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.

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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!!