with Lobachevskian space. A saddle shaped surface is a Lobachevskian space at the center of the saddle, but a true Lobachevskian space does not have a center. Other Lobachevskian models distort shapes and sizes. There is no representation of a Lobachevskian surface that shares the virtues of a sphere in having no center, no singularities (i.e. points that do not belong to the space), and in allowing figures to be moved around without distortion in shape or size. Three dimensional non-Euclidean spaces of course cannot be modeled at all using Euclidean space.

This raises two questions: 1) what can we spatially visualize? (a question of psychology) And 2) what can exist in reality? (a question of ontology). We cannot visualize any true Lobachevskian spaces or any non-Euclidean spaces at all with more than two dimensions--or any spaces at all with more than three dimensions. Also we can only visualize a positively curved surface if this is embedded in a Euclidean volume with an explicit extrinsic curvature. "Curvature" was thus a natural term for intrinsic properties because there always was extrinsic curvature for any model that could be visualized. Why are there these limits on what we can visualize? Why is our visual imagination confined to three Euclidean dimensions? It is now common to say that computer graphics are breaking through these limitations, but such references are always to projections of non-Euclidean or multi-dimensional spaces onto two dimensional computer screens. Such projections could be done, laboriously, long before computers; but they never produced more, and can produce no more, than flat Euclidean drawings of curves. If such graphics are expected to alter our minds so that we can see things differently, this is no more than a prediction, or a hope, not a fact. And considering that non-Euclidean geometries have been conceived for almost two centuries, the transformation of our imagination seems a bit tardy, however much help computers can now give to it. Mathematicians don't have to worry about these questions of visualization because visualization is not necessary for the analytic formulas that describe the spaces. The formulas gave meaningfulness to non-Euclidean geometry as common sense never could.

The Euclidean nature of our imagination led Kant to say that although the denial of the axioms of Euclid could be conceived without contradiction, our intuition is limited by the form of space imposed by our own minds on the world. While it is not uncommon to find claims that the very existence of non-Euclidean geometry refutes Kant's theory, such a view fails to take into account the meaning of the term "synthetic," which is that a synthetic proposition can be denied without contradiction. Leonard Nelson realized that Kant's theory implies a prediction of non-Euclidean geometry, not a denial of it, and that the existence of non-Euclidean geometry vindicates Kant's claim that the axioms of geometry are synthetic. The intelligibility of non-Euclidean geometry for Kantian theory is neither a psychological nor an ontological question, but simply a logical one--using Hume's criterion of possibility as logically consistent conceivability. Kant does not say non-Euclidean geometry is logically impossible, but that is only because he does not claim that any geometry is logically true; geometry in his view is synthetic, not analytic. And Kant's belief that Euclidean geometry was true, because our intuitions tell us so, seems to me to be either unintelligible or wrong.

If we are unable to visualize non-Euclidean geometries without using extrinsically curved lines, however, the intelligibility of Kant's theory is not hard to find. The sense of the truth of Euclidean geometry for Kant is no more or less than the confidence that centuries of geometers had in the parallel postulate, a confidence based on our very real spatial imagination. If Kant's claim is "unintelligible," then Gray has not reflected on why everyone in history until the 19th century believed that the parallel postulate was true. That is the psychological question, not the logical or ontological one. The sense of ancient confidence can be recovered at any time today simply by trying to explain non-Euclidean geometry to undergraduate students who have never heard of it before. We might say that attempts to prove the postulate show that people were uneasy about it; but the universal expectation was that the postulate was really a theorem, and no one cashed in their unease by trying to construct geometry with a denial of it. Saccheri denied it, but only because he was constructing reductio ad absurdum proofs. Non-Euclidean geometry did not change our spatial imagination, it only proved what Kant had already implicitly claimed: the synthetic and axiomatically independent character of the first principles of geometry. It could well be the case that Kant is right and that we will never be able to imagine the appearance of Lobachevskian or multi-dimensional non-Euclidean spaces, or to model them without extrinsic curvature, however well we understand the analytic equations. This is purely a question of psychology and not at all one of logic, mathematics, physics, or ontology. Mathematicians are free to ignore the limitations of our imagination, although they then run the risk of wandering so far from common sense that the frontiers of mathematics will never be intelligible to even well-informed persons of general knowledge. Furthermore, since Kant believed that space was a form imposed by our minds on the world, he did not believe that space actually existed apart from our experience. This leads us to the ontological question: what can exist in reality? Non-Euclidean geometry was no more than a mathematical curiosity until Einstein applied it to physics. Now the whole issue seems much deeper and complex than it did in Kant's day, or Riemann's. If our imagination is necessarily Euclidean, hard-wired into the brain as we might now think by analogy with computers, but Einstein found a way to apply non-Euclidean geometry to the world, then we might think that space does have a reality and a genuine structure in the world however we are able to visually imagine it.

In light of the distinction between intrinsic and extrinsic curvature, we must consider all the kinds of ontological axioms that will cover all the possible spaces that Euclidean and non-Euclidean geometries can describe. If the limitations imposed by our imaginations present us with features of real space, we would have to say that intrinsic curvature, despite being analytically independent of extrinsic curvature, can only exist in conjunction with extrinsic curvature and so with an embedding in higher dimensions. This could be called the axiom of ortho-curvature, according to which there would actually be no true non-Euclidean geometry, for non-Euclidean geodesics would necessarily have extrinsic curvature and so would never be the actual straight lines that we need ex hypothese to contradict Euclid. The geometry of the surface of a sphere would thus involve ortho-curvature because its intrinsic straight lines, the great circles, must be simultaneously visualized and understood to be curved lines in three dimensional Euclidean space. On the other hand, it may be that intrinsically curved spaces can exist in reality without extrinsic curvature and so without being embedded in a higher dimension. This could be called the axiom of hetero-curvature, and it would make true non-Euclidean geometry possible, since lines with non-Euclidean relations to each other would be straight in the common meaning of the term understood by Euclid or Kant.