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 selfevident. 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 selfevident. 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 (18621943) 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 [nonArchimedean 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 nonLegendrian 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 semiEuclidean 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 nonEuclidean ones.
Philosophy of Science, Space and Time
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 LeibnizClarke 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.
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. And we also must note that not all optical transformations work this way. Lenses flip images right to left but also top to bottom. Curiously, mirrors and lenses do not work the same way.
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/c^{3})^{1/2}], or the "reduced" Plank Length, 1.6160(12) x 10^{35} m [(G/c^{3})^{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).
Leibniz cannot be excused from involvement with this argument, for he himself said that, if all bodies in the universe doubled in size overnight, we would notice no difference the next morning [cf. Hal Hellman, Great Feuds in Science, John Wiley and Sons, 1998, p.59]. This is an ambiguous challenge, since it might mean that the linear dimensions of all bodies would double, or that the actual volume of all bodies would double. Either way, the result would be both noticeable and dangerous. Our bodies would have subtantially more mass, and, the Earth being larger, its gravity would be stronger. The proportions of our bodies are not built for that, and so we would be hampered, immobilized, or killed as a result. Even worse, the velocity of the Earth in its orbit would not be sufficient to maintain the orbit, given the increased mass of the Sun. The Earth would begin to fall towards the Sun. Probably not into it, but towards a perihelion point that would substantilly increase the radiation received by the Earth. This would not be good for life on earth. Thus, Leibniz was not aware and would not believe that arbitrary changes in volume would make any physical difference, which is precisely the issue in an argument from scale about space.
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
Philosophy of Science, 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.163164]
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 nonEuclidean 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 nonEuclidean geometry. These issues are discussed in detail elsewhere.
Philosophy of Science, Space and Time
It now common in philosophical conventional wisdom for people to say that the very existence of nonEuclidean 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 nonEuclidean geometries, and some one hundred years later, Einstein showed that physical space was in fact nonEuclidean. [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 (18021860), Nikolai Lobachevskii (1792–1856), and Carl Gauss (1777–1855), who were responsible for the first nonEuclidean geometry, "Lobachevskian" geometry. Boghossian doesn't even bother to explain why the discovery of nonEucliean geometries would refute Kant, though the implication is clear that Kant should have predicted the impossibility of nonEuclidean 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 nonEuclidean." Einstein's nonEuclidean 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.
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 worldseries 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 coexisting 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 coexisting 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 nonexistence; 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.
Philosophy of Science, Space and Time
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. 
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
Philosophy of Science, Physics
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.