*The following is an introductory chapter to a mathematical manuscript I am compiling from old notes written in the 1980s on the subject of disorder. This chapter deals with the set theoretic categories I use to develop the idea of disorder. However, I'm also including it as a footnote to my Thinknet series as it touches on the subject of "self-awareness" and incomputability.*

# Foundational Issues

**1**

**Introduction**

For me the
subject of disorder starts with a simple model: I imagine a set of items where
these items are not envisaged to have any particular orientational or
sequential relation to one another; as in a set of database records the order in
which the items occur, if any, is unimportant, irrelevant in fact. I then
imagine a variety of properties being assigned or distributed over these items
thereby leading to

*classes*of items. This item-property model is essentially a literal interpretation of the Venn diagram. The points in a Venn diagram are thought of as concrete items which can be collected together in various classes according to the properties those items have: In Venn notation we draw a circle round a set of points to represent a class, where each item in the class has some defining property. If we draw two circles based on two classes each defined by their particular property then we can then represent the relationship between these two classes by the extent of the overlap, if any, between the two classes; an idea familiar to anyone who has seen a Venn diagram. If we add more property defined classes we may end up with a complex picture looking something like:*Figure 1: Many overlapping classes*

This picture of
the Venn diagram gives us a simple concrete model of set theory which
immediately circumvents Russell’s paradox. It does this by distinguishing
sharply between classes and the items of which the classes are composed; that
is, classes are collections of items but the items are not themselves classes. This
means that it is not possible to create a class of classes, a construction from
which the paradox arises; in the Venn picture one can only form classes from
items. This is a very easy and elementary way of preventing Russell’s paradox,
although there are of course big mathematical disadvantages in disallowing
classes of classes. Von Neumann’s version of axiomatic set theory for instance,
which is a more sophisticated solution to Russell’s paradox, makes a
distinction between classes which are not elements and classes which are elements.
This prevents Russell’s paradox without sacrificing the possibility of allowing
classes of classes.

However, there
is a way of extending the Venn model in order to allow classes of classes
without incurring the penalty of Russell’s paradox. We take a second Venn
diagram that maps the classes in the first diagram to the points (= items) of
this second diagram and then uses the properties of these second order items
(which in fact includes classes mapped from the first diagram) to group them
into classes. Provided only allow mappings from the first Venn diagram to the
second Venn diagram, and not the other way round, this hierarchal strategy for
eliminating Russell’s paradox works because in the hierarchal model a class can
never be a member of itself; a class can only be a member of a higher order
class in another Venn diagram.

The foregoing solution
to Russell’s’ paradox reminds me of a picture used by Karl Popper at a lecture
I attended by him at Kent University in 1971. He imagined a man drawing a map
of the room in which he was sitting. To be thorough the man would have to eventually
draw his own map as it is part of the room and this would include the very
marks on the map! If you think about it
this leads to an infinite regress: The map would include a picture of the map
which would include a picture of the map etc etc. This is not necessarily a very
tractable solution to Russell’s paradox, but it’s a possibility that can be conceived.
Thus the system which can boast full self-awareness must be infinite!

**2**

**Self-description, conceptual feedback and contradiction**

If a class is
permitted to be a member of itself then it opens the possibility of
contradictory self-description. (a form of

*self-reference*). In contradictory self-description it is possible for the act of self-description to render the description false. For example consider the following sentence: “*This sentence contains a reference to itself*”. The latter sentence is self-referencing in that it talks about itself; it is telling us that it contains a reference to itself which of course in this case it does. This is in fact a case of*consistent*self-description; that is, this sentence contains non-contradictory self-description and the sentence states a truth about itself. On the other hand compare the following phrase: “*Hello world!*”. Nothing wrong with that of course but it is clearly a phrase which contains no reference to itself. OK, so if it contains no reference to itself let’s affirm this and rewrite it as it as: “*Hello World*,*this sentence contains no reference to itself*”? This latter sentence now contains a reference to itself, but as this reference is denying that it contains a reference to itself it thereby contradicts itself…. It is not possible, without contradiction, for a sentence to contain no reference to itself and at the same time this sentence describe itself as lacking in self-reference.
As in the case
of self-descriptive sentences, when we open up the possibility of a class being
a member of itself self-description and therefore self-contradiction becomes a
liability. We can see this from the following analysis: A class is defined by a
property; thus a class is a way of describing the items in that class because
the class is telling us that each item has some property, let us call that property
“X”. Therefore if a class, let’s call it C, is member of itself then C is
effectively describing itself because it is telling us that C must also have
property X. But whether or not the class C as defined by X can be formed
without contradiction now depends on the nature of X. For if X is something
like “

*The class of classes that don’t contain themselves*” then we are quickly faced with Russell’s oscillating contradiction: Viz: If C isn’t a member of itself then X requires that we must put C inside C thus making it a member of itself. But if C is now a member of itself it then violates the stipulation that X =*all classes that aren’t members of themselves*. And so we oscillate between putting C inside itself and them taking it out. This kind of conceptual behavior can be likened to a form of unstable feedback. This is something I developed in this blog post.
As I have
already said, employing a second level Venn diagram is one way, if an unconventional
way, of solving Russell’s problem; here one can only form classes of classes using
a higher or meta-level Venn diagram whose points (or items) represent classes
in the lower level diagram. Thus it becomes impossible for a class to be self-descriptive
because classes are never members of themselves thereby preventing the
possibility of an unstable oscillatory conceptual feedback loop. In the
hierarchal system where one Venn diagram looks down on and describes another
Venn diagram the information only moves one way, namely bottom up, thus
preventing contradictory “feedback loops” forming.

This method of
using a hierarchy of Venn diagrams is probably not a convenient solution for
conventional mathematics, but it works for open ended real world ontologies
where systems are not necessarily in isolated self-containment as required by
axiomatic mathematical systems. In the real world ontologically separate
systems can take up the job of describing other systems without getting into
unstable/contradictory “feedback loops”. As we shall see in the next section this
realization helps solve the conundrum raised by Penrose in his two books “

*The Emperor’s New Mind*” and “*Shadow Minds*”.**3**

**Penrose: Is the human mind an incomputable process?**

In his books “

*The Emperor’s New Mind*” and “*Shadow Minds*” Roger Penrose concludes that the human mind is an incomputable process. I dealt with this matter more thoroughly in the blog posts here and here, but in the following I rehash some of my reasoning against this conclusion.
Penrose defines
a class of algorithms using the symbolism C

*(*_{q}*n*) – this notation represents the*q*^{th}computation acting on a single number*n*. In fact it must be stressed that the set C*(*_{q}*n*) is taken to enumerate*all possible algorithms*that take the number*n*as a parameter. Penrose then goes onto demonstrate a version of the halting theorem. This theorem tells us that there is no general algorithm which can*in all cases*be used to correctly test the halting status of any other algorithm and flag this status by itself halting if the tested algorithm tested doesn’t halt. As Penrose then shows this is because the algorithm which tests for halting status must itself be one of the C*(*_{q}*n*) and therefore when the halting-test algorithm is submitted to itself typical contradictions of self-description arise. Viz: If the halting-test algorithm halts when testing itself it would be trying to tell us that it doesn’t halt when testing itself! Another way of saying that is: When submitted to itself the halting-test algorithm is supposed to halt if it doesn’t halt! So, in the very act of trying to tell the truth about itself the halting-test algorithm invalidates that “truth”; this is a typical self-descriptive conundrum.
Penrose submits
that the way to remove this contradiction is to propose that the halting-test
algorithm fails to stop when it is submitted to itself. But this means that it
is thereby unable to determine the truth about itself. According to Penrose the
set defined by C

*(*_{q}*n*) is universal enough to include any algorithms that are running in the human mind, so here’s the twist: If we as humans can see this algorithm doesn’t halt, but the universal halting-test algorithm fails to reveal it, this must mean that the human mind is doing something an algorithm can’t. Ergo (according to Penrose) the human mind must be using an incomputable process.
The real world
is open ended and its ontologies are not self-contained: One ontology can
always be described by another ontology thus removing the potential for contradiction
when a single ontology attempts to describe itself. My proposal is that the
contradictions in Penrose’s example arise through self-description. So, even
though the human mind may be running one of the C

*(*_{q}*n*)*in form*the ontology of the mind is a*meta-ontology*separate from the symbolic manipulations of Penrose’s self-contained algorithmic enumeration. Because the human mind is effectively one-up the descriptive hierarchy it therefore does not become entangled with the self-descriptive contradiction that Penrose finds to exist*inside*the set of C*(*_{q}*n*). Therefore because the mind is meta to the ontology of the C*(*_{q}*n*) the contradiction which arises out of the self-description in the halting-test algorithm does not arise in the human mind. In this instance the human mind is not attempting to describe itself; rather it is describing something which is*other*than itself. Hence being ontologically at a higher level the human mind isn’t hamstrung by the contradictory self-referencing loop which occurs inside Penrose’s algorithmic set. But... and this is the big ‘but’ …. if the human mind turns in on its own algorithms and*attempts to make descriptions about itself*we find that the conceptual feedback loops which are liable to create self-descriptive contradictions and which result in the halting theorem reassert themselves. In short the human mind isn’t above algorithmic self-referencing limitations and I am therefore inclined to reject Penrose’s conclusion that human thinking is an incomputable process.
## No comments:

Post a Comment