***** FROM HERE ON I WILL BE WRITING WITH SYMBOLS AND WORDS INTERCHANGEABLY.

***** WHENEVER YOU SEE "AW:" YOU CAN READ IT AS "AWARENESS OF..."

***** I WILL USE PARENTHESIS "( )" TO CONTAIN SOME CONCEPTS OR GROUPS OF OBJECTS.

***** THE "()" ACT AS A WAY TO REFER TO A SINGLE OBJECT/THING/OR CONCEPT AS A GROUP.

***** YOU WOULDN'T NECESSARILY READ THE "()"

***** WHENEVER YOU SEE A SEMI-COLON ";" YOU CAN READ IT AS "IN THE FORM OF..."

***** OR YOU CAN READ IT AS "AS..." OR "AS A..."

***** WHENEVER YOU SEE AN EQUALS "=" YOU CAN READ IT AS "IDENTICAL TO..."

***** OR READ IT AS "THE SAME AS..."

***** WHENEVER YOU SEE AN EQUIVALENCE "≈" YOU CAN READ IT AS "EQUIVALENT TO..."

***** OR READ IT AS "INTERCHANGEABLE WITH..."

***** WHENEVER YOU SEE A SEMI-COLON ";" YOU CAN READ IT AS "IN THE FORM OF..."

***** OR YOU CAN READ IT AS "AS..." OR "AS A..."

***** WHENEVER YOU SEE AN EQUALS "=" YOU CAN READ IT AS "IDENTICAL TO..."

***** OR READ IT AS "THE SAME AS..."

***** WHENEVER YOU SEE AN EQUIVALENCE "≈" YOU CAN READ IT AS "EQUIVALENT TO..."

***** OR READ IT AS "INTERCHANGEABLE WITH..."

***** WHENEVER YOU SEE AN ARROW -> YOU CAN READ IT AS "BECOMES..." OR "TRANSITIONS TO ..."

***** ARROWS CAN ALSO BE READ "AS.." EG. X -> X;Y "X AS, X AS Y"

***** THE WAY YOU CHOOSE TO PHRASE THE SYMBOLIC RELATIONSHIPS IS UP TO YOU.

***** SOME WAYS MAY SEEM MORE NATURAL OR APPROPRIATE THAN OTHERS.

Using flowers to show how representation and awareness works may not be too useful for some people. So let's show how it works with math.

2 is the same as an awareness of 2:

2 = AW:2

2 is one of those things in reality that is certainly real. But where is it? What sort of atoms make up 2? None of course; because, 2 is an abstraction. Which means what? It only exists in our minds? Then how can you and I and Aristotle all have the same awareness of 2? Using the function of awareness, it doesn't really matter where 2 is. Or even what 2 is. 2 is. And wherever you find a 2, you find an awareness of 2.

2 = AW:2

Try it out, go looking for 2 or any other number that is NOT your awareness of it. Certainly you may find numbers that you are mistaken about. Such as the results of some calculation, but to be mistaken does not mean awareness is erroneous, it means the representation of objects is inconsistent. Math errors are inconsistencies of representation. Numbers in awareness are not inconsistent as themselves.

Look at these numbers and how they relate to awareness.

2 = AW:2 10 = AW:10 5 = AW:5

Each of the numbers above does correspond to a different awareness of them. And your awareness of 5 or of 10 is not the same as an awareness of 2.

AW:2 ≠ AW:10 ≠ AW:5

Then what does 10/5 = ??

10/5 = AW:(10/5)

In your awareness, is 10/5 the same as your awareness of 2? Or, are you aware of both 10/5 and aware of 2 separately? Awareness of 10/5 is not the same as as awareness of 2.

We know this, because when we learn math we learn about identity. And at every stage of math we encounter this idea of identity. It's most famously encountered in algebra because we start dealing with variables. But at every stage of learning math, we are learning about numbers and how they are the same in different configurations.

10/5 is one such configuration. And we can distinguish between 10/5 and 2 in our awareness. Because we do distinguish between them, we MAKE a distinction. We learn about identities when learning math.. Because "in math reality" 10/5 is 2. We MAKE an association between 10/5 and 2

eg. 10/5 = 2.

But our awareness of 10/5 is not the same as 2. That math identity is something you learn. Even after it's learned, you are still aware of 2 as being different and separate from 10/5. It is an act of association thats puts them together. It is the act of representation. Mathematical identity is a representation.

It looks like this:

AW:2 = 2

2 = 2;(10/5) = (10/5)

(10/5) = AW:(10/5)

2 is interchangeable with 2 in the form of 10/5

2 in the form of 10/5 is interchangeable with 10/5

In math, all 2's can be expressed as 10/5 and vice versa. It's an identity.

but in awareness it's not, exactly, the same.

AW:2 = 2

AW:2 = AW:2;(10/5)

AW:2;(10/5) = AW:(10/5)

AW:(10/5) = (10/5)

2 is interchangeable by steps, with an awareness of '2 in the form of 10/5'.

an awareness of '2 in the form of 10/5', is interchangeable with 10/5. But 2 and 10/5 are not the same things. There are multiple objects of awareness. 2, 10/5, and 2;10/5

In awareness, what you actually see happening is not an identity but a sort of switching. a switching based on equivalence. In the "objective world" we say 2 is identical to 10/5. But in in the "subjective world" of awareness we are aware of them as different but interchangeable. In awareness, 10/5 is interchangeable with 2, so we can switch them. We learned they are interchangeable, so we switch them.

This isn't just how we learn math. It's how we do math. It's how we apply math to physics, and how math gets applied to everything else. We switch one object of awareness for another object of awareness. But it's not math that does this switching. It's the function of awareness and representation. We constantly do this switching. It is how our awareness "moves around".

But remember, there really is no objective and subjective world. That's a conceit. The switching is done "all in awareness". but it must match the non-awareness side of the function. That is the only way to explain how representations are "part of the universe". It is the only way to explain how there is an "objective" world where representations "work".

2 is the same as 10/5, in math terms. It's not just something we made up to be the same. But "in our heads", 10/5 is not 2, because we are aware of 10/5 and we are aware of 2 and that awareness is distinct, ergo, the objects are distinct.
***** SOME WAYS MAY SEEM MORE NATURAL OR APPROPRIATE THAN OTHERS.

Using flowers to show how representation and awareness works may not be too useful for some people. So let's show how it works with math.

2 is the same as an awareness of 2:

2 = AW:2

2 is one of those things in reality that is certainly real. But where is it? What sort of atoms make up 2? None of course; because, 2 is an abstraction. Which means what? It only exists in our minds? Then how can you and I and Aristotle all have the same awareness of 2? Using the function of awareness, it doesn't really matter where 2 is. Or even what 2 is. 2 is. And wherever you find a 2, you find an awareness of 2.

2 = AW:2

Try it out, go looking for 2 or any other number that is NOT your awareness of it. Certainly you may find numbers that you are mistaken about. Such as the results of some calculation, but to be mistaken does not mean awareness is erroneous, it means the representation of objects is inconsistent. Math errors are inconsistencies of representation. Numbers in awareness are not inconsistent as themselves.

Look at these numbers and how they relate to awareness.

2 = AW:2 10 = AW:10 5 = AW:5

Each of the numbers above does correspond to a different awareness of them. And your awareness of 5 or of 10 is not the same as an awareness of 2.

AW:2 ≠ AW:10 ≠ AW:5

Then what does 10/5 = ??

10/5 = AW:(10/5)

In your awareness, is 10/5 the same as your awareness of 2? Or, are you aware of both 10/5 and aware of 2 separately? Awareness of 10/5 is not the same as as awareness of 2.

We know this, because when we learn math we learn about identity. And at every stage of math we encounter this idea of identity. It's most famously encountered in algebra because we start dealing with variables. But at every stage of learning math, we are learning about numbers and how they are the same in different configurations.

10/5 is one such configuration. And we can distinguish between 10/5 and 2 in our awareness. Because we do distinguish between them, we MAKE a distinction. We learn about identities when learning math.. Because "in math reality" 10/5 is 2. We MAKE an association between 10/5 and 2

eg. 10/5 = 2.

But our awareness of 10/5 is not the same as 2. That math identity is something you learn. Even after it's learned, you are still aware of 2 as being different and separate from 10/5. It is an act of association thats puts them together. It is the act of representation. Mathematical identity is a representation.

It looks like this:

AW:2 = 2

2 = 2;(10/5) = (10/5)

(10/5) = AW:(10/5)

2 is interchangeable with 2 in the form of 10/5

2 in the form of 10/5 is interchangeable with 10/5

In math, all 2's can be expressed as 10/5 and vice versa. It's an identity.

but in awareness it's not, exactly, the same.

AW:2 = 2

AW:2 = AW:2;(10/5)

AW:2;(10/5) = AW:(10/5)

AW:(10/5) = (10/5)

2 is interchangeable by steps, with an awareness of '2 in the form of 10/5'.

an awareness of '2 in the form of 10/5', is interchangeable with 10/5. But 2 and 10/5 are not the same things. There are multiple objects of awareness. 2, 10/5, and 2;10/5

In awareness, what you actually see happening is not an identity but a sort of switching. a switching based on equivalence. In the "objective world" we say 2 is identical to 10/5. But in in the "subjective world" of awareness we are aware of them as different but interchangeable. In awareness, 10/5 is interchangeable with 2, so we can switch them. We learned they are interchangeable, so we switch them.

This isn't just how we learn math. It's how we do math. It's how we apply math to physics, and how math gets applied to everything else. We switch one object of awareness for another object of awareness. But it's not math that does this switching. It's the function of awareness and representation. We constantly do this switching. It is how our awareness "moves around".

But remember, there really is no objective and subjective world. That's a conceit. The switching is done "all in awareness". but it must match the non-awareness side of the function. That is the only way to explain how representations are "part of the universe". It is the only way to explain how there is an "objective" world where representations "work".

-----

So what's going on?

All these different kinds of "numeral manifestations" have to be associated to the abstractions of quantities and then the abstractions get related to each other. All the external, perceptual stimulus, have to be associated to pure abstractions that only exist "in the mind".

And every instance of a "numeral manifestation" or how many fingers and hands we have, the very idea of 2 and 5 and 10 and "how many" and division and addition, and putting things in the right sequence - each of those things is something different and unique in awareness.

you can ask a kid to count the number of 2's on a chalkboard and they can do it with ease. They can then divide them by 2. But none of the 2's on the board are the "real 2". They are themselves, representations of 2. If the kid saw two roman numerals together on the board, they could count II as 2, too. And would they count "too", "to", and "two" as 2's too? It's a good question that produces different kinds of answers based on what you mean by to/two/2!

How does a kid do this? She does it, because the variety of objects of her awareness have been associated together, not just in her mind, but in "real" life. And a "2" of awareness and the 2 of "real life" are connected as if they are an identity. They are associated together in the same way "to" and "two" and "2" are associated to a sound. All the objects of awareness are constantly getting associated and represented together in a variety of ways.

The problem is, can we model all this activity? Can we extend this way of modeling awareness, objects of awareness, and the variety of human experience with just these basic relationships?

AW:X = X and that X = X;Y and AW:X -> X;Y