— The Mathematics of Change —
At the heart of virtually modern-day science is the ability to model the behavior of real events using mathematics.
For this we can thank Isaac Newton. Newton saw the world as being in a constant state of “Flux”, a world constantly changing. In the simplest possible terms, what Newton realized was that, if one knows the “Amount of Change” that occurs over time, then one can calculate the appropriate “Rate of Change”. So for instance, if we know that we have traveled 150 miles in 3 hours, we can easily figure out that our average “Rate of Change of Position” (i.e. our average speed) has been 50 mph. Calculating “Rates of Change” is known as “Differential Calculus”.
“Differential Equations” are the objects of Calculus. Differential equations are mathematical equations, which contain Rates of Change. Now, the beauty of calculus is not the calculation of Rate of Change from a given Amount of Change, but actually the real beauty of calculus is in the reverse operation. Solving a differential equation is the calculation of the Amount of Change from a given Rate of Change and is known as “Integral Calculus”.
And so the simple expression, speed = 50 mph, is a differential equation defining, as it does, the rate of change of position. When we can write down a differential equation, we can solve this equation which contains a Rate of Change to get to the Amount of Change. So for example, if we know that our average speed is 50 mph, then we know that we can calculate the number of miles traveled by multiplying the velocity by the amount of time we have been traveling. Thus solving the differential equation gives us the underlying Amount of Change:
Amount of miles traveled = (average speed) × (the number of hours traveling)
= (50 mph) × (3 hours)
= 150 miles
Differential equations are Newton’s so-called “Equations of Change”. These Equations of Change irrevocably changed the world and are ultimately responsible for the technological paradigm we live in today.
Calculus effectively gave us the tools to mathematically determine the effect of forces by incorporating them into mathematical equations. It was mankind’s first true understanding of The Nature of Change described in the mathematical language of the universe. And the primary tool of this mathematical language of change is the base of the natural logarithm “e”…