A Rational Cosmology
G. Stolyarov II
A Journal for Western Man-- Issue XLI-- September 16, 2005
Note: This essay is the third chapter of Mr. Stolyarov's new comprehensive filosofical treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, which can be ordered in electronic format for only $2.50 at http://www.lulu.com/content/140855. Free previews, descriptions, and information on A Rational Cosmology can be found at http://rationalargumentator.com/rc.html.
There is no such thing as “space.” In order to be defined as an entity, space would need to meet the first ontological corollary and be the sum of its qualities. In order to pass this test, space must have some qualities in the first place. But it lacks any qualities whatsoever. “Space” cannot be said to have mass or a finite volume (as previously proved, there is no finite boundary at which “space” officially ends, nor is there a finite shape that “the entirety of space” can be fit into.) Moreover, though separate stretches of what can be termed “space” are measurable (such as the distance between Entity A and Entity B), linear measurements in three dimensions cannot be attributed to the totality of space. As an example, it would be absurd to propose that the entirety of space is twelve billion kilometers long, three billion kilometers wide, and sixteen billion kilometers high.
We have affirmed that space is not an entity. But what can be logically meant by the referent “space”? There are in fact two referents concealed in one, each of which has a different purpose with which it is used. Here, they shall be termed space-as-absence and space-as-relationship.
The term “space-as-absence” is synonymous with “void,” “emptiness,” and “nothing.” “Space-as-absence” denotes merely the non-presence of entities. It is essential to note that space-as-absence is not an existent. As follows from the axiom of existence, something is, but nothing is not.
As the thinker Manfred Schieder demonstrates in his treatise, “Ayn Rand, I, and the Universe,” it is not even possible to logically use the word “is” following a term such as “space-as-absence.” Schieder describes two premises that are essential for an accurate description of the universe and of “space-as-absence.”
1) "What is, is"
After the enumeration of 2) a blank space has been left on purpose, to better convey the sense of the premise immediately resulting from the first one, which is: "What is not, is not". As said before, our language is so object oriented that it cannot describe what is meant by the statement "What is not, is not" in any other way than by not saying it, since "what" already implies an object and "not" is the negation of either something existing or of negating the action of something existing.
Since “space-as-absence” does not exist, neither as an entity, nor as a quality, nor as a relationship, nor even as a totality of entities like the universe, it is fruitless to discuss it further. There is nothing more to be discovered about nothing!
The term “space-as-relationship” is synonymous with “distance” and “separation.” In order to have any meaning, it cannot be a metafysical primary. Rather, it must involve two or more distinct entities, or a single entity capable of motion and having its current position compared with respect to some earlier or later position.
It is self-evident (ubiquitously observable by all human sensory faculties) that not all distinct entities touch each other. There exist abundant examples of particular entities whose boundaries are not adjacent to the boundaries of other entities. As an illustration, the boundary of the entity “Pluto” does not contact the boundary of the entity “Big Ben.” The entity “Taj-Mahal” also does not contact the boundary of the entity “Big Ben.” Yet it is also self-evident that the entity “Taj-Mahal” would not need to alter its location to as substantial a degree as the entity “Pluto” would in order for its boundary to be immediately adjacent to that of the entity “Big Ben.” Thus, the degree to which the boundary of one entity can be separated from that of another can differ in magnitude. This variable separation is the reason for man’s need to use the term “space-as-relationship.”
Moreover, let us presume that the entity “dog” is running in such a manner as to alter its position over time. At time X, it will be farther from its starting location than it was at time (X-1). The dog at its starting time is separated from the dog at time (X-1), and even farther separated from the dog at time X. The magnitudes of these two separations also differ. Thus, it has been demonstrated that the use of space-as-relationship is also necessary when relating an entity to that same entity at a different time, provided, of course, that this entity is capable of altering its position in any manner.
If a single homogeneous entity, like a singularity, were all that existed, however, “space-as-relationship” would be useless, as this entity would not be capable of any motion whatsoever (this was explained in our earlier discussion of homogeneous entities’ inabilities to alter their qualities). The fact that “space-as-relationship” has its self-evident and demonstrable applications to describing the universe, and that it could never have come to describe a universe with only one homogeneous singularity, further verifies the impossibility of the universe being created by such a singularity.
Space-as-Relationship in Three Dimensions
Space-as-relationship is not, moreover, a single relationship. Rather, it is a threefold relationship, describable by three parameters, known as dimensions. This is primarily deduced not from the nature of the relationship “space,” but from the nature of all entities as such. Here we find the need to define several qualities which must be possessed, in some quantity, by any entity. We shall call these the ubiquitous qualities of entities.
1) Matter- Matter is otherwise known as the constituent quality of entities. Matter is, simply, that, which entities are made of, and without which they cannot have any other qualities. It is not the province of ontology or cosmology to describe what the fundamental “building blocks of matter” (i.e. the entities that would represent Democritus’s concept of “atomos”) are. The specific-observational sciences must discover whether such fundamental building blocks exist, how many types of them there are, and how they look like and behave. Cosmology has only to point out that matter exists, and exists as a quality of every entity.
It may be asked here, “What, then, are such things as freedom, beauty, and peace, which are not in themselves composed of matter?” These are indeed not entities, but rather relationships between entities that are composed of matter. Freedom cannot exist without the individual who is free, and the individual is a material entity. Beauty, whether it be in a painting or a piece of music, cannot exist without the material canvas that holds the painting, nor without the instruments which emit the music. Peace cannot exist except among material individuals who decide not to relate to each other in a certain aggressive manner. These are highly abstract and complex relationships, which, for the sake of word economy, men often speak of as having certain “qualities” of their own. However, these qualities do not pertain to the relationships in themselves, but rather to every entity that undertakes these relationships. It might be said, for example, “Freedom has the quality of the non-existence of government economic regulation.” In the context of word economy, of course, this is an acceptable expression, provided that one knows what one is truly talking about. The words for which this intellectual shorthand stands are more numerous:
“The existence of individuals who partake in the relationship, ‘freedom,’ and who simultaneously partake in the relationship, ‘government economic regulation,’ is impossible.”
The quality “matter” can be measured, and the measurement of matter is called mass. It is, of course, self-evident that one entity can have a greater or smaller mass than another. This mass can conceivably be of any finite magnitude, but must be of some finite magnitude.
2) Volume- Volume is an entity’s expanse. Anything possessing the quality, matter, must have an expanse that corresponds in some proportion (though it could correspond in a variety of proportions) to the amount of the quality ”matter” that the entity has. That is, if the quality “ matter” exists in an entity, it must have a real manifestation; this manifestation is volume. If the quality “matter” and the quality “volume” did not coexist and were not inextricably connected, we would encounter absurdities. Volume without matter does not describe anything whatsoever. It would be just an arbitrarily picked region of space-as-absence, the latter being nothing whatsoever. Matter without volume, too, describes what cannot exist. This would be tantamount to the quality, matter, existing nowhere, i.e., not existing, and the consequences would be the same: space-as-absence. It is self-evident that both qualities must be present, in some magnitude and combination, in every entity.
3) Linear Measurements: Length, Width, and Height- A line, in Euclidean geometry, denotes the shortest conceivable path which an entity would need to travel in order to reach any location from any other location. The linear measurements of an entity are the measurements of those qualities which express the separation of various parts and boundaries of that entity with respect to the shortest conceivable path between them. There are three independent linear measurements, which are mutually perpendicular. Any other linear measurement is in fact some combination (a vector sum) of any or all of these three mutually perpendicular linear qualities, which are known as length, width, and height (or, in the three-dimensional Cartesian coordinate system, as the x, y, and z-axes). Length, width, and height, as qualities, can also be termed dimensions.
It is important to note that these dimensions do not exist independently, but rather pertain to the entities that exhibit them. Each entity must have a certain maximum length, width, and height, though these measurements may vary in some relation to each other, i.e., depending on the particular region of the entity one examines. (For example, an entity may have a certain height somewhere along its length, and have its height increase or decrease farther along its length.) In relation to one of the three dimensions, an entity can conceivably have any measurements in the other two dimensions, but must have some measurements.
As a primary, it is not space-as-relationship that is three-dimensional (as relationships cannot exhibit qualities qua relationships), but rather every single entity that exists or can conceivably exist. It has already been demonstrated that different entities can be separate in their boundaries, and the degree of this separation is precisely what space-as-relationship denotes. Because, moreover, all entities exhibit the three dimensions as qualities, their separation can only be expressed as a combination of three measurement parameters. After all, one entity can be separated from another by a distance A in the X direction, as well as in either the Y or the Z direction. In each of these three cases, the relationships are not the same, and were there four entities thus positioned (including the original entity and the three entities separate from it), each would occupy a distinct position and would be separated from every other.
To render it simpler to relate any one of the multitude of entities in existence to any other among such entities, it is possible to devise a coordinate system based on three numerical parameters which measure each of the three qualities known as dimensions. The measurement interval deemed a unit in the coordinate system must necessarily be an arbitrary product of human decision, since no such thing as space exists, and thus no absolute markers on it are provided to determine what the one true unit must be. However, this arbitrarily selected interval must uniformly denote a unit in all instances in which this coordinate system is used. It is impermissible to have a given interval declared the stretch from position 0 to position 1, and then have position 2 pronounced to be thrice as far from position 0 as position 1.
Moreover, the coordinate system must assume an arbitrary starting point, or origin, in which each of the three dimensional parameters has value zero. This point could be located on an entity or outside it, however human convenience may dictate, so long as due caution is exercised not to mistake such a point, or any point for an entity in itself. Every point on the coordinate system, including the origin, is merely a part of a mental model used by man to interpret the real relationship between entities in three dimensions.
The necessity of points is evidenced in the fact that it is conceivable for any entity to assume any degree of proximity or distance with respect to any other entity. It is possible for the entity “dog” to be separated from the bouncing entity “ball” by distances of (2,3,4) units in each dimension. The dog then endeavors to approach the ball and bounce it upward against its head, somewhere in the process achieving a separation of (0,0,10) units from the ball. It is also conceivable for a spaceship to then pick up this ball and carry it far beyond the dog’s access, reaching a separation of (1050, 1053, 2040) units between dog and ball. Any combination of finite, rational numbers, however large or small, can express the degree of separation between real entities, and thus must be available via an accurate model of said separation. Thus, the idea of a “point,” some hypothetical position denoting a particular degree of separation from some other such hypothetical position, becomes necessary.
This does not, however, mean that the point is an actual existent, nor that the point can contain, in itself, an entire entity. Entities must, by the very fact of their existence, have some measurements in all three dimensions, and thus cannot be constrained to a single dimension, and, even more so, to a non-dimensional point. However, it may be proper to state that a given point may identify precisely with one of the positions along an entity’s outermost boundary, i.e., describe the precise extent of an entity’s measurements of either length, width, or height.
The impossibility of a point ever containing an entity is illustrated every time a mathematician seeks to represent a point on a piece of paper. It is impossible to draw a “point” on paper that does not have some measurement in each of the three dimensions. This “point” might be a millimeter long and a millimeter wide, and the grafite from the pencil used might extend to a height of a tenth of a millimeter, but some length, width, and height are inevitably possessed by the grafic representation of the point. Any such specific grafic representation could be considered an entity, but not the concept of the “point” which it is supposed to depict.
This is further proof of the impossibility of the existence of “singularities,” a proposition upon which the Big Bang and Big Crunch theories rest. A singularity conceived of as a sole point containing mass, but mass without volume, i.e., a point-entity, is a contradiction in terms. We have already explored, via numerous perspectives, the truth that mass and volume must be mutually present in every conceivable entity.
The Euclidean System: Points, Lines, and Planes
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, 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 his 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 ≠ 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 fysical isolation of such a quality, which remains firmly integrated into actual entities, and is inseparable from them.
The Euclidean plane, a two-dimensional construct, enables the study of an even vaster and more complex interplay of qualities than does the Euclidean line. The plane is, in effect, a mental model isolating for study all the possible variations that can exist in the combination of any of two of the three linear dimensions. Two-dimensional shapes, curvatures, and patterns may be the results of such variations, which can be found as emergent qualities (qualities whose existence is based on a certain interplay of more basic qualities) in entities. Circles, for example, are a quality possessed by the entity, “cylinder,” which, being three-dimensional, can exist in reality. Each of the properties of shape and curve constructs on a Euclidean plane will hold if these shapes and curves are qualities of a given entity; the sum of the angles on the surfaces of a triangular prism will always measure 180 degrees, given that this prism possesses the quality, “triangles.” A three-dimensional projectile will still follow a parabolic path in two of three dimensions (and will not alter its parameters in the third). A cylinder’s rim will measure 2∏ times the radius of its surface. Thus, we see how the findings of a Euclidean investigation of the isolated interplay of two dimensions can be applied, with perfect accuracy, to actual, three-dimensional entities.
Moreover, elementary and ubiquitously accessible empirical observation yields the conclusion that, though entities can never be purely two-dimensional, there is nothing barring the surfaces of entities from being such. The entity, “cube,” for example, is three-dimensional, and, presuming that man possesses a technology precise enough to refine the faces of a real cube so that no ridges, creases, or miscellaneous imperfections may remain on them, the resulting perfectly smooth surface would be two-dimensional. No matter which point one picks on the side of the cube, it would have the same numerical coordinate in one certain dimension that does not vary on the two-dimensional surface. (Rather, such a dimension would constitute the cube’s depth, and the measurement of this dimension would be necessary in order to describe those regions of the cube which are beneath its surface.)
Whether or not ideal two-dimensional surfaces have yet been observed in nature or obtained via man’s technological precision is not the province of cosmology to judge. Cosmology only informs man that such surfaces are conceivable as existing in reality, as parts of real entities. Of course, not all surfaces are two-dimensional. Surfaces may be three-dimensional, as the surface of a sfere, cone, or any other entity with non-planar contours will demonstrate.
Moreover, whenever Euclidean geometry ventures to describe three-dimensional relationships and shapes, it begins to address the entire interplay of linear measurements necessary in comprising an entity. Sferes, cubes, cylinders, and prisms, for example, are all conceivable as actual entities. Of course, in order to be such, they would also need to be composed of the quality, “matter,” which Euclidean geometry does not directly address. Thus, three-dimensional geometry can express, with perfect accuracy, the entirety of the linear measurements applicable to an entity, and study these qualities in isolation from the remainder of the entity’s qualities, such as matter. Though immensely realistic, three-dimensional geometry, like all mathematics, remains a model, not an actual existent.
G. Stolyarov II is a science fiction novelist, independent filosofical essayist, poet, amateur mathematician and composer, contributor to organizations such as Le Quebecois Libre, Enter Stage Right, the Autonomist, and The Liberal Institute. Mr. Stolyarov is the Editor-in-Chief of The Rational Argumentator. He can be contacted at firstname.lastname@example.org.
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