Mandelbrot Fixed points, Non-Self and Poverty

Have I never blogged on this before?

Firstly what have fixed points to do with poverty? It's to do with Self. It is hypothesised that the experience of "self" is an artefact that occurs when the brain tries to process its own output. At the singularity you have Descartes and similar meditations where the direct output like "thoughts" or even "consciousness" is being taken as input. It is quite an illusion even the great Descartes took the self to be a real thing.

I struggle with Heiddeggar's Dasein. For him "mineness" and "thereness" are essential features of Dasein. There is truth to this, but we must be careful not to go too far and instate a "self" that is "me" and "here." "Thereness" suggests, if we are not careful, that there is something "here." And "mineness" suggests, if we are not careful, that there is an owner or a thinker or doer-- the same mistake made by Descartes.

The truth is that we don't need to add anything to the world it is exactly as it already is. The whole problem with Western Civilisation is that it feels the need to "add" something. That something is a mistake.

This is most obvious when we think about our own death. For me the essence of the mistake is the sense that death "takes us away from the party." We think "oh no when I die I never get to see the world again, I must say good bye to everyone, and I will never get to see or enjoy all the things I look forward to in the future." He's a silly example. I want to see Halley's Comet but there is a good chance I will die before it comes around again. In this sense Death will rob me of seeing Halley's Comet. But this is the essence of the mistake that the "mental fixed point" creates. If we look very carefully where am I? So we can study Buddhism and we can intellectually understand that there is no self - I mean point at it? Its no where. And yet the mystery thing still owns everything. It owns that realisation, it owns qualities like its a success or a failure, it owns memories and experiences and it even owns its consciousness. In Buddhism the phantom thing its even looking forward to getting Enlightened! What a ridiculous dream that is. How can something that doesn't exist get enlightened! And even when we think great I understand that I don't exist am I enlightened yet, we are still looking to own. The whole world is illuminated by this lighthouse of shines from "My Self." But look again the light doesn't come from the lighthouse: it comes from the World (sort of - that doesn't produce light either but 1st things first). There is no need to invent a Private Personal Lighthouse to illuminate the world, it just does it all by itself. It is not ours. There is no need for a Self! Its an illusion that comes from fixed points. So Death robs us of nothing cos nothing was owned in the first place. And my future and the things I look forward to that will never happen when I die were never mine in the first place. There is no loss, because there was never anyone gaining. We firmly just let the world be without adding anything. All those monuments, and gravestones to people as though "someone" was actually lost are a mistake. Indeed the person is no longer with us, but there is no "loss." That sense of loss comes from a fixation on a fixed point. The fixed points of self is not just ours, but we project onto other people too. We think inside them is an unchanging singularity that fixes them in the world, around which their world rotates. An absolute point by which space and time are measured relative to them. Its an illusion created by fixed points.

And what has that to do with poverty and this blog. Well if there is no one to own anything then there is no such thing as wealth and poverty. There is just the world. 

And so to Mandelbrot fixed points. A quick rendition of the famous set. C -> C^2+P where C and P are complex numbers (a+b i) with P being a constant which becomes the position in the 2D array of the set. Black are points which are stable or remain finite, and the colours are the number of cycles taken for C to explode out beyond 2.

If we initialise C=P then the 1st order fixed point solves: P= P^2+P. In other words simply the root of P^2=0.
If we expand this:
a^2 - b^2 + 2 a b i
So the root solves the system: {a= a^2 - b^2, b=2ab} which has just one value (0,0).
Indeed b=ai also solves this but we want only real values.
Now 2nd Order solves (iterating 1ce) gives a second Fixed point at (-1, 0).
3rd Order ((P^2+P)^2+P)^2+P=P:

(0, 0), (-1.75488, 0), (-0.122561, -0.744862), (-0.122561, 0.744862)

Here are the 5 fixed points found so far up to Order 3 circled in white.

As can be seen there is a bifurcation with each iteration. It is a bifurcation because this is a square function and so the root proliferate by an order of 2. The fixed points are the stable points. I wonder if this root finding exercise was continued to infinity it would find all the points in the set (the black points).

All the real roots/fixed points of the Mandelbrot 6th iteration. Solving the equation:

(((((a + b i)^2 + (a + b i))^2 + (a + b i))^2 + (a + b i))^2 + (a + b i))^2 + (a + b i) == 0

But what everyone is most interested in, cos of the pretty patterns, is the behaviour of the points that are unstable and escape to infinity (the coloured points). There is continuous behaviour here as they seem linked by contours of same escape value (the bands) that orbit the fixed points. I'll just note here that these patterns are created around the fixed points and the need to obey the fixed points is what gives fractals their structure. And so I hypothesis it is with much more complex dynamic systems like the brain. The fixed points that occur when self-reflecting provide the structure around which thought and human experience orbit. But they are just artefacts of a dynamic system. And that gives us an insight into not taking them so seriously. Things like the fixed point of self are just artefacts they are not fundamental.