Finally SRH break through

So I was looking at this the wrong way it seems.

The concept of interest is "closure."

In the figure are 3 regions A,B,C. B is enclosed by the red circle, but importantly it is still within the domain of A. Any point within B can be referenced on the same co-ordinate system as A. This is how we define B in fact as the locus of points within A that are within the area bounded by the circle.

Region C is different. While it exists in the same plane as regions A and B, I've deliberately removed the co-ordinate grid from around it to suggest that in fact the points within C cannot be referenced relative to A or B. I'm using the word "closure" here to mean that the points within C are relative to C and nothing else. Closure here means to separate like a Monad or Solipsism.

In computer programming we might illustrate the same thing with these 3 programs.

A
10 Print "hello"
20 Goto 10

B
10 Input a$
20 Print a$
30 Goto 10

C
10 a$ = "hello"
20 Goto 10

Program A has an output. We can run it and observe as it interacts with the world "outside" it. Program B interacts even more as it take an input, during which time it halts, and then it supplies an output. The system is very open. Program C however has no output and no input, it does nothing.

We can write the programs in many ways. Program A might be written like D:

10 Print "h"
20 Print "e"
30 Print "l"
40 Print "l"
50 Print "o"
60 Goto 10

D is a much longer program but as far as the outside world goes it is the same as A. We can in fact say that programs A and D are functionally equivalent. They may work differently but as far as the outside world is concerned they perform the same function.

Program C is interesting however. It is a special type of program that does nothing. As a result we can say that all such programs are equivalent. There is only one type of program that takes no input and produces no output.

We can say that Type C programs are "closed." They exist entirely within their own "world." We don't know what a particular Type C program is doing, it might be extraordinarily complex. But because it is closed it is the same as all other type C programs.

This was my first experience of SRH in fact. Pursuing artificial "consciousness" I wanted a program that was aware of itself. So I first made a program that could discover patterns in a matrix and then planned to map (isomorphically) the state of the machine to a matrix. This way input was the system state itself. In terms of this blog I realised that once the syste, had "closure" and the input was entirely limited to the system itself then the resulting patterns whatever they might be would be meaningless. Having no relationship or point of reference with the outside world would make the operation of this solipsistic world entirely meaningless. I realised since that there would actually be patterns meaningful to an observer, as such systems produce fractal output, with the system structure being formed around fixed-points. Fixed points as hypothesised in this Blog are a metric that breaks closure, and cannot be expressed within the system. The reason they cannot be expressed within the system is that the system does not operate on them, and you need to be outside a system to generate such meta-data that the function made no change to a given input. To put that another way a fixed-point is unchanged by a system, yet for the system needs to change somewhere to identify a fixed point, so isomorphically there isn't room in the system to perform its function AND comment on when it does nothing. Suppose there is a program that finds a fixed points of given function F(f) -> fp such that f(fp) = fp. If we apply the function to itself and it finds a fixed-point then F(F) -> F. Then like Godel and Turing we would be able to construct a contradctory function who secretly knows the truth and breaks it ... todo)

In the geometric example above we can say that all regions that cannot be referenced from a particular co-ordinate system, or space are also equivalent. There is only one such space.

So this is the SRH point. There is only one type of entity that does not interact with a particular system, there is only one type of "closure."

What the original formulation of the SRH was driving at was that any system which had a particular nature must have an outside. In the new formulation, to be something distinct we can't be closed.

So that means that the "totality of knowledge" or the "totality of existence" (i.e. the Universe) cannot be closed.

Now we might think that an ant living in space C is living within a closed system. But the ant can never know the limits of space C without reference to something outside space C. An ant living within space C might walk to one boundary and then measure across to the other boundary. And they would discover the size of space C. But at exactly the same time they would have created a co-ordinate system that extends outside space C and they would know the limits of space C at the same time they knew that there must be something outside space C.

And this is where the Self-Reference part of the Hypothesis came from. When you refer to yourself, you must be an open system first. A closed system cannot define itself. From which we can infer that anything with self-reference is open.

The original formulation said that "true" self-reference was impossible. The intuition here was that trying to refer to ones "totality" was impossible as it implied openness, and so self-reference must also come at the cost stepping outside the boundaries of what is referenced. And since we must step "outside" the boundary, it isn't "true" self-reference. Like the ant living in space C, to define the boundaries the ant much create a metric that extends outside space C. When Godel defined the Godel Statement within Principia Mathematic he created a Diagonalisation metric that was able to number sentences that were outside the system. When Cantor extended the numbers into boundless infinities he created the idea of set Cardinalities that stepped outside the limits of numbers. In each case a particular metric is the vehicle by which we first define and reference what we have, but then at the same time can extend beyond the limits that we have.

And this is true of the human ego. It is a metric that firstly defines ourselves. But in the very moment that we are defined as an ego we have a way to define other people, and we instantly occupy the space of a community.

SRH was also called the God principle. Because it means that any system that becomes "self-aware", or self-referencing does so by virtue of the fact that it is open. God being that constant awareness that there is no boundary to existence and knowledge, that we are always embedded in a mystery that both illuminates the known and obscures the unknown at the same time.

Returning to a few side shoots to see if they work. The issue of a box containing itself. Initially it was the show the absurdity of "true" self-reference. A box can contain a representation or reference to itself, but never its actual self. But formally why is that? Well if we want to reference ourselves in entirety we can't be closed, which contradicts the idea of enclosing!

Another important example of SRH is the data/code distinction in the use/mention example. And also the concet of "meta", usually attributed to Douglas Hofstadtler. So the sentence "The word 'word' is 4 letters long" uses the symbol "word" in two ways. One is part of the sentence to indicate a word, and the other is a mention or reference to a particular word namely "word." In a reverse use of "closure" the two uses of the symbol "word" are in closure to each other. The metric (context in this case) of dictionary words is an entirely different space to that of words in use. There is closure as you cannot move from one to the other. This is how Russell and Whitehead would have liked to keep it with their theory of types, because when a metric is found that spans these worlds you get contradictions (Godel Numbering/Diagonalisation in Godels theorems). So SRH is actually turned around here and here is a todo.

Few other notes:

ExEy | x is closed & y is closed
Shouldn't this strictly be a contradiction since how can x and y be different if they are indistinguishable. And if they are distinguishable then they are expressing something unique (an essence?) and so are not closed.

y = set of x
Ax-Ex | x == y

The "closed" set cannot be a member of itself, because "closed" here means non-literally "self-contained" and if it really did contain itself them it would have to be open by the new SRH.

Some more thought to make this rigorous but this seems to be the correct way to think about it now.