Note: This essay is the fifth 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://www.oocities.org/rational_argumentator/rc.html.

Motion is the change in the three spatial dimensions facilitated by change in the one temporal dimension.

Having now an understanding of the four different qualities we know as dimensions, three spatial and one temporal, which entities must exhibit all four of and cannot conceivably exhibit any more than four of, we may proceed to describe a fenomenon which is a subcategory of the broader term, “change,” namely, motion. When we observe motion, what we truly perceive is the change of measurements of the three spatial dimensions pertaining to an entity. The entity’s uniform accumulation of the quality, time, is, of course, what makes this change, like all other changes, possible.

We can verify the occurrence of the motion of a given entity when we note that its spatial separation from other entities has changed in some manner, that is, the threefold magnitude of the relationship, “space,” between that entity and certain other entities has been altered. Such relations of multiple entities’ past and present positions are sufficient to assert beyond doubt that motion has happened, but they do not in themselves define what motion is. This is so because entities’ relative positions to one another can change due to a multitude of different events taking place. A can move toward B until they are separated by X units, or B can move toward A until they are separated by X units. These two events are not one and the same, though defining motion solely in terms of relationships among different entities would conflate them. The sole means of avoiding this pitfall is to define motion solely in terms of the entity said to be moving, which would entail the use of an indispensable spatial coordinate system.

Due to uniformly accumulating measurements of the quality of time, it is possible to observe spatial separation in certain entities not only with respect to other entities, but also with respect to themselves at past times. For example, we can state that, 24 hours ago, a ball’s center was located at point A. Presently, it is located at point B, 50 meters away. By relating the ball’s position when it has age X (in hours) at present to its position when it had age (X-24), we can claim with certainty that the ball’s center has indeed moved a net distance of 50 meters during 24 hours. By defining motion as a relationship between an entity’s present and past configurations of qualities, it is possible to refer to motion on all surfaces, in all environments, and in contexts where all entities other than the one explicitly analyzed can behave in any of the entire range of conceivable ways. Points A and B could be located beyond the Earth’s orbit, in an ocean, on a ramp, or slightly above a floor, and, if one configuration of a ball’s qualities entails its presence at point A, while another necessitates its presence at point B, the ball could be said to have moved from A to B. A and B need not be entities themselves, however; they are merely reference points. This is where a coordinate system is necessary to identify them as such and to relate the entity’s past and present states to each other, as well as to the spatial measurements of all the entities in the universe, by means of a set of uniform parameters.

The question may arise as to how one might precisely define the location of departure for a given moving object (i.e. the location from which motion was initiated) and its location of arrival (i.e. the location at which motion ends). That is, how, given two different sets of spatiotemporal coordinates for an object at point A and the same object at point B, how can one state that the object moved from A to B and not vice versa? The answer to such an inquiry would be that, out of all configurations of spatiotemporal coordinates pertaining to a given event of motion, the configuration with the smallest measurement of the entity’s quality, time, also pertains to the location of departure, while the configuration wherein the entity’s age is greatest out of the set pertains to the location of arrival, since arrival must take place after departure, and an object must accumulate age uniformly throughout the motion, as it would in stasis.

Thus far, we have spoken of an object’s net displacement, i.e., the ultimate change in its spatial qualities as a result of motion during a given time interval. However, an object’s motion from A to B in a straight line will be a different type of motion from motion in a zig-zag pattern from motion along a curve. If the object moves continuously (which term we have yet to define), this will be a different motion from motion that is interrupted somewhere along the way by interludes of stasis. The mathematical endeavors of Sir Isaac Newton have been able to produce a valid model for us to analyze the differences among these types of motion. This treatise shall aim to demonstrate how the model of Newtonian calculus can be interpreted strictly in terms of the entities that exist and their actual qualities, so that this model might be used in coordination with the proper generalizations that its correctness presupposes. 

Continuous and Intermittent Motion

First, we shall address the question of continuous versus discontinuous motion, as Newton’s calculus provides the most direct investigation of the former of these. If we were to define continuous motion, we would need to take into account the fact that, if continuous motion is the opposite of intermittent motion, it is motion not interrupted by periods of stasis between an entity’s departure from and arrival at, respectively, the two points of reference selected by the observer. If there are no periods of stasis involved in continuous motion, then, by extension, this must mean that, were we to select any amount of particular combinations of spatiotemporal parameters pertaining to a continuously moving object, we would never see the correspondence of the same spatial parameters to different temporal parameters unless the object’s motion entails passing through the same point several times, as would be characteristic of an object traveling in a loop, for example. In that case, if the amount of times a continuously moving object passes through a given point C is n, we can never encounter more than n sets of the spatial coordinates of C plus some temporal coordinate, different for each set.

In contrast with continuous motion, were we to examine intermittent motion during which an object has stopped at point C, we could find any number of sets which each have the spatial coordinates of C plus some temporal coordinate, different for each set. Let us presume, for example, that the object stops at C for a mere second, 4 seconds after it has initiated its motion. We shall also let (Cx, Cy, Cz) be the set of point C’s spatial coordinates. Though the object lingers at C for only a finite amount of time, it is possible to take an indefinite amount of valid spatiotemporal coordinates for such a condition.

(Cx, Cy, Cz, 4.1), (Cx, Cy, Cz, 4.01), (Cx, Cy, Cz, 4.001), (Cx, Cy, Cz, 4.0000000000001) are all correct spatiotemporal parameters describing the object during the static stage interrupting its motion from A to B. Seeing as it is possible to conceive of an indefinite number of decimals between any two integers, or any two rational numbers, and it is possible to create a time scale based on units of any conceivable magnitude, whenever an object is not involved in continuous motion, there is no exhausting the valid spatiotemporal parameters that might describe it.

However, we must also add that motion which is intermittent using some two points of departure and arrival as points of reference will be continuous using, in this manner, some other two points of departure and arrival. For example, an object moving from A to B and stopping only once in the process at C can be said to move intermittently from A to B, but continuously from A to C and from C to B. The fenomenon of intermittent motion is, therefore, nothing more than continuous motion interspersed with rest, and, during the process of intermittent motion, the time intervals over which the object actually moves cannot be characterized by anything but continuous motion.  Thus, continuous and intermittent motion are not truly contrary or mutually exclusive states. The latter designation is better attributed to continuous motion and rest, or spatial stasis. Nevertheless, the fenomenon of “intermittent motion” is still one that needs to be described within the context of the proper frame of reference. If we are concerned about a car traveling from Chicago to New York, we will treat the situation differently if the car undertakes the ride without ever halting until it reaches New York than we will if the car makes a stop in Cleveland, or two stops in Cleveland and Philadelphia. Some parameters are different among these entire experiences of the car along its trip, and this difference is the amount of times, if at all, that the car comes to rest. (Another difference might be, how long the car remains at rest in each case.)

In summation:

An object in continuous motion may be described as exhibiting any three particular spatial parameters only a limited amount of times during its motion.

An object at rest, be this rest a part of some fenomenon of intermittent motion or pertaining to an entirely spatially static object, may be described as exhibiting some three particular spatial parameters an inexhaustible amount of times during its state of rest.  

The Calculus and Continuous Motion

The calculus is a mathematical system that enables one to distinguish between not only different magnitudes of continuous motion (i.e. some objects moving faster than others) but also the temporal trends that these magnitudes follow (i.e. acceleration and deceleration). When one knows the equation modeling an entity’s position as a function of time, differential calculus permits one to find a model for its velocity (first derivative) and acceleration (second derivative) as a function of time. If one knows any of the latter two, plus values for initial velocity and/or position of the entity in motion, integral calculus can assist one in creating an accurate model for the entity’s position at any time at which it is moving. The mathematical structures entailed in the calculus are well known and can be found in any comprehensive textbook on the subject. What shall concern this treatise in regard to the calculus is similar to what had concerned it in regard to Euclidean geometry. Euclidean geometry, though in itself merely a model not equivalent to the entities it describes, is nevertheless capable of describing all entities’ spatial qualities perfectly. Thus, on the matter of the calculus, the subject of our investigation is the manner in which this mathematical model is capable of describing with perfect accuracy the motion of entities while remaining a mere model not equivalent to said motion.

The derivative of a position equation, as a function of time, expresses an object’s so-called “instantaneous velocity,” or velocity at a given point in spatiotemporal coordinates. We can, however, conclude that, if an entity is said to be in motion, it cannot be said to be in motion for only a single instant. That is, an entity’s motion cannot occupy only one point in spatiotemporal coordinates, just as its position cannot occupy only one point in spatial coordinates. Just as the Euclidean model of an entity’s position necessitates that any real entity occupy an inexhaustible amount of points, though some of these specific points can be said to lie on the entity’s outermost boundary or its center of mass, so does Newtonian calculus necessitate that any entity in motion must be describable by an inexhaustible amount of different spatiotemporal parameters (though motion, as we have discussed previously, places limitations on how many times one can find the same sets of spatial coordinates in even a limitless array of parameters describing a moving entity’s position).

Therefore, no one instantaneous velocity, nor one instantaneous acceleration, nor one nth derivative of a position function, can completely describe an entity’s motion. The question before us, then, is, can it accurately describe said motion within the limited point of view that such an approach necessarily entails?

When discussing the necessity of the point model in Euclidean geometry, I wrote that “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.” Correspondingly, within the realm of motion, any combination of finite, rational numbers, however large or small, can express the degree of spatiotemporal separation between two states of a moving entity. The addition of the fourth coordinate, time, to this consideration, implies, in particular, the possibility variance in the time separations between two of a moving entity’s states. Thus, an entity could travel between two spatial points in one second, or in ten, or in 10⁴⁴. But this variance is just as true for points that are separated by 1000 units of distance as it is for points that are separated by only 0.001 such units. No matter how small the interval of spatial separation between the two points used as a reference frame in the model becomes, it remains conceivable for an entity to arrive from one point to the other while its measure of the quality, time, increases by any of an inexhaustible range of quantities.

In Newtonian calculus, the derivative of a position function is obtained by means of taking a limit. That is, as we continue to indefinitely decrease our reference frame of an object’s motion, and “narrow” this reference frame so that it continually approaches a given point (though it can never quite get there, since there is no sense in describing motion from a point to itself), what can we state about the entity’s motion? The derivative function for an entity’s instantaneous velocity can always be used, in combination with our knowledge of the time of an entity’s presence at the given point, to provide a numerical value for speed. This value allows us to state how an entity behaves as it moves through a given point at a given time, yet, in itself, motion through a point implies motion from some other point and to some point still, the same able to be said about motion at a given instant in time. Within the context of such an insight, an instantaneous velocity can only be interpreted as the constant rate of motion that an entity would follow had it undergone its motion in precisely the manner in which it is known to have undergone motion through a given point at a given time. Velocities are always given in units of distance per units of time (meters per second, for example), and, an instantaneous velocity answers the question: if an object continued to travel just as it has traveled through a given point at a given time, how many meters would it traverse in a second? (Other units may be used here where appropriate, depending on one’s chosen time scale and coordinate system.)

Precisely what an instantaneous velocity describes is no mere technicality—it is essential to our knowledge of what motion is and how to take account of it. The human perception of time is analog: men view time’s accumulation as a continuous flow rather than a series of discrete instants—all the better for it, of course, because there is an inexhaustible amount of temporal coordinates between any two points on a linear time scale, and, were we to perceive time in discrete quanta (as some empiricist-positivist devotees of that twentieth-century pseudoscience known as “quantum mechanics” suggest), it would take us an infinite amount of time to perceive any finite span of time, however small, an evident logical contradiction. Because we cannot perceive any single instant of time, or what happens during it, we can only explain an entity’s state during said instant by the model of Newtonian calculus, which extrapolates that behavior onto an analog interval of time, which is accessible to human comprehension. This is the reason why the graf of a function’s derivative at a given point is the straight line tangent to that point. The tangent most often does not correspond with the graf of the motion itself, as rates of motion tend to change for most moving objects, but it allows us insight into what course the object follows at the point along the graf to which the tangent is drawn.

In the real universe, fundamentally, all that exists is entities, and entities have measurements in all four dimensional qualities. The human mode of perception, absolutely and undeniably correct, indeed fathoms entities as having measurements in all four dimensions, and, in the case of time, accumulating age in a uniform, analog fashion. The Euclidean model of geometry, however, further verifies the accuracy of the human ability to comprehend these properties by allowing, through the use of points, a description of any part of an entity’s spatial position. Similarly, Newtonian calculus reinforces the correctness of human perception of motion by allowing, by means of instantaneous velocities, and, by extension, accelerations, changes in acceleration, and even changes in those changes, so on indefinitely, the observation of any part of an entity’s motion. Due to the discoveries of Euclid, no point comprising an entity need be unaccounted for, and, due to the ideas of Newton, no time interval during which an object moves need be left unexplained.

But it cannot be overemfasized that, however indispensable in mathematics and human cognition, points and instants are not real existents. Entities are real existents, and entities can never be confined to points, nor their behaviors to instants. Whatever special tools and techniques its analysis might require, reality remains, and shall always remain, analog and four-dimensional.

Differences in Paths of Motion

Thus far our discussion has concerned itself with an explanation for motion trends’ variation with time and the manner in which the model of calculus elucidates these. However, if these variations were the sole ones possible in motion, all of them could occur with respect to an entity moving along some particular path, such as a straight line. However, it is ubiquitously observable that an object moving between some two points, A and B, can conceivably follow one among an indefinite variety of paths, be they curved, looped, bent, or any conceivable combination thereof.

Having merely the entity’s points of arrival and departure as our reference frame, we cannot adequately describe its particular path. Thus, other considerations are necessary. As with motion trends with respect to time, a narrowing of reference frame will result in a more accurate understanding of the entity’s path, and, if the path between the chosen two points happens to be perfectly linear, the approximation mirrors reality precisely, and net displacement, combined with an understanding of the object’s velocity and changes therein, will suffice for a true description of its motion. However, if the path is not linear, it would be necessary to refer, again, to the model of Newtonian calculus.

Newtonian calculus may be applied as a model for the description of entities’ precise paths in a similar manner to its ability to describe their rates of motion. However, instead of relating a coordinate of spatial position to a coordinate of time, calculus used in the description of paths relates a coordinate of spatial position to another coordinate of spatial position. The relationships involved still compare measurements in one dimensional parameter to those in another, however, both dimensions involved (or all three of them, given a multivariable equation) are of necessity spatial, since any path but a line requires two or three dimensions to accurately describe. A derivative of a position equation entirely in spatial variables, again a constant when the precise spatial parameters of the point at which it is being analyzed are known and substituted into the expression for the derivative, gives the trend of an object’s motion through the given point in spatial coordinates, i.e., the manner in which the object would have moved had the relationships of quantitative change among its spatial coordinates maintained throughout the entire motion the same nature as they possess when the object is moving through the given point. Thus, no matter what point along an object’s path one examines, one can state precisely how the entity is moving through that given point, and, it therefore follows that, via this model, no part of the entity’s spatial motion need be left inexplicable.

It would be fitting here to note that, even when the position equation of an object is one that never has a linear equation for a derivative (such as y=e^x, whose nth derivatives are all equal to e^x as well), the particular derivative (as well as the particular position coordinates of an object at a given point) will always have a numerical constant for a value, since x is presumed to be a known value of a point measured relative to the three-dimensional coordinate system we must necessarily use to accurately describe spatial qualities. Moreover, the question may arise as to objects in motion along paths that combine in themselves a multiplicity of functions (such as a “v-shaped” or “absolute value graf” path) and could be said to have different derivatives for the same point, depending on the function in relation to which the derivative is taken (i.e., at a point of intersection between the two segments constituting a “v-shaped” path, one could conceivably take a derivative of either the positively or the negatively sloped segment). Nevertheless, the presence of this multitude of possible derivatives need not be contradictory, provided that each is confined to its proper context. The derivative of the function describing the path leading to the point of intersection describes the object’s motion as it enters the point, whereas the derivative of the function describing the path leading away from the point of intersection characterizes the object’s motion as it leaves the point. During the instant at which the object moves through the point, its path changes without its motion being halted. Since the human (and correct) mode of perception of spatiotemporal fenomena is analog, men cannot directly grasp happenings encompassing an instant of time and a point in space (as these are zero-dimensional by all standards), but, nevertheless, the human analog perception is capable of encompassing that instantaneous change and, via the model of calculus, pinpointing exactly where it occurs. The presence of multiple possible derivatives at a point is a sign provided by the model of calculus that the given point is indeed a location for instantaneous change in paths of motion.

Finally, the all-pervasive question of this exploration may be put forth: if the model of calculus may describe a moving object’s trends through spatial points and temporal instants, and human beings cannot by nature perceive points and instants, does the model of calculus nevertheless describe precisely what humans perceive? The answer is yes, for, indeed, all the inexhaustible variety of potential paths and rates of motion that calculus might account for can be perceived by the human eyes automatically, and needs not take an infinite amount of time to be thus fathomed.  Indeed, the model of calculus employs points and instants to narrow the field of human investigation from what is normally perceived rather than broaden it, thus maintaining the fenomena thereby described entirely within the realm of perception, with the realm of perception being the broadest possible one as pertains to motion. That is, there is no motion that the human senses cannot perceive given the proper reference frame, considering, of course, that magnification technology may well be required to furnish such a reference frame. As for points and instants, they are not real existents (which are entities with analog measurements), but merely convenient tools employed by the model of calculus for keeping track of real existents to the level of precision desired.

Thus far, the empiricist-positivist advocates of the pseudoscience known as “relativity,” which has dominated the fysics of the 20^th century, have been shown to be utterly mistaken in postulating that there can be no absolute definition for motion, as this treatise has been able to not only formulate such a definition but also to explicate precisely the manner in which existing mathematical models are able to convey an accurate description of this motion, and verify that this motion occurs in precisely the manner in which it is perceptible by and accessible to the human senses.   

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 gennadystolyarovii@yahoo.com.

Order Mr. Stolyarov's new comprehensive treatise, A Rational Cosmology, explicating such terms as the universe, matter, space, time, sound, light, life, consciousness, and volition, at http://www.lulu.com/content/140855. You only pay $2.50 for 81 electronic pages of in-depth reasoning. Free previews, descriptions, and information on A Rational Cosmology can be found at http://www.oocities.org/rational_argumentator/rc.html.