History Of Change - I
There have always been those last questions which remain unanswerable. Right from knowing whether certain axioms in geometry are true to how does the curvature of continuum actually work, making them the barrier between the frontier of our understanding and ignorance. One such frontier was understanding change. The earliest accounts of why understanding change was important was to decode the mystery behind motion. Above all, motion was more or less a change in speed (acceleration), a change in momentum (force) and a change in location, a mixture of the three.
In the fifth century BC, Zeno of Elea a philosopher, a brilliant mind, formulated a ton of paradoxes, described by some as "subtle and profound". But at the end of the day he was a philosopher, in today's world philosophy is close to being obsolete. In the fifth century though it was of great interest. One of Zeno's paradoxes with regards to motion was according to his reasoning, when an arrow shot from a bow cannot actually move in reality. If one observes the arrow during its flight at particular point in time, it also occupies a very specific place in the air. It must be at rest in this place, otherwise it wouldn't be in a place. What is valid for one point in time must be valid for all points in time during the flight, which must signify that the arrow is constantly at rest, implying motion is paradoxical.
As Zeno nicely put it, “What is in motion moves neither in the place it is nor in one in which it is not.”
Now, obviously that doesn't seem precise. Summoning motion to be impossible is not only vague, it also doesn't agree with our daily life experiences. Today we understand change in a very precise manner, thanks to calculus. But why would Zeno, given that he can think very well, make such a point. Take this for instance, I drive a car on a track of about eight kilometers, I take about a minute to cover that distance. So, that is an average speed of about 37 kilometers per hour, but that evidently is not going to be a constant 37 kilometers per hour, considering all the curves and slopes and air resistance on the track. Further, it's not difficult to reckon the fitting average speeds for each individual kilometer. Or for all eight-hundred-meter intervals. Or even for every individual meter, centimeter, or millimeter. Nevertheless, these are still average values that tell me how quickly I completed a certain distance. But what if I want to know how quick I am at a very specific moment? Then we are back with Zeno's contradictory story of the flying arrow. In a single moment, I do not cover any distance, there is no line along which I have moved with a tangible speed.
From a mathematical perspective, it ushers to the question of infinity. The distance for which we want to calculate an average speed smaller and smaller and simply need to divide this distance by the time it takes us to cover it. As long as the length of the distance is not zero, that's a straightforward task. At some point between “zero” and “arbitrarily small,” we meet infinity and this was mathematically difficult to comprehend in the seventeenth century. In order to calculate the speed of an object during a very specific moment, the distance has to be made infinitely small. But how?
The answers as we know today came out in the following century with the advent of Newton and Leibniz.