The Metaontology of Universe

Euclid's parallel postulate, in its modern reformulation,				(corresponding to spacetime) but geodesics in plane
holds that, on a plane, given a line and a point not on				space (corresponding to space).
the line, only one line can be drawn through the point				Intrinsic curvature, which was introduced by Riemann
parallel to the line. Gerolamo Saccheri (1667-1733)				to explain how straight lines could have the
brilliantly attempted to prove this through a reductio				properties associated with curvature without being
ad absurdum argument. There were two ways to				curved in the ordinary sense, is now used to explain
contradict the postulate: space could have 1) no				how something which is obviously curved, e.g. the
parallel lines (straight lines in a plane will always meet				orbit of a planet, is really straight. Something has
if extended far enough), or 2) multiple straight lines				gotten turned around. If the curvature of spacetime
through a given point parallel to a given line in the				is evident to us in extrinsically curved lines in three
plane. These become non-Euclidean axioms. Saccheri				dimensional space, then the form of the analogy
convincingly achieved his reductio for the first				forces us to posit the "higher" or extrinsic dimension,
possibility with the innocent assumption that straight				into which the straight lines are curved, as a spatial
lines are infinite [cf. Jeremy Gray, Ideas of Space				one, not the temporal one. If three dimensional space
Euclidean, Non-Euclidean, and Relativistic, Oxford,				is not extrinsically curved into time according to the
1989; p. 64]. Later David Hilbert (1862-1953) would				axiom of open ortho-curvature, then it must be time
point out that the same reductio proof could be				that is extrinsically curved into the dimensions of
achieved by assuming that given three points on a				space. In the model, where before the surface of
line only one can be between the other two [David				the sphere was analogous to solid space, now the
Hilbert and S. Cohn-Vossen Geometry and the				surface must be analogous to two dimensions of
Imagination (Anschauliche Geometrie--better				space plus time, with the third dimension of space as
translated Intuitive Geometry), Chelsea Publishing				that into which the geodesics of spacetime are
Company, 1952; p. 240]. For the second possibility,				extrinsically curved. Switching the role of time
however, Saccheri did not achieve a good proof. And				suddenly makes the model very non-intuitive, but it is
it was using just such an axiom that the first				compelled by the feature of the model that the
complete non-Euclidean geometries were achieved by				geodesic is on the surface of the sphere. It does not
Bolyai (1802-1860) and Lobachevskii (1792-1856).				help the philosophical issue to eject the complications
If by "flat" we mean a plane of straight lines as				of the axiom of open ortho-curvature and simply
understood by Euclid, then true non-Euclidean				take the four dimensions of spacetime as satisfying
manifolds (i.e. areas, volumes, spacetimes, etc.), in				hetero-curvature; for this loses sight of Kant's
order to really contradict Euclid, who was talking				Antinomy of Space, which we hope to answer, and
about straight lines, would have to be flat. They could				of the circumstance that even in Relativity the
not be curved. Straight lines would be Euclidean				dimension of time is not exactly the same as the
straight, but the properties specified by non-Euclidean				dimensions of space. That is the most intuitively
axioms would be satisfied. Nevertheless, since				obvious in the "separation" formula: s2 = t2 - (x2 +
Bernhard Riemann (1826-1866), non-Euclidean				y2 + z2)/c2. Here the Pythagorean formula for
manifolds are said to be "curved," and only Euclidean				changes in spatial location, divided by the velocity of
space itself is called "flat." Contradiction #1 above				light squared, is subtracted from the change in time
produces "positively" curved space ("spherical" or				squared, to give the spacetime "separation" in units
"elliptical" geometry, first described by Riemann				of time. Thus time is not treated as simply another
himself), and contradiction #2 "negatively" curved				spatial dimension. Thus we must consider the
space ("hyperbolic" or Lobachevskian geometry). To				differences between space and time, and the axiom
Euclid, this doubtlessly would seem to prove his point:				of open ortho-curvature alone allows for this.
the parallel postulate is about straight lines, so using				The result of attributing extrinsic curvature to time is
curved lines hardly produces an honest non-Euclidean				also suggested by the peculiarity of using "curved
geometry. "Curvature" in this respect, however, is				space" alone to explain gravity, as is common in
used in an unusual sense. Euclidean geodesics				museums and textbooks around the world; for
"straight" and generalized straight lines "geodesics".				curved space conjures up images of hills and valleys
"Flat" spaces of more than three dimensions may be				through which moving objects describe curved paths.
called "Euclidean" because of their lack of curvature;				However, those images presuppose motion, and
but this is an extension of geometry that would have				motion is the very thing to be explained. Gravity
very much been news to Euclid, and I wish to retain				does not just direct motion; it causes it. An object
the historical connection between "Euclidean" and				passing by the earth is accelerated towards the
Euclid]. What "curvature" would have meant to Euclid				earth and thereby acquires a velocity along a vector
is now "extrinsic" curvature: that for a line or a plane				where it previously may have had no velocity at all.
or a space to be "curved" it must occupy a space of				An object placed at rest with respect to the earth,
higher dimension, i.e. that a curved line requires a				with no initial velocity in any direction, will be
plane, a curved plane requires a volume, a curved				accelerated with a velocity towards the earth. If
volume requires some fourth dimension, etc. Now				there are no "forces" acting on the body, as Einstein
"intrinsic" curvature has nothing to do with any higher				says, then the only change that takes place is the
dimension. But how did this happen? Why did				body's movement along the temporal axis; and if the
"curvature" come to have this unusual meaning? Why				body is thereby displaced in space, it must be
should we confuse ourselves by saying that "intrinsic"				displaced by its movement along that axis. The
straight lines, geodesics, in non-Euclidean spaces have				temporal axis can displace the object if the axis is
curvature? This happened because non-Euclidean				itself curved; so the curvature of spacetime in a
planes can be modeled as extrinsically curved				gravitational field must result from the curvature of
surfaces within Euclidean space. Thus the surface of				time, not of space. The extrinsic dimension of
a sphere is the classic model of a two-dimensional,				ortho-curvature, into which the straight lines curve, is
positively curved Riemannian space; but while great				a dimension of ordinary Euclidean space. This can be
circles are the straight lines (geodesics) according to				intuitively shown, not so much in our non-Euclidean
the intrinsic properties of that surface, we see the				models, but simply in a graph plotting time (t) against
surface as itself curved into the third dimension of				one dimension of space (r). An accelerating body will
Euclidean space. A sphere is such a good				describe a curved line that changes its coordinate in
representation of a non-Euclidean surface, and				the r axis as its coordinate in the t axis changes. If
spherical trigonometry was so well developed at the				the acceleration comes from spacetime itself, then
time, that it now is a little surprising that it was not				the coordinate grid will itself be curved: the t axis
the basis of the first non-Euclidean geometry				lines will curve, displacing themselves against the r
developed [cf. Gray ibid. p.171]. However, as noted,				axis (spatial location), while the r axis lines will not
such a geometry does contradict other axioms that				curve. The curvature of time itself is hidden from us
can easily be posited for geometry. Accepting				because, indeed, we intersect only one point on the
positively curved spaces means that those axioms				temporal axis. Consequently, how do we know we
must be rejected. Also, and more importantly, these				are being accelerated by gravity? In free fall we are
models in Euclidean space are not always				being displaced with space itself, and so we move
successful.with Lobachevskian space. A saddle shaped				with our entire frame of reference and would not be
surface is a Lobachevskian space at the center of				able to detect that locally. Indeed, we cannot. It is
the saddle, but a true Lobachevskian space does not				Einstein's own "equivalence" principle of General
have a center. Other Lobachevskian models distort				Relativity that we cannot tell the difference between
shapes and sizes. There is no representation of a				free fall in a gravitational field and free floating in the
Lobachevskian surface that shares the virtues of a				absence of a gravitational field. The motion induced in
sphere in having no center, no singularities (i.e. points				us by the curvature of time is evident only because
that do not belong to the space), and in allowing				we can observe distant objects that are not subject
figures to be moved around without distortion in				to our local acceleration. When we are not in free fall,
shape or size. Three dimensional non-Euclidean spaces				e.g. standing on the surface of the earth, we feel
of course cannot be modeled at all using Euclidean				weight, just as according to the equivalence principle
space.				when we are being accelerated by a force (e.g. a
This raises two questions: 1) what can we spatially				rocket engine) in the absence of a gravitational field.
visualize? (a question of psychology) And 2) what				These are indeed equivalent because in each case
can exist in reality? (a question of ontology). We				we are moving relative to space according to our
cannot visualize any true Lobachevskian spaces or				own frame of reference. When we are accelerated
any non-Euclidean spaces at all with more than two				by a rocket we say that we move in the stationary
dimensions--or any spaces at all with more than three				reference of external space; but when we are
dimensions. Also we can only visualize a positively				accelerated standing on the surface of the earth, it is
curved surface if this is embedded in a Euclidean				space itself that is displaced (by time) relative to us.
volume with an explicit extrinsic curvature.				Either we move through space, or space moves
"Curvature" was thus a natural term for intrinsic				through us. That is the experience of weight.
properties because there always was extrinsic				A question remains about the global character of
curvature for any model that could be visualized.				spacetime. Gravitational fields are locally positively
Why are there these limits on what we can visualize?				curved, but Einstein and his philosophical successors
Why is our visual imagination confined to three				evidently expected that spacetime as a whole would
Euclidean dimensions? It is now common to say that				be positively curved, since a finite but unbounded
computer graphics are breaking through these				universe is aesthetically more satisfying--and it
limitations, but such references are always to				answers Kant's Antinomy of Space. Now, however,
projections of non-Euclidean or multi-dimensional				the geometry of cosmological spacetime is usually
spaces onto two dimensional computer screens. Such				tied to the dynamical fate of the expanding universe.
projections could be done, laboriously, long before				Open, ever expanding universes, are regarded as
computers; but they never produced more, and can				having Lobachevskian or even Euclidean geometry
produce no more, than flat Euclidean drawings of				and only closed universes, headed for ultimate
curves. If such graphics are expected to alter our				collapse, positive Riemannian curvature. The
minds so that we can see things differently, this is no				observational evidence at the moment is for an open
more than a prediction, or a hope, not a fact. And				universe, and "inflationary" models even have reasons
considering that non-Euclidean geometries have been				to prefer a Euclidean over a Lobachevskian
conceived for almost two centuries, the				geometry. These possibilities, however, introduce
transformation of our imagination seems a bit tardy,				considerable trouble; for Euclidean and Lobachevskian
however much help computers can now give to it.				spaces are both infinite, and it is a much different
Mathematicians don't have to worry about these				proposition to say that an infinitely dense Big Bang
questions of visualization because visualization is not				starts at a finite singularity, into which a finite
necessary for the analytic formulas that describe the				positively curved space can be packed, than it is to
spaces. The formulas gave meaningfulness to				say that an infinite homogeneous and isotropic
non-Euclidean geometry as common sense never				universe, which must have begun infinite, starts from
could.				an infinitely dense Big Bang. An infinitely dense
The Euclidean nature of our imagination led Kant to				singularity can have a finite mass, but an extended
say that although the denial of the axioms of Euclid				infinite density, even in a small finite region of space,
could be conceived without contradiction, our intuition				cannot.
is limited by the form of space imposed by our own				In a recent cosmological article in Scientific American,
minds on the world. While it is not uncommon to find				"Textures and Cosmic Structure" (March 1992), the
claims that the very existence of non-Euclidean				authors, Spergel and Turok, speak of the universe
geometry refutes Kant's theory, such a view fails to				(they do not say "the observable universe") starting
take into account the meaning of the term				from an "infinitesimally small point" or of the universe
"synthetic," which is that a synthetic proposition can				being at one time the size of a "grapefruit," as
be denied without contradiction. Leonard Nelson				though that would hold true for all model universes.
realized that Kant's theory implies a prediction of				The infinite universes are not even considered, and
non-Euclidean geometry, not a denial of it, and that				so the questions about density can be happily
the existence of non-Euclidean geometry vindicates				ignored. The problem is compounded here because
Kant's claim that the axioms of geometry are				there are actually two infinities competing with each
synthetic. The intelligibility of non-Euclidean geometry				other: there is the infinite volume of space, and there
for Kantian theory is neither a psychological nor an				is the infinite shrinkage, or compression, represented
ontological question, but simply a logical one--using				by the big bang singularity. However much you shrink
Hume's criterion of possibility as logically consistent				an infinite space, it is still infinite. On the other hand,
conceivability. Kant does not say non-Euclidean				any finite region within infinite space, however large,
geometry is logically impossible, but that is only				can be compressed to a single point at the big bang.
because he does not claim that any geometry is				There is no conflict between the two infinities so
logically true; geometry in his view is synthetic, not				long as you specify just what it is that you are
analytic. And Kant's belief that Euclidean geometry				talking about.
was true, because our intuitions tell us so, seems to				The problem here, however, is not visualization, it is
me to be either unintelligible or wrong.				the hard logical truth that an infinite space remains
If we are unable to visualize non-Euclidean geometries				infinite and that the big bang for an infinite space,
without using extrinsically curved lines, however, the				although it can be described as a singularity in relation
intelligibility of Kant's theory is not hard to find. The				to any finite region of space, cannot be a finite
sense of the truth of Euclidean geometry for Kant is				singularity.
no more or less than the confidence that centuries				Einstein himself introduced his Cosmological Constant
of geometers had in the parallel postulate, a				to preserve a static universe, before Hubble's
confidence based on our very real spatial imagination.				evidence of the red shift. He thus seems to have
If Kant's claim is "unintelligible," then Gray has not				been thinking that a global positively curved
reflected on why everyone in history until the 19th				geometry for spacetime was not necessarily tied to
century believed that the parallel postulate was true.				some dynamical evolution of the universe. This is still
That is the psychological question, not the logical or				a possibility. Three dimensional space can still be
ontological one. The sense of ancient confidence can				conceived as having an inherent hetero-curvature
be recovered at any time today simply by trying to				apart from the gravitational fate of the universe:
explain non-Euclidean geometry to undergraduate				non-Euclidean without the need to regard time or
students who have never heard of it before. We				anything else as a fourth dimension into which space
might say that attempts to prove the postulate				needs to be extrinsically curved. This makes for a
show that people were uneasy about it; but the				finite Big Bang regardless of the dynamical fate of
universal expectation was that the postulate was				the universe, where that fate is tied to the effect of
really a theorem, and no one cashed in their unease				the curvature of time, locally positively curved but
by trying to construct geometry with a denial of it.				globally possibly Lobachevskian or Euclidean. However,
Saccheri denied it, but only because he was				a theory of global hetero-curvature then stands
constructing reductio ad absurdum proofs.				separate from the mathematical Relativistic theory of
Non-Euclidean geometry did not change our spatial				gravity and becomes a theory in metaphysical
imagination, it only proved what Kant had already				cosmology more than a theory in physical cosmology.
implicitly claimed: the synthetic and axiomatically				A positively hetero-curved universe happens to suit
independent character of the first principles of				the most commonly used cosmological model of all:
geometry. It could well be the case that Kant is right				the inflating balloon, where motion is added to our
and that we will never be able to imagine the				spherical model of non-Euclidean geometry. The
appearance of Lobachevskian or multi-dimensional				surface of the balloon remains spherical regardless of
non-Euclidean spaces, or to model them without				whether the balloon is blown up forever or whether
extrinsic curvature, however well we understand the				it eventually is allowed to deflate. As a model the
analytic equations. This is purely a question of				balloon therefore actually posits five dimensions, with
psychology and not at all one of logic, mathematics,				the surface representing the three dimensions of
physics, or ontology. Mathematicians are free to				space, time as the fourth, but as a fifth the third
ignore the limitations of our imagination, although they				spatial dimension into which the surface is curved and
then run the risk of wandering so far from common				through which the surface moves in time. A positively
sense that the frontiers of mathematics will never be				hetero-curved universe, however, does not need
intelligible to even well-informed persons of general				that fifth dimension. Space would be non-Euclidean
knowledge. Furthermore, since Kant believed that				without higher dimensions, even while it moves along
space was a form imposed by our minds on the				a temporal axis that is locally ortho-curved into an
world, he did not believe that space actually existed				apparently hetero-curved spacetime because of the
apart from our experience. This leads us to the				curvature of time. The balloon model therefore can
ontological question: what can exist in reality?				represent a different kind of theory than it was
Non-Euclidean geometry was no more than a				intended to, but a most suggestive one, where the
mathematical curiosity until Einstein applied it to				global structure of the isotropic and homogeneous
physics. Now the whole issue seems much deeper				universe may allow us to avoid an infinite Big Bang
and complex than it did in Kant's day, or Riemann's. If				independent of the dynamical fate of the universe
our imagination is necessarily Euclidean, hard-wired into				and fulfill the hope of the philosophers that Einstein
the brain as we might now think by analogy with				answered Kant's Antinomy of Space.
computers, but Einstein found a way to apply				Â§4. Conclusion
non-Euclidean geometry to the world, then we might				Just because the math works doesn't mean that we
think that space does have a reality and a genuine				understand what is happening in nature. Every
structure in the world however we are able to				physical theory has a mathematical component and a
visually imagine it.				conceptual component, but these two are often
In light of the distinction between intrinsic and				confused. Many speak as though the mathematical
extrinsic curvature, we must consider all the kinds of				component confers understanding, this even after
ontological axioms that will cover all the possible				decades of the beautiful mathematics of quantum
spaces that Euclidean and non-Euclidean geometries				mechanics obviously conferring little understanding.
can describe. If the limitations imposed by our				The mathematics of Newton's theory of gravity
imaginations present us with features of real space,				were beautiful and successful for two centuries, but
we would have to say that intrinsic curvature,				it conferred no understanding about what gravity
despite being analytically independent of extrinsic				was. Now we actually have two competing ways of
curvature, can only exist in conjunction with extrinsic				understanding gravity, either through Einstein's
curvature and so with an embedding in higher				geometrical method or through the interaction of
dimensions. This could be called the axiom of				virtual particles in quantum mechanics.
ortho-curvature, according to which there would				Nevertheless, there is often still a kind of deliberate
actually be no true non-Euclidean geometry, for				know-nothing-ism that the mathematics is the
non-Euclidean geodesics would necessarily have				explanation. It isn't. Instead, each theory contains a
extrinsic curvature and so would never be the actual				conceptual interpretation that assigns meaning to its
straight lines that we need ex hypothese to				mathematical expressions. In non-Euclidean geometry
contradict Euclid. The geometry of the surface of a				and its application by Einstein, the most important
sphere would thus involve ortho-curvature because				conceptual question is over the meaning of
its intrinsic straight lines, the great circles, must be				"curvature" and the ontological status of the
simultaneously visualized and understood to be				dimensions of space, time, or whatever. The most
curved lines in three dimensional Euclidean space. On				important point is that the ontological status of the
the other hand, it may be that intrinsically curved				dimensions involved with the distinction between
spaces can exist in reality without extrinsic curvature				intrinsic and extrinsic curvature is a question entirely
and so without being embedded in a higher dimension.				separate from the mathematics. It is also, to an
This could be called the axiom of hetero-curvature,				extent, a question that is separate from
and it would make true non-Euclidean geometry				science--since a scientific theory may work quite well
possible, since lines with non-Euclidean relations to				without out needing to decide what all is going on
each other would be straight in the common meaning				ontologically. Some realization of this, unfortunately,
of the term understood by Euclid or Kant.				leads people more easily to the conclusion that
A further ontological distinction can be made. Even if				science is conventionalistic or a social construction
the ortho-curvature axiom is true, a functionally				than to the more difficult truth that much remains to
non-Euclidean geometry would be possible if a higher				be understood about reality and that philosophical
dimension that allows for extrinsic curvature exists				questions and perspectives are not always useless or
but is hidden from us. We must consider whether				without meaning. Philosophy usually does a poor job
only the three dimensions of space exist or whether				of preparing the way for science, but it never hurts
there may be additional dimensions which somehow				to ask questions. The worst thing that can ever
we do not experience but which can produce an				happen for philosophy, and for science, is that people
intrinsic curvature whose extrinsic properties cannot				are so overawed by the conventional wisdom in
be visualized or imaginatively inspected by us. Thus				areas where they feel inadequate (like math) that
we should distinguish between an axiom of closed				they are actually afraid to ask questions that may
ortho-curvature, which says that three dimensional				imply criticism, skepticism, or, heaven help them,
space is all there is, and an axiom of open				ignorance.
ortho-curvature, which says that higher dimensions				These observations about Einstein's Relativity are not
can exist. This gives us three possibilities:				definitive answers to any questions; they are just an
That, with the axiom of closed ortho-curvature,				attempt to ask the questions which have not been
there are no true non-Euclidean geometries (and no				asked. Those questions become possible with a
spatial dimensions beyond three), but only				clearer understanding of the separate logical,
pseudo-geometries consisting of curves in Euclidean				mathematical, psychological, and ontological
space;				components of the theory of non-Euclidean
That, with the axiom of open ortho-curvature, there				geometry. The purpose, then, is to break ground, to
are no true non-Euclidean geometries but we may be				open up the issues, and to stir up the complacency
faced with a functional non-Euclidean geometry in				that is all too easy for philosophers when they think
Euclidean space whose external curvature is				that somebody else is the expert and understands
concealed from us in dimensions (more than the				things quite adequately. It is the philosopher's job to
three familiar spatial dimensions) not available to our				question and inquire, not to accept somebody else's
inspection--this is an apparent hetero-curvature;				word for somebody else's understanding.
And that, with the axiom of hetero-curvature, there				Grappling with the causes of inertia, Newton imagined
are real non-Euclidean geometries whose intrinsic				two buckets partially filled with water. The first
properties do not ontologically presuppose higher				bucket is left still, and the surface of the water is
dimensions (whether or not there are more than				flat. The second bucket is spun rapidly, and the
three spatial dimensions).				surface of the water is concave. Why?
It is necessary to keep in mind that these axioms				The naive answer is centrifugal force. But how does
are answers to questions concerning reality that				the second bucket know it is spinning? In particular,
would be asked in physics or metaphysics and are				what defines the inertial reference frame relative to
logically entirely separate from the status of				which the second bucket spins and the first does
geometry in logic or mathematics or from our				not? Berkeley [!] and Mach's answer was that all the
psychological powers of visual imagination. The				matter [which Berkeley didn't believe in] in the
second axiom leaves open the question whether				universe collectively provides the reference frame.
"hidden" dimensions are just hidden from our				The first bucket is at rest relative to distance
perception or actually separate from our own				galaxies, so its surface remains flat. The second
dimensional existence. With these ontological				bucket spins relative to those galaxies, so its surface
alternatives in mind, we can now examine the				is concave. If there were no distant galaxies, there
philosophical implications of Einstein's use of				would be no reason to prefer one reference frame
non-Euclidean geometry.				over the other. The surface in both buckets would
Â§3. Geometry in Einstein's Theory of Relativity				have to remain flat, and therefore the water would
Einstein's general theory of relativity proposes that				require no centripetal force to keep it rotating. In
the "force" of gravity actually results from an intrinsic				short, there would be no inertia. Mach inferred that
curvature of spacetime, not from Newtonian				the amount of inertia a body experiences is
action-at-a-distance or from a quantum mechanical				proportional to the total amount of matter in the
exchange of virtual particles. If we view Einstein's				universe. An infinite universe would cause infinite
philosophical project as an answer to Kant's Antinomy				inertia. Nothing would ever move. [p. 92, comments
of Space--to explain how straight lines in space can				added]
be finite but unbounded--the introduction of time				Whatever the "naive" explanation may be, it is not
reckoned as the fourth dimension suggests that we				the one used by Newton. The argument made by
may separate the intrinsic curvature of spacetime				Luminet et al., Berkeley, and Mach is actually the
into curvature based on the relationship between				argument originally made by Leibniz (and just recycled
space and time: we can think of Einstein's theory as				by Berkeley, who believed in space less than in
one that satisfies the axiom of open ortho-curvature,				matter) against Newton's idea that space was real.
with the peculiarity that it is indeed time, rather than				For Newton, the rotating bucket was rotating in
a higher dimension of space, that is posited beyond				relation to space itself. Evidently, it is now such
our familiar three spatial dimensions. This is a				"conventional wisdom" that space itself provides no
metaphysically elegant theory, since is gives us the				inertial frame of reference, since Einstein, that it
mathematical use of a higher dimension without the				doesn't occur to anyone that the kind of reference it
need to postulate a real spatial dimension beyond our				provides vis Ã vis rotation is rather different from
experience or our existence. Time is a dimension that				what it fails to provide to establish absolute linear
is present to us only one spatial slice at a time, just				motion. The argument that, in empty space, with no
as the third dimension is only intersected at one				"distant galaxies," there would be no centrifugal force
(radial) point by the curved surface of a sphere in				in the bucket and the water in one would be just as
our previous model of a positively curved space.				flat as in the other is not a necessary conclusion, but
Our spherical model for non-Euclidean spacetime,				only a theory. And not a theory easily tested without
however, is not quite right; for on the analogy, the				an empty universe available.
intrinsic lines in space should be the geodesics and so				On the other hand, the question can still be asked
should appear straight to us. They should appear				how the bucket can "know" that the "distant
curved only from the perspective of the higher				galaxies" are out there. There must be a physical
dimension, as the great circles on the sphere appear				interaction for that (the range of gravity is infinite);
curved from our three dimensional perspective. That				yet Einstein, again, said that no physical interaction
is not true in terms of astronomical space, where the				can travel faster than the velocity of light, and in an
lines drawn by freefalling bodies in gravitational fields				"inflationary" universe (which Mach didn't know about)
are most evidently curved to our three dimensional				light can have reached us from only a finite part of
imaginations, even while they are understood to be				the universe, even in an infinite universe. Thus the
geodesics only in terms of their form in the higher				argument of Luminet et al. fails, for a infinite universe
dimension of spacetime. That is exactly the opposite				would make for infinite inertia only if the whole
of the case in the model: Freefalling paths ("world				universe could physically affect a location. If only a
lines") are geodesics in spacetime but extrinsically				finite part of the universe, infinite or otherwise,
curved lines in space, while in the model great circles				affects a location, then there will still only be finite
are extrinsically curved lines in solid space				inertia.

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