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