Anaxagoras

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Has anyone created a mathematical model for Anaxagoras's model? That is, everything has a dominant overall substance, but it heterogeneous and can be divided into smaller parts, some of which are a different nature. So the basic mathematical model would be having a set, and it has a dominant nature as some property. Then it can be divided into smaller subsets which are also Anaxagoras measurable, which can have the same nature as the whole, but some must have a different nature. Ultimately everything is made of infinitesimal parts which are a homogeneous element, which could be the elements of an uncountable set, and the Anaxagoras measure of a set would be dominant element in the set. The hard part is defining measure for determining how much of an element is in a set. The heterogeneous nature of the sets means that the pure elemental parts of the sets would be unmeasurable by most measures.

Suppose a measurable set of points in the plane is the countable union of measurable subsets. Associate each of the subsets with a "kind". The total measure of the set is the countable sum if the measures of the measurable subsets. And there you have it.

It's not going to work, because there is no countable union of measurable sets from some base case as they can be divided into an uncountable number of heterogeneous subsets.

Okay, I just realized I came up with a half-solution in the past while working on an entirely different problem (spacefilling curves): I've found a method to arrive at values for points inductively assuming measures for the sets. So let's a assume were have a square of points, and it's over all element is fire. We'll assume that for this particular set, it can be broken down into special subsets which are either fire, water, earth or air, and they can all be broken down into special subsets in the same manner. It is as follows:
F
_
F E F
A W A
F E F

This represents breaking down a fire subset into nine subjects of the four elments arranged geometrically in that manner. Here are how the other three subsets are arranged:

E
_
E F E
W A W
E F E

W
_
W A W
E F E
W A W

A
_
A W A
F E F
A W A

Following this method of dividing a countable amount of points can be determined to be of a particular element, as each point is associated with an infinite sequence of positions in the grid and thus element values, and if the last element value just repeats infinitely, then it can be concluded to be of that element. The problem that creates with associating measure with points is that means that only some of the points contribute to the measure based on their position, which means the measure of a set could be changed through translation, and thus this method is insufficient.

Inductively the fire, earth, water and air measures should be able to be calculated algebraically for the original set assuming it's total measure is one. Let F be the fire measure, A the air measure, W the water measure, and E the earth measure. So F + A + W + E = 1. Earth and air are symmetric as far as fire is concerned, it follows that A = E, and the basic equation can be simplified to F + 2E + W = 1. Next we know F = 4F/9 + 2A/9 + 2E/9 + W/9, as the proportions each element makes up in fire are the same proportions fire makes up in the respective element, and fire has four fire parts, two of each air and earth parts, and one water part. This can in turn be simplified to F = 4F/9 + 4E/9 + W/9. However, I've been unable to come up with the last equation needed to solve the series. All I can come up with is the inequality F > E > W. Regardless, the larger problem is that this would only give a measure for sets with the measure defined recursively and have a fractal nature, it doesn't give anyway to generalize the measure to the points which make up the sets.

Okay, maybe I get the math to work using just two element. So we got fire and water.

F
-
F W F
W F W
F W F

W
_
W F W
F W F
W F W

Then we can defined F as the portion of fire which is fire, and W as the portion of water which is water. So F + W = 1, and F = 5F/9 + 4W/9. W = 1 - F. F = 5F/9 + 4(1 - F)/9 = 5F/9 + 4/9 - 4F/9 = F/9 + 4/9, 8F/9 = 4/9, F = 1/2. Which means W also equals 1/2. Thus there is no way to actually differentiate between fire and water, at least not with this model.

Try leptons and quarks...

Please shut up.

The Real Number set is the union of semi-open intervals bounded by integers. That is a countable union of measureable sets. in the semi-open intervals are of the form [n, n+1) you have a union of measurable sets each set (interval) having measure 1.

But we're only talking about one integer interval anyway, THAT is what need to be divided infinitely.

https://www.amazon.com/Creative-Whacks- ... 0911121056

Not quite sure what this has to do with Anaxagoras.

The pieces seem to fit the pattern you are describing. I have one on my desk.

They don't. The patterns in those puzzles are atomic, while Anaxagoras was an anti-atomist. According to Anaxagoras things can be infinitely divided. That's easy enough to model mathematically, any line segment can be split infinitely many times. The confusing part of his philosophy is the claim that while any substance is predominantly one element, but contains parts which are other elements. It's hard to describe beyond that, which is part of the reason I'm trying to find a mathematical model for it.

_

So I figured that with any model where there is a fixed ratio between types of parts in a part, the ratio between the elements is always going to be one half, and thus there is no meaningful qualitative difference different the different types of parts. However, it's still possible to create different arrangements of elements down to the point level which can be associated with different total elements, even if technically the cardinality of the different elements is the same. Which is where the measure problem comes in. All I've got so far is that with the previous model, a set is fire if it's corner points are fire, and water if it's corner points are water. But if we were to define a homogeneous set aside from it's corner, then what basis is there to saying it's the majority rather than the corners?