incomplete
SRH is a major feature of this blog. Time for a review. Will be incomplete as just dumping some ideas from the morning.
It has been noted that again and again problems arise through self-reference. My first encounter with this was a rather pretentious AI experiment called AIME in 1996 which was just a pattern recognition (compression) algorithm whose state was then mapped into the problem space. I thought it might shed light on "consciousness" that I then used to think was an emergent property of "self awareness." Once a system started to get a concept of itself, then magically it would start to get consciousness. The two seemed inseparable.
This is another thread of the blog. In fact they are separable. You can be conscious without "self" conscious. But conscious is always by definition conscious it is conscious. However its not a loop, it's in its nature. Current understanding is that conscious is informed by something more fundamental which we could call "reality." The important thing to realise is that "things" IN reality are different from Reality. Modern science tends to try and think of consciousness as arising in bodies: which is does. But the important part of conscious that everyone is interested in:--the "reality" aspect--is more fundamental than bodies. When we ask how does consciousness create a sense of reality we're asking the wrong question. Obviously Reality is already there! Sense Consciousness is just a part of that Reality that is about sensing the "world".
That last part "sensing the world" is subtle. When we say "world" what do we mean? In the more important sense we mean what we sense! So it's circular. "World" is just whatever we sense. The other sense is to do with the "model" we create from the senses.
Its important to see that both the "sense" of the world and the "model" of the world sit side-by-side in Reality. We are conscious of both! Some researchers (like my past self) think that "sense is processed to create model" and that somehow involves consciousness. But if we are conscious of sense before even processing then this cannot be true! This is why consciousness is closely tied to Reality. We are conscious to some extent what our brains are doing: that is the Reality of our brains! So people ask well what when we are asleep? This is the extraordinary thing: while sense consciousness fades in sleep, the Reality part is still there. That Reality part is always there even after death when our senses and brain have decayed. This is hard to see because we are habituated to the brightness of sense consciousness. We are also habituated to another problem that we tend to associate all this with an owner at the centre of it all. So many researchers are ultimately looking for a "thing" which belongs to a "person" or "body" when they explore consciousness. This is part true. But there are many layers and quickly this is not true. To understand consciousness fully we need put self and world together. Neither is before or below the other in consciousness.
Consciousness aside however, it also became clear that AIME was going to run into another problem. She was going to create a fractal and that was it. Essentially the image compression algorithm was a function and mapping it to the problem space and recursively applying it was simply going to search for a fixed point, diverge or go into chaos. At the time I didn't think of Fixed Points and just thought about the ensuing meaningless chaos. Far from being a reasonable conscious being, AIME was just going to be churning meaningless numbers. And so AIME ended and SRH started.
Things applied to themselves show a narrowing as they constrain themselves and become less functional. Douglas Hofstadter notes this in "Godel, Esher, Bach."
Gödel's Theorems
So Gödel's Two Theorems state that in a logic which can represent itself then a valid statement can be constructed which contradicts itself. Beq() in his demonstration.
Beq() is based upon a function that lists proof. That is to say in a consistent logic every statement is provable from the axioms. Provably simply means by applying the established rules.
That is like establishing the rules of chess, placing all the pieces on the board and then working out what moves get you from the opening position (axiom) to that position. If no path, or list of rules, exists we say that the position cannot be proved and does not belong to a valid chess game.
So Gödel shows that through self-reference we can create a chess position by applying the normal moves in chess which while being valid says (once we assign meaning to the pieces) that it is not a valid chess position. So we have a problem:
(1) If we accept it is a valid position because it used only valid moves then we have a contradiction. This means Chess is logically inconsistent. Which means we can prove falsehoods.
(2) If we don't like that,
I NEED GO BACK TO DETAILS AS RUNNING OUT OF TIME
Anyway SRH is very similar but it does not construct a particular proof. It simply says that:
(1) Given self reference you can construct a contradiction.
Hypothesis because it's not proven or understood, only conjectured from many, many cases.
Now logical contradictions through self-reference are isomorphic with recursive loops. This is a second part of SRH.
(2) Any logical contradiction can be represented as infinitely recursive functions.
2 is interesting because Turing Halting Problem and Proof is based on SRH.
Self-reference in Algorithms is achieved by calling the function from within itself. This is how to achieve an infinite loop and so stop an algorithm from Halting.
But its important to note that many recursive function have "bailout" terms that ensure that all recursive paths ultimately end. So its not enough to see recursion to decide whether something Halts.
Indeed Turing's Proof is that there is no function that can decide whether a given function halts.
He proves this by Contradiction from Self-Reference (CSR). Essentially were such a function to exist then what would it say of itself? It's undefined. Well it needs to halt to give an answer, so it cannot say it doesn't halt. Problems can be seen here. His proof uses the function to create a function that cannot be decided because it loops if it doesn't halt. The algorithmic version of a contradiction.
So what SRH wants to do is decide whether Self-Reference leads to Halting.
We run into the same problem of it being undecided.
Godel definitely concluded in ALL system with self-reference you can generate a contradiction (#TODO CHECK THIS). By (2) above this means in all systems with recursion you can create a Non-Halting program.
BUT
What is interesting about SRH is that its Twice self-reference. Other SRH cases like Godel or Turing use CSR to show that a particular statement is a contradiction because it allows for self-reference and then contradicts itself. The most famous example is "This is a lie."
But SRH is not making a particular statement like H(x) decides if x Halts. SRH(S) simply says that a contradiction is possible in system S because self-reference is present.
(3) can SRH(S) decide if S has self-reference without contradiction. Well SRH(SRH)?
How do we express the system? Godel says "all logics" of which Principia Mathematica is one, but he means any.
So SRH(L) where L is a logic. How would you express SRH in a Logic?
So its double self-referential because
its a self-referential statement about the systems self-reference property.
Beq() was within Principia. Godel isomorphically showed it for all logic with self-reference.
SRH is logic independent. It says that given Self-Reference you have a problem.
You need a way to express Self-Reference which is Logic independent. OR given an example of Self-Reference extract/deduce the logic.
SRH is essentially Godel's Theorem but with the shift to make a statement about the role of Self-Reference.
Previously I've said that Godel Numbering is the SRH. The mapping of statements about numbers to numbers/indexes by listing them is the key move that sets up self-reference.
All self-reference of this type is about listing.
But Turing's proof is not about listing. He generates Self-Reference by recursion.
However (2) above suggests that a recursion can be expressed as a listing.
F(x) where x is the index of a function, and then make x = F, is similar to:
F(x) {
F(x)
}
(2.1) can you find any other means of Self-reference other than indexing, recursion?
Immediately there is Smullyan's SELF and Quine:
"yields falsehood when preceded by its quotation" yields falsehood when preceded by its quotation
So this is using position, and use/mention to identify something. It uses the language term "its" which is like "this" to generate the self-reference.
So we can add "indicating" to this list then. Any system which has a means of indicating an element can indicate itself and so generate self-reference.
Recursion is like indicating in that binding to a function name or variable is really indicating. Its been noted before that "binding" is needed by the system, but cannot be expressed in the system.
There are things outside the system.
(4) This is the other version of SRH that no system can fully contain itself. If it did then it would map into itself, which means parts of itself would map out beyond itself thus proving that it was incomplete.
Indeed can we extend (4) to the proof of SRH.
Self-reference means a mapping from something "into" itself. This means that parts of itself not mapped can be mapped outside itself.
The ability of something to map to things it is not is where the problems start.
Much TODO. Definietly out of time.
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