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 than 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]. The basic analysis here follows Robert Paul Wolff, Kant's Theory of Mental Activity [Harvard, 1963]. 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.
In terms of recent physics, we get the Alice in Wonderland features of quantum mechanics such that empty space is not really empty. It's full of virtual particles that have real physical effects  particularly that they mediate the transmission of the forces of nature. And there are going to be more of them in larger volumes of space. This may pose a dilemma to Leibniz. Does the vacuum allow for the virtual particles? Or do the virtual particles, which cannot actually be detected (but could be said to be part of the representation in the Monads), define the existence of a space where space otherwise does not exist independently? The reality of space thus would default to the question of the nature of the reality of virtual particles. And virtual particles, of course, rely on the principle that we can steal energy from nothing as long as we return it in a certain length of time  which looks like a fudge on the principles of the conservation of energy and of mass. While the metaphysics of this seems to yet defy a coherent theory, it nevertheless bespeaks the entanglement of physical reality with consciousness  because it is a function of uncertainty, a characteristic of knowledge  which is something that makes some physicists unhappy. On the other hand, there is an alternative to virtual particles to explain fundamental forces, namely the device of Einstein that a "force" is actually a curvature of spacetime. Empty space thus mediates the transmission of the forces of Nature. The accusation of Leibniz that Clarke and Newton make space more real than substances can then be embraced by some (of the same) physicists as no less than the simple truth.
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.
When we look at an image reflected in a mirror, as just above right, the top of the image lines up with the top, the bottom with the bottom, the right with the right and the left with the left. Nevertheless, we see the the image reserved in the mirror, so that, as we look at it, the right has become left, and the left right. This is what seems odd.
Of course, the way these images are presented is not quite right. The mirror does not hold an image like a movie screen. If we are off looking at what is reflected, in this case the letter "R," we are not going to see it reflected in the mirror, unless the mirror is turned just the right way (they do this in the movies, where, of course, the camera cannot be occupying the position of what is reflected). But here this is going to be treated like a thought experiment, where the mirror will hold the image and we can move around and consider what is actually seen from only one point of view. So the image above shows, on the right, the "R" that is to be reflected, and the backwards "R" in the mirror as it would been seen from the point of view of the "R," just as anyone sees their face directly in the bathroom mirror.
Given those parameters, we can imagine how the pair of images will look if we stand behind the object being reflected, and can look around it. As at left, we see that the initial "R" seems to be reversed also, while its relationship to the reflected image has no changed at all (i.e. top to top, bottom to bottom, left to left, and right to right). Of course, if this were a person, and not a large letter "R," we would be seeing their back, while the mirror image would be of their front. But if front and back are identical, or we consider that only the front of the person is available to be reflected, then the image and its relfection are identical.
We see a similar effect if we go around behind the mirror; and if the mirror both catches the image and is transparent, we now see both images in their correct orientation.
So what goes on here? We may think of the reversal of the image in a mirror as a kind of rotation, but real rotation in these images is of the viewpoint of the viewer. Looking towards the mirror, we are looking in one direction, but looking through the mirrow towards the image, we are seeing things from the opposite direciton. We have rotated.
If we flip the "R" top to bottom rather than right to left, this also changes its handedness, which we can see in the image at left, where the "R" is first flipped, and then, at right, is rotated in the same plane. The produces the same effect and the flip from right to left. The flip from top to bottom is something we can see if a mirror is lying on the floor, or we see things reflected in the water of a lake.
This may be the answer to the silly question. If mirrors flipped things top to bottom as well as left to right, this would actually accomplish two lefttoright flips, which would cancel out. This means that the transition to a mirror image, from left to right, is accomplished by any rotation through a dimension at right angles to a two dimensional object (or a surface). A mirror gives the appearance of a rotation, without a physical rotation actually taking place, although we can imagine, as with the thought experiment above, playing with the rotation by rotating our own point of view. With asymmetrical three dimensional objects, like amino acids and proteins, mirrors produce the same apparent rotation without a physical rotation being possible, since there is no extra dimension at right angles to three dimensional objects available for such a rotation. This is a truly remarkable effect.
In a good indication of the philosophical commitments that underlie science at any one time, physicists long nourished the hope, conviction, or expectation that Leibniz was correct about handedness. One blow against this was the discovery that many of the molecules involved in life  amino acids and proteins  included right and left handed versions, only one kind of which figures in the chemistry of living things. After H.G. Wells wrote a science fiction story in which a man appeared to have been flipped into a mirror image of himself, to no otherwise ill effects, Isaac Asimov subsequently understood, as Wells in his day did not, that such a person would starve, since his body chemistry would be unable to metabolize the proteins in food. So Asimov wrote his own story about a man in that situation, whose nutrition required the laboratory synthesis of the mirror image amino acids and proteins that do not occur in nature.
Organic chemistry, however, did little to dent the conviction of physicists that handedness would not exist at the fundaments of nature. That came when it was discovered that the Weak Nuclear Force violated "Parity," i.e. the equivalence of right and left. It turned out that only lefthanded (1/2 spin) particles & righthanded (+1/2 spin) antiparticles can "see" the Weak Force. This deeply shook the physics world but nevertheless seemed to pass unnoticed among philosophers talking about space. They missed their chance to offer Wittgenstein as the multidimensional magician who could simply flip a righthanded particle into a lefthanded one, so that all particles could (counterfactually) "see" the Weak Force. Sometimes Nature just doesn't measure up to our own (or at least Wittgenstein's) genius.
In implicit agreement with 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  theories we see pondered in, of all things, the movie Animal House [1978]. In Men in Black [1997], we see an entire galaxy kept in a piece of jewelry, or (for our own) a marble. The FiftyFoot Woman [1958, 1993] may be big and slow, but otherwise her body works like that of an ordinary sized woman. We also get the franchise that began with Honey, I Shrunk the Kids [1989] where scale didn't matter in the functioning of tiny human bodies. However, these are all physical impossibilities. Atoms and solar systems work very differently. A galaxy small enough to be in a marble would be imploding into a Black Hole. The bones of the FiftyFoot Woman would break under her weight, which is why elephants and Brontosauri have legs like tree trunks and why people like André the Giant (André René Roussimoff, 19461993) have such physical problems that usually they don't live very long. Once the kids are shrunk, they would rapidly lose body heat and die of hypothermia  small mammals like mice have elevated metabolisms that compensate. Meanwhile, the metabolism of the FiftyFoot Woman would be too much for her size, and she would experience hyperthermia and heat prostration. Elephants compensate with a slow metabolism. But for there to be physical differences of scale, as occur in these cases, 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. Scale and size alone make for physical differences.
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. Scale is literally one of the differences, and a major one, between an elephant and a mouse. With their differences in size and mass, they simply cannot have the same structure. Thus, if we looked at models of a mouse and of an elephant, we could estimate, from their structure alone, their absolute size. That is not something that Leibniz's metaphysics can explain, even as every volume contains an equally infinite number of Monads. Similarly, if we floated in our cube at right an amoeba, or galaxies (as we see here), its scale would become apparent. These are things that actually do not exist at the opposite extremes of scale, i.e. we do not find galaxies associated with amoebas.
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. For such a thought experiment, that difference doesn't make a real difference, since the bodies will be significantly larger in either case. On the other hand, a correspondent has pointed out that the proposal is also ambiguous about what happens to the mass. Is the mass conserved and thus will remain the same as the body doubles (in size or dimension)? Or are we talking about a body that increases proportionally in mass also? Of course, changing the mass at all contradicts the premise that we are only talking about space and its relative size. But either way, whether we conserve the mass or scale it up, the result would be both noticeable and dangerous, since a larger body of the same mass or a larger body with scaled up mass will be physically different in either case from the smaller body.
That is because as a linear dimension increases, volume increases as the cube of that dimension. This is purely a spatial parameter and thus cannot be explained by Leibniz's view that space is simply a matter of the relations between bodies or features. Thus with the cube at right, asking how large it is supposed to be  from microscopic to cosmological  is a meaningless question for Leibniz. Only a relation to some other figure or body would define anything about it. But, at different scales, our actual bodies would function very differently, whether they retained the same mass or whether they had subtantially more mass. And, if we are remaining on the Earth  the Earth being larger (with necessarily proportionally greater mass, as a solid body, unless we change the laws of nature)  its gravity would be stronger. The proportions of our bodies would not be suitable for that, and so we would be hampered, immobilized, or killed as a result. Bones could break, either from being unnaturally elongated, with a loss in strength, or from being subject to too much weight.
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, to say that the size of everything could be doubled without a difference, Leibniz did not consider and would not believe, on the basis of his metaphysics, that arbitrary changes in volume would make any physical difference, which is precisely the issue in an argument from scale about space.
While the forces of Nature draw our attention to the question of empty space in recent physics, quantum mechanics figures otherwise in the question of scale. I have already noted that the Planck Length defines the smallest scale of physical reality; but an older question in quantum mechanics involves a more general question about scale. That is the problem we see in the familiar paradox of Schrödinger's Cat. Thus, Planck's Constant is a very small number, and quantum effects are things that we see in the microscopic  the atomic and subatomic  domain. However, Erwin Schrödinger realized that in principle there was nothing to prevent quantum effects in the macroscopic world. Thus, locked in its box and subject to the risk of death, a cat could be viewed at any point as simultaneously both dead and alive in a state of quantum superpositon. Since Schrödinger regarded the possibility of a cat both dead and alive at once as an absurdity, his paradox was intended as a reductio ad absurdum of this feature of quantum mechanics. While as a physical theory quantum mechanics has appeared to most to have been justified by its success, and Schrödinger's Cat tends to be regarded as a delightful curiosity, there is still no principle to define the boundary between the quantum microscopic world and the macroscopic world where a cat must either be dead or alive, but not both. There are perhaps really two parts to this matter. If the cat is understood as a sentient being, then it could qualify as an "observer" under the principles of Niels Bohr's quantum interpretation; and that would collapse the wave function, meaning that the cat really is dead or alive. On the other hand, with no living things present, does this clear the way for macroscopic quantum effects, perhaps in stars? Leibniz, of course, would need to think that physical processes would be the same at any scale, so Schrödinger's Cat presumably would not bother him. On the other hand, since every Monad consists of a representation of the entire universe, it's not clear how every one would not qualify as an "observer," entirely eliminating things like quantum superposition.
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. This is what we do see in Nature.
Three Points in Kant's Theory of Space and Time
Einstein's Equation for Gravity
Note on the Metaphysic of Space; the Paradoxes of the Ether
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]
...Einstein's relativistic physics refuted Kant's claim that Euclidean geometry expresses synthetic a priori knowledge of space, thereby not only depriving Kant of an account of geometrical knowledge, but also, and more importantly, putting his entire account of synthetic a priori knowledge under a cloud of suspicion.
Jerrold Katz, The Metaphysics of Meaning, A Bradford Book, the MIT Press, 1990, p.292; an unfortunate confusion in a good philosopher.
Impressed by the beauty and success of Euclidean geometry, philosophers  most notably Immanuel Kant  tried to elevate its assumptions to the status of metaphysical Truths. Geometry that fails to follow Euclid's assumptions is, according to Kant, literally inconceivable.
Frank Wilczek, "Wilczek's Universe: No, Truth Isn't Dead," The Wall Street Journal, June 2425, 2017, p.C4; Wilczek's "philosophers" would be people like Leibniz and Hume, not Kant; and he needs to look up the meaning of "synthetic" and its relation to "inconceivable."
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 is why spherical trigonometry existed for centuries without anyone thinking of it as a nonEuclidean geometry. These issues are discussed in detail elsewhere.
It can be argued however, that Einstein answered Kant by proposing a nonEuclidean (Riemannian) universe that is finite but unbounded (i.e. without an edge). This would be an elegant and beautiful resolution of Kant's dilemma, but unfortunately the observational evidence is against it. The mass of the universe may be just enough to make space flat and Euclidean. It is still possible to imagine a finite but unbounded universe beyond the horizon of the observable universe, but this is largely a matter of speculation, constituting a project that I call "Save the Balloon."
Note on the Metaphysic of Space; the Paradoxes of the Ether
The Ontology and Cosmology of NonEuclidean Geometry
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 Jules Henri Poincaré, Roger Penrose, and even Jerrold Katz making this mistake  and I have added a recent epigraph by Frank Wilczek to the same effect. 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 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 is against it (not enough mass), and the general opinion in cosmology now is that space is "in fact" Euclidean  unless it takes a different form beyond the horizon of the observable universe. 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.
The epigraph from Jerrold Katz above also uses an appeal to Einstein as providing grounds for refuting Kant's theory of space. Since Katz does not explain how this works, we must speculate about his reasoning and must supply for ourselves the points where he has gone wrong and misconstrued Kant, and perhaps Einstein also.
Unlike Boghossian, Katz does not say that Einstein "showed that physical spece was in fact nonEuclidean," but only that somehow the existence of Einstein's physics "refuted" Kant's theory. Perhaps Einstein's use of nonEuclidean geometry in a successful physical theory was enough to credibly contradict Kant's expectation that we know space a priori as Euclidean. Einstein's theory doesn't need to be true, only possible, to falsify the implication of Kant's theory that a physical nonEuclidean space would be impossible.
In this, Katz has actually skipped completely over the nature of Kant's theory about geometry and its relation to nonEuclidean geometry. This leaves out an essential step in the whole business, which is what it meant for Kant to say that the axioms of geometry are synthetic. But if "synthetic" means that a proposition can be denied without contradiction, then logically consistent alternative geometries could be constructed with the denials of Euclidean axioms. That is the essence of nonEuclidean geometry. We see David Hilbert discussing many variations. Leonard Nelson is still the only philosopher I have seen who seems to understand this, even though the principle is not that complicated or obscure.
The next step is to consider why Kant thought Euclidean geometry to be true, i.e. upon what foundation the synthetic axioms of geometry are cognitively grounded. Since the answer to that in Kant is the "pure intuition" of space, i.e. how we are able to imagine it, we must ask whether nonEuclidean geometry can be imagined in the same way that Euclidean geometry is. Honesty, it cannot be. Edward Frenkel says:
The human brain is wired in such a way that we simply cannot imagine curved spaces of dimension great than two; we can only access them through mathematics. [Love and Math, The Heart of Hidden Reality, Basic Books, 2013, p.2; see here]
Thus, the "straight lines," the geodesics, of "curved," nonEuclidean spaces cannot be imagined as straight lines without contradicting their properities, i.e. that a geodesic in a positively curved space will return on itself, like Great Circles on the surface of the Earth. It is kind of rare for mathematicians or philosophers to be as perspicacious, or honest, as Frenkel. Quite a few confuse the use of models or projections with the thing itself. See the examination of projections of multidimensional Platonic Solids.
What does this mean for Einstein's physics? First, it could mean that Einstein's theory, although using nonEuclidean geometry, does not imply that physical space has such a structure, only that mathematics can elegantly represent the force of gravity in this way. The mathematics, as was urged on Galileo, is just "a device for calcuation." Of course, Einstein himself passionately affirmed the Realism of scientific theories; but what has actually been a lot more popular among recent philosophers, and some scientists (like Stephen Hawking), is a "Positivism" that denies any metaphysical truth to science, with theories only meaning, and only justified, by the predictions they make. This would mean that with Relativity, and nonEuclidean geometry itself, whether or not they have anything to do with external physical reality is something we can never know.
Since that is an appalling attitude to have toward scientific knowledge, we can hope it is not true. However, the second altenative about Einstein's physics is that Kant's "pure intuition" of space is more itself a function of the brain, as Frenkel says, and may not actually match the nature of physical reality, especially on the cosmological scales that would have been invisible during human evolution and that therefore, in Darwinian terms, we actually would not expect to be wired into the brain anyway. Since Kant did not believe that space and time applied to thingsinthemselves, this leaves them in a kind of metaphysical limbo; and it opens the possibility that, whatever the nature of what we can visualize, physical reality may differ in many ways from what common sense and our visual imagination can handle.
The third alternative is that space ends up being in fact Euclidean. This is now a real possibility, since, as far as anyone can tell, the universe consists of "flat," i.e. Euclidean, space. This is a contingent matter, and we can hardly rely on a perhaps temporary situation in science to stand as the vindication of Kant. At the same time, Inflationary models of the universe mean that a large, unknown portion of it lies outside the horizon of what can be observed  i.e. the point where the Red Shift becomes the velocity of light. Since both Euclidean and Lobachevskian spaces are infinite, either one creates difficulties for the idea of a Big Bang as a finite event. Thus, without quite admitting what the problem is, there is a strong motivation for construing all that external unobservable space as positively curved, as original conceived by Einstein, which preserves the comfort of a finite Big Bang. As much as such inquiries are reasonable, the element of dishonesty or selfdeception in the whole business is troubling.
A fourth alternative about space is what has been explored at this website. Thus, Einstein's theory is not really about the curvature of space, but about the curvature of a four dimensional spacetime. This means that we can actually analyze the curvature as due to time, not space. Indeed, this makes a lot more sense. Motion is about a displacement in space in relation to the temporal axis. If gravitational motion is due to the structure of spacetime itself, and not to a Newtonian "force," then the displacement follows a curved temporal axis, as we see at right. This leaves space itself as Euclidean. You need four dimensions for a nonEuclidean geometry.
With all this fun, there is nothing left here that would put Kant's "entire account of synthetic a priori knowledge under a cloud of suspicion." Instead, we have no more than additional reminders that even good philosophers can misunderstand the implications of something so basic as Kant's definition of "synthetic" truth, which undermines their entire analysis of Kant's theories of space and geometry.
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, since the light carrying the subsequent states of the clock would never catch up to the tram (which, hopefully, was not about run over anyone). 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. The lore about this now is that Einstein saw some roofers at work in 1907. It occurred to him that if a roofer fell off the roof, a not unusual event in the trade, he would experience weightlessness during the fall. Was this weightlessness physically identity to that of free fall in space?
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.

Curiously, these cases to not cover the experience of weight during rotation. Both physically and metaphysically this is a signficant feature of the business. This is part of the more general question of angular momentum in physics.
Three Points in Kant's Theory of Space and Time
The ClarkeLeibniz Debate (17151716)
Einstein's Equation for Gravity
Some Metaphysics of Angular Momentum and Gravity
Note on the Metaphysic of Space; the Paradoxes of the Ether
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 determine the absolute motion of the earth against the ether. Not only did this fail (the velocity of light in a vacuum 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.