Saturday, December 23, 2006

by Damjan Bojadziev, from here

Formal self-reference in Gödel's theorems has various features in common with self-reference in minds and computers. The theorems do not imply that there can be no formal, computational model of the mind, but on the contrary, suggest the existence of such models within a conception of mind as subject to similar limitations as formal systems. If reflexive theories do not themselves suffice as models of mind-like reflection, reflexive sequences of reflexive theories could be used.

* Introduction
* Self-reference in Gödel's theorems
o Implications of Gödel's theorems
o Non-implications of Gödel's theorems
* Formal models of the mind
o The basic incompleteness argument
o Mind over machine omega:1 ?
o Reflexive sequences of reflexive theories
* Self-reference in computers
* Self-reference in minds
* Conclusion
* References

At first sight, the designation of the topic of this special issue, "MIND <> COMPUTER", also transcribed as "Mind NOT EQUAL Computer", looks like a piece of computer ideology, a line of some dogmatic code. But there are as yet no convincing artificial animals, much less androids, and computers are not yet ready for the unrestricted Turing test. Although they show a high degree of proficiency in some very specific tasks, computers are still far behind humans in their general cognitive abilities. Much more, and in much more technical detail, is known about computers than about humans and their minds. Thus, the required comparison between minds and computers does not even seem possible, much less capable of being stated in such a simple formula.

On the other hand, it could be argued that it is precisely because we do not know enough about ourselves and our minds that we can make comparisons with computers and try to design computational models. This is especially so because we also do not know exactly what computers are incapable of, although we have some abstract, general results about their limitations, such as Turing's theorem about the inability of an idealized computer to determine for itself whether its computation terminates or not. This theorem, and related results by Gödel and Church, are frequently used in arguments about the existence of formal models of the mind; interestingly enough, they have been used to argue both for and against that possibility. As a preliminary observation, it can be noted that the "negative" use of limitative theorems, as these meta-mathematical results are called, is less productive in the sense that the faculty by which mind is supposed to transcend "mere" computation remains essentially mysterious. The "positive" use of the theorems promotes a more definite, less exalted view of the mind as something which has its own limitations, similar to those which formal systems have. The present paper argues for this latter view, exploring the common feature of all these theorems, namely self-reference, and focusing on Gödel's theorems.

Self-reference in Gödel's theorems
The application of Gödel's theorems to fields outside meta-mathematics, notably the philosophy of mind, was initiated by Gödel himself. He had a strong philosophical bent towards realism/platonism which also motivated his (meta)mathematical discoveries [Feferman 88, p. 96], [Weibel & Schimanovich 86]. Gödel first thought that his theorems established the superiority of mind over machine [Wang 90, pp. 28-9]. Later, he came to a less decisive, conditional view: if machine can equal mind, the fact that it does cannot be proved [Weibel & Schimanovich 86], [Casti 89, p. 321]. This view also parallels the logical form of Gödel's second theorem: if a formal system of a certain kind is consistent, the fact that it is cannot be proved within the system. Gödel's more famous first theorem says that if a formal system (of a certain kind) is consistent, a specific sentence of the system cannot be proved in it.

Gödel's theorems are actually special, self-referential consequences of the requirement of consistency: in a consistent system, something must remain unprovable. One unprovable statement is the statement of that very fact, namely the statement which says of itself that it is unprovable (first theorem): you cannot prove a sentence which says that it can't be proved (and remain consistent). Another unprovable statement in a consistent system is the statement of consistency itself (second theorem). In addition, if the formal system has a certain stronger form of consistency, the sentence which asserts its own unprovability, called the Gödel sentence, is also not refutable in the system. Rosser later constructed a more complicated sentence for which simple consistency is sufficient both for its unprovability and for its unrefutability. Similar sentences were constructed by others (Rogers, Jeroslow [Boolos 79, pp. 65-6]), showing that consistent formal systems cannot prove many things about themselves. On the other hand, a formal system can retain all the insight into itself that is compatible with consistency: thus, although it cannot prove its Gödel sentence, if it is to remain consistent, it can prove that very fact, namely the fact that it cannot prove its Gödel sentence if it is consistent [Robbin 69, p. 114].

Implications of Gödel's theorems
The fact that a particular sentence is neither provable nor disprovable within a system only means that it is logically independent of the axioms: they are not strong enough to either establish or refute it - they don't say enough about it one way or the other. Saying more, by adding additional axioms (or rules of inference) might make the sentence provable. But in Gödel's cases, this does not work: even if Gödel's sentence is added as an additional axiom, the new system would contain another unprovable sentence, saying of itself that it is not provable in the new system. This form of self-perpetuating incompleteness might be called, following [Hofstadter 79, p. 468] and [Mendelson 64, p. 147], essential incompleteness.

Gödel's theorems uncover a fundamental limitation of formalization, but they say that this limitation could be overcome only at the price of consistency; we might thus say that the limitation is so fundamental as to be no limitation at all. The theorems do not reveal any weakness or deficiency of formalization, but only show that the supposed ideal of formalization - proving all and only all true sentences - is self-contradictory and actually undesirable:

* what good is a formalization that can prove a sentence which says that it is not provable (first theorem)?
* what good is a formalization that can prove its consistency when it would follow that it is not consistent (second theorem)?

On the positive side, the theorems show that certain formal systems have a much more intricate, reflexive structure then formerly suspected, containing much of their own meta-theory.

Gödel's theorems show that the notions of truth and provability cannot coincide completely, which at first appears disturbing, since, as Quine says,

we used to think that mathematical truth consisted in provability [Ways of Paradox, p. 17].

Gödel's theorems undermine the customary identification of truth with provability by connecting truth with unprovability: the first theorem presents a case of
not provable -> true

(if the sentence asserting its own unprovability is not provable, then it is true); the second theorem presents a case of
true -> not provable

(if the sentence asserting the consistency of the system is true, then it is not provable). However, the notion of truth has a problem of its own, namely the liar paradox, of which Gödel's sentence is a restatement in proof-theoretic terms. Thus, Gödel's theorems do not actually establish any disturbing discrepancy between provability and truth. Furthermore, the implication (*) above is an oversimplification: assuming consistency, Gödel's sentence is not simply true, because it is not always true i.e. not in all interpretations. If it were, it would be provable, by the completeness theorem (also proved by Gödel), as noted in [Penrose 94, p. 116, note 5]: provability is truth in all interpretations). The first theorem shows that if the system is consistent, it can be consistently extended with the negation of the Gödel sentence, which means that the sentence is actually false in some models of the system. Intuitively, without going into details, this could be explained by saying that in those models the Gödel sentence acquires a certain stronger sense of unprovability which those models do not support [Bojadziev 95a, p. 391]. Gödel's theorem thus shows that there must always exist such unusual, unintended interpretations of the system; as Henkin says, quoted in [Turquette 50]:

We tend to reinterpret Gödel's incompleteness result as asserting not primarily a limitation on our ability to prove but rather on our ability to specify what we mean ... when we use a symbolic system in accordance with recursive rules [Gödel & the synthetic a priori].

Similarly, Polanyi says, though only in connection with the second theorem:

we never know altogether what our axioms mean [Personal Knowledge, p. 259]. We must commit ourselves to the risk of talking complete nonsense if we are to say anything at all within any such system [p. 94].

This characterization of formal language sounds more like something that might be said about ordinary, natural language. Thus, if we take as a characteristic of ordinary language its peculiar inexhaustibility and the frequent discrepancy between intended and expressed meaning ("we never know altogether what our sentences mean; we must risk talking nonsense if we are to say anything at all"), Gödel's theorems would show that, in this respect, some formal languages are not so far removed from natural ones. Certain similarities between the self-reference in natural language and in Gödel's sentence and theorems have also been noticed at the lexical and pragmatic level (indexicals [Smullyan 84], performatives [Hofstadter 79, p. 709]). This line of thought, namely that the self-reference which leads to Gödel's theorems makes a formal system more human, so to speak, will be followed here to the conclusion that such systems are indeed suitable for modelling the mind.

Non-implications of Gödel's theorems
Certain authors, especially some of those who attempt to apply Gödel's theorems to disciplines other than meta-mathematics, are handicapped by a more or less severe misunderstanding of the theorems. For example, Watzlawick, Beavin and Jackson state:

Gödel was able to show that it is possible to construct a sentence G which

1. is provable from the premises and axioms of the system, but which
2. proclaims of itself to be unprovable.

This means that if G be provable in the system, its unprovability (which is what it says of itself) would also be provable. But if both provability and unprovability can be derived from the axioms of the system, and the axioms themselves are consistent (which is part of Gödel's proof), then G is undecidable in terms of the system [Pragmatics of Human Communication, p. 269].

Of course, this is completely garbled, but the authors nevertheless have very interesting ideas about applications of Gödel's theorems.

Formal models of the mind
Gödel's (first) incompleteness theorem can be expressed in the form: a sufficiently expressive formal system cannot be both consistent and complete. With this form, the attempt to use such formal systems as models of the mind invites the following brush off:

Since human beings are neither complete nor consistent, proving that computers can't be both doesn't really help [Roger B. Jones, "quoted" from sci.logic, May 1995].

A different intuition was followed by Wandschneider: the limitations of formalization revealed by Gödel's theorems prevent the use of formal systems as models of the mind [Wandschneider 75]. Most authors, however, accept the comparison between mind and formal systems of the kind considered by Gödel, but reach different conclusions. For example, according to Haugeland

most people are agreed ... that [Gödel's] result does not make any difference to cognitive science [Mind Design, p. 23].

According to [Kirk 86], arguments against mechanism based on Gödel's theorems are agreed to be mistaken, though for different reasons; cf. [Dennett 72] and especially [Webb 80]. These arguments try to establish the superiority of mind by suggesting that mind can reach conclusions which a formal system cannot, such as Gödel's sentence.

The basic incompleteness argument
Arguments about the relative cognitive strength of minds and machines usually invoke only the first Gödel theorem, although the second theorem also establishes the existence of a sentence which, if true, is not provable. The premise of both theorems is consistency, and it frequently appears neglected in the basic version of the argument from incompleteness: since any formal system (of a certain kind) contains a true but unprovable sentence, mind transcends formalism because mind can "see " that the unprovable sentence is true. This conviction can be traced, in various forms, from [Penrose 94], [Penrose 89], through [Lucas 61] back to [Nagel & Newman 58, pp. 100-1]. For example, Lucas says:

However complicated a machine we construct, it will ... correspond to a formal system, which in turn will be liable to the Gödel procedure for finding a formula unprovable-in-that-system. This formula the machine will be unable to produce as being true, although a mind can see it is true. And so the machine will still not be an adequate model of the mind [Minds, Machines and Gödel].

The consistency premise is not very prominent here, but some suspicious phrasing is: 'producing as being true', 'seeing to be true', instead of the simpler and more to the point 'proving'. This way of comparing cognitive strength in men and machines leaves out an obvious symmetry while emphasizing a dubious asymmetry. The symmetry is that, just as a formal system cannot prove a sentence asserting its own unprovability, unless it is inconsistent, so can a mind not do so, if it is consistent; cf. [Casti 89, p. 321]. The doubtful asymmetry between mind and machine concerns their possession of the notion of truth. The mind is supposed to have this notion in addition to the notion of provability, and is supposed to have no problems with it (but it does, namely the liar paradox). On the other hand, the machine is only supposed to be able to prove things (as its only means of establishing truth) without having, and apparently without being able to have, an additional notion of truth. But this is not so: for expressing the truth of the Gödel sentence (as opposed to proving it), even the most restricted definition of the truth predicate true1(x), covering sentences containing at most one quantifier, is sufficient [Webb 80, p. 197].

Mind over machine omega:1 ?
A more intricate version of the argument from incompleteness considers adding a "Gödelizing operator" to the system. This form of the incompleteness argument was also first advanced by Lucas:

The procedure whereby the Gödel formula is constructed is a standard procedure ... then a machine should be able to be programmed to carry it out too ... This would correspond to having a system with an additional rule of inference which allowed one to add, as a theorem, the Gödel formula of the rest of the formal system, and then the Gödel formula of this new, strengthened, formal system, and so on ... We might expect a mind, faced with a machine that possessed a Gödelizing operator, to take this into account, and out-Gödel the new machine, Gödelizing operator and all [Minds, Machines and Gödel].

The sound part of this argument is already contained in the notion of essential incompleteness: a Gödel operator only fills a deductive "lack" of the system by creating a new one. Adding the Gödel sentence of a system as a new axiom extends the notion of provability and thereby sets the stage for a new Gödel sentence, and so on. Thus, a Gödel operator only shifts the original "lack" of the system through a series of displacements, without ever completing the system.

The Lucas argument, especially in the form advanced by Penrose [Penrose 94], now centers on how far into the transfinite can a Gödel operator follow the mind's ability to produce the Gödel sentence of any system in the sequence
S1 = S0 + G(S0)
S2 = S1 + G(S1)
Somega+1 = Somega + G(Somega)
A relevant result here is the Church-Kleene theorem which says that there is no recursive way of naming the constructive ordinals [Hofstadter 79, p. 476]. This would mean that a Gödel operator could only follow the mind's ability to produce Gödel sentences through the recursively nameable infinite; cf. [Penrose 94, p. 114]. [...] On the other hand, as Webb says,

there is not the slightest reason to suppose that ... a machine could not model the 'ingenuity' displayed by a mind in getting as far as it can [Mechanism, Mentalism and Metamathematics, p.173].

But for the purposes of this paper it is more interesting to observe that it does not seem plausible that the argument about the formalizability of mind should be decided by the outcome of the race between mind and machine over remote reaches of transfinite ordinality. And even if it makes sense to conceive of mind as always being able to out-reflect a reflective formal model, it would seem that the ability to perform the self-reflection is more important than the question of how far does this ability (have to) reach.

Reflexive sequences of reflexive theories [update]
A further possibility in the direction of making reflexive formal models is to make the progression of reflexive theories itself reflexive. The usual ways of extending a reflexive theory by adding its Gödel sentence, or the statement of consistency (Turing), or other reflection principles (Feferman) are themselves not reflexive: what is added to a theory only says something about that theory, and nothing about the one which its addition produces. Thus, what is usually added to a theory does not take into account the effect of that very addition, which is to shift the incompleteness of the original theory to the extended one. Of course, certain things about the extended theory cannot be consistently stated; for example, the sentence stating that its addition to a theory produces a consistent theory would lead to contradiction, by the second Gödel theorem. But the sentence which is added to a theory could make some other, weaker statement about the theory which its addition produces. If the procedure of theory extension operated not only on the theory it is to extend but also on a representation of itself, it could build on its own action and improve its effects. It might thus produce in a single step an extension which is much further down the basic sequence of extensions, produced by linear additions of Gödel sentences; the size of this ordinal jump could then be taken as a measure of the reflexivity of the procedure. This kind of procedure, operating on something which contains a representation of that procedure itself, is already familiar from the construction of the Gödel sentence: the process of diagonalization operates on a formula containing a representation of (the result of) that very process [Hofstadter 79, p. 446], [Bojadziev 90]. Another example of a reflexive procedure of this kind would be the Prolog meta-circular interpreter, which can execute itself, though only to produce statements of iterated provability [Bratko 90, p. 536].

Self-reference in computers
In saying of itself that it is not provable, the Gödel sentence combines three elements: the representation of provability, self-reference and negation. In computer science, self-reference is more productive in a positive form, and in programs, programming systems and languages more than in individual sentences. The first ingredient in Gödel's sentence, the representation of provability, corresponds to the explicit definition of the provability predicate of a logic programming language in that same language. In the simplest case, specifying Prolog provability in Prolog, the definition consists of just a few clauses [Bratko 90, p. 536], comparable to those which express the conditions on the provability predicate under which Gödel's theorems apply. This definition of Prolog provability is then used as a meta-circular interpreter to extend the deductive power of the basic interpreter, for example by detecting loops in its proof attempts. This use of the meta-circular interpreter could be compared to the work of the Gödel operator on extending the basic, incomplete theory. Meta-circular interpretation is also applicable to other programming languages, notably LISP [Smith 82].

Generalizing meta-circular interpretation, provability can be specified in a separate meta-language, and reflection principles defined for relating and mixing proofs in both languages. Such meta-level architectures [Yonezawa & Smith 92] can be used to implement reflective or introspective systems, which also include an internal representation of themselves and can use it to shift from normal computation about a domain to computation about themselves [Maes & Nardi 88] in order to achieve greater flexibility. Meta-level architectures are useful for knowledge representation, allowing the expression and use of meta-knowledge, and opening the possibility of computational treatment of introspection and self-consciousness [Giunchiglia & Smaill 89, p. 128]. For example, Perry suggested an architecture of self-knowledge and self in which indexicals mediate between bottom level representations, in which the organism is not itself represented, and higher levels at which it is represented generically, as any other individual [Perry 85].

Self-reference in minds
The basic lesson of Gödel theorems, namely that the ability for self-reflection has certain limits, imposed by consistency, does not seem to be less true of minds than it is of formal systems. Applied to minds, it would translate to some principled limitation of the reflexive cognitive abilities of the subject: certain truths about oneself must remain unrecognized if the self-image is to remain consistent [Hofstadter 79, p. 696]. This formulation recalls the old philosophical imperative which admonishes the subject to know himself. If this were simple or possible to do completely, there would be no point to it; the same goes for the explicit interrogative forms: who am I, where am I going, what do I want, ... Hofstadter rhetorically asks:

Are there highly repetitious situations which occur in our lives time and time again, and which we handle in the identical stupid way each time, because we don't have enough of an overview to perceive their sameness? [Gödel, Escher, Bach, p. 614].

Such an overview can be hard to achieve, especially in regard to oneself, as Laing's knots in which minds get entangled show [Laing 70]. In a similar vein, Watzlavick, Beavin and Jackson suggest that the limitative theorems show the mathematical form of the pragmatic paradoxes to which humans are susceptible in communication [Watzlavick et al 67, p. 221]. [update]

It may be that, as Webb says, the phrase 'the Gödel sentence of a man' is an implausible construction [Webb 80, p. x], but certain interpretations might be imagined, such as self-falsifying beliefs. On a humorous note, the Gödel sentence for a human could work like a recipe for self-destruction, activated in the process of its comprehension or articulation; by analogy with the terminology of performatives we might call it a (self-)convulsive, asphyxiative, combustive, ... A more elaborate interpretation, as the paralysing effect of some self-referential cognitive structure, is presented in Cherniak's story [Hofstadter & Dennett 81, p. 269]. The history of logic itself records lethal cases (Philetas) and cases of multiple hospitalization (Cantor, Gödel). Of course, this is all anecdotal, speculative and inconclusive, but it does suggest that the apparent gap between minds and machines could be bridged, in two related ways:

* the vulnerability of minds to paradoxes of self-reference
* the implementation of self-referential structures in machines

The mind-machine gap could thus be reduced by emphasizing the formal, machine-like aspects of the mind and/or by building mind-like machines.

Finally, taking speculation one literal step further, the self-reference in Gödel's sentence can be compared to a formal way of self-recognition in the mirror, by noticing the parallelism between things (posture, gesture, movement) and their mirror images. The basis for this comparison is the way the Gödel code functions as a numerical mirror in which sentences can refer to, "see" themselves or other sentences "through" their Gödel numbers. The comparison, developed in [Bojadziev 95b], covers the stages of construction of Gödel's sentence and relates them to the irreflexivity of vision and the ways of overcoming it. The comparison attempts to turn arithmetical self-reference into an idealized formal model of self-recognition and the conception(s) of self based on that capacity. The motivation for this is the cognitive significance of the capacity for self-recognition, in mirrors and otherwise. The ability to recognize the mirror image, present in various degrees in higher primates and human infants, has been proposed as an objective test of self-awareness [Gregory 87, p. 493]. Self-recognition in the mirror is a basic, even paradigmatic case of self-recognition, the general case being the recognition of effects on the environment of our own presence in it. Self-recognition in this wider sense is the common theme of Dennett's conditions for ascribing and having a self-concept and consciousness [Hofstadter & Dennett 81, p. 267]. Self-recognition is also the common theme of the self-referential mechanisms which, according to [Smith 86], constitute the self:

* indexicality (self-relativity of representations)
* autonimy (recognizing one's own name)
* introspection (recognizing one's own internal structure)
* reflection (recognizing one's place in the world)

The comparison between formal and specular self-reference and self-recognition might also connect these contemporary attempts to base the formation of a self(-concept) on the capacity for self-recognition with the long philosophical tradition of thinking about the subject in optical terms.


It is not possible to see oneself completely, in the literal, metaphorical ("see=understand"), formal and computational sense of the word. Gödel's theorems do not prevent the construction of formal models of the mind, but support the conception of mind (self, consciousness) as something which has a special relation to itself, marked by specific limitations.


Blogger LauLuna said...

Please, correct the following statement:

"The fact that a particular sentence is neither provable nor disprovable within a system only means that it is logically independent of the axioms"

That is so only in first order systems. The axioms of second order Peano Arithmetic have all the true sentences in its language as logical consequences and even so any consistent second order system of arithmetic contains undecidable sentences. This happens, of course, because second order logic is not complete relative to the logical consequence.


7:33 PM  

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