. It seems to me that no serious argument can be based upon what we can or cannot imagine; but I should emphatically deny that we can imagine space with nothing in it. You can imagine looking at the sky on a dark cloudy night, but then you yourself are in space, and you imagine the clouds that you cannot see. Kant’s space is absolute, like Newton’s, and not merely a system of relations. But I do not see how absolute empty space can be imagined.
The third metaphysical argument says: “Space is not a discursive, or, as is said, general concept of the relations of things in general, but a pure intuition. For, in the first place, we can only imagine [sich vorstellen] one single space, and if we speak of ‘spaces’ we mean only parts of one and the same unique space. And these parts cannot precede the whole as its parts . . . but can only be thought as in it. It
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[space] is essentially unique, the manifold in it rests solely on limitations.” From this it is concluded that space is an a priori intuition.
The gist of this argument is the denial of plurality in space itself. What we call “spaces” are neither instances of a general concept “a space,” nor parts of an aggregate. I do not know quite what, according to Kant, their logical status is, but in any case they are logically subsequent to space. To those who take, as practically all moderns do, a relational view of space, this argument becomes incapable of being stated, since neither “space” nor “spaces” can survive as a substantive.
The fourth metaphysical argument is chiefly concerned to prove that space is an intuition, not a concept. Its premiss is “space is imagined [for presented, vorgestellt] as an infinite given magnitude.” This is the view of a person living in a flat country, like that of Königsberg; I do not see how an inhabitant of an Alpine valley could adopt it. It is difficult to see how anything infinite can be “given.” I should have thought it obvious that the part of space that is given is that which is peopled by objects of perception, and that for other parts we have only a feeling of possibility of motion. And if so vulgar an argument may be intruded, modern astronomers maintain that space is in fact not infinite, but goes round and round, like the surface of the globe.
The transcendental (or epistemological) argument, which is best stated in the Prolegomena, is more definite than the metaphysical arguments, and is also more definitely refutable. “Geometry,” as we now know, is a name covering two different studies. On the one hand, there is pure geometry, which deduces consequences from axioms, without inquiring whether the axioms are “true”; this contains nothing that does not follow from logic, and is not “synthetic,” and has no need of figures such as are used in geometrical text-books. On the other hand, there is geometry as a branch of physics, as it appears, for example, in the general theory of relativity; this is an empirical science, in which the axioms are inferred from measurements, and are found to differ from Euclid’s. Thus of the two kinds of geometry one is a priori but not synthetic, while the other is synthetic but not a priori. This disposes of the transcendental argument.
Let us now try to consider the questions raised by Kant as regards space in a more general way. If we adopt the view, which is taken for granted in physics, that our percepts have external causes which
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are (in some sense) material, we are led to the conclusion that all the actual qualities in percepts are different from those in their unperceived causes, but that there is a certain structural similarity between the system of percepts and the system of their causes. There is, for example, a correlation between colours (as perceived) and wavelengths (as inferred by physicists). Similarly there must be a correlation between space as an ingredient in percepts and space as an ingredient in the system of unperceived causes of percepts. All this rests upon the maxim “same cause, same effect,” with its obverse, “different effects, different causes.” Thus, e.g., when a visual percept A appears to the left of a visual percept B, we shall suppose that there is some corresponding relation between the
