Euclid's Axioms and Postulates

One interesting question about the assumptions for Euclid's system of geometry is the difference between the "axioms" and the "postulates." "Axiom" is from Greek axíôma, "worthy." An axiom is in some sense thought to be strongly self-evident. A "postulate," on the other hand, is simply postulated, e.g. "let" this be true. There need not even be a claim to truth, just the notion that we are going to do it this way and see what happens. Euclid's postulates, indeed, could be thought of as those assumptions that were necessary and sufficient to derive truths of geometry, of some of which we might otherwise already be intuitively persuaded. As first principles of geometry, however, both axioms and postulates, on Aristotle's understanding, would have to be self-evident. This never seemed entirely quite right, at least for the Fifth Postulate -- hence many centuries of trying to derive it as a Theorem. In the modern practice, as in Hilbert's geometry, the first principles of any formal deductive system are "axioms," regardless of what we think about their truth -- which in many cases has been a purely conventionalistic attitude. Given Kant's view of geometry, however, the Euclidean distinction could be restored:  "axioms" would be analytic propositions, and "postulates" synthetic. Whether any of Euclid's original axioms are analytic is a good question.

• First Axiom: Things which are equal to the same thing are also equal to one another.

• Second Axiom: If equals are added to equals, the whole are equal.

• Third Axiom: If equals be subtracted from equals, the remainders are equal.

• Fourth Axiom: Things which coincide with one another are equal to one another.

• Fifth Axiom: The whole is greater than the part.

• First Postulate: To draw a line from any point to any point.

• Second Postulate: To produce a finite straight line continuously in a straight line.

• Third Postulate: To describe a circle with any center and distance.

• Fourth Postulate: That all right angles are equal to one another.

• Fifth Postulate: That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side of which are the angles less than the two right angles.

Hilbert's Axioms of Geometry

Given below is the axiomatization of geometry by David Hilbert (1862-1943) in Foundations of Geometry (Grundlagen der Geometrie), 1902 (Open Court edition, 1971). This was logically a much more rigorous system than in Euclid.

• I. Axioms of Incidence:
  1. For every two points A, B there exits a line a that contains each of the points A, B.
  2. For every two points A, B there exists no more than one line that contains each of the points A, B.
  3. There exist at least two points on a line. There exist at least three points that do not lie on a line.
  4. For any three points A, B, C that do not lie on the same line there exists a plane [alpha] that contains each of the points A, B, C. For every plane there exists a point which it contains.
  5. For any three points A, B, C that do not lie on one and the same line there exists no more than one plane that contains each of the three points A, B, C.
  6. If two points A, B of a line a lie in a plane [alpha], then every point of a lies in the plane [alpha].
  7. If two planes [alpha], [beta] have a point A in common, then they have at least one more point B in common.
  8. There exist at least four points which do not lie in a plane.

• II. Axioms of Order:
  1. If a point B lies between a point A and a point C, then the points A, B, C are three distinct points of a line, and B then also lies between C and A.
  2. For two points A and C, there always exists at lest one point B on the line AC such that C lies between A and B.
  3. Of any three points on a line there exists no more than one that lies between the other two.
  4. Let A, B, C be three points that do not lie on a line and let a be a line in the plane ABC which does not meet any of the points A, B, C. If the line a passes through a point of the segment AB, it also passes through a point of the segment AC, or through a point of the segment BC.

• III. Axioms of Congruence:
  1. 1. If A, B are two points on a line a, and A' is a point on the same or on another line a' then it is always possible to find a point B' on a given side of the line a' through A' such that the segment AB is congruent or equal to the segment A'B'. In symbols AB = A'B'.
  2. If a segment A'B' and a segment A"B", are congruent to the same segment AB, then the segment A'B' is also congruent to the segment A"B", or briefly, if two segments are congruent to a third one they are congruent to each other.
  3. On the line a let AB and BC be two segments which except for B have no point in common. Furthermore, on the same or on another line a' let A'B' and B'C' be two segments which except for B' also have no point in common. In the case, if AB = A'B' and BC = B'C' then AC = A'C'.
  4. Let angle(h,k) be an angle in a plane [alpha] and a' a line in a plane [alpha]' and let a definite side of a' in [alpha]' be given. Let h' be a ray on the line a' that emanates from the point O'. Then there exists in the plane [alpha]' one and only one ray k' such that the angle(h,k) is congruent or equal to the angle(h',k') and at the same time all interior point of the angle(h',k') lie on the given side of a'. Symbolically angle(h,k) = angle(h',k'). Every angle is congruent to itself, i.e., angle(h,k) = angle(h,k) is always true.
  5. If for two triangles ABC and A'B'C' the congruences AB = A'B', AC = A'C', angleBAC = angleB'A'C' hold, then the congruence angleABC = angleA'B'C' is also satisfied.

• IV. Axiom of Parallels:
  1. (Euclid's Axiom) Let a be any line and A a point not on it. Then there is at most one line in the plane, determined by a and A, that passes through A and does not intersect a.

• V. Axioms of Continuity:
  1. (Archimedes' Axiom or Axiom of Measure) If AB and CD are any segments, then there exists a number n such that n segments CD constructed contiguously from A, along the ray from A through B, will pass beyond the point B.
  2. (Axiom of Line Completeness) An extension of a set of points on a line with its order and congruence relations that would preserve the relations existing among the original elements as well as the fundamental properties of line order and congruence that follow from Axioms I-III, and from V,1 is impossible.

Hilbert's Comments:

If Archimedes' Axiom is dropped [non-Archimedean geometry], then from the assumption of infinitely many parallels through a point it does not follow that the sum of the angles in a triangle is less then two right angles. Moreover, there exists a geometry (the non-Legendrian geometry) in which it is possible to draw through a point infinitely many parallels to a line and in which nevertheless the theorems of Riemannian (elliptic) geometry hold. On the other hand there exists a geometry (the semi-Euclidean geometry) in which there exists infinitely many parallels to a line through a point and in which the theorems of Euclidean geometry still hold.

From the assumption that there exist no parallels it always follows that the sum of the angles in a triangle is greater than two right angles. [Foundations of Geometry, p.43]

Hilbert's comments should serve to remind us that not only the Parallel Postulate can be denied without contradiction. Many other logically possible geometries exist besides the most familiar non-Euclidean ones.

Philosophy of Science, Space and Time

Philosophy of Science

Home Page

Copyright (c) 1996, 1999, 2001 Kelley L. Ross, Ph.D. All Rights Reserved

The Clarke-Leibniz Debate,
1715-1716

An exchange of letters between Samuel Clarke, defending Isaac Newton's conception of space and time, and Gottfried Wilhelm Leibniz, who disputed Newton's ideas about space [cf. The Leibniz-Clarke Correspondence, With Extracts from Newton's Principia and Opticks, H.G. Alexander, Manchester University Press, 1956, 1965, 1970]. Clarke only had a good response to Leibniz's second argument. Kant came up with a decisive response to the third. Kant, however, actually agreed with Leibniz's first argument.

• Leibniz's First Argument:
  Newton had said that space was "God's boundless uniform sensorium," which makes it sound like some kind of organ of perception. Leibniz says, first, that God does not need a "sense organ" to perceive objects and, second, that space cannot be an absolute reality, or it would possess a greater reality than substances themselves. As Robert Paul Wolff says [Kant's Theory of Mental Activity, Harvard, 1963, pp. 4-8], "...the postulation of an infinite, subsistent non-substance (an "unthing" as Kant later called it) is simply a monstrosity."

• Leibniz's Second Argument:
  Motion and position are real and detectable only in relation to other objects. Motion or position cannot be detected in relation to space itself, since space itself represents no object. Therefore empty space, a void, and so space itself is an unnecessary hypothesis.

• Clarke's Reply:
  Motion is detectable in relation to space itself, for an object accelerating or rotating alone in a void betrays the effect of forces (inertial and centripetal) that exist in relation to no other object. This is really the only solid point that Clarke scored against Leibniz, and efforts have been made since then to fix up Leibniz's argument. One proposal, by Ernst Mach (1838-1916), was to postulate that inertia is the result of an interaction with all the rest of the mass of the universe. This is discussed in more detail elsewhere. Curiously, Leibniz's modern partisans in this respect tend to be Positivists, for whom Leibniz's own metaphysics would be absurd or meaningless.

• Leibniz's Third Argument:
  There would be no reason, and so no sufficient reason, for God to create the universe one way rather than as any one of its spatial counterparts, i.e. up rather than down, right rather than left, or east rather than west. Therefore, spatial relations are symmetrical relations among objects that are equivalent and do not exist apart from objects.

• Kant's Reply:
  Asymmetrical objects and their mirror-imaged counterparts (i.e. right-handed and left-handed "incongruous counterparts") are genuinely and physically different. No rotations in three-dimensional space, e.g. of right and left hands, can turn one into the other. Since the objects differ only in their spatial relationship -- i.e. they could be rotated into each other through a fourth spatial dimension (as noted by Wittgenstein, who, however, thought that this refuted Kant, not understanding that objects do not rotate through a fourth dimension in Nature) -- they reveal that space itself is real and independent of the objects. Kant splits the difference between Newton and Leibniz by holding that space is empirically real, agreeing with Newton, but that it does not exist among things-in-themselves, agreeing with Leibniz.

  On a silly television show some years ago, I saw one of the young characters ask his father why mirrors reverse images from left to right but not from top to bottom. This totally perplexed the father (as it was intended to). But, as it happens, this is a very profound question. The answer is simply that there is a geometrical difference between left and right but not between top and bottom. Mirror images make things spatially different, and this only works in terms of handedness. Why space would be this way is a good question also, but it is a difference that makes for physical differences in the world.

• Another Reply:
  Similar to Kant's reply to Leibniz would be an argument from scale. Thus, we might ask, what is the size of the cube at right? Well, it could be any size. In terms of geometry alone, a cube with dimensions of a few nanometers is identical in form and function to one with dimensions of a few light years. Because of this, people often think that scale physically doesn't make any difference. The solar system could be an atom in some larger kind of matter. Atoms could be little solar systems. In Men in Black [1997], we see an entire galaxy kept in a piece of jewelry, or a marble. The Fifty-Foot Women may be big and slow, but otherwise her body works like that of an ordinary sized woman. However, these are all physical impossibilities. Atoms and solar systems work very differently. A galaxy small enough to be in a marble would implode, probably into a Black Hole. The bones of the Fifty-Foot Woman would break under her weight, which is why elephants and Brontosauri have legs like tree trunks and why people like André the Giant have such physical problems that usually they don't live very long. For there to be physical differences of scale, there must be a physical difference between different volumes of space. This means that space as such must be a physical thing, just as Newton or Kant, but not Leibniz, would have thought.

  There is an absolute and a relative sense in which this is true. The absolute sense is that the Laws of Nature only operate above the scale of the Planck Length, which is 4.0510 x 10^-35 m [(hG/c³)^1/2], or the "reduced" Plank Length, 1.6160(12) x 10^-35 m [(G/c³)^1/2]. The relative sense of scale is in terms of density. A cubic meter of water, if contained, will simply sit there. A cube of water, however, that was the radius of the Earth's orbit on a side would immediately undergo gravitational collapse and become a very, very massive star. A cubic meter of water dispersed in a cube that was the radius of the Earth's orbit (an Astronomical Unit) on a side would again be inactive, and in fact hardly detectable. Density is the ratio of mass to volume. Without the physical reality of space, mass would be the only physical factor; and, according to Leibniz, the physical characteristics of the mass would only be a function of their relative arrangement. The actual volume, however, changes all that. The mass can be identical, and the relative arrangement unchanged, but the actual volume will make for very different densities. That is literally one of the differences, and a major one, between an elephant and a mouse. That is not something that Leibniz's metaphysics can explain (every volume contains an infinite number of Monads).

  Note that the argument from scale has a significant difference from Kant's reply to Leibniz. Handedness is a characteristic of geometry in its own terms. It is the same at all scales. Scale, however, is not a geometrical characteristic at all. It introduces space as a physical reality, where different volumes, however identical they are geometrically, are physically different quantities and make for different physical realities.

Three Points in Kant's Theory of Space and Time

Immanuel Kant (1724-1804)

Philosophy of Science, Space and Time

Philosophy of Science

Metaphysics

Home Page

Copyright (c) 1996, 1999, 2001, 2009, 2010, 2012 Kelley L. Ross, Ph.D. All Rights Reserved

Three Points in Kant's
Theory of Space and Time

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

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

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

Trying to reconcile the metaphysics of Newton and Leibniz, Kant proposed that space and time exist at one level of reality but not at another. The value of this depends on the nature and credibility of Kant's Transcendental Idealism. Such a theory, however, makes possible a Kantian interpretation of quantum mechanics.

1. What space and time are:
   Kant proposes that space and time do not really exist outside of us but are "forms of intuition," i.e. conditions of perception, imposed by our own minds. This enables him to reconcile Newton and Leibniz: agreeing with Newton that space is absolute and real for objects in experience, i.e. for phenomenal objects open to science, but agreeing with Leibniz that space is really nothing in terms of objects as they exist apart from us, i.e. with things in themselves.

2. How space is known:
   Kant does not believe that the axioms of geometry are self-evident or true in any logically necessary way. They are logically "synthetic," which means that they may be denied without contradiction. That is a significant claim because it would mean that consistent non-Euclidean geometries are possible (which would involve the denial of one or more of the axioms of Euclid, as Bolyai, Lobachevskii and Riemann actually accomplished). Nevertheless, Kant did believe that the axioms of geometry are known "a priori," i.e. that they are known to be true prior to all experience, because Euclidean axioms depend on our "pure intuition" of space, namely space as we are able to imaginatively visualize it. Only if non-Euclidean space can be visualized would Kant be wrong.

   Kant is not wrong. Those who think he is can only cite models and projections of non-Euclidean geometries as visualizations [note]. There is no model or projection of Lobachevskian (negatively curved) space that does not distort shapes and sizes. The best model of a positively curved Riemannian space, the two dimensional surface of a sphere, nevertheless only has lines that are intuitively curved in the third dimension (and would be intuitively curved even in that space just by shortening the lines). The surface cannot be visualized without that third dimension. This why spherical trigonometry existed for centuries without anyone thinking of it as a non-Euclidean geometry. These issues are discussed in detail elsewhere.

3. The cosmology of space and time:
   Kant does not think we can know, or even imagine, the universe as either finite or infinite, in space or in time, because space and time are only forms of perception and cannot be imagined or visualized as absolute wholes. The universe, as the place of things in themselves, is not in space or in time and so is neither finite nor infinite in space or in time. Thus there cannot be an a priori, rational or metaphysical, cosmology. Kant's Antinomies are intended to show that contradictory metaphysical absolutes can be argued and justified with equal force, meaning that neither can actually be proven. It can be argued however, that Einstein answered Kant by proposing a non-Euclidean (Riemannian) universe that is finite but unbounded (i.e. without an edge).

Immanuel Kant (1724-1804)

Kant Index

Philosophy of Science, Space and Time

Philosophy of Science

Metaphysics

Home Page

Copyright (c) 1996, 1999, 2001, 2004, 2008, 2011 Kelley L. Ross, Ph.D. All Rights Reserved

Three Points in Kant's
Theory of Space and Time, Note

It now common in philosophical conventional wisdom for people to say that the very existence of non-Euclidean geometry refutes Kant's theory. This usually involves multiple confusions. For example, in an otherwise sensible recent book, we find Paul Boghossian (of New York University) saying:

Kant's own claim about geometry came to grief:  soon after he made it, Riemann discovered non-Euclidean geometries, and some one hundred years later, Einstein showed that physical space was in fact non-Euclidean. [Fear of Knowledge, Clarendon Press, Oxford, 2006, p.40]

It is discouraging how poorly informed Boghossian is. Bernhard Riemann wasn't even born until 1826. "Soon" after Kant (d.1804) came János Bolyai (1802-1860), Nikolai Lobachevskii (1792–1856), and Carl Gauss (1777–1855), who were responsible for the first non-Euclidean geometry, "Lobachevskian" geometry. Boghossian doesn't even bother to explain why the discovery of non-Eucliean geometries would refute Kant, though the implication is clear that Kant should have predicted the impossibility of non-Euclidean geometry. Since, however, synthetic propositions can be denied without contradiction, and Kant believes that the axioms of geometry are synthetic, one wonders what part of that Boghossian, or others, does not understand. Einstein, of course, did not "show" anything, much less that "space was in fact non-Euclidean." Einstein's cosmology was an elegant resolution of Kant's First Antinomy, but unfortunately the observational evidence was against it (not enough mass), and the general opinion in cosmology now is that space is "in fact" Euclidean. Einstein could still be right, but only if we detach the geometry of the universe from its dynamics, something otherwise unrelated to Einstein's theory.

Return to Text

Kant's First Antinomy,
of Space and Time

Critique of Pure Reason, pp. A 426-429,
Norman Kemp Smith translation

Kant's Antinomy of Space and Time is the first of four Antinomies. The meaning of the Antinomies and the possibility of expanding them is considered elsewhere.

Thesis	Antithesis
The world has a beginning in time, and is also limited as regards space.	The world has no beginning, and no limits in space; it is infinite as regards both time and space.
Proof	Proof
If we assume that the world has no beginning in time, then up to every given moment an eternity has elapsed, and there has passed away in that world an infinite series of successive states of things. Now the infinity of a series consists in the fact that it can never be completed through successive synthesis. It thus follows that it is impossible for an infinite world-series to have passed away, and that a beginning of the world is therefore a necessary condition of the world's existence. This was the first point that called for proof. As regards the second point, let us again assume the opposite, namely, that the world is an infinite given whole of co-existing things. Now the magnitude of a quantum which is not given in intuition [i.e. perception] as within certain limits, can be thought only through the synthesis of its parts, and the totality of such a quantum only through a synthesis that is brought to completion through repeated addition of unit to unit. In order, therefore, to think, as a whole, the world which fills all spaces, the successive synthesis of the parts of an infinite world must be viewed as completed, that is, an infinite time must be viewed as having elapsed in the enumeration of all co-existing things. This, however, is impossible. An infinite aggregate of actual things cannot therefore be viewed as a given whole, nor consequently as simultaneously given. The world is, therefore, as regards extension in space, not infinite, but is enclosed within limits. This was the second point in dispute.	For let us assume that it has a beginning. Since the beginning is an existence which is preceded by a time in which the thing is not, there must have been a preceding time in which the world was not, i.e. an empty time. Now no coming to be of a thing is possible in an empty time, because no part of such a time possesses, as compared with any other, a distinguishing condition of existence rather than of non-existence; and this applies whether the thing is supposed to arise of itself or through some other cause. In the world many series of things can, indeed, begin; but the world itself cannot have a beginning, and is therefore infinite in respect of past time. As regards the second point, let us start by assuming the opposite, namely, that the world in space is finite and limited, and consequently exists in an empty space which is unlimited. Things will therefore not only be related in space but also related to space. Now since the world is an absolute whole beyond which there is no object of intuition, and therefore no correlate with which the world stands in relation, the relation of the world to empty space would be a relation of it to no object. But such a relation, and consequently the limitation of the world by empty space, is nothing. The world cannot, therefore, be limited in space; that is, it is infinite in respect of extension.
These proofs really only use one argument, that an infinite series cannot be completed ("synthesized") either in thought, perception, or imagination. That was roughly Aristotle's argument against infinite space.	There are two arguments here: First, that there is no reason for the universe to come to be at one time rather than another, where all points in an empty time are alike. Second, that objects can only be spatially related to each other, not to empty space, which is not an object.

Stephen Hawking says that Kant's arguments for the thesis and antithesis of the antinomy of time are effectively the same (p. 8 in A Brief History of Time), but note that they are really based on quite different principles. The argument for the thesis is based on the impossibility of constructing an infinite series, while the argument for the antithesis is an argument from the Principle of Sufficient Reason, a kind of argument first used (on just this subject and to this effect) by Parmenides. Although Hawking says that both arguments are based on an "unspoken assumption" of infinite time, he actually agrees with the argument of the thesis that time is not infinite.

Aristotle believed that space was finite because of the impossibility of an actual infinite quantity. The way this would work is, if we are unable to imagine an infinite quantity, and if the most real is the most knowable, then the unknowability of an actual infinite quantity means that it cannot be real. On the other side, the Skeptics argued that space cannot be finite because we can imagine space on the other side of any boundary. This means that where the boundary is is arbitrary, which violates the Principle of Sufficient Reason, i.e. there is no reason why the boundary should be where it is. More vividly, they imagined Hercules punching out the boundary. We could use Arnold Schwarzenegger.

Immanuel Kant (1724-1804)

Kant Index

Philosophy of Science, Space and Time

Philosophy of Science

Metaphysics

Home Page

Copyright (c) 1996, 1999, 2001, 2008 Kelley L. Ross, Ph.D. All Rights Reserved

Einstein's Equivalence Principle

According to the "Relativity" of Galileo, there is no physical difference between "rest" and constant velocity, although he and Newton assumed there was some frame of reference based on an absolute velocity of "rest". Trying to physically identify such a frame of reference led to difficulties [note]. In response, Albert Einstein denied that there is a frame of reference that is absolutely at "rest," but he did propose another absolute velocity:  not one of rest but one just the opposite -- the velocity of light. This stood common sense on its head, but one reason for it was perfectly conventional:  The velocity of light could be derived from Maxwell's Equations, implying that it was always the same, regardless of frame of reference. An absolute "rest" frame of reference would relativize the velocity of light, contradicting Maxwell's Equations. Einstein simply chose to accept this implication and followed the consequences. That gave the "Special Theory of Relativity" of 1905. In the "General Theory of Relativity" of 1915, Einstein examined the implications of equating inertial mass with gravitational mass. Newton had assumed these were the same, but he had not considered that they were exactly the same, i.e. in some sense physically operated the same way.

Inertial mass resists changes in velocity. A frame of reference moving, or "resting," at a constant velocity is thus called an "inertial frame of reference." Gravitational mass exerts and responds to gravitational accelerations. Newton assumed these two kinds of mass were the same thing. Einstein made this a postulate of General Relativity, the Equivalence Principle. According to this principle, since cases 1) and 2) below are experienced in the same way, without weight, they are the same. Similarly with cases 3) and 4), with weight. It is cases 1) and 4), however, and 2) and 3), that seem to match up on the criterion of the absence, or presence, of motion, respectively. The identities of the Equivalence Principle will hold if it is space itself, in a gravitational field, that is doing the accelerating in 2) and 4), carrying the inertial frames of reference, insensibly, along with it. Weight is produced by the application of an inertial force: by a rocket engine in 3) but by the surface of the earth in case 4). In relation to space itself, the surface of the earth is accelerating and pushing on us in 4). The acceleration of space itself is the "curvature" of spacetime.

1) floating in free fall in the absence of a gravitational field, as in deep space. a) no weight. b) no gravity c) no change in velocity.	2) floating in free fall in the field of a large gravitating body, such as the earth. a) no weight. b) gravity c) change in velocity. Return
3) accelerating through the application of a force in the absence of a gravitational field, as in deep space. a) weight. b) no gravity c) change in velocity.	4) standing on the surface of a large gravitating body, such as the earth. a) weight. b) gravity c) no change in velocity. Return

A Metaphysic of the Forces of Nature in Multiple Dimensions

Philosophy of Science, Physics

Philosophy of Science

Home Page

Copyright (c) 1996, 2001 Kelley L. Ross, Ph.D. All Rights Reserved

Einstein's Equivalence Principle, Note

Although Einstein's theory is called "Relativity," it is noteworthy (1) that it contains the absolute velocity of light and (2) that in the context of Relativistic cosmology an absolute frame of reference for motion has actually been discovered: the Cosmic Background Radiation allows the absolute velocity of the earth in relation to the Universe as a whole to be determined. Back with Galileo and Newton, although space was thought of as providing an absolute frame of reference, this frame of reference could not actually be determined. The Michelson and Morley experiment attempted to determined the absolute motion of the earth against the ether. Not only did this fail (the velocity of light was the same whatever direction it was measured), but, if it was supposed to measure the absolute velocity of the earth, it presupposed that the ether was at rest in relation to space itself. Galilean and Newtonian mechanics would thus seem to embody the "true" Relativity.

Return to text