The Corpuscular Theory of Dice: An Overview
The following two articles already get my points across, essentially. You can read them here in order to follow the lineage of my line of thought:
The ensuing article is intended to be a streamlined and finely distilled overview, so that there is no confusion as to my overall points and arguments.
*Disclaimer: These concepts for both the “Wave View” and “Corpuscular View” that will be discussed will apply to not only two rolls, but to any number of rolls.*
What is the Corpuscular Theory of Dice?
It is quite simple. When considering two rolls of a fair six-sided die (for the sake of simplicity), we are abandoning the approach of multiplying the various possibilities together, also known as a “Cartesian product”.
The traditional rules of Probability Theory would dictate that you must group all of the possibilities of rolls into ordered pairs (a Cartesian product) in order to calculate the total number of outcomes.
For example, (1,2) would be considered to be a different result or outcome than (2,1).
In total, when doing the traditional rules of Probability Theory, there are 36 different ordered pairs in two rolls:
(1,1) (2,1) (3,1) (4,1) (5,1) (6,1)
(1,2) (2,2) (3,2) (4,2) (5,2) (6,2)
(1,3) (2,3) (3,3) (4,3) (5,3) (6,3)
(1,4) (2,4) (3,4) (4,4) (5,4) (6,4)
(1,5) (2,5) (3,5) (4,5) (5,5) (6,5)
(1,6) (2,6) (3,6) (4,6) (5,6) (6,6)
This traditional approach to calculating Dice Probability I will refer to as the “Wave View”, where you would conclude there to be 36 possibilities or outcomes in total.
The more formal terminology would be 36 “Events”.
When viewed statistically, a Bell Curve is formed from these ordered pairs of the Cartesian product of two rolls (or any number of multiple rolls):
In truth, in 2024 and as I have deeply contemplated this over the past 7 or so years, I am not saying that this traditional approach or “Wave View” is incorrect.
Actually, I am instead introducing the “Corpuscular View” into the discussion as an equally valid and alternative approach, which utilizes the rules of Set Theory to make its logical arguments.
In this approach, you are no longer treating two rolls as an ordered pair or Cartesian product (x,y).
You are instead treating two rolls as Singleton Sets {x},{y}.
In Set Theory, you are allowed to establish what your Sets are, you are allowed to create Sets within Sets, and the order or arrangement of the Sets is also irrelevant.
This means that {x},{y} for two rolls is not a distinct “Event” that can be distinguished from {y},{x}.
They are simply TWO distinct “Events”, and the order in which they happen is irrelevant.
In summary, I have established {x} and {y} to be Singleton Sets, where {x} and {y} each represent a single physical die, and turning them into Cartesian ordered pairs would fundamentally destroy their Singleton status (and would also dictate that the order of the arrangement is relevant, rather than irrelevant).
Therefore, we cannot turn {x} and {y} into a Cartesian product when calculating the total number of possibilities.
So if we were to place {x} and {y} within a larger Set we will call Z (just for organizational purposes), where Z represents two distinct individual fair six-sided dice labeled as {x} and {y}, it can be expressed as:
Z = {{x},{y}}
How are we now to calculate how many possibilities there are between two dice rolls {x} and {y} without destroying their Singleton status?
Keep in mind that turning the two rolls into an ordered pair not only destroys their Singleton status, but that it’s also irrelevant to do so in Set Theory.
So instead of multiplying a Cartesian product of 6x6=36 total outcomes, the Corpuscular Theory of Dice would now dictate rather that you would add, not multiply, the total number of possibilities together.
Since {x} and {y} each represent a single die within Z, you simply count up the sum of how many possibilities that there are in total.
Z:
{x} has 6 possibilities
{y} has 6 possibilities
6+6=12.
Thus, when utilizing some of the available rules in Set Theory and turning each die into a Singleton Set, there are now instead 12 possibilities in total within Z:
{x} = (1) (2) (3) (4) (5) (6)
+
{y} = (1) (2) (3) (4) (5) (6)
This new and original approach that is of my invention I like to refer to as the “Corpuscular View”.
When viewed statistically, each roll takes on a uniform distribution of a single roll that extends linearly per additional roll that is added on to the tally. There is no Bell Curve because there are no more ordered pairs in two rolls (or any number of rolls):
As you can see, the probability distribution for a single die roll takes on a fundamentally different form than two rolls (or any multitude of rolls), where the possibilities remain in uniform and an even probability distribution rather than taking on the form of a Bell Curve.
The Corpuscular View is stating that with any collection of rolls, whether one, two, or a million rolls, there maintains the uniform equal probability distribution of 1/6 per roll, because we have established each roll to be a Singleton Set, and we are adding, not multiplying, the total number of possibilities together.
It is true that the more traditional approach of the “Wave View” also acknowledges the independence and equal probability of 1/6 per roll, but where it differs from the “Corpuscular View” is that it treats two rolls as one Event (x,y) rather than as two distinct Events {x},{y}.
This crucial difference is what separates relative, apparent, and common “independence” of rolls from what I refer to as Absolute Independence of rolls.
Finally, when you bring the orthodox “Wave View” of Probability Theory together with the new “Corpuscular View” that utilizes the rules of Set Theory, what you now have here is the realization of a genuine “Wave-Particle Duality” of Dice Probability distribution.
However, the only way to recognize this “Wave-Particle Duality” of Dice Probability is to first recognize the legitimacy of:
The Corpuscular Theory of Dice
-Thrice Hist Morphs
