In the previous post, I explained how the second postulate, like the third postulate, is not really needed, if one understands that the assumptions it makes explicitly are really a consequence of the logic of the first postulate; That is, if it is assumed that the units of space-time are all that exist, and that they exist as reciprocals, and the only way a non-unit ratio of these units can be formed is if the scalar "direction" of their increase can change to a scalar decrease, and thereafter alternate between an increase and a decrease of scalar magnitude, then it follows that the algebra of the magnitudes so produced must be ordered, commutative and associative, by definition, which is really all the second postulate states.

That there is no other possibility follows from the nature of scalar quantities themselves. The mathematics of scalar numbers with two "directions" corresponds to a point, even though a point has no extent and therefore cannot have "direction." However, the potential for "direction" is still there. This potential direction is expressed as a zero power in the number's exponent. Since each of the three dimensions of geometry also has two "directions," this leads to a correspondence between numbers of base 2 with exponents of 3 or less.

In other words, this leads us to the numerical pseudoscalars: 2¹ (corresponding to geometric lines), 2² (areas), 2³ (volumes). However, an expanding (contracting) pseudoscalar still has only two "directions," the expanding "direction" and the contracting "direction," and thus it is scalar in that sense, even though it has extent. It follows then that the algebra of these pseudoscalars is also scalar; that is, it follows that it is ordered, commutative and associative.

When the algebras of these multi-dimensional pseudoscalars is examined, however, one finds that there is a limit at dimension four (counting 0 as the first dimension.) Mathematicians say that these four algebras are the only known normed division algebras. Something called the Bott periodicity theorem proves mathematically that there is no new phenomena beyond the three (four counting 0) dimensions of the number 2.

Just how remarkable is this? It is very remarkable. We find the same numerical limit in algebra as we do in geometry. Three dimensions (four counting 0) is the limit of nature, geometrically and mathematically speaking. But how does this relate to non-Euclidean geometry? Euclidean versus non-Euclidean geometry has to do with parallel lines. Can the natural limit of three geometric and mathematical dimensions (four), each with two "directions," be applied to the parallel, or so-called fifth postulate, which we find so problematic in Euclidean geometry?

I think it can in the following way: the development of non-Euclidean geometries, though they can be shown to be mathematically and logically consistent, are usually limited to less than three dimensions. For example, the foundation of hyperbolic geometry starts with the 2D surface of a sphere, while the foundation of projective geometry starts with the 2D surface of a plane. However, if we consider these in the context of the 3D volume of a ball, we can make some interesting observations.

In the case of hyperbolic geometry, the apparently flat and parallel line segments of the surface actually converge (diverge) at the poles, seemingly disproving the fifth postulate of Euclid geometry, but the geometry of the radii stemming from the center of the ball and penetrating the surface is not considered. Clearly these vertical lines are not parallel, regardless of how large the ball and "flat" its spherical surface may be.

Similarly, the point on the horizon of projective geometry, where the "parallel" lines converge, is not considered, as constituting the center of a ball, where the lines are clearly not parallel, and not only converge on the point, but necessarily diverge on the other side of it.

Moreover, if we chose any one of these non-parallel lines reprenting the radii of the ball, then we can say without doubt that there are only two other lines in the infinite set of them, that are at right angles to it. There cannot be any others, and each of them reveals an additional dimension. This means then, that in a three-dimensional universe, motion not confined to a given dimension, must be in another dimension, and the number of dimensions cannot be more than three.

Finally, if we draw a line parallel to any one of the radii, we know that the two lines will never meet within the ball, since the parallel line cannot pass through the center point, by definition. Thus, we conclude that the universe must be flat, or that its geometry must be Euclidean, just as has now been observed, with all this implies.