Philosophy and Physics of Space and Time
PHIL/PHYS 433

Alternative Geometries and Conventionalism

1.The Euclidean Model

>From 300 B. C. to the 19th century, Euclid's geometry reigned supreme as the last (and only) word about geometry. It was also construed as a model of how human knowledge should be organized: clear principles leading to particular theorems or results.

As far as we know, Euclid's main contribution was to systematize a number of results already known about spatial measurements and relations. He organized these results in the form of an axiomatic and postulational system. In such a system, a small number of axioms or postulates were taken to be foundational truths in no need of proof. The other results were then derived from these axioms and postulates by means of clear, explicit and unambiguous rules of inference. It is now known that many of the results in Euclid's treatment rely on implicit assumptions and unstated rules. Euclid's treatment is thus incomplete and inadequate. Nevertheless, it stands as a model for axiomatic treatments of a subject matter.

MORAL:In the axiomatic development of a subject matter, NOTHING can be taken for granted.

Euclid's Postulates have several (equivalent) formulations. One such is the following:

Euclid's Postulates:

P1:A straight line can be drawn between any two points.

P2:A finite straight line can be extended continuously in a straight line.

P3:A circle can be drawn with any center and any radius.

P4: All right angles are equal to one another.

P5:Given a straight line and a point not on that line, there is one and only one line through that point parallel to the given line. (That is, parallel lines will never meet even if they are extended ad infinitum)

From ancient times, some of the postulates have seemed more evident than others. The parallel postulate [P5], with its reference to what happens at infinity, was particularly suspect.

What is the status of the axioms? Are they empirical truths or somehow mathematical necessities? The majority view until the 19th century was that the axioms are self-evident truths. However, they are not analytically true, that is, their denial is not self-contradictory. In addition, the parallel postulate struck almost no-one as self-evident.

Kant's view, developed at the end of the 18th century, was that the axioms were a priori (that is, no experience could possibly justify rejecting them) but synthetic (as opposed to "analytic"). They are conceivably false but, as a matter of fact, they are not. Kant's view is tied to a particularly suspicious metaphysics and has been criticized along those lines.

Long before Kant, suspicions about parallel postulate generated a long history of attempts to prove the parallel postulate as a theorem from the remaining axioms. These attempts all ended in failure. In the 19th century, the development of non-Euclidean geometries showed that P5 was indeed an independent assumption which could be rejected without engendering a contradiction or inconsistency.

2. Three Alternative Geometries

Non-Euclidean geometries are formed by rejecting the parallel postulate and substituting an alternative in its place. For Euclid, given any straight line L and a point, p, not on the line, there is one and only one line through p and parallel to L. There are two basic alternatives: there can be NO parallels through such points or MANY. Those with MANY parallels are called Bolyai-Lobachevskian (or Hyperbolic) geometries after the two men who first worked them out. Geometries with NO parallels are called Riemannian (or Elliptical) geometries.

The postulate systems for these alternative geometries are as follows:

Lobachevsky's Geometry (hyperbolic)

P1:A straight line can bedrawn between any two points.

P2:A finite straight line can be extended continuously in a straight line.

P3:A circle can be drawn with any center and any radius.

P4: All right angles are equal to one another.

P5L:Given a point p not on a straight line L, there is more than one line through p which is parallel to L.

Riemann's Geometry (spherical)

P1:A straight line can be drawn between any two points.

P2R:Any two lines have two distinct points in common.

P3:A circle can be drawn with any center and any radius.

P4: All right angles are equal to one another.

P5R:There is no line parallel to any other line.

Elliptical Geometry

P1:A straight line can be drawn between any two points.

P2:A finite straight line can be extended continuously in a straight line.

P3:A circle can be drawn with any center and any radius.

P4: All right angles are equal to one another.

P2E:Any two lines have a unique intersection.

These are metric geometries [distances are defined for them] with constant curvature. They are the only geometries which satisfy the Axiom of Free Mobility, which says, in effect, that objects do not change their shape merely in virtue of moving from place to place in the space. [When we come to study the General Theory of Relativity which takes Gravitational Influences into account we shall see that this axiom
is not satisfied, in general, by GTR spaces!]

There are standard Two-Dimensional Models of these Non-Euclidean Spaces:

Lobachevskian Geometry:

L-points: {p| p inside circle O}

L-lines: { arcs of orthogonal circles within O }

L-parallels:{ Two L-lines are parallel if they never meet}

L-distance: PQL = k/2 log(PA/PB x QA/QB)

L-triangles: An L-triangle is a closed figure bounded by 3 L-lines.

Every L-theorem can be translated into an E-theorem.

Example:

L-theorem:The sum of the interior angles of an L-triangle are < 180

E-theorem:The sum of the interior angles of a {Euclidean figure bounded by the arcs of circles orthogonal to a fixed circle} < 180.

This inter-translatability shows that if L-geometry is inconsistent, then so is E-geometry. This is (the sketch of) a proof of relative consistency.

Riemannian Geometry

The surface of a Euclidean sphere S is a two dimensional model of Riemannian geometry.

R-points:{p| p is on S}

R-lines:Arcs of great circles on S

R-triangle: A closed figure bounded by 3 R-lines.

R-theorem:The sum of the interior angles of an R-triangle is > 180.

E-theorem:The sum of the interior angles of a {spherical Euclidean triangle} is > 180.

As for Lobachevskian geometry, every R-theorem is inter-translatable with some E-theorem. This shows that R-geometry is consistent if E geometry is.

The differences between these geometries can be neatly summarized by means of the following chart

Euclidean Bolyai-
Lobachevskian Riemannian/ Elliptical Number of
Parallels

2-Dimensional
Model

Triangle Theorem

Circumference/
Diameter Ratio
1

Plane

… = 180°

c/d = ¼ Many

… < 180°

c/d > ¼

0

Sphere

… > 180°

c/d < ¼
The key to the 2-dimensional models is that we redefine "straight line" to mean the same as "geodesic." A geodesic on the surface of sphere or on a saddle is the shortest distance between two points that remains on the surface.

3.Relative versus absolute consistency

The 3 alternative geometries are relatively consistent. That is, if any are free of contradiction, then all are. This follows from the fact that there is a translation of every theorem in any one geometry into a theorem in any other geometry. The fact that wecan find "models" for each geometry shows that they are consistent IF Euclidean geometry is consistent. We assume Euclidean geometry IS consistent. On that condition, the Lobachevskian and Riemannian geometries are consistent as well.
But suppose euclidean geometry is not consistent. Then the consistency of the non-euclidean geometries is likewise suspect. Can we ever be sure that the euclidean axioms will never lead us to a contradiction or inconsistency? That is, can we ever get an absolute proof of the consistency of euclidean geometry? The surprising answer is ...NO! The consistency of euclidean geometry cannot be proved using geometrical techniques alone (Gödel's Incompleteness Theorem entails this result). To prove the absolute consistency of geometry we would have to use the techniques of a different mathematical theory whose own consistency would need to be proved by a further theory . . .

The effect of the discovery of the consistency of non-Euclidean geometries was striking. First, Euclidean geometry lost its privileged status. Second, and MOST IMPORTANT: The question "What is the geometrical structure of physical space" now becomes an open question to be determined empirically! Applied geometry becomes a branch of physics.

4.Pure and Applied Geometries

We are now in a position to distinguish between pure and applied geometries. A pure geometry is an axiomatic mathematical system. The theorems that logically follow from the axioms are "true" if the axioms are "true." Experience is not relevant to determining their validity. All that is required is that the results FOLLOW from the axioms. A physical geometry, on the other hand, is an empirical hypothesis about the structure of some physical structure such as space and time. In order to determine how to apply the pure geometry, we must supply "coordinating definitions" that link up the concepts of the pure geometry with concepts which are physically defined. Thus, we might characterize a "physical" straight line between two points as the path of a straight edge connecting the points. Alternatively, we might define a "physical" straight line as the path of a laser beam between the two points. It is not obviously true (i. e., not "self-evident") that these definitions always coincide. A physical geometry with one set of coordinating definitions might be "true" of the physical world whereas an alternative physical geometry (with the same underlying pure basis) might be false of the physical world with a different set of definitions.

An axiomatic system is given an application when its terms are given an interpretation via co-ordinative definitions or interpretation rules of some sort. These rules indicate how the terms of the system are to be physically realized. Thus, we might ask how we are to understand what counts as a physical realization of an E-line. One way to do would be to say that any physical line drawn along a ruler counts as a straight line. Another way might specify the laser path between two points as the physical E-line connecting them.

Given this, in what sense are the non-euclidean and euclidean geometries "incompatible" with each other? If we adopt the "standard" Euclidean coordinating definitions of "straight line," etc., then the axioms of the alternative systems ARE clearly incompatible with each other and lead to conflicting results. On the other hand, suppose the Earth were a perfect sphere. We now ask: what is the geometry of the surface of the earth? The sketch of the relative consistency of E- and R-geometry suggests that we may choose to say that we live on a Euclidean sphere or we may choose to say that we live on a Riemannian surface depending upon the extent to which we are willing to alter certain linguistic conventions. So, Euclidean and Riemannian geometry ARE NOT incompatible!

Is there a conflict here? Not really. The uninterpreted axiom systems by themselves are not incompatible. If we decide on one set of interpretation rules for the term "line" in all the theories, then the interpreted systems will be incompatible in the sense that no physical model will be able to simultaneously satisfy the axioms of both the E- and the R-systems. If we are willing to use one set of interpretation rules with the E-system and a different set of interpretation rules for the R system, then the same physical model will be able to simultaneously satisfy the axioms of both the E- and the R-systems. The determination of the geometrical structure of physical objects involves elements of convention as well as fact.

5.The Geometry of Physical Space

How can we test alternative theories of physical geometry? One way would be to empirically determine which theorems are true in the real world and which are not. We use the triangle theorem as the touchstone since each of the 3 alternative geometries we are now considering make different predictions about the sum of the interior angles of physical triangles. In the 19th century, Gauss is alleged to have suggested (and possibly carried out) the following experiment to test the geometrical structure of physical space. Set up 3 lanterns on the tops of 3 nearby mountains (we could use laser beams for greater accuracy). Measure the angles between the beams and sum them up. If you do the experiment, you discover that … = 180°. Gauss concluded that physical space is Euclidean.
The French mathematician, Henri Poincare, was not so sure. Suppose, he asked, the result had been otherwise. He argued that in that case the result would not prove that physical space was not Euclidean. It might be that light does not travel in "straight" lines. What is being tested, he pointed out, is not the (pure) Euclidean geometry, but the combination [Pure geometry + Some physical assumptions (embedded in the coordinating definitions)]. So the argument looks like this:

P1:IF Euclidean Geometry & Physical Assumptions, THEN … = 180°

P2: … ‚ 180° [Assumed negative test result]

Ä C:EITHER Space is not Euclidean

OR Some physical assumption is wrong.

It is always open to us, Poincare argued, to reject one or more of the physical assumptions that we invoked to set up the test. Poincare concluded that the geometrical structure of physical space is a convention. This claim is sometimes referred to as the POINCARE/REICHENBACH thesis.
Poincare thought that we would always choose Euclidean geometry as the simplest assumption. In 1905, Einstein argued that this was not so. He felt that a simpler assumption, all things considered, is that the geometrical structure of space is non-Euclidean.

QUESTION: What result does one obtain if one redoes the GAUSS experiment using synchronous satellites at interplanetary distances? What do we conclude about the geometry of physical space and time?

6.What is fact and what is convention in the determination of the geometry of physical space?

We may distinguish trivial from non-trivial claims of conventionality.

Some trivial senses:

TC-1:The meanings of all linguistic noises are conventional. This is the sense in which we say "blue" and the Germans say "blau."

TC-2:The choice of standards of measurement is conventional. We use "feet" where almost everyone else uses "meters."

TC-3:It is a matter of convention (i. e., of our choice of interpretation rules) whether we say the surface of S is the surface of a Euclidean sphere or the a non-Euclidean surface.

TC-4:The meter is (or was at one time) defined as the length between two marks on a standard bar kept under standard conditions in a vault in Paris. We chose to call this distance "one meter." We could easily have shifted the marks closer together or moved them farther apart and called that distance "one meter."

Some non-trivial senses:

NTC-1:Space (time) has no intrinsic metric, i. e., if we move a standard rod (clock) from point A to a different point B, then it is a matter of convention and not a matter of fact that the length of the rod (tick of the clock) at A is equal to the length of the rod (tick of the clock) at B.

NTC-2:We are free to define the congruence of non-coincident spatial or temporal intervals as we please (i. e., by using whatever interpretation rules we wish).

Having defined "congruence at a distance" by convention, several factual issues arise.

F-1:In the case of lengths (times) it is a matter of fact and not of convention that non-standard rods (clocks) agree with the standards. When they do conflict, which we choose is a matter of convenience ( one may be easier to use; one may result in simpler physics).

F-2Is "Is the geometry of the universe Euclidean or not" an empirical question?

A) It is empirical BUT only after we have conventionally determined when two intervals at a distance are congruent.

B) Back to the Gauss experiment: If we decree that light rays travel along geodesics, then it is a matter of fact that the local geometry around the surface of the Earth is (roughly) Euclidean. The geometry of interstellar space (as determined by performing the Gauss experiment with stars, e.g., as the corners of the triangle) turns out, as a matter of fact (given the decree) to be non-Euclidean.