SRH - onward

Just watched this. Its kind of obvious what is missing. Godel as they say got himself in his own mess by trying to break out from formal system by first isolating himself in formal systems. They also point to the experience of revelation, they even quote Turing as saying that the notion of computers came in “a vision”, this is exactly what the behaviour of formal systems is not. It is as though peering into the mirror we have forgotten that we are only in the reflection because we are not! Formal systems only exist because we are not! Penrose goes to lengths to clarify that incompleteness does not apply to the mind but to the formal systems that may be “run” in the mind. My favourite enemy (after Dawkins) D.Dennett argues that the human brain runs a virtual computer – the Turing in all our minds that enables us to sequentially work our way through systems of rules, working out what will happen tomorrow if certain things are true. I’d say that Turing machines start up whenever we start thinking within a set of fixed rules, rather than have some permanent software running in our brains that handles Turing machines. It matters not, the point is that we are outside the systems that we imagine. We create these things as Turing did to investigate formality but not because we are formal. Rules as I’ve analysed endlessly are the foundations of meaning but we don’t have to obey them. We don’t have to be meaningful (within a fixed singular system of meaning). Just watched “Golden Child” as I lay here ill these past two days. The Tibetan monk says you need to know when to break the rules – when to step outside the system. We aren’t going to get anywhere if we refuse to work within any system (which would be my failing – I refuse employment, religion, science, marriage, sexuality, politics etc) – we need to know when to work within and when to work outside the systems – or more accurately we able to chose the appropriate system – that is wisdom that is mind.

Humans can chose which system of rules to use, even create their own systems of rules (maths, culture, politics). Computers I don’t know. There are sophisticated adaptive genetic algorithms etc that I know nothing about but I guess they will always be limited by the physical hardware on which they run which itself is limited by incompleteness. I guess we humans chose computer architectures and softwares for our own purposes so I suppose we are choosing the systems of rules just as we do in our own heads.

Someone in the random comments after this video said something about evolution. Dennett loves the idea that evolution is a mechanism that could have given rise to humans. Referring to the random comment, evolution is not computable, like the three body problem we can only approximate (I am wrong here – need to understand more). I don’t remember Dennett covering this in Consciousness Explained but computability is very limited and the human mind must be beyond computability for us to entertain half the idea we have.

Just fold one thing else into the mix of this blog. In GEB I read the bit about complementary systems. The example was the prime numbers. They are defined negatively (not the numbers with factors other than 1 and themselves). So we have two sets of numbers those that have factors other than 1 and themselves and the prime numbers that don’t. Defining the former set is easy we just check for factors. Defining the prime is impossible it is just made up of the leftovers when the non-prime are removed from the natural numbers. Interesting that along with all my other hopeless tasks (given that my maths is so bad) I should have some scribbled notes this year on this problem also (why I bother given that no-one has got there and no-one includes the best minds ever in maths bewilders me, the problems just seem to want solving). So Hofstadter points out that the complement of a set doesn’t mean that the logic that created it is complemented also. This was a bit of a revelation because my whole approach to all of this is to do with the relationship between sets and complements of sets.

And, that is my answer to the Dangerous Knowledge doco. The mind doesn’t exist in any of the sets or imaginations that might occur within it. And once these imaginations have been created we may seek to get outside them like Godel. But as Buddha would remind us in the Shurangama sutra, given an imagination the mind exists neither inside nor outside, nor inside and outside, nor not inside and outside. The point is that the mind underpins the very possibility of that imagination at all. We accept a system of rules, how are we to understand our freedom to chose these rules if we are bound within the system. But at the same time how are we to understand our freedom without the rules: freedom requires first of all imprisonment. Godel sort the greatest freedom by first of all creating his prison so solidly that he couldn’t escape (a battle of will more than anything – he could have just given up!). We may become the best chess player in the world, or we can just toss the pieces on the floor. The skill (after Golden Child) is knowing when to play – I do not possess this skill, I don’t play anything yet, not even Buddhism! So the SRH is seeking naively but sternly a formal proof that self cannot be defined in terms of self that forces all systems to be defined in terms of things that are non-self. This forces all systems to acknowledge their complement as integral to their construction. This doesn’t break out of systems but forces them to see within themselves the isomorphism with their “other” or complement. Thus totality is created just as yin-yang is the symbol for totality between opposites that bear each other within. It seems so simple yet I have no idea where to turn. Hofstadter may have put the cat amongst the pigeons however because if the complement is not derivable formally in a way isomorphic with the system it represents something “new” to the system.

Also and this goes right back to the start of this enquiry in 2007 how do we define the complement of a system? It depends upon our universal set (the set from which we chose our system) and this itself is chosen…until when can’t we chose…and what is the complement set of a formal system for example? I know Turing had stuff to say about the limits of the symbol set for Turning machines… but the what of the rules, what is the complementary set of a system of rules?

It is worth reminding myself also that my 1984 experience of the inverting light circle effectively was the process of seeing and isomorphism between something visible (the circle of light) and its complement the rest of the infinite mind something which is far from isomorphic with the circle of light. Just writing this fills me with fear, yet I know it was the most beautiful experience. What am I afraid of?

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So from the stand point of a formal system there is an indefinable world beyond it that coincides with the notion of the “choice” of that system. The very restrictions that were put in place to create formality depend upon an informal world. This is analogous to the religious idea of the gods creating/choosing the world. The same problem in physics of why are things chosen the way they are from the infinite possibilities, indeed transinfinite possibilities – because they are infinite within any tuple of numbers, but there are infinite tupes to chose from in describing a universe. So the popular version of Godel is not that there are worlds beyond the mind, but worlds beyond any consistent fixed system created to model the world. Now what can we say about the informal world from inside the formal world? Well we have only the complement to go by… what is the complement of a formal system?

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btw on Cantor. Why did all of us miss this in elementary maths?… the tangent function was the first function we saw which maps a subset of the Reals onto the whole of the Reals. I remember playing around with it a lot but never thought of this: Every number in the interval (-pi/2,pi/2) is mapped onto every number between (-inf,inf) by the schoolboy function. And conversely every number in the Reals can be mapped onto the interval (-pi/2,pi/2) by its inverse the arctan function. How odd!! We know that there are an infinite number of (-pi/2,pi/2) intervals in the Reals so we automatically have an infinite number of intervals which each embody the infinite real numbers!! But wait doesn’t this structure get mapped into the (-pi/2,pi/2) interval also… so we have a fractal ad infinitum. Choose a different function and the fractal is different which suggests that it is arbitrary how the infinites are related.

Take the Exponent function at A-level. This maps all the Reals to the positive Reals. Again this mapping of the Reals into half themselves can be repeated so it’s another fractal, but a different one from above!.

==update 1/9/10

Need to be careful with above. The uncountably infinite reals map into a real sub-interval. There are only countable sub-intervals however. These countable sub-intervals form the fractal within the reals with at the limit the whole of the uncountable reals mapping into a countably infinitesimal interval (not an uncountably infinitesimal interval). This isn't news however tho would have woken up my O-level brain if I'd noticed it then however!