A Rational Cosmology: The Calculus as a Model for 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 has 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 number 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? We shall answer this question as our exploration of motion unfolds.

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