The Beauty and Power of Discrete Calculus

Learn the inner workings of calculus by understanding a more general version and watch as the beauty unfolds

Image from Wikimedia Commons

Not too long ago I was writing an article about the Planck length — a hypothetical shortest length in our Universe. Then it occurred to me that maybe the reason why our modern physics of gravity breaks down at tiny scales is that it is all about calculus (continuity and smoothness) but applied to real physical units like space and time.

In calculus, we make variables go to zero all the time including differences. But what if that doesn’t make any sense for physical units?

I started to develop a calculus from scratch where instead of taking limits, I only worked with discrete intervals. My hope was that we could then apply it to physics using the various Planck units to get some insight into what quantum gravity may look like.

I will show you my attack on this physical problem in another article because first, we need the mathematical theory described above which is completely independent of physics and is a beautiful mathematical subject on its own as you will see. It also seems that we get some very nasty and perhaps unsolvable (discrete) differential equations when trying to solve Einstein’s field equations…