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.
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 Cq(n) – this notation represents the qth computation acting on a single number n. In fact it must be stressed that the set Cq(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 Cq(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 Cq(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 Cq(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 Cq(n). Therefore because the mind is meta to the ontology of the Cq(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.