Snowflakes and Calculus
Or how things grow and die: an Intro
Everything you touch and see is an inhabitant of a matrix. Not "The Matrix" - that's a fun head trip for another day. What I'm alluding to, is that reality is composed of data points, which keep changing constantly.
- What makes them change?
- What is the nature of this change?
- Is information truly immutable?
Turns out these simple questions lead us to profound ideas, and almost every answer ends up as a Matrix.
The laws of physics remarkably lead us to the conclusion that everything that exists is indeed just the solution to some differential equation. These solutions in turn can become the variables of some other equation, or even feed back into the original one. What is a differential equation? To put it as simply as we can:
> A differential equation turns changing variables into algebraic equations
Consider the simple 1D equation:
This equation describes a situation where the rate of change of a variable x depends on the initial quantity of the same. And
is a coefficient that determines "how" the change proceeds.
If
, then we have exponential growth. And for negative values, we have exponential decay.
Okay, this toy model is easy enough to understand, but that doesn't ensure it is useful by default. Or does it?
In nature, we observe exponential growth and decay almost everywhere. Population explosion, propagation of rumors, pandemics, mycelial growth, and on and on. You'd be amazed that social networks try and leverage this sort of behavior through what is known as "The Network Effect", which, as you might've guessed, connect this behavior with Networks of Graphs.
Wait, wasn't the Brain a product of this sort of Network? Yes. When an idea takes shape in the brain, it often utilizes such behavior, albeit with some twists. There's always some adjustment due to external factors. These could be positive or negative factors, and they determine the "curve" of the solution space.
Solution space. So we're talking about many solutions now? Yes, we are. And we must.
What if the previous equation was indexed? What if in an ensemble of different variables, each variable had a separate equation determining its growth/decay? Or, what if we had different coefficients? Turns out, these cases could be simplified by a common construct: Matrix.
I'll leave you to ponder over it, and read the fantastic book "Differential Equations with Applications and Historical Notes" by G.F. Simmons. Specifically start thinking about how nature is modelled with differential equations. Matrices are an ubiquitously simple way to collect and operate on information, and that's what we'll be learning next.
I'll end with this food-for-thought:
> Computation is just Linear Algebra.