Fixed Points, Loops, Chaos, Woozle Searching vs Owl.

Just formally extending interest in Fixed Points to loops, and noting they have nothing to do with self-reference by themselves.

Given a function with a range that is a subset of the domain we can evaluate it recursively. A fixed point is the value with a loop of 0. That is f(a) = a. But actually it need not be achieved in one iteration. f(f(f(a))) = a: this will do for a fixed point. And the key thing is once a fixed point is found further iterations cannot change this. A loop is thus defined as one we find our way back to the start we are no longer able to leave the already traced path and are bound by our previous behaviour. To deviate would be to contradict our own definition and so the only solution must be to trace the existing path. The decision of what shall I do becomes what did I do. 

This could be called Woozle Searching. While a woozle is a kind of "weazle like" creature, in fact a far more interesting definition would be the path traced until you rejoin your steps. On a Woozle Search once you rejoining your steps you are fixed into a loop and the path is defined. The system cannot now change the path, or to put another way the "nature" or "eigen" (to use the German) is now expressed. In the Pooh books a woozle search gives different results for every run. Essentially its a random walk that follows itself once re-encountered.

The interesting thing here, and which captured my childhood imagination, was that the "thing" you were searching for turns out to be yourself. Yet, and this is of interest regarding Anatta, you had yourself at the start: you were the one doing the search so you have uncovered nothing new. Yet realising that you are searching for yourself is something new. "You" change "value" from being the one doing the search to being the one being searched. You the subject becomes you the object; you experience a change in grammatical tense. This is brilliantly exploited in film "Angel Heart" where the detective pursuing a murderer finds out it is himself. Finding yourself changing tense and meaning is quite an interesting thing if you have a fixed "atta" concept of yourself.

This "woozle" following what turns out to be "your" footprints leads to the idea that recursive functions are a kind of self-reference where the function responds to "itself."

But actually this is WRONG. Owl is needed to see the Woozle Path! A Woozle Search is only expressive enough to trace a path. You need Owl at the centre to apply a meta function that gives the self-reference! In the case of Woozles: to point out that the footprints are your own.

This means actually that Fixed Points are not expressiveness enough to code self-reference. This is actually obvious. A Fixed Point is just a value of the domain which is not changed by the function. It is interesting in that recursion also makes no difference. The presence of Fixed Points proves only one thing that the range of a function must intersect the domain it says nothing about "self" reference. You need a Godel mapping of functions into the domain for that to happen. It happens at a whole different "meta", isomorphic level.

So these kinds of fractal patterns with space being contorted around fixed points, do not have anything to do with "self" reference. Yes they include loops of recursive functions, and once a value executes a "Woozle Path" through the function it cannot escape to infinity and is bound into the set, but the Mandelbrot set does not itself know anything about this. It is just billions of Poohs and Piglets searching for Woozles.

An Owl on the other hand can look at all these paths being traced and see the spirals and "Applemen."

This was noted before in the Blog but being reversioned here. A fixed Point does not "know" it is a fixed point. A Woozle Path likewise. They needs an Owl to become fixed points or Woozle Paths.

This is another version of SRH, indeed the original 1996 insight.

Now if we were to encode Owl into a system we have something interesting.

But we can also note the recent Normative observation. An Owl cannot be subject to an Owl themselves. For a Woozle Path to exist requires an Owl sitting in the tree to see it. But an Owl does not need another Owl to see it!

We enter Plato's Third Man Fallacy here. If Owl needs another Owl in order to function then there is no such thing as an Owl.

Owl(x) must be sufficient to do what Owl does to x.

If Owl(x) is of the form 

value Owl(x) {

    ...

    Owl(x)

    ...

}

Then we are saying that Owl(x) is not sufficient to perform the operation of Owl. In which case applying more Owl(x) to the process cannot help!

This links with the previous post and the idea of "expressiveness." If Owl does not already have the power to do what Owl does then adding more owl cannot help.

Its an devastating interpretation of recursion. The presence of recursion actually reduces the power of the function. It is saying that the function does not have the power by itself.

Yet Recursion is appealing here. Applying a function recursively can provide hugely broader expressiveness: 

Thing to note here is that--like with all fractals--the application of simple function recursively to a simple seed creates an infinitely complex form which is really just copies of the same seed on infinite levels. Its been noted that Nature uses this a lot as DNA is a limited store of information and so recursion is almost always used by nature to expand it "expressiveness."

You might say Ah ha we have an Owl being defined in terms of lots of little Owls.

But no. The seed is given. The result is just the result of transforming infinite copies of this. There is nothing "new" here. So its not "full recursion." A trivial example is this recursion for the factorial:

Factorial (n) {

    if (n == 0)

        return 1;

    else

        return n * Factorial(n-1)

}

The point is that there is a clause which stops the recursion and returns an absolute non-recursive "seed" in this case the value 1.

So for n=4 we have doing replacement:

Factorial (4) = 4 * Factorial(3) = 4 * 3 * Factorial(2) = 4 * 3 * 2 * Factorial(1) = 4*3*2*1*Factorial(0) 4*3*2*1*1 = 24.

The last term is an absolute expression with no recursion and an absolute result.

Now this is a tricky thing because it suggest that all things are based upon an absolute "seed." And this is true, sort of.

Within a domain this is true. This Factorial is only defined over the domain of Positive Integers. But if we were to apply it to -1 then we get an unexpected result:

 Factorial (-1) = -1 * Factorial(-2) = -1 * -2 * Factorial(-3) = -1 * -2 * -3 * Factorial(-4) ...

As can be seen this never delivers a result and the recursion continues forever. This function never Halts! And if we inspect the partial results it looks like it approaches positive or negative infinity, a very unsatisfactory result indeed!

By different methods the Gamma function can be found which gives results for the Real Domain (and Complex Domain) as well.

So this Factorial function above only works for a specific domain. When we say Absolute we also need to define the Domain. In this sense it is not "Absolute" it is relative to the Domain.

I used to say "All things are Relative" and the Platonic criticism of this is to make the point above that "relativity" demands recursion.

Likewise we were arguing that Owl cannot be relative. If Owl depends upon Owl then Owl has no power. A recursion that is without limit has no power.

But if we do define a limit to a recursion then we also limit that recursion to the domain of the limit.

The Ultimate Domain to which all other Domains belong cannot be arrived at through recursion, because it cannot be the "seed" of a recursion, and a recursion which never returns a seed is infinite and does not Halt.

So "All things are Relative" is true in that a Domain is needed. Nothing exists for definite outside a given Domain, which is the set from which the "seed" is taken.

Owl cannot be saved by recursion. If Owl is insuffient then no compounding Owl can help.

This puts a limit on "emergence" too. Buddha says in Diamond Sutra that since everything is made of components that are not itself then everything is an illusion. This actually says that Owl is made from components that are not Owl. And if Owl is one of the components then nothing emerges from the other components. If you put a cake into the ingredients list then you can just bin all the other ingredients and keep the cake.. obviously.

SRH is saying that Owl cannot be made from components that include Owl.

Or we can say that were a function that defined Owl contained Owl we can say that the rest of the function serves no purpose.

If a function contains itself it must be the case that the rest of the function serves no purpose and is extraneous.

This is all very obvious.

One small thing missed. A Fixed Point is a value that maps to itself from domain to range. Now if the functions are countable and the domain is ordered and countable then functions map to the domain. (Time run out) but I believe there is no connection between the fixed point of this function and self-reference. That ends the blogs interest in fixed points.

===

Wanted to add the odd feature of recursion that it tends to lead to fixed points. For Cos(x) = x it very rapidly homes in on x = 0.739085133 which is the fixed point. It is a "locally stable" fixed point as values around it reduce to it under recursion.

This is the general feature of many recursions and self-reference that it seems to limit a function and force it towards some constraint.

Even a function like x = x^2 which diverges is forced towards a limit of a kind in this case infinity.

Some limits are loops or cycles like most of the values in the Mandelbrot set, and by forming loops the actual number of unique solutions must be considerably limited like in the 100 Prisoners puzzle. I wonder if that is an exponential level of restriction to shift cardinality from Reals to Countably infinite? Pure random speculations.

And then some "loops" are chaotic and while not going to infinity or zero also never cross their own path. This seems impossible but these functions are not continuous and move in steps so can jump around in Real space and never run out of new places to fall.  

Which looks like this:

The fixed point is around 1.16573 but starting at 0.6 the Woozle Search never intersects itself and never loops. For reasons I don't understand yet the region around this fixed point diverges... something to do with its differential absolutely no doubt.

Starting at 1.16573 with a precision of 5 dps the result is still not stable. This fixed point is not stable at all and regions around it diverge into Woozle Searches:

Which can be used to make a fun never repeating chaotic orbit in any dimensions e.g. 2D with a0 = 0.6, b0 = 0.7

So in this instance recursion is creating apparently infinite complexity, and appears to increase the expression of the system. I know this has had quite an allure with infinite complexity apparently being reducible to just a few parameters like in this example. And many like Stephen Wolfram believe that the complexity of Nature likely being recursive can be reduced to much simpler chaotic parameters. Never-the-less at root there must still be absolute forms which cannot be reduced further. So the question remaining here is what is the difference between  the outcome of a recursion and the absolute form upon which the recursion operates. In other words is the fractal owl created from repeated operations on a seed Owl actually any different?

But just flagging this way in which "energy" of a system seems to be usually limited by recursion, and analogously self-reference.

This idea of an "energy" of a system that grows weaker around stable attractors, but fuelled in systems with unstable attractors that are never able to stabilise. And then the 3rd type of those that explode to infinity. In chaotic systems these fixed points are even called "Strange Attractors" even suggesting a kind of force or attraction/repulsion within the system.

Equally there is a kind of analogous "force" in logical systems that sets statements like "I am false" into instability. There is no fixed point here, no solution that remains untouched by the statement for it to be stable. This is true of all statements that negate themselves in some way. Particularly problematic with "∀" statements that include a Not as there is no way to "escape" the statement if the statement falls into its own scope.

Anyway rambling now. Point made.

===

Tho I should be careful:

general self-referential lemma,^[6] which says that for any formula F in a theory T satisfying certain conditions, there exists a formula ψ such that ψ ↔ F(°#(ψ)) is provable in T [ref]

That is the existence of fixed-points IS the existence of self-reference. And yet what the above is interested in is that such expressions only exist because the system is not operating on the "self." Self-Referential statements are actually those statements which are independent of the system.

Perhaps that is interesting. "Self" even in humans is ironically our way out of the system. We think of self as being separate and cut off from the world. It appears to be independent of the world. It is unaffected by the world. In deep meditation states the self reveals itself without reference to the phenomena of the world. It is like a "fixed point" that is not operated on by the world. But we don't normally realise that this "self" is the same for all people, it is the true nature of reality, the "value" unaffected by the system that is the world.

So its not "self-reference" in that the system cannot express itself. Rather it can only fail to operate on certain elements of its domain - the fixed points - which also make the "selves."

This needs formalisation, just brainstorming.