Though there never were a circle or triangle in nature, the truths demonstrated by Euclid would for ever retain their certainty and evidence.

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] []

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 a
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 the 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.

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]. 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

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

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:

- 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;
- 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; - 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:
s^{2}
= t^{2}
- (x^{2}
+ y^{2}
+ z^{2})/c^{2}.
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.

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.

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 tha 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. At 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 intertial 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 intertia. 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 (http://www.friesian.com/space.htm#clarke)* (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 ā 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.

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 **de**celerated 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. It also may clear up another pecularity about
"standard" cosmology that had been swept under the rug.
That
is, all closed universes, where decleration would be enough to produce
a colapse into the "Big Crunch," prefered 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 (http://www.friesian.com/time2.htm)*
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 reshifts. 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.