This post will interpret what Kant means by ‘intuition’ and then explain his singularity argument for space’s being a pure intuition.
Intuition is an immediate relation between a mode of knowledge and an object, to which all thought is directed (A19). By ‘immediate’ we mean that intuitions do not relate to by means of some other thing (e.g. another representation, conceptual or otherwise). Rather, if I am intuiting some object , my thought is immediately directed to the -ness of ; my mind is directly aware of . We intuit an object only insofar as the object is ‘given to us; an object cannot be given to us unless our mind is affect in the right way (A19). We are given objects via sensibility (B34). This means only sensibility can give rise to intuitions, for an intuition is a direct relation to an object and objects are only given through sensibility. There is a post before you: you stand in an immediate awareness relation to it: you are intuiting an object, namely my Kant post.
Intuition is to be distinguished from concept. My concept of refers to (s) indirectly — that is, ‘mediately by means of a feature which several things may have in common’ (B377). Call it the generality criterion. My concept of an elf refers to Galadriel (and Legolas and Haldir) by means of feature(s) they share, e.g. pointy ears or forest-frolicking, (and not by my representation of the totality of elves). So I can indirectly refer to Galadriel and Haldir by considering and representing the concept of pointy eared forest-frolickers.
This means that intuitions must be singular representations, in the sense that they always present a particular, single object. Why? Because in intuition the relation between knowledge and object is direct unmediated. If your relation were indirect, then you would be representing features that a set of objects share, not that particular object as it appears to your sense organs (and so this could not be called a singular representation). Your intuition of an object is brought about from a particular object affecting your sense organs, consequently it must be a representation of that object producing those sensations — that is to say, it must be a singular representation. Call this criterion for intuition ‘the singularity criterion’.
There is always an object of thought — something to which the thought is directed (A19?). This means that all thought must, directly or indirectly, relate to intuitions (B34). For objects are given to us only through sensibility, and sensibility alone produces intuitions. So if a thought could not relate, in some way, to an intuition, then there could be no object of the thought — and a contentless thought is no more a thought than a blank square of glossy paper is a photograph.
An object affecting our faculty of representation is a sensation (B34). An intuition is empirical iff relates to an object of sensation. (The [undetermined] object of an empirical intuition is an appearance [B34].) So in experience there is an appearance which affects my sense faculties, producing sensations of, e.g. browness, bitterness, warmth. (A cup of coffee?)
That quality of an appearance which allows its (that is, the appearance or sensations) being ordered in a certain way is the form of appearance (B34). Sensation is not the form/arrangement/ordering of sensation — to say that sensations are ordered is not to say that sensations are an ordering or arrangement. Because the form of appearance is divorced from the sensation of the appearance, the form of appearances must given to us a priori. For objects affect our sense-faculties and are given through sensibility, giving rise to sensation, but sensations are not orderings — only effects of objects on our mind. So the ordering cannot come from experience of objects — it cannot be a posteriori — and so it must be a priori, in the mind (B34).
A pure or a priori intuition is an intuition-sans-sensation. (Consequently, pure intuitions are present even in the absence of all appearances.) That is, a pure intuition contains nothing (or rather, is of nothing) but the form of sensibility. The form of sensibility is a feature of our minds which determines the manner in which we necessarily must represent things. It gives rise to two pure intuitions, (1) the representation of space and (2) the representation of time. We say representation of space/time because pure intuitions, as containing nothing but the form of sensibility, are mind-dependent and not mind-independent features of the world.
We’re now in a position to explain Kant’s singularity argument for space’s being a pure intuition, as opposed to a general concept (e.g. a general concept of the particular spatial relations of things). Recall that intuitions are singular representations: they are not representations of features which a set of objects might share; they are always representations of singular/particular objects. Kant’s aim is to show that space is a particular representation which contains nothing but the form of sensibility.
Kant’s first premise is that ‘we can represent to ourselves only one space’. By ‘space’ here, Kant has in mind the representation of the single all-embracing space. That is, the space in which all our intuitions of outer sense take place or are seemingly represented. (If Kant did not mean the ‘all-embracing space’ then this first premise would not make sense, because we represent particular objects in particular spatial locations and arrangements, and so there would in fact be multiple spaces we represent, contrary to his opening premise.)
Because we can represent to ourselves only one space, the representation of space must be an intuition. Why? If we can only represent one all-embracing space, then our representation of space can only be of one, single thing. This means that it is a representation of a particular thing. Representations of particular things are always intuitions, by the singularity criterion. Our representation of space is not a conceptual representation of general features of space that all spaces or spatial things share, rather it is the particular manifold upon which my sensations of particular objects take place (B34).
Prima facie, this looks like a misapplication of the singularity criterion. Consider our representation of God (a maximally perfect being). God is certainly not an empirical intuition (nor is She an a priori intuition). Rather, our representation of God is formed from conceptual representations finitely and relatively perfect beings (presumably we relate these conceptual representations in such a way as to form a concept bearing maximal perfection as a feature. In a similar way, you might think that our representation of all-embracing space is formed from the conceptual representation of a finite space (or spaces). That is, the concept of our all-embracing space is formed from considering and representing the aggregate of all objects falling under the general concept of space, in order create a sort of infinite ‘all-embracing’ space (B40).
But the subsequent portion of the singularity argument aims to overcome this objection. A part of space cannot be prior to the singular, all-embracing space. Therefore, parts of space are not constituents composing the all-embracing space, but rather we can only think of parts of space as being in the all-embracing space (A25). Consequently, we cannot build a general concept of space out of the concept of a space. Why is the all-embracing space prior to a part of space? For Kant, space is ‘essentially one’ (A25). All things in space (parts of space, empirical intuitions, etc…) depend on placing ‘limitations’ on space. To bring this out, consider your representation of a particular region of space. In representing this, you must also represent it as having bounds. In representing it as having bounds, you represent it as being surrounded by . The only plausible candidate for is the all-embracing space. Therefore, the representation of all-embracing space is necessary for any particular representations within its manifold. This entails that the intuition of all-embracing space is a condition on our representing particular spaces (and things in them).
So space must be an a priori intuition, for two reasons. (1) Our representation of all-embracing space is singular, so it must be an intuition. (2) Our representation of space is the condition on our having empirical intuitions, but the representation of space itself does not depend on any particular mind-independent objects and as such cannot be empirical. Therefore, it is a priori.