A Brief Defense of the Kantian Conception of Space as the a
Priori Form of Outer Intuition
At one time, the certainty of Euclid's propositions, and their universality, were thought to prove that Kant had been right to assume that space was a quality which we imposed on our perceptions. Since geometry was so rigorous, it was held up as the perfect example of a priori knowledge; that is, knowledge which requires nothing of experience to prove or obtain. No one (well, no one sensible) denied that experience occasioned the discovery of this knowledge, whether that experience was that of the ancient Greeks measuring the earth, or the axiom-dulled student staring at chalked diagrams in the classroom.
Space was a counter-part to all this geometric certainty. Geometry, after all, was how we measured it. "...[I]n order that certain sensations be referred to something outside me...and similarly that I may represent them as outside and alongside one another...the representation of space must be pre-supposed" Kant states in his Critique of Pure Reason. Geometric knowledge and the a priori quality of space were intertwined, and thought to be perfect.
All this changed with the advent of non-Euclidean (then anti-Euclidean or pangeometry or Absolute) geometry. Geometers had long despised Euclid's fifth Postulate, known as the Parallel Postulate, which states that
if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than two right angles.
It needed proof, they said, which was true, but no one had yet succeeded. Proclus, Nasir-Eddin, Vitale, and Wallis tried and failed. Then came the non-Euclideans.
The Parallel Postulate was unnecessary, they claimed. And as they experimented in a geometry without it, they found new forms of space. In non-Euclidean geometry, space can be curved. Now, no one had every thought of space itself as having properties; it was just there so that things could be alongside one another, as time was just there so that things could change. This discovery was a revolution. As in many revolutions, the baby was thrown out with the bath-water.
[T]herefore space is only a special case of a three-fold extensive magnitude. From this, however, it follows of necessity, that the propositions of geometry cannot be deduced from general magnitude-ideas, but that those peculiarities through which space distinguishes itself from other thinkable threefold extended magnitudes can only be gotten of experience. [italics mine] Hence arises the problem, to find the simplest facts from which the metrical relations of space are determinable--a problem which from the nature of the thing is not fully determinate; for there may be obtained several systems of simple facts which suffice to determine the metrics of space; that of Euclid as weightiest is for the present aim made fundamental. These facts are, as all facts, not necessary, but only of empirical certainty; they are hypotheses.
Riemann, in his inaugural dissertation "Ueber die Hypothesen welche der Geometrie zu Grunde liegen". If knowledge of space can only be gotten through experience; if space as we perceive it is only one of a number of thinkable threefold magnitudes (tenfold, string theorists!): how can it be called a priori?
Easily enough, as it turns out. Simply because experience is the occasion upon which we learn which magnitude space follows, does not mean that experience defines which magnitude space follows. Space is still, presumably, universal (everyone lives in the same space, which follows the same rules) and necessary (no one has outer intuition without it). If universal and necessary, then a priori.
What non-Euclidean geometry had done was to explain the faults in Euclid, not in geometry. Space may well be more complex than the featureless void we had imagined it to be, but why then should we have assumed it so simple? The a priori may well be exceedingly complex. As someone once pointed out, if you assume that space does not condition things, take a knife and a potato and cut a seven-edged solid. I'll go further: if you assume space does not condition things, what good is it? If it is the form of outer intuition, we ought not to be surprised when it insists on forming things its own way.
Kant may not have conceived of a space as odd as the one in which we live. But his arguments remain convincing. Moreover, please note that he makes no claims as to where the a priori forms of intuition come from. He never says that they descend from some spirit world; they might well have simply evolved. It's impossible to say.
