(Image 1 taken from https://sites.nicholas.duke.edu/statsreview/continuous-probability-distributions/)
(Image 2 taken from https://www.scribbr.com/statistics/standard-normal-distribution/)
Above I have given both the functional and graphical representation of the normal distribution. I will not elaborate on the history of the normal distribution or how it was discovered. Instead I want to write about the philosophical basis of the normal distribution.
The first thing to note about the normal distribution is that empirically objects and measurements do follow its probabilistic determinations. Heights, blood pressure and IQ scores among countless other naturalistic phenomena follow the normal distribution.
The second thing to note is that it is a symmetrical function, completely mirroring itself over the y-axis.
The third thing is that both transcendental numbers, pi and e, play a prominent role in determining the probability density.
How do we reconcile and synthesize these three facts of the normal distribution. In order to do so we require Platonic philosophy and mathematics. I will state an axiom (something indisputable but which there is compelling evidence to believe): All naturalistic objects have their origin in unchanging Platonic forms, and once instantiated are subject to the laws of generation and destruction. Why should you believe in the Forms? Insofar as there are patterns in the universe, it must suggest there are original forms which all patterns base themselves after. Without an original transcendental object, patterns could not exist, and all things would behave chaotically as opposed to orderly. We may call this Ideal existence. Now my original contribution to philosophy is to suggest that the essence of Real Existence, or Instantiated Existence, is imperfection. The only difference between the realm of thought and the realm of reality is that reality can change i.e. subject to generation and destruction, or becoming more of something and less of something. Real existence essence is deviation from the source, or mean.
This explanation explains the first and second facts of the normal distribution. Naturalistic objects are found within empirical, real reality and hence obey the laws of instantiation. Moreover, the reason there are two sides of the normal distribution, or the reason they are symmetrically opposite is because they represent respectively generation and destruction. There are only two ways to deviate from a mean, either going above it or below it. Reality bifurcates itself in this way.
The third fact is the most interesting. The reason e is found within the equation becomes clear when we understand its essence. The essence of e is that objects change (derivative) according to themselves (integral=derivative), and e is nothing more or less than time itself. Time will act on an object according to its current state. And since each current state is related to the next, the derivative is continuously equal to itself at the stage it is at.
Pi is more mysterious. But I would wager that it represents the function of space somehow. Pi is a perfect relationship between circumference and diameter, suggesting the universal circular shape of all matter.
This is about as deep as I could delve into the normal distribution. Perhaps with more time and philosophy and mathematics, we will be able to penetrate all its secrets.