The Ontology and Cosmology of Non-Euclidean Geometry


David Hume, An Enquiry Concerning Human Understanding, Section IV, Part I, p. 20 [L.A. Shelby-Bigge, editor, Oxford University Press, 1902, 1972, p. 25] [note]

If we can show that the denial of a proposition does not contradict the consequences of certain other propositions, we have then found a criterion of the logical independence of the proposition in question. In other words, the logical independence of this Euclidean axiom [the Parallel Postulate] of the other axioms would be proved if it could be proven that a geometry free of contradictions could be erected which differed from Euclidean geometry in the fact, and only in the fact, that in the place of the parallel axiom there stood its negation. That is just what Gauss, Lobachevski, and Bolyai established: the possibility of erecting such a noncontradictory geometry which is different from the Euclidean...

What is important to us here is this: The results of modern axiomatics are a completely clear and compelling corroboration of Kant's and Fries's assertion of the limits of logic in the field of mathematical knowledge, and they are conclusive proof of the doctrine of the "synthetic" character of the mathematical axioms. For it is proved that the negation of one axiom can lead to no contradiction even when the other axioms are introduced... And this was just the criterion that Kant had already specified for the synthetic character of a judgment: the uncontradictory character of its negation.

Leonard Nelson, "Philosophy and Axiomatics," 1927, Socratic Method and Critical Philosophy [Yale, 1949, Dover 1965, pp.163-164]

The scientist who says, "The only way to explain this is to show you the math," either doesn't want to explain the question, and so is brushing you off, or he cannot explain the question. If he doesn't want to explain the question, either he cannot because he doesn't actually understand it, or he is a Positivist who doesn't think that it needs to be or can ever be explained. Either way, if he seems annoyed, rude, or hostile, one's suspicions are reasonable aroused.

Enklinobarangus ()

The human brain is wired in such a way that we simply cannot imagine curved spaces of dimension great than two; we can only access them through mathematics.

Edward Frenkel, Professor of Mathematics, University of California, Berkeley, Love and Math, The Heart of Hidden Reality [Basic Books, 2013, p.2]

§1. Introduction

Until recently, Albert Einstein's complaints in his later years about the intelligibility of Quantum Mechanics often led philosophers and physicists to dismiss him as, essentially, an old fool in his dotage. Happily, this kind of thing is now coming to an end as philosophers and mathematicians of the caliber of Karl Popper and Roger Penrose conspicuously point out the continuing conceptual difficulties of quantum theory [cf. Penrose's searching discussion in The Emperor's New Mind, chapter 6, "Quantum magic and quantum mystery," Oxford 1990]. The Paradox of Schrödinger's Cat is sometimes now even presented, not as a wonderful exciting implication of the theory, but for what it originally was: a reductio ad absurdum argument against the "new" quantum mechanics of Heisenberg and Bohr. Schrödinger shared the misgivings of Einstein and others.

A fine statement about all this can be found in Joseph Agassi's foreword to the recent Einstein Versus Bohr, by the dissident physicist Mendel Sachs (Open Court, 1991):

When I was a student of physics I was troubled by the difficulties presented and aired by Sachs in this book. Both the physicists and the philosophers of science to whom I confessed my troubles -- as clearly as I could -- showed me hostility rather than sympathy. I had earlier experienced the same from my religious teachers, so that I was not crushed by the hostility, but I was discouraged from pursuing my scientific interests. This book has returned me to those days and reminded me of the tremendous joys I experienced then, reading Einstein and Schrödinger and meeting Karl Popper and Alfred Landé. All four expressed, one way or another, the same sentiment as Sachs: feeling difficulties about current ideas should be encouraged, not discouraged.

It is amazing that such things need to be said, and it is particularly revealing that the responses Agassi got to his questions reminded him of the intolerance of religious dogmatism.

Nevertheless, there is still rarely a public word spoken about the philosophical intelligibility of Einstein's own theory: the Relativistic theory of gravitation. That theory rests on the use of non-Euclidean geometry. There are still many good questions to ask about non-Euclidean geometry; but in treatment after treatment in both popular expositions and in philosophical discussion, the questions consistently seem pointedly not to get asked. A good example of this may be found in two articles published in Scientific American in 1976. J.J. Callahan's article, "The Curvature of Space in a Finite Universe" in August, makes the argument that Riemann's geometry of a positively curved, finite and unbounded space, which was used by Einstein for his theory, answers the paradox of Kant's Antinomy of Space, avoiding both finite space and infinite space as they had been traditionally understood. This is the philosophically satisfying aspect of Einstein's theory that clearly continues to exercise profound influence on contemporary physicists like Stephen Hawking, as may be seen in his recent Brief History of Time. On the other hand, in March of 1976, Scientific American also published an article by J. Richard Gott III (et al.), "Will the Universe Expand Forever?" This article detailed the evidence then available indicating that the universe was not positively curved, finite and unbounded, as Einstein, and everyone since, has wished. Instead, the universe is more likely to be infinite, either with a Lobachevskian non-Euclidean geometry, or even with a Euclidean(!) geometry after all.

Now, my point is not that scientific theory is in flux. It usually is. My problem is that the philosophical implications of the likelihood that observation will continue to reveal an infinite universe (despite "missing mass," "dark matter," etc.) have not been explored. The clear implications of the observational March article for the significance of the philosophical August article have never been considered in any other cosmological article in Scientific American -- or anywhere else that I have ever seen, inside or outside of philosophy. It is as though everyone is waiting around in the hope that the "missing mass" turns up. Meanwhile there is virtually a conspiracy of silence. If we just don't think -- ostrich-like -- about facing an infinite universe again, then it won't happen. This is not intellectually or philosophically honest. But it is of a piece with much of the way non-Euclidean geometry and its related cosmological issues have been dealt with for a long time. The closest we seem to have come to a more open consideration of these matters is when both Stephen Hawking and Karl Popper [Karl Popper, Unended Quest, Open Court, 1990; p.16] point out that Einstein, whether or not he successfully answered Kant's Antinomy of Space, did not answer the Antinomy of Time: despite decades of everyone glorifying in the philosophical revelation of a finite but unbounded universe, they simply didn't notice that the solution proposed for space didn't work with time. It is to Hawking's great philosophical credit that he faces this question squarely.

In what follows I will attempt to ask questions about non-Euclidean geometry that I do not often, or ever, see asked. In section three I will then briefly attempt to suggest how the philosophical implications Einstein's application of geometry in his theory of gravitation may be reconsidered.

§2. Curved Space and Non-Euclidean Geometry

Euclid's parallel postulate, in its modern reformulation, holds that, on a plane, given a line and a point not on the line, only one line can be drawn through the point parallel to the line. Gerolamo Saccheri (1667-1733) brilliantly attempted to prove this through a reductio ad absurdum argument. There were two ways to contradict the postulate: space could have 1) no parallel lines (straight lines in a plane will always meet if extended far enough), or 2) multiple straight lines through a given point parallel to a given line in the plane. These become non-Euclidean axioms. Saccheri convincingly achieved his reductio for the first possibility with the innocent assumption that straight lines are infinite [cf. Jeremy Gray, Ideas of Space Euclidean, Non-Euclidean, and Relativistic, Oxford, 1989; p. 64]. Later David Hilbert (1862-1953) would point out that the same reductio proof could be achieved by assuming that given three points on a line only one can be between the other two [David Hilbert and S. Cohn-Vossen Geometry and the Imagination (Anschauliche Geometrie -- better translated Intuitive Geometry), Chelsea Publishing Company, 1952; p. 240]. For the second possibility, however, Saccheri did not achieve a good proof. And it was using just such an axiom that the first complete non-Euclidean geometries were achieved by Bolyai (1802-1860) and Lobachevskii (1792-1856).

If by "flat" we mean a plane of straight lines as understood by Euclid, then true non-Euclidean manifolds (i.e. areas, volumes, spacetimes, etc.), in order to really contradict Euclid, who was talking about straight lines, would have to be flat. They could not be curved. Straight lines would be Euclidean straight, but the properties specified by non-Euclidean axioms would be satisfied. Nevertheless, since Bernhard Riemann (1826-1866), non-Euclidean manifolds are said to be "curved," and only Euclidean space itself is called "flat." Contradiction #1 above produces "positively" curved space ("spherical" or "elliptical" geometry, first described by Riemann himself), and contradiction #2 "negatively" curved space ("hyperbolic" or Lobachevskian geometry). To Euclid, this doubtlessly would seem to prove his point: the parallel postulate is about straight lines, so using curved lines hardly produces an honest non-Euclidean geometry. "Curvature" in this respect, however, is used in an unusual sense. "Intrinsic" curvature is distinguished from "extrinsic" curvature. A space can possess "intrinsic" curvature yet contain lines ("geodesics") that will be straight according to any form of measurement intrinsic to that space. A geodesic is "straight" in relation to its own manifold. Euclidean straightness thus characterizes the geodesic of a three dimensional space with no intrinsic curvature, and it is simply a matter of convention and convenience that we call Euclidean geodesics "straight" and generalized straight lines "geodesics" [Note that my references to "Euclidean" space will always mean three dimensional space as understood by Euclid himself (or Kant). "Flat" spaces of more than three dimensions may be called "Euclidean" because of their lack of curvature; but this is an extension of geometry that would have very much been news to Euclid, and I wish to retain the historical connection between "Euclidean" and Euclid]. What "curvature" would have meant to Euclid is now "extrinsic" curvature: that for a line or a plane or a space to be "curved" it must occupy a space of higher dimension, i.e. that a curved line requires a plane, a curved plane requires a volume, a curved volume requires some fourth dimension, etc. Now "intrinsic" curvature has nothing to do with any higher dimension. But how did this happen? Why did "curvature" come to have this unusual meaning? Why should we confuse ourselves by saying that "intrinsic" straight lines, geodesics, in non-Euclidean spaces have curvature?

This happened because non-Euclidean planes can be modeled as extrinsically curved surfaces within Euclidean space. Thus the surface of a sphere is the classic model of a two-dimensional, positively curved Riemannian space; but while great circles are the straight lines (geodesics) according to the intrinsic properties of that surface, we see the surface as itself curved into the third dimension of Euclidean space. A sphere is such a good representation of a non-Euclidean surface, and spherical trigonometry was so well developed at the time, that it now is a little surprising that it was not the basis of the first non-Euclidean geometry developed [cf. Gray ibid. p.171]. However, as noted, such a geometry does contradict other axioms that can easily be posited for geometry. Accepting positively curved spaces means that those axioms must be rejected. Also, and more importantly, these models in Euclidean space are not always successful. The biggest problem is 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 [note].

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 [Leonard Nelson, "Philosophy and Axiomatics," Socratic Method and Critical Philosophy, Dover, 1965; p.164], as quoted above. 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. Something of the sort is admitted with hesitation by Jeremy Gray:

As I read Kant, he 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. [Gray, Ibid. p. 85]

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.

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 closest thing to confronting this conflict that I have seen is a passage in The Matter Myth by Paul Davies and John Gribbin (Touchstone, 1992):

I suppose infinity always dazzles us, and I have never been able to build up a good intuition about the concept. 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.

Well, I can say all this in words, and I know I can make mathematical sense of it, but I confess that to this day I cannot visualize it. (p.108)

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.

§5. Postscript

The logjam of conformity and complacency that irritated me for so many years before originally writing this paper, and since, may now be breaking. A new article in Scientific American, "Is Space Finite?" [Jean-Pierre Luminet, Glenn D. Starkman, & Jeffrey R. Weeks, April 1999, pp. 90-97] finally divorces the geometry of the universe from its dynamics. A teaser at the beginning of the article says, "Conventional wisdom says that the universe is infinite" [p. 90]. Really? This is "conventional wisdom" now? What does Stephen Hawking have to say about that? The text says, "The question of a finite or infinite universe is one of the oldest in philosophy. A common misconception is that it has already been settled in favor of the latter" [p. 91]. Perhaps it is time for a new edition of A Brief History of Time.

Acknowledging that the density of matter in the universe does appear to be too low to "close" the universe gravitationally, which means it is dynamically open, and so, as we have been given to understand, infinite, the article says:

One problem with the conclusion is that the universe could be spherical yet so large that the observable part seems Euclidean, just as a small patch of the earth's surface looks flat [a common idea in "inflationary" theories]. A broader issue, however, is that relativity is a purely local theory [!]. It predicts the curvature of each small volume of space -- its geometry -- based on the matter and energy it contains. Neither relativity nor standard cosmological observations say anything about how those volumes fit together to give the universe its overall shape -- its topology. [p. 92, comments added]

So all this time, all of the angst about the dynamics of the universe wasn't necessarily about the large scale structure of the universe at all. A little bit of this in the 70's would have been quite nice. If Scientific American had actually printed the letter, as they said they might, that I wrote them in 1976, I would be in an excellent position to say "I told you so" (although my concern was the difference that an extra dimension would make, not the kind of topological questions now opened). In fact, I did tell them so, but I fear it did not make it into the public record. Nor was this essay noticed by anyone in particular when it was posted on the Web in 1996. It probably still won't be, to dampen or affect any of the latest enthusiasms.

Another point in the article may be worth noting. Luminet et al. say that the universe may be finite because of Mach's argument about the source of inertia.

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, including the gravitational force, can travel faster than the velocity of light, and in an "inflationary" universe (which Mach didn't know about) light and gravity 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.

Apart from a shake-up over the geometry of space, there has been another surprise in recent cosmology. An article in the January 1999 Scientific American, "Surveying Space-time with Supernovae" [Craig J. Hogan, Robert P. Kirshner, and Nicholas B. Suntzeff, pp. 46-51], discusses observational data that seems to indicate that the expansion of the universe has accelerated over time, not decelerated as it should under the influence of gravity alone. This implies the existence of Einstein's "Cosmological Constant" or some other exotic force that would override the attraction of gravity -- this has now come to be called "dark energy." It also may clear up another pecularity about "standard" cosmology that had been swept under the rug. That is, all closed universes, where deceleration would be enough to produce a collapse into the "Big Crunch," preferred by cosmologists like Stephen Hawking, would have to be younger than 2/3 of the Hubble Time (1/H). This would also mean that no objects in the universe could have a red shift larger than 2/3 of the velocity of light (c), since the red shift gives us the distance in proportion to the Hubble Radius (c/H), and also the age in proportion to the Hubble Time. Thus, in the diagram at right, all the universes under the green curve are closed, and all those above the green curve are open. Now, many quasars have red shifts larger than 2/3 c. Many are even over 90% of c. This has been prima facie evidence since the 70's that the universe was open, but nobody of any influence seems to have noticed. Now, however, if the universe is accelerating, then all possible universes are above the straight red line in the diagram which indicates the Hubble Constant. They will all be older than the Hubble Time. This suddenly makes it quite reasonable that very old objects, like many quasars, would have very, very large red shifts. Indeed, the Big Bang itself would appear to be receding faster than the velocity of light -- it would have an infinite red shift. So again we have an object lesson in the history of science, that a careful examination of the implications of a theory is sometimes neglected by professional science. Inconsistencies can be revealed by even a lay examination.

None of this has made things any clearer for cosmologists. The universe is dominated by dark matter and dark energy, whose natures are no more than speculative. An infinite univere starting from a Big Bang singularity involves paradoxes that I have never seen addressed in popular or semi-professional media. And academic philosophers continue to assert that Kant's axioms of geometry are suppose to be self-evident or that the existence of non-Euclidean geometries refutes his philosophy of mathematics -- where quite the opposite is the case for both. They seem to have missed the most elementary definition of "synthetic," which was actually already familiar in Hume. Yet even Hume's statements about geometry ("the truths demonstrated by Euclid would for ever retain their certainty and evidence") are ignored.

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The Ontology and Cosmology of Non-Euclidean Geometry, Note 1

The reader is thus put on notice that it is the empiricist "skeptic" Hume, not the rationalistic Kant, who thinks that the axioms of geometry are self-evident, and Hume who denies that their truth owes anything to experience.

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The Ontology and Cosmology of Non-Euclidean Geometry, Note 2

A correspondent has asserted that non-Euclidean geometries can be represented in Euclidean space with straight lines, using the rules of perspective adopted for images since the Renaissance. His examples of what this looks like, however, are the diagrams at right. Yes, there are straight lines, and, as in the square, there are four of them in each diagram. However, they are not connected together to make polygons. So each figure, the one for "elliptical" geometry and one for "hyperbolic," actually consist of a single line that simply turns three times before ending (although one line crosses the other in the "hyperbolic" diagram). The properties of each geometry are represented by the angles of those turns (right, obtuse, and acute, respectively) and the length of the line segments. However, the latter feature misses the point. Since positively curved space is supposed to contain more space then Euclidean, and negative curved space less, the line segments are correspondingly longer and shorter than in the Euclidean square. But this does not do the trick. The point about the non-Euclidean spaces is that lines of the same length are supposed to enclose greater and smaller areas, respectively. Since the lines don't connect up into polygons, they don't actually enclose any spaces. That being the case, it is presumably why the expedient of lengthening and shortening the lines was adopted. So these figures are not projections, models, representations, or perspectives of squares in non-Euclidean space. They don't even contain all the elements of a proper non-Euclidean representation, since the lines are not the correct length.

More to the point are the alternative diagrams at right. Here the lines are the same length, but curved. The "elliptical" square contains angles greater than 360o (where 360 = 4 x 90) and more area than the Euclidean square, while the "hyperbolic" square contains angles less than 360o and less area than the Euclidean square. Thus, the curved lines enable us to reproduce the properties of the non-Euclidean figures. In these terms, it is not surprising that non-Euclidean spaces should be called "curved," even though the lines are geodesics and so "straight" according to their intrinsic geometry.

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