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.
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.
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.
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. 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.
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.
Kant is not wrong. Those who think he is can only cite models and projections of non-Euclidean geometries as visualizations. 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. 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.
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.
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.
|
2) floating in free fall in the field of a large gravitating body, such as the earth.
a) no weight.
|
| 3) accelerating through the application of a force in the absence of a gravitational field, as in deep space.
a) weight.
|
4) standing on the surface of a large gravitating body, such as the earth.
a) weight.
|
A Metaphysic of the Forces of Nature in Multiple Dimensions
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.
