A Rational Cosmology: The Euclidean Line

The work of the ancient Greek mathematician Euclid has been perhaps the greatest leap in human history toward the understanding of real spatial relationships among entities.

The geometry that Euclid laid the foundation of (today known as Euclidean) functions perfectly as a model to study the dimensional qualities of entities, provided that it is always remembered that the tools used by the Euclidean model, as by all mathematics, are just aides for the human cognition, and do not represent things in themselves.

The sum total of Euclid's findings and derivations need not be explicated here, as they are easily accessible in any elementary treatise on mathematics, and their systematic elaboration is not the purpose of cosmology. Rather, cosmology seeks to discover in what manner Euclid's system is capable of representing reality using constructs, such as points, lines, and planes, which cannot possibly represent any real entities qua points, lines, and planes.

Since the subject of points has already been extensively covered in "Coordinate Systems," we move now to the matter of lines, or one-dimensional constructs.

Though no entity could have only a single dimension (as this would deny it the quality of volume), it must be recalled that each of the dimensions is a quality representable by a linear measurement, a line being the shortest distance between two distinct locations.

To measure dimensions in any other manner but linearly is absurd and standardless. When one admits measurements of arched dimensions, parabolic dimensions, zigzag dimensions, or dimensions twisted and curved in any manner one fancies, one is allowing one's whim, not any objective fact of reality, to decide the magnitude of a given separation. Moreover, one commits the contradiction of claiming as one dimension what inevitably requires two parameters to describe. Since A=A, and 1 does not equal 2, dimensions are linear.

To isolate a line and investigate whatever pertains to such a construct, as Euclidean geometry undertakes, is merely to examine one of the qualities possessed by entities and to study what this quality is and how it is made manifest. This does not render the qualities of length, or width, or height, which can be examined through a study of lines, independently existing, as all qualities can only exist as derived from the entities that exhibit them.

The Euclidean model focuses upon the study of qualities that pertain to entities, and can do so without necessarily analyzing the entire entities that have such qualities.

For example, it is possible, in reality, to encounter the necessity of determining how wide the separation between two boxes of identical shape and volume is. These two boxes are on a level floor, aligned with one another, and have no other parameters separating them except one.

It is quite permissible to use the model of a line on which two points can be designated the extremities of one box, and two further points-the extremities of the other, and thus compare the boxes' position with respect to the sole quality which differentiates them, separation in the dimension of width. All other qualities the boxes possess are simply irrelevant in the context of this study, but the Euclidean model can still perfectly represent the quality that we do wish to examine.

Once again, it must be remembered that the mental isolation of the quality in question that man's mind performs is in no manner akin to a physical isolation of such a quality, which remains firmly integrated into actual entities, and is inseparable from them.

Read other parts of "A Rational Cosmology" by clicking here.