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Stepping across levels: Curry’s paradox and Gödel’s incompleteness theorems
I have tried to make clear in the previous paragraph that Bateson’s problem is a relevant topic for proposals from different theoretical outlooks. Blurring of levels, often made visible through self-referential constructions, seems to be hardly avoidable, yet it leads to considerable even if unnoticed difficulties, which one may expect hampers the straightforward development of theory. Is there a way out? To investigate this question I shall look at two topics as a further elaboration of Bateson’s research. The first concerns Curry’s paradox, the second Gödel’s first incompleteness theorem.
Apart from the blurring of levels, Bateson seemed to believe that also the logical operation of negation was responsible for the possible emergence of paradox, for he wrote that ‘no paradox can be generated [in iconic[4]communication] because in purely [...] iconic communication there is no signal for “not” ’ (Bateson 1972, 291). However, Haskell Curry discovered a paradox, an elaboration of Russell’s that does not use negation. Several versions of the paradox exist (e.g. Curry, Feys & Craig 1958, 258-59; Visser 1989, 643), but they have in common that an object is treated as an equivalent of an implication in which the same object acts as an antecedent. Any arbitrary statement can then be derived using the implication, a paradoxical result since the arbitrariness means that for any derived statement also its opposite can be derived.[5]Curry’s paradox only aggravates the difficulties encountered already with Russell’s, since the selfinclusion or self-reference of certain objects cannot always be avoided in social life. For example, if the objects and people referred to by the term ‘a culture’ can be studied at all, it cannot be excluded that people who might be included in the reference of that term study those objects and people themselves (see the discussion of Brumann 1999 above). But then the idea of explaining or deriving facts by using ‘culture’ (or similar concepts) as a term in social-scientific theories is doomed to fail because by Curry’s paradox such explanations are totally arbitrary. The proposal that ‘culture does not exist’ does not seem to be helpful because this would mean that the term ‘culture’ really refers to nothing at all in whatever interpretation, a rather trivial way out of an interesting question of semantics. Viewed in the light of Curry’s paradox, the consequence of the existence of such collective items like culture would be that these items are actively preventing the formulation of any straightforward and consistent explanatory theory about themselves. A possibility to be investigated is that one considers certain parts of the collective items in question, and proposes (hopefully consistent) theories about those parts. The problem arising then is immediately relevant for our research on the possibility of unifying the social sciences; one would wish to avoid the result that the facts we encounter in social life would all be explainable separately with separate theories while at the same time collectively with an overall theory they would not.
We have seen that Bateson stressed the importance of the theory of types in order to avoid paradoxes in mathematics and theories in the social sciences. There does exist a formal procedure in mathematics, however, which violates the principle that ‘the name should not be included in the thing named’, but nevertheless does not produce paradoxes or inconsistencies. I am hinting at Kurt Gödel’s method of using numbers to code for expressions of Peano arithmetic and then use them in that same Peano arithmetic.[6]Peano arithmetic (‘arithmetic’ for short) is an axiomatic system that defines the natural numbers (0, 1, 2 etc.). It had been a problem for mathematicians to find out whether this system could be shown to be complete, i.e. to demonstrate that from the axioms making up arithmetic, all true statements of arithmetic could be derived (and all false statements could be disproved). Gödel showed in 1930 that this is not possible. He demonstrated that there exists at least one statement of arithmetic, which is true but cannot be derived from the axioms (Gödel 1988 [1931]).[7]
A simple example will suffice here to see what Gödel’s method entailed. Suppose we write down some sentence in the language of arithmetic, like:
2 + 2 = 5.
This sentence, which is false, must be considered not provable from the axioms of arithmetic. But the statement ‘ “2 + 2 = 5” is not provable’ constitutes a statement about the sentence ‘2 + 2 = 5’, not in the language of arithmetic itself, but in a meta-mathematical language. Gödel showed, however, that the notion ‘provable’ could itself be written as an arithmetical operation. Gödel demonstrated further that arithmetical expressions could be coded for using natural numbers (in fact, the coding numbers are now called ‘Gödel numbers’).[8]Expressions like provable and coding number of are themselves expressible in the coding device Gödel established. Gödel then showed there exists a sentence in arithmetic with coding number h, its corresponding meta-mathematical statement running like:
The sentence with the associated coding number h is not provable.
The sentence cannot be proved from the axioms of arithmetic, which is what it indeed asserts itself, but meta-mathematically we know that the sentence is true. In this way arithmetic is shown to be incomplete. In fact, if the sentence would be provable, arithmetic would be inconsistent and useless as a mathematical theory.
Gödel’s result provides an interesting possible elaboration of Bateson’s insightful investigations. In the ‘Epilogue 1958’ of Naven, Bateson introduces the following state of affairs: ‘Take [...] a system S of which we have a description with given complexity C’ (1958 [1936], 299) and then writes almost at the end of the book: ‘Here is the central difficulty which results from the phenomenon of logical typing. It is not, in the nature of the case, possible to predict from a description having complexity C what the system would look like if it had complexity C + 1’ (1958 [1936], 302). Indeed, prediction is likely to be impossible, but with Gödel one does have a glimpse of complexity C + 1 when looking from C. Gödel avoided paradox through his ingenious numbering system in which the distinction between the language of arithmetic and the meta-language is kept intact. But by actually showing that such a numbering system is indeed possible, Gödel’s method provides an alternative for Bateson’s (1979, 229) use of the theory of types as a prohibition to present a name as of equal (or higher) logical type as the object named. Even though Gödel numbers should not be regarded as ‘denoting’ the arithmetical sentences (Shanker 1988, 216), it is still the case that they both code for sentences and can be incorporated in sentences. In fact, in Gödel’s unprovable sentence one is incorporated in an object of the code for that very object, without any paradoxical consequence of the ‘Russell’ or ‘Curry’ sort. There is also the recognition that the expression ‘This sentence is not provable’ can itself be read as a meta-mathematical judgement about its corresponding arithmetical sentence, which we know, through the coding system, to be stating its own unprovability. The truth of that judgement can be established with meta-mathematical considerations, but not with the devices provided by arithmetic itself. However, when the meta-mathematical language would be formalised, this would in turn contain unprovable sentences (i.e. only provable by yet another more powerful language, and so on).
This recognition has given rise to debates as to whether or not the human brain, or its functioning, can be modelled with fully formalisable theories (see e.g. Penrose 1990 [1989], 407-09). It is impossible to give here a discussion of these debates (which still seem to be undecided), but at least it shows that Gödel’s result is relevant for social sciences dealing with cognition and learning. For example, evolutionary psychologists tend to adhere to a view of the human mind as a computational device (Samuels 2000, 15), clearly a statement for which Gödel’s discovery has relevance. But Gödel is also of interest for the present article in its relevance for what it might have to say about the possibility of unifying the social sciences.