Am I being stupid? Isn't SRH simple? Self-Reference

How hard is this.

So as argued in blog last year there is some use to deciding by random. So let me decide whether to start the next sentence with "Amphoteric" if the coin is heads or "Blastocyst" if it is tails:     

"Amphoteric" reached its maximum usage in 1929 according to Google NGram. Okay I actually cheated there cos I was supposed to use the word and instead I used its quotation. But then there is no agreed way in English grammar to express "use the quotation" so its actually ambivalent, a word interestingly which has the same root as "amphoteric" where "ampho" is Greek for both.

Anyway the coin could have come down as a tail:

And that is it. Apparently if you ask "Yes/No" questions what can go wrong. And since anything can be expressed as a "Yes/No" question you can decide anything this way.

But what if you extend the questioning from simply worldly things like what shall I do, to meta statements about coin flips.

Well a "near" meta statement is: shall I never use coins to decide anything again (assume Head = Yes)

Well that is okay cos the coin is deciding never to use coin flips again. And if we flip again:

That is fine we just ignore it.

But what about: Shall I obey the next coin flip?

So the decision is that I should not obey this coin flip. But if I do not obey it and ignore it then I am actually obeying it. I'm stuck.

So its incredibly easy to cause a contradiction through self-reference. In the real world its just a system that can't get a stable answer. Suppose a Turing Machine with the following rules:

δ(q0, 0) = (q1, 1, →)

δ(q0, 1) = (q1, 0, →)

δ(q1, 0) = (q0, 0, ←)

δ(q1, 1) = (q0, 1, ←)

Or in this online TM the following program:

0 0 1 r 1

0 1 0 r 1

1 0 0 l 0

1 1 1 l 0

Recording the last move in the state means its easy to step back from the second position and loop.

So that is essentially what a contradiction is. A system that keeps contradicting itself and never stabilising. It doesn't Halt.

Now Turing was able to show that there is no Turing Machine that can decide whether another Turing machine will Halt. The problem is that it can't tell if it will Halt itself, and worse if there was such a TM you could use it to loop if it said it should not loop. That is literally isomorphic with the coin flip above or "I am false" liar paradox.

So that maps onto saying that there is no system that can tell if another system is a contradiction because it can't tell if it would be a contradiction (more accurately if it could, then you can engineer a contradiction).

So all that SRH is saying is that Self-Reference always leads to loops. Now worth exploring: do loops mean Self-Reference. Is the Turing Machine above an example of self-reference? That is, is it iso-morphic with a self-referential system? @TODO

For that consider Godel: he was able to create self-reference by listing and numbering sentences in Principia Mathematica. Once each formula had a number then a formula could operate on itself.

Now it was noted early in this blog that this "self-reference" was not "within the system", it exists in the creativity of an observer hermetically locked outside the system. Everyone knew that PM had functions that operated on elements that were understood to be numbers. But it took Godel to show that there was a diagonalisation that mapped formula to numbers thus giving sentences a self-reference meaning, and then used this to create a contradiction. But that all lay outside the system: something that viewers could decide for themselves without interacting with the system at all. Shall we call this "Extra Self-Reference". This is very different from the Self-Reference experienced by humans which appears to be "Intra Self-Reference" where the "self" knows that it is self-referential. That is not just awareness of the self, but knowledge that this actual self is the one being self aware. There is no "outside" to this situation (its also an illusion potentially as the "self" here is not a exactly "self" but rather a Universe).

Anyway must go. But how hard is this? But if SRH is going to have greater significance than this, formal systems, mechanisms and isomorphism will have to be the limit of this part of the investigation.

So really no need to get complicated. Now with such a simple formulation can you define exactly what the conditions for loop are. Are loops possible without self-reference (under any isomorphism).

And immediately we can say that were the conditions for looping possible then we have Halting Proof cos you could use these conditions to loop when you should not be looping. So that is impossible.

So the conditions under which contradictions and looping occur cannot be decided universally.

BUT and this is where SRH is interesting this proof uses self-reference. And we are saying that self-reference is the condition.

So 

(1) is self-reference the condition for looping (see above)... looping <=> self-reference

(2) can we decide whether self-reference is involved without using self-reference, i.e. can we decide whether loops happen just by looking at self-reference.

Now obviously a universal statement must apply to itself and so be self-referential. BUT what if the statement has an SRH caveat that (knowing that self-reference is a problem) excludes the case of self-reference?

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The thing to do first is to find all the conditions of the coin flips that lead to loops. What is the structure of a loop in something as simple as the coin flips?

It is something like Tarski's Undefinability Theorem, but we are using Truth/False more abstractly to be anything "meta" within the system which is "suitably" meta to effect the outcome of the operation.

Some rules for this coin flip game:

1. The outcomes are binary Yes/No instead of True/False. But any binary outcome will do.
2. Heads maps to Yes.
3. All input must be a Yes/No question, or interpreted as such.
4. A "Program" is the chaining of Yes/No decisions, notated within {} brackets.
5. Decisions can be based upon previous answers.
6. Decisions can effect future statements (in program scope, unlimited scope)

Proposition>This system can make any decision from a countable set. Suppose we wanted to pick an N, we could just set up an infinite number of questions until we got a Yes.  

So in the first example above "ignore future coin flips?" is the greatest extent of Rule 6 and its the one that causes the problems.

We can define 3 types of questions.

Basic: eat chocolate or strawberry ice-cream

Meta1: stop playing this game.

Meta2: ignore this statement  

Meta2 have scope over the decision itself. SRH.

If we remove Rule 6 loops are loops still possible?

So 3 things to be clear on:

(1) What exactly is the condition for contradiction? Call that C.

(2) Is C the same as self-reference

(3) Is C the same as fixed point.

Now (4) is a response to the Woozle Loops post. If (3) is true then the condition for contradiction is actually a system which does not have "power" over all its elements. To phrase crudely. A Fixed Point is that element which remains unaffected by the System, like an Eigen-Vector it is unchanged, and represents a higher "meta" feature of the system. It represents the limits of the system "power." 

In fact for a Matrix to have a particular "transform" means it must be bound in a particular way and that is characterised by its Eigen-Vectors. Such meta statements interestingly are invariant under the transform. For a system to have a nature, automatically implies it has a limit. That limit defines its nature. Like a boundary on a geometric figure, that boundary defines the shape, and produces its "limit" also.

So Fixed Points actually define the limits of systems because they lie outside the systems, as does Self.

So in the post on Woozle Paths the thing being tackled is that TRUE "self-reference" itself is a contradiction. A "self" is an eigen-vector of a system and so cannot be part of the system. It is a "meta" feature of the system which lies outside it. And for a system to "be" something or have a "nature" it must have a boundary which provides the limit, which makes it what it is. But that boundary cannot be expressed "in" the system, since the system is defined in terms of the boundary!

The nature of something is what makes it what it is. It cannot then define its nature! And so a system can never actually refer to itself. This may have been the 1st version of SRH in fact that TRUE self-reference is impossible: in the sense of a system being able to encode itself, or operate on itself, or know itself. Whatever "power" or "nature" it has must be "greater" than the system. Its fixed points cannot be discovered by the system itself.

So not abandoning fixed-points after all. But rather they prove that TRUE immanent self-reference is impossible! Which was original SRH.

Derived SRH then says that Extrinsic Self-Reference like Godel Numbering (which only denotes elements within the system so that an element can refer to itself) but importantly must be done "extrinsically" by an external analyser of the system like "Godel" himself, using diagonalisation and isomorphisms, this type of SRH states that Self-Reference is the condition for contradictions.

Godel says his theorems hold for systems where self-reference is possible, but we're saying that self-reference is the problem from which his and many other paradoxes arise. We should understand the self-reference itself!