A Rational Cosmology: Instantaneous Velocities as a Means for Describing Motion

This is Essay XXXV 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.

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 number 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 doctrine known as "quantum mechanics" suggest), it would take us an infinite amount of time to perceive any finite span of time, however small, which would result in 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, an interval that is accessible to human comprehension.

This is the reason why the graph of a function's derivative at a given point is the straight line tangent to that point. The tangent line most often does not correspond with the graph 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 graph to which the tangent line is drawn.