A Rational Cosmology: The Role of Limits in Describing Motion

Essay XXXIV

This is Essay XXXIV of Mr. Stolyarov's series, "A Rational Cosmology," which seeks to present objective, absolute, rationally grounded views of terms such as universe, matter, volume, space, time, motion, sound, light, forces, fields, and even the higher-order concepts of life, consciousness, and volition. See the index of all the essays in "A Rational Cosmology" here.

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?