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.

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.

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.

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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.

Three Points in Kant's Theory of Space and Time

Immanuel Kant (1724-1804)

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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)

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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 -- I have found Roger Penrose and even Jerrold Katz making this mistake. 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 (17921856), and Carl Gauss (17771855), 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 non-Euclidean 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.

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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.

ThesisAntithesis
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.
ProofProof
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.

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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 was built into 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. There is now a lore about this, with Einstein considering throught experiments as he rode the trams back and forth to his job at the Swiss Patent Office in Bern. If the tram were going the velocity of light, the time on the clock at the end of the street would appear to have frozen. 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.

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.

Rotation as Gravity

A Metaphysic of the Forces of Nature in Multiple Dimensions

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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.

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