Damjan Bojadziev

Self-reference in phenomenology and cognitive science

Expanded HTML version; drawings by Edo Podreka [more].

ABSTRACT: Self-reference in formal arithmetic can be compared to a formal way of self-recognition in the mirror. The basis for this comparison is the role of the Gödel code in arithmetical self-reference. The comparison is developed in a series of diagrams which show the stages of construction of a self-referential sentence and relate them to the irreflexivity of vision and ways of overcoming it. The aim of the comparison is to turn arithmetical self-reference into an idealized formal model of self-recognition and the conception(s) of self based on this capacity.

Gödel's theorem about the incompleteness of formal arithmetic has been frequently used in philosophy of mind, mainly in attempts to show the superiority of mind over machine. The concensus on such attempts, surveyed e.g. in (Webb, W. 1980), is that they are mistaken, based on various misunderstandings of the theorem. A neutral conclusion, drawn from the failure of these attempts, is that Gödel's theorem is not relevant to cognitive science (Haugeland, J. 1981, p. 23). On the positive side, Gödel's theorem has been used precisely as a formal model of the mind, notably by Hofstadter in Gödel, Escher, Bach (Hofstadter, D. 1979). The idea was that the self-reflection of formal systems, exemplified by Gödel's theorem, can be used to model the reflexivity, associated with self, consciousness, subjectivity, (self-)awareness, ... What such notions designate is commonly supposed to involve the capacity for self-reference, at least as a necessary ingredient; this capacity is also commonly thought to be necessarily limited, incomplete. These are also the main elements of Gödel's theorem: it depends on the construction of a sentence which says something about itself and is true (in the intended interpretation) but unprovable in the system in which it is constructed, if that system is consistent. Hofstadter compared this limitation of the self-reflection of a formal system to our inability to see our faces with our own eyes (Hofstadter, D. 1979, p. 697).

More generally, we cannot see our heads with our own eyes, as noted by the man who "lost" his head in Harding's story (Hofstadter, D. and Dennett, D. 1981, p. 23); more specifically, and more necessarily, we cannot see our own eyes with our own eyes. This limitation of vision as a basic phenomenological, or meta-phenomenological fact is not made much of in phenomenology itself; buddhism seems more susceptible to it, along with traditional metaphysics, which regards the self as the origin of the perceptual field, or a point of view on the world (Evans, G. 1982, p. 222). In representationist cognitive science, the irreflexivity of vision could be related to the postulated lowest level of representation, at which the physical carrier of representations is not itself represented, "does not exist" (Perry, J. 1985).

Comparing the limitation of self-reflection in a formal system with our inability to see our own eyes (faces, heads, ...) seems to suggest an obvious remedy: just as we use mirrors to enable us to see what we cannot see directly, so some analogous device might be thought to help in the case of formal systems. Such a construction, with two systems, each proving the "missing" theorems of the other, has indeed been attempted (Wandschneider, D. 1975), but does not work, for technical reasons (Church's theorem). But what is important here, for the purposes of this paper, are not so much these reasons themselves but the fact that the idea of such a construction is fundamentally misguided, namely the idea that a complete formal system should be built in order to be able to serve as a 'cybernetic model of the mind', as Wandschneider says. The incompleteness of a formal system does not make it unsuitable for building models of the mind, but, on the contrary, as Hofstadter's comparison suggests, it is precisely its incompleteness which makes a formal system suitable for modeling the mind. Furthermore, it can be said that a mirror-like device cannot be used to overcome incompleteness because it has already been used in producing it. Namely, the self-reference of a sentence on which arithmetical incompleteness depends is indirect, by way of a numerical code: the sentence refers to a certain number which belongs to that sentence itself under some system of numeric coding of sentences as numbers. This Gödel code functions as a numerical mirror in which a sentence can refer to itself, "see itself". To re-use Hofstadter's comparison, just as I cannot see my face, in particular my eye(s), except in a mirror, so sentences of arithmetic cannot refer to, "see" themselves, except in a numerical mirror. More generally, this is how a sentence of arithmetic can refer to another formula: by refering to its numerical image, Gödel number. The code could be pictured as

where F is a formula, f is its Gödel number, and the dotted line indicates a mapping between expressions (left) and numbers (right). The reference of sentences to formulas would then be pictured as

where f is a term denoting the number f and the arrow marks the relation of reference. The code thus functions as a numerical mirror for indirectly refering to expressions as numbers, the way mirrors are used for seeing objects as virtual images. This is how we use mirrors to see things indirectly, eg. around a corner or behind us:

> In general, we can see things outside the normal field of vision by seeing their mirror images within it; arithmetical terms can refer to things outside the normal field of reference by refering to their numerical images within it. As noted, a distinguished "thing" outside the visual field is the point of its origin; attention and reference will now be directed toward its mirror image. We thus move from the general, neutral use of mirrors as looking- glasses to their special, self-reflective use.

The standard procedure which leads to arithmetical self-reference is the substitution of a term in a formula:

This is comparable to looking into a mirror sideways, the way we look into rear-view or traffic mirrors:

That happens only when the process of diagonalization is itself reflected in the formula to which it is applied:

The metaphor of mirroring has been frequently used in non-technical expositions and discussions of Gödel's results, notably in (Nagel, E. and Newman, J. 1985) and in (Hofstadter, D. 1979), but in different ways and with a narrower coverage. The analogy as developed here is of course limited: with real mirrors, I can see myself without realizing that I see myself, and I can note the parallelism without drawing that conclusion ("a demon imitates me behind the glass"). This is the situation of the headless buddhist in Harding's story, who does not recognize himself in the mirror (Hofstadter, D. and Dennett, D. 1981, p. 28); the significance of such self-alien experiences is examined in (Baars, B. 1988, p. 332).

Self-recognition in the mirror seems to be a particularly simple, basic, even paradigmatic case of self-recognition, the general case being the recognition of effects on the environment of our 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 (Dennett, D. 1981, p. 267). Self-recognition is also the common theme of the self-referential mechanisms which, according to Smith, constitute the self: autonimy (recognizing one's own name), introspection (recognizing one's own internal structure) and reflection (recognizing one's place in the world) (Smith, B. 1986). The mirror analogy might then also serve to 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: ..., Eckhart, Locke, Fichte, Brentano, Husserl, Wittgenstein, ...