addendums…

I must make earnest moves toward preparing my speech for a wedding next month and also with the GCSE exams approaching I should put down these topics until June. Hopefully then I will be able to do some proper intensive learning and get up to scratch on mathematical logic etc altho it is still my hope to get through the SRH without being too embedded in the subject (it is only a tool), for people like Godel, Zermelo etc it was their whole life!

The SRH/Plus-One/Horatio Hypothesis

It occurred to me that at first sight the “complement of a formal system” lent little to a “proof” itself that systems exist outside themselves – it only serves to show that a system is built from the outside rather than the inside. The SRH was based upon the loose intuition that a full scale of a system could not be “known” within the system, and so the very existence of a system of knowledge meant that it could always expand its knowledge by adding itself. This I see forms a new hypothesis the plus one or Horatio hypothesis after Hamlet saying “There is more in Heaven and Earth than is accounted for in your philosophy Horatio”. I need a clear and un-distracted mind to approach this and I see that putting off the wedding preparations has actually been distracting me!

I think the best formulation of the SRH will be of the form that any system can always accommodate another “term” (quite probably “itself”) and so will always be short of “totality”. It is the notion of capturing Totality in a formal system that is being attacked here from the start. The God principle is that any formal system will always fall short of capturing the “entirety” so that some notion of a god must always be included to include what is “not known” by necessity. In other word a system can never capture “itself”. But what does capture actually mean? A lead for June not now!!!

My Muse

It occurred to me after watching Stardust last night that I do still believe in “love” even after having thought I had totally removed all relevance of it and even recently of sexuality. I realise that my mission to escape my love for “my muse” which has been running since the day I met her and when I had that single thought “yes”, from that moment it was a lost cause. Given this I have to simply accept that I love her and there is nothing I can do about it. Reality seems to be like a broken space in which she is simply obscured from view by some folds, that the division between the void and existence, or between life and death doesn’t seem so absolute anymore and that peer over the edge and you can see into the realms of non-existence just as easily as we see into the realms of existence. Anyway I have to just love with this now and move on.

OK that is the last update on this blog for a few weeks, in a probably vain attempt, to focus on some pressing issues that require actual action rather than words and theory.

=== just clearing up notes and buffers

> SRH - what is produced by the world becomes the exactly the only raw materials for the next cycle. It feeds on itself. But this violates the SRH so Time (& space) are necessary so that the start and the end of the cycle are distinct. Suggestion that SRH could be an axiom from which time and space are derived!

> Ouroboros

The Viper/serpent is the original symbol of the unchanging because it flows through all things! But cf with the Devil in Xianity who led Man to the Tree of Knowledge and brought us to eat via the woman! Knowledge in Jewish thinking is opposed to God, or at least leads Man to challenge God. Interesting that the serpent also feeds on its own tail to show the eternal recycling of the universe (the Universes efficiency and completeness). - indeed the "things" are made from the universe only so the "things" are illusions and the universe real, the knowledge is of "things" (emergent properties), the serpent = the universe that seems to flows through the things because the "things" are changing so the serpent seems to flow, but really it doesn't, and it only seems to feed itself, because the "things" are flowing.

> Opposites both violently oppose one another but also enable balance when their oppositions are matched. The scales can only be equal because of opposing forces! This is the essential nature of dialectic!!

> Sexy Son Hypothesis

There is a certain self-fulfilling nature of genetics, but I can’t remember quite what I meant by this note:

> SRH sexy son = if i do then other girls do if not no them either

> “My” Child’s genotype

I was also interested by the probability of someone else’s child being more genetically similar to me than my own. Taking gene frequencies of each gene in a population there is a probability that someone else’s children could be more genetically related to me than my own. Want to look at this quantitatively… would certainly throw the genetic argument for having children.

Complement of a formal system

In set theory because it is easier for me.

I remember reading that Turing argued that a symbol set (alphabet) must be finite for the symbols to have meaning. Does this also apply to the complement of an alphabet? I may have a universal set E composed from the 3 members who I will name 1,2,3. I chose these from an infinite number of possible symbols (one for each counting number). So any universal set is going to be finite but with no upper bound to its possible size (maybe the number of atoms in the universe). The TransUniversal Set (T) will be the Union of all the possible universal sets. T will then also be finite! We can define the complement of a universal set E thus:

E U E` = T

Where E is formed by positive choice, E` is equally defined by negative choice. The same actions define E and E`.

Now I am deviating from standard set theory because I am dealing with meaningful symbols here. {x in N} is not allowed here. Need to work out details why.

So we can then construct our hierarchical system as the union of an infinite recursion with the Power set function. S = …P(P(E)) U P(E) U E. This set is now infinite.

Now if we use some system of Godel numbering then each set maps to a Natural number, and conversely each Natural number maps to a set.

e.g. I’ll use 2 bit bytes for simplicity. The = symbol means corresponds in the metasystem.
{ = 11
} = 00
, = 01

With an empty universal set we can then construct the natural numbers as cardinals of the sets as usual. Each set corresponds to the given code which can be interpreted as a binary number as given in decimal.
{}    = 1100 = 12
{ {} } = 11110000 = 240
{ {},{{}} } = 111100011111000000 = 247744
{ {},{{}},{{},{{}}} } = 11110001111100000111110001111100000000 = 259780452096

In reverse using the meta mapping, the number 15 = 1111 corresponds to the string {{ which isn’t a set in our system.

Worse the random number 267453 = 1000001010010111101 can’t even be mapped.

There are three levels then at which the system maps to Natural Numbers:
Level 1 – S1 – the set of numbers which correspond to valid set strings – this is S. {…,240,…}
Level 2 – S2 – the set of numbers which correspond to strings – which includes nonsense strings. {…,15,…,240,…}
Level 3 – S3 = N – which includes even numbers that cannot be mapped to strings at all. {N}

I imagine that S2 and S3 can be merged with a suitable “choice” of meta system. I’ll assume this and call it S*.

So the complement of S can be defined as:

S U S` = S*

So a system has a complement at two levels. The complement of the elements (character set) and the complement of the set of possible sets of these elements. And now I’ve run out of steam this morning… but can’t these two sets be put into correspondence?

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?

===

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?

===

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!

GEB & SRH the deal!!

I am being put off from reading this because it is so huge (and life isn’t that long… and so is this blog!!! will weed through it an condense it down to less than 1000 words when I get to the destination!!!), but I get the feeling that Hofstadter has trail blazed through the very land which I have been looking towards all these years. That said I don’t think there is a proof of the SRH or given the work last week that he even has it right! Dipping in last night for the first time I came to this:

However, it is important to see the distinction between perceiving one-self, and transcending oneself. You can gain visions of yourself in all sorts of ways – in a mirror, in photos or movies, on tape, through the descriptions of others, by getting psychoanalysed, and so on. But you cannot quite break out of your own skin and be on the outside of yourself. .. A computer program can modify itself but it can’t violate its own instructions-it can at best change some parts of itself by obeying its own instructions. p478 GEB.2000 edn

Now this is exactly the SRH, but Hofstadter has made a very fundamental mistake. If we logically cannot get outside our ”self” then what exactly are we talking about? In other words what we think is “ourself” is exactly not ourself because such a view would break the SRH. What we think is “Ourself” is always just a trivial construct within our “true self”. Ergo we can’t know self, for the very idea of self defeats itself in a viscous circle. That is the SRH.

Hofstadter seems (at this point in the book) to keep the idea of self as a fixed “reality” that somehow he magically knows about even while admitting that we can’t step outside ourselves. He seems satisfied somehow that this isn’t self-defeating.

This is Descartes in a way: “I think therefore I am” doesn’t construct an individual self it only implies a larger self – that self is exactly what allows B.Russell and A.J.Ayer to say, “I think therefore there are thoughts” – if there is are thoughts then there is also a world and then there is also be a self for they are the same!

I can’t remember what is in notes and what is blogged at this stage but last week it became clear that the only necessary object of study is the circle.

[while just looking for some reference for what I was about to write I looked at http://www.themathcircle.org/coursenotes/settheory1.pdf and saw a very simple demonstration of NULL as discussed before in this blog: {All x such that x is not in a set} which they decided was identical with {}. It is not quite the discussion of True/False/NULL but I blog it for future reference]

Circles are interesting for two reasons:

(1) They are isomorphic with the notion of set theory via Venn diagrams. So to define the set of points “within” a circle (using set theory) is at the same time isomorphic with defining what a set is (via Venn Diagrams)!! E.G. I may draw two circles and put names in each meaning that there are two sets. But I may also write a set constructor which defines two circles. It works both ways. Can I use this? Also what about the boundaries – is the line around the set in Venn Diagrams part of A or ¬A. Also the issue of open/closed sets. If A is an open set then ¬A is closed and vice versa. The points of the “edge” condition belongs to either A or ¬A but the condition cannot be “in” the set. I.E. the elements may conform to a rule, but the rule cannot be in the set itself. The points of the boundary cannot be the rule!!! This needs to be explored
  
as well as I know set constructor logic at the moment : it is the set of points within a circle (the unit disk), a subset of the Euclidean plane where x and y are members of the real numbers. It’s an open set without a boundary.

The boundary is given by

So the  union of the two sets is the closed set of the unit disc with a boundary. There are a number of points I’m not analysing now just splattering down on the scrapbook: but one is whether the set of the boundary can be used to construct the closed circle… basically impredicativity.

 

(2) They are also interesting because all this is becoming identical to that experience I had as a child with the angle-poise lamp. Last night I realised that everything here is heading to that one point.

The conclusion last week was that to construct the set of points within a circle we need to agree on a system of defining points. But as soon as we do that we define a space larger than we need. Within this canvas we then create a rule to define the subset which will be our circle. Thus a circle presupposes what is not the circle. To have a circle means that we must already have not a circle. This is the same as saying that every set has a complement (I wonder what the official proof of that is … explore). “A” seems to imply in some sense “A U ¬A”, or at least is presupposes … I have a bad cold and not much sleep at the moment and my brain isn’t up to all this today (I just had difficulty following the plot of The Departed – not good !!) … this is Hegelian Logic and inside here somewhere is this realisation that an entity implies (at some level) its opposite – not literally but at a deeper level. Is this isomorphism between something and it’s opposite expressible in some logic. That was the goal of the SRH.

===

Returning to GEB I have issues with the the concept of perceiving oneself too. How many photos can there be of “oneself”? There is no limit. How many selves can there be? By definition the latter question can be only one. Thus no picture (of which there are endless potential copies) actually “contains” in any way something that can be called self (of which there is only one). It is at best just a name for the self. “My Muse” had an imaginary friend called George. Just because George has a name doesn’t mean that George exists. In fact just because “My Muse” has a name doesn’t mean that she exists either. That relation which by definition must remain singular to qualify as “self” cannot be experienced; if it is perceived it is through intellect only not through sensory data. The search for a definition of self in symbols has been the failed quest for the SRH. Given a symbolic model of “self” the rest of the SRH should follow simply.

The best example of self found so far is Smullyan’s “norm” based on Quine. To recall the norm of a sentence is that sentence followed by itself in quotations. It is a purely mechanical function. Thus,

1.0 norm(A) = A”A”

1.1 The norm of.

1.2 The norm of “The norm of”

Identifies the norm of the sentence “The norm of” (1.1). The norm of 1.1 is 1.1 followed by its quotation which is 1.2. So 1.2 identifies itself.

I know there is still something interesting about the self-reference because of the way that a system can be committed to an inescapable contradiction or an inescapable truth – that inescapable relationship is interesting. However it is not “very” interesting anymore. I feel like Gilbert Ryle removing the mystery of knowing one’s own thoughts “privately”. The set of things that an axe can cut in half includes anything wood. It even includes its own handle in a general sense and the tree that formed the handle. The only reason that these two examples become a problem are not because of something intrinsically mysterious about “self” but simply because those system contain a contradiction. But, there is not anything intrinsically mysterious about contradiction and self is only one way to create contradictions. (This is more the way of my mind after last weeks analysis). One may create a system where an axe is shared by two lumber jacks who accidentally agree to use the axe on the same day. They can’t both use the axe at the same time: this is just a factual limitation of “being one thing”. There is no issue of self here just issues of existence as an individual thing. That is what seems magical about “self”.

So the SRH had been much more precisely demarked and errors in my original conception finally exposed. It is not actually about self very much at all! Rather to do with self being a conception that presupposes non-self. “Self” can never be everything, no construction can ever be everything, and also that every self implies a greater self which is composed from the old self and its antithesis exactly like Hegel. Hofstadter over looks the fact that for a system to know itself it must actually be not itself!!

SRH - Real progress - Boundaries

I went to the library with the intention of getting somewhere with this and realised that there are really 2 possible proofs of SRH:

(1) The Boundary proof
This proof aims to show that a system cannot contain its own boundary.


(2) The dialectic proof.
This proof aims to show that a system cannot define itself without reference to things which are not within the system.

I realise today that they are the same thing. The SRH is not really about self-reference this has been a red-herring: there are lots of examples of self-reference. The problem is to do with self-definition.

Now if something cannot define itself then it must be defined in terms of things that are not part of the system. This has two implications

(1) A system is never complete. No system can construct "All Things", so there is no system called "Totality"

(2) A system is really "not itself" since it is defined with elements that are not itself. The Yin-Yang.

It is easy to see that a circle boundary is not contained "in" a circle. But there is work tio do

Identity is isomorphic with membership (SRH)

interesting point to highlight from last post: Identity is isomorphic with membership.

When we define something in maths we can do so simply by identifying it in a set to which it is a member. Thus we might say that there exists an x such that x is in the set of natural numbers and x > 4 and x < 6. This is true when x=5 so we are talking about x being 5.

But this is ironic also. Because to be a member of the natural numbers is to define some property (or predicate) of x which it shares with all other natural numbers – hardly making it unique and individual. So to find my identity in being “black skinned” is hardly making me an individual but really just making me a member of black skinned people in which I would lose my identity. We must have some other notion of identity which is separate from the predicate to ensure that we gain some identity when we become a member! I believe that this unconditional entity which we feel enables us to add a unique 1 to what is an homogenous membership is what Buddha considers an illusion.

In maths I can’t find a way to “add 1” to a predicatively defined set, since all members that fit the definition are automatically members and so there is no way that some entity can be added.

However “countability” is an interesting concept. If the set can be ordered – like numbers are usually ordered according to the property of their magnitude 3 being larger than 2 comes after 2 – then each number has a unique place in the set. Suddenly its uniqueness is assured within the membership.

So we humans feel that there is some analogy to countability within the human race or the black race or the Arsenal football stand that preserves our identity while allowing us to be a member.

Interesting that some memberships have their own countability like height or bank balance. People can order themselves in a room according to height and people order themselves according to how rich they are. But again there is some individual countability that places each person as an individual within the set of human beings, or friends or whatever. This property for now must be the property of being a self.

Now I sat today to get a step closer to the SRH but have been forestalled. Yet here have run into it again. The hypothesis is that self-reference is impossible. I have been forced to modify this as self-reference is quite easy. Impredicativity was a step further but draws in the idea of constructivism. Am I saying that you cannot construct a self-reference because you don’t know what to predicate before it is built.

Fractals repeat a form which must be completely built in order to build any part of them so this is not a problem. The assumption here is that the fractal exists and we merely have algorithms which find the points in the set. The set is defined then by conditions which allow a solution to exist which repeats itself.

So it does seem that self-reference may exist and we are only producing expressions to represent this. While the SRH is saying that nothing can exist which depends upon itself.

Looks like I need to clarify my notion of “existence”!

Making

 

Occurred to me last night that this concept is incredibly complex. It relies upon that well examined feature here now called “emergent property”. When we make something we are relying upon the feature of the world that what we end up with is different from what we started with. Yet in material terms this is absurd since you can’t make matter or energy. We conclude rather paradoxically that when we make something we actually make nothing!

This is the source of the ancient observation that the “world” is made from nothing. This because the “world” is not to be counted as the matter or stuff (in its modern conception) in the universe but rather the emergent forms into which this is made.

Did God then make the stuff of the universe or the forms?

Man can make forms. One assumes that God makes matter.

From Buddhism however we know that there is no such thing as matter – at best it is a form (idea) made in the mind. After all how can you separate the matter of the universe from the form? What is pure matter (matter without form) is like asking what shape is water?