The first of the four questions Kant sets himself in the preamble is "how is pure mathematics possible?" If math consists of synthetic a priori cognitions, we must be able to draw connections between different concepts by means of some form of pure intuition. The word translated as "intuition" is the German word Anschauung, meaning literally a point of view or way of seeing. For Kant, intuition connects the two distinct concepts that are joined in synthetic judgments. Kant distinguishes between empirical intuitions and pure intuitions. Empirical intuition is what we normally call sense perception: in the synthetic proposition, "my cat has brown fur," my sense experience, or empirical intuition, leads me to connect the concept of "my cat" with the concept of "has brown fur" (this is not Kant's example).
Since math consists of synthetic a priori cognitions, there must be some form of pure intuition innate within us that allows us to connect different concepts without reference to sense experience. Kant's answer is that space and time are not things in themselves, to be found in the world, but are what he calls the "form of sensibility": they are innate intuitions that shape the way we perceive the world. Prior to any sense experience, we have no concept of the objects we find in space and time, but we still have the concepts of space and time themselves. Geometry is the a priori study of our pure intuition of space, and numbers come from the successive moments of our pure intuition of time. If space and time were things in themselves that we could only understand by reference to experience, geometry and math would not have the a priori certainty that makes them so reliable.
Neither space nor time, nor the objects we perceive in space and time, are things in themselves: the objects we perceive are mere appearances of things in themselves, and space and time are empty forms that determine how things appear to us. If space were actual and not built into of our mental framework, two things with all the same properties would be in every way identical. However, Kant points out, our left and right hands have all the same properties, but they are not identical: a left hand glove will not fit on a right hand. This suggests that space is not independent of the mind that perceives it.
These conclusions lead Kant to three final remarks. First, he points out that we can have a priori certainty of geometry, and thus of our understanding of spatial relations, only because we have a pure intuition of space. Our certainty comes because we are only examining our own mental framework, and not things in the world. Second, he responds to the potential accusation that he is engaging in idealism. Idealism claims that there are no objects in the world, only minds, and that everything we see is just a construction of the mind. Though Kant has argued that we cannot perceive things in themselves, but only appearances of things, he still maintains that things in themselves, independent of our perception, exist, and that they are the source of what we do perceive. Third, he points out that appearances cannot be deceptive. I can misinterpret what I see, and be deceived in this way, but I cannot be mistaken about the appearances themselves. If space and time were things in themselves, then we could misinterpret our perception of them and be deceived regarding them. However, since they are mere appearances, they are a priori certain.
In the preface to the second edition of the Critique of Pure Reason, Kant claims his system has caused a "Copernican revolution in philosophy." The revolution he refers to is a reversal of our concept of space and time. Until Kant, it had been assumed that space and time were properties of the world, into which the objects of sensory experience were placed. Kant's radical reversal consists in claiming that space and time are not properties of the world but are rather properties of the perceiving mind. Space and time are like mental spreadsheets that organize how information is organized in our minds. Bertrand Russell explains this idea: "If you always wore blue spectacles, you would be sure of seeing everything blue . Similarly, since you always wear spatial spectacles in your mind, you are sure of always seeing everything in space."
Kant's argument for this position starts from the assumption that geometry and mathematics consist of synthetic a priori cognitions. To make synthetic judgments a priori, we must have some sort of pure intuition that allows us to draw concepts together without making any reference to experience. Geometry, for instance, gives us a priori knowledge about space, so our knowledge of space must be built into our minds. Therefore, Kant concludes, our concept of space is not something we learn from experience, but it is something we have prior to experience. Our concept of space is a feature of our minds and not a feature of reality. Kant believes he can make a similar argument about our concept of time with reference to our synthetic a priori knowledge of arithmetic.
If computers had existed in Kant's time, he would have had a useful metaphor for explaining the relationship between things in themselves, appearances, and our perceiving mind. We can compare things in themselves to data. Data in itself is invisible, and yet the programs we run are nothing more than data being interpreted. We can "read" data only once it's been through a processor and then projected onto a monitor. What we see on the monitor is not the data itself, but the "appearance" of the data. The processor and the monitor are like the pure intuitions of space and time: we cannot understand the thing in itself (data) until it has been made comprehensible by these intuitions. We do not perceive things in themselves but appearances of things. Our minds do not have the capacity to understand things in themselves just as we cannot understand data in itself by staring at a microchip.
Kant's discussion of space as an a priori form of intuition is meant to settle a debate between Newton and Leibniz regarding the nature of space. Newton maintains that space is absolute: it exists already as a thing in itself independently of the things that are in it. Leibniz holds to a relational theory of space, according to which space is a relational property that holds between objects. Space is not absolute, but dependent on the objects that are in it.
Both positions share the assumption that space is mind-independent. Though Leibniz does not, like Newton, believe that space is absolute, he believes that space only depends on the relations between objects and not on the minds that perceive space. Kant's example of how two internally identical hands cannot fit the same glove is meant to contradict Leibniz.
At the beginning of the twentieth century, with Einstein's theory of relativity, Kant's theory of space became the subject of heated controversy. According to Kant, our knowledge of space comes prior to experience, and Euclidean geometry can tell us everything we need to know about space. General relativity shows that the universe does not in fact conform to the laws of Euclidean geometry and that space and time are far more complicated than we think. Space and time, far from being pure intuitions that we can know a priori, are quite different from what our intuition tells us they are.
Kantians reply to this objection by saying that Kant is not talking about time and space in themselves but just about our cognition of time and space. Though space-time may be a four-dimensional curved space, our mind perceives space as flat, three-dimensional, and independent of time. Kant is not making a statement about how the world is, but about how the mind perceives the world.
The arguments for both sides are complicated, but over the course of the century, the defenders of Kant have diminished in number. The new physics of relativity and quantum mechanics have increasingly showed that any "intuitions" we might have about the nature or structure of reality are liable to be mistaken. Though Kantian constructions are arguably possible, this new physics makes a lot more sense if we assume that space-time exists independently of the mind.