A further ontological distinction can be made. Even if the ortho-curvature axiom is true, a functionally non-Euclidean geometry would be possible if a higher dimension that allows for extrinsic curvature exists but is hidden from us. We must consider whether only the three dimensions of space exist or whether there may be additional dimensions which somehow we do not experience but which can produce an intrinsic curvature whose extrinsic properties cannot be visualized or imaginatively inspected by us. Thus we should distinguish between an axiom of closed ortho-curvature, which says that three dimensional space is all there is, and an axiom of open ortho-curvature, which says that higher dimensions can exist. This gives us three possibilities:

1. That, with the axiom of closed ortho-curvature, there are no true non-Euclidean geometries (and no spatial dimensions beyond three), but only pseudo-geometries consisting of curves in Euclidean space;
2. That, with the axiom of open ortho-curvature, there are no true non-Euclidean geometries but we may be faced with a functional non-Euclidean geometry in Euclidean space whose external curvature is concealed from us in dimensions (more than the three familiar spatial dimensions) not available to our inspection--this is an apparent hetero-curvature;
3. And that, with the axiom of hetero-curvature, there are real non-Euclidean geometries whose intrinsic properties do not ontologically presuppose higher dimensions (whether or not there are more than three spatial dimensions).

It is necessary to keep in mind that these axioms are answers to questions concerning reality that would be asked in physics or metaphysics and are logically entirely separate from the status of geometry in logic or mathematics or from our psychological powers of visual imagination. The second axiom leaves open the question whether "hidden" dimensions are just hidden from our perception or actually separate from our own dimensional existence. With these ontological alternatives in mind, we can now examine the philosophical implications of Einstein's use of non-Euclidean geometry.

§3. Geometry in Einstein's Theory of Relativity

Einstein's general theory of relativity proposes that the "force" of gravity actually results from an intrinsic curvature of spacetime, not from Newtonian action-at-a-distance or from a quantum mechanical exchange of virtual particles. If we view Einstein's philosophical project as an answer to Kant's Antinomy of Space--to explain how straight lines in space can be finite but unbounded--the introduction of time reckoned as the fourth dimension suggests that we may separate the intrinsic curvature of spacetime into curvature based on the relationship between space and time: we can think of Einstein's theory as one that satisfies the axiom of open ortho-curvature, with the peculiarity that it is indeed time, rather than a higher dimension of space, that is posited beyond our familiar three spatial dimensions. This is a metaphysically elegant theory, since is gives us the mathematical use of a higher dimension without the need to postulate a real spatial dimension beyond our experience or our existence. Time is a dimension that is present to us only one spatial slice at a time, just as the third dimension is only intersected at one (radial) point by the curved surface of a sphere in our previous model of a positively curved space.

Our spherical model for non-Euclidean spacetime, however, is not quite right; for on the analogy, the intrinsic lines in space should be the geodesics and so should appear straight to us. They should appear curved only from the perspective of the higher dimension, as the great circles on the sphere appear curved from our three dimensional perspective. That is not true in terms of astronomical space, where the lines drawn by freefalling bodies in gravitational fields are most evidently curved to our three dimensional imaginations, even while they are understood to be geodesics only in terms of their form in the higher dimension of spacetime. That is exactly the opposite of the case in the model: Freefalling paths ("world lines") are geodesics in spacetime but extrinsically curved lines in space, while in the model great circles are extrinsically curved lines in solid space (corresponding to spacetime) but geodesics in plane space (corresponding to space).

Intrinsic curvature, which was introduced by Riemann to explain how straight lines could have the properties associated with curvature without being curved in the ordinary sense, is now used to explain how something which is obviously curved, e.g. the orbit of a planet, is really straight. Something has gotten turned around. If the curvature of spacetime is evident to us in extrinsically curved lines in three dimensional space, then the form of the analogy forces us to posit the "higher" or extrinsic dimension, into which the straight lines are curved, as a spatial one, not the temporal one. If three dimensional space is not extrinsically curved into time according to the axiom of open ortho-curvature, then it must be time that is extrinsically curved into the dimensions of space. In the model, where before the surface of the sphere was analogous to solid space, now the surface must be analogous to two dimensions of space plus time, with the third dimension of space as that into which the geodesics of spacetime are extrinsically curved. Switching the role of time suddenly makes the model very non-intuitive, but it is compelled by the feature of the model that the geodesic is on the surface of the sphere. It does not help the philosophical issue to eject the complications of the axiom of open ortho-curvature and simply take the four dimensions of spacetime as satisfying hetero-curvature; for this loses sight of Kant's Antinomy of Space, which we hope to answer, and of the circumstance that even in Relativity the dimension of time is not exactly the same as the dimensions of space. That is the most intuitively obvious in the "separation" formula: s² = t² - (x² + y² + z²)/c². Here the Pythagorean formula for changes in spatial location, divided by the velocity of light squared, is subtracted from the change in time squared, to give the spacetime "separation" in units of time. Thus time is not treated as simply another spatial dimension. Thus we must consider the differences between space and time, and the axiom of open ortho-curvature alone allows for this.

The result of attributing extrinsic curvature to time is also suggested by the peculiarity of using "curved space" alone to explain gravity, as is common in museums and textbooks around the world; for curved space conjures up images of hills and valleys through which moving objects describe curved paths. However, those images presuppose motion, and motion is the very thing to be explained. Gravity does not just direct motion; it causes it. An object passing by the earth is accelerated towards the earth and thereby acquires a velocity along a vector where it previously may have had no velocity at all. An object placed at rest with respect to the earth, with no initial velocity in any direction, will be accelerated with a velocity towards the earth. If there are no "forces" acting on the body, as Einstein says, then the only change that takes place is the body's movement along the temporal axis; and if the body is thereby displaced in space, it must be displaced by its movement along that axis. The temporal axis can displace the object if the axis is itself curved; so the curvature of spacetime in a gravitational field must result from the curvature of time, not of space. The extrinsic dimension of ortho-curvature, into which the straight lines curve, is a dimension of ordinary Euclidean space. This can be intuitively shown, not so much in our non-Euclidean models, but simply in a graph plotting time (t) against one dimension of space (r). An accelerating body will describe a curved line that changes its coordinate in the r axis as its coordinate in the t axis changes. If the acceleration comes from spacetime itself, then the coordinate grid will itself be curved: the t axis lines will curve, displacing themselves against the r axis (spatial location), while the r axis lines will not curve. The curvature of time itself is hidden from us because, indeed, we intersect only one point on the temporal axis. Consequently, how do we know we are being accelerated by gravity? In free fall we are being displaced with space itself, and so we move with our entire frame of reference and would not be able to detect that locally. Indeed, we cannot. It is Einstein's own "equivalence" principle of General Relativity that we cannot tell the difference between free fall in a gravitational field and free floating in the absence of a gravitational field. The motion induced in us by the curvature of time is evident only because we can observe distant objects that are not subject to our local acceleration. When we are not in free fall, e.g. standing on the surface of the earth, we feel weight, just as according to the equivalence principle when we are being accelerated by a force (e.g. a rocket engine) in the absence of a gravitational field. These are indeed equivalent because in each case we are moving relative to space according to our own frame of reference. When we are accelerated by a rocket we say that we move in the stationary reference of external space; but when we are accelerated standing on the surface of the earth, it is space itself that is displaced (by time) relative to us. Either we move through space, or space moves through us. That is the experience of weight.

A question remains about the global character of spacetime. Gravitational fields are locally positively curved, but Einstein and his philosophical successors evidently expected that spacetime as a whole would be positively curved, since a finite but unbounded universe is aesthetically more satisfying--and it answers Kant's Antinomy of Space. Now, however, the geometry of cosmological spacetime is usually tied to the dynamical fate of the expanding universe. Open, ever expanding universes, are regarded as having Lobachevskian or even Euclidean geometry and only closed universes, headed for ultimate collapse, positive Riemannian curvature. The observational evidence at the moment is for an open universe, and "inflationary" models even have reasons to prefer a Euclidean over a Lobachevskian geometry. These possibilities, however, introduce considerable trouble; for Euclidean and Lobachevskian spaces are both infinite, and it is a much different proposition to say that an infinitely dense Big Bang starts at a finite singularity, into which a finite positively curved space can be packed, than it is to say that an infinite homogeneous and isotropic universe, which must have begun infinite, starts from an infinitely dense Big Bang. An infinitely dense singularity can have a finite mass, but an extended infinite density, even in a small finite region of space, cannot.

In a recent cosmological article in Scientific American, "Textures and Cosmic Structure" (March 1992), the authors, Spergel and Turok, speak of the universe (they do not say "the observable universe") starting from an "infinitesimally small point" or of the universe being at one time the size of a "grapefruit," as though that would hold true for all model universes. The infinite universes are not even considered, and so the questions about density can be happily ignored.  The problem is compounded here because there are actually two infinities competing with each other: there is the infinite volume of space, and there is the infinite shrinkage, or compression, represented by the big bang singularity. However much you shrink an infinite space, it is still infinite. On the other hand, any finite region within infinite space, however large, can be compressed to a single point at the big bang. There is no conflict between the two infinities so long as you specify just what it is that you are talking about.

The problem here, however, is not visualization, it is the hard logical truth that an infinite space remains infinite and that the big bang for an infinite space, although it can be described as a singularity in relation to any finite region of space, cannot be a finite singularity.

Einstein himself introduced his Cosmological Constant to preserve a static universe, before Hubble's evidence of the red shift. He thus seems to have been thinking that a global positively curved geometry for spacetime was not necessarily tied to some dynamical evolution of the universe. This is still a possibility. Three dimensional space can still be conceived as having an inherent hetero-curvature apart from the gravitational fate of the universe: non-Euclidean without the need to regard time or anything else as a fourth dimension into which space needs to be extrinsically curved. This makes for a finite Big Bang regardless of the dynamical fate of the universe, where that fate is tied to the effect of the curvature of time, locally positively curved but globally possibly Lobachevskian or Euclidean. However, a theory of global hetero-curvature then stands separate from the mathematical Relativistic theory of gravity and becomes a theory in metaphysical cosmology more than a theory in physical cosmology.

A positively hetero-curved universe happens to suit the most commonly used cosmological model of all: the inflating balloon, where motion is added to our spherical model of non-Euclidean geometry. The surface of the balloon remains spherical regardless of whether the balloon is blown up forever or whether it eventually is allowed to deflate. As a model the balloon therefore actually posits five dimensions, with the surface representing the three dimensions of space, time as the fourth, but as a fifth the third spatial dimension into which the surface is curved and through which the surface moves in time. A positively hetero-curved universe, however, does not need that fifth dimension. Space would be non-Euclidean without higher dimensions, even while it moves along a temporal axis that is locally ortho-curved into an apparently hetero-curved spacetime because of the curvature of time. The balloon model therefore can represent a different kind of theory than it was intended to, but a most suggestive one, where the global structure of the isotropic and homogeneous universe may allow us to avoid an infinite Big Bang independent of the dynamical fate of the universe and fulfill the hope of the philosophers that Einstein answered Kant's Antinomy of Space.

§4. Conclusion

Just because the math works doesn't mean that we understand what is happening in nature. Every physical theory has a mathematical component and a conceptual component, but these two are often confused. Many speak as though the mathematical component confers understanding, this even after decades of the beautiful mathematics of quantum mechanics obviously conferring little understanding. The mathematics of Newton's theory of gravity were beautiful and successful for two centuries, but it conferred no understanding about what gravity was. Now we actually have two competing ways of understanding gravity, either through Einstein's geometrical method or through the interaction of virtual particles in quantum mechanics.

Nevertheless, there is often still a kind of deliberate know-nothing-ism that the mathematics is the explanation. It isn't. Instead, each theory contains a conceptual interpretation that assigns meaning to its mathematical expressions. In non-Euclidean geometry and its application by Einstein, the most important conceptual question is over the meaning of "curvature" and the ontological status of the dimensions of space, time, or whatever. The most important point is that the ontological status of the dimensions involved with the distinction between intrinsic and extrinsic curvature is a question entirely separate from the mathematics. It is also, to an extent, a question that is separate from science--since a scientific theory may work quite well without out needing to decide what all is going on ontologically. Some realization of this, unfortunately, leads people more easily to the conclusion that science is conventionalistic or a social construction than to the more difficult truth that much remains to be understood about reality and that philosophical questions and perspectives are not always useless or without meaning. Philosophy usually does a poor job of preparing the way for science, but it never hurts to ask questions. The worst thing that can ever happen for philosophy, and for science, is that people are so overawed by the conventional wisdom in areas where they feel inadequate (like math) that they are actually afraid to ask questions that may imply criticism, skepticism, or, heaven help them, ignorance.

These observations about Einstein's Relativity are not definitive answers to any questions; they are just an attempt to ask the questions which have not been asked. Those questions become possible with a clearer understanding of the separate logical, mathematical, psychological, and ontological components of the theory of non-Euclidean geometry. The purpose, then, is to break ground, to open up the issues, and to stir up the complacency that is all too easy for philosophers when they think that somebody else is the expert and understands things quite adequately. It is the philosopher's job to question and inquire, not to accept somebody else's word for somebody else's understanding. .

Grappling with the causes of inertia, Newton imagined two buckets partially filled with water. The first bucket is left still, and the surface of the water is flat. The second bucket is spun rapidly, and the surface of the water is concave. Why?

The naive answer is centrifugal force. But how does the second bucket know it is spinning? In particular, what defines the inertial reference frame relative to which the second bucket spins and the first does not? Berkeley [!] and Mach's answer was that all the matter [which Berkeley didn't believe in] in the universe collectively provides the reference frame. The first bucket is at rest relative to distance galaxies, so its surface remains flat. The second bucket spins relative to those galaxies, so its surface is concave. If there were no distant galaxies, there would be no reason to prefer one reference frame over the other. The surface in both buckets would have to remain flat, and therefore the water would require no centripetal force to keep it rotating. In short, there would be no inertia. Mach inferred that the amount of inertia a body experiences is proportional to the total amount of matter in the universe. An infinite universe would cause infinite inertia. Nothing would ever move. [p. 92, comments added]

Whatever the "naive" explanation may be, it is not the one used by Newton. The argument made by Luminet et al., Berkeley, and Mach is actually the argument originally made by Leibniz (and just recycled by Berkeley, who believed in space less than in matter) against Newton's idea that space was real.

For Newton, the rotating bucket was rotating in relation to space itself. Evidently, it is now such "conventional wisdom" that space itself provides no inertial frame of reference, since Einstein, that it doesn't occur to anyone that the kind of reference it provides vis à vis rotation is rather different from what it fails to provide to establish absolute linear motion. The argument that, in empty space, with no "distant galaxies," there would be no centrifugal force in the bucket and the water in one would be just as flat as in the other is not a necessary conclusion, but only a theory. And not a theory easily tested without an empty universe available.

On the other hand, the question can still be asked how the bucket can "know" that the "distant galaxies" are out there. There must be a physical interaction for that (the range of gravity is infinite); yet Einstein, again, said that no physical interaction can travel faster than the velocity of light, and in an "inflationary" universe (which Mach didn't know about) light can have reached us from only a finite part of the universe, even in an infinite universe. Thus the argument of Luminet et al. fails, for a infinite universe would make for infinite inertia only if the whole universe could physically affect a location. If only a finite part of the universe, infinite or otherwise, affects a location, then there will still only be finite inertia.