Meaning, metalanguage, and internal/external criteria

Putnam's 'twin world' example only works, I think, if we allow ourselves to articulate the possibility of seriously confused interlocutors - a possibility that cannot be articulated within any conversation they are having with each other. It's a language/meta-language confusion.

We can talk about the twin worlds, and their confused (?) inhabitants, but they cannot - without resolving the ambiguity his example depends on. In order for them to talk to one another, the water/twater issue would have to be resolved.

We can't hypothesise that we may, in some systematic way, be making water/twater mistakes, or be subject to water/twater ambiguities - we might as well hypothesise that we are confused about our hypotheses.

The conversation we are having now is always at the top of whatever meta-hierarchy is coherently conceivable. We can hypothesise about the confusions of users of lower level languages, but we can't extrapolate upwards from these hypotheses. And: interpretations which render the native confused are always corrigible (Kripke again ...). We can only (and we must) make incorrigible mutual attributions of intention in a shared conversation.

Davidson's idea that we must always be interpreting each other also has to be considered in this light: we do not have an internal conversation within which we make 'interpretive judgements' about what an external interlocutor means. We are able to make correct judgements about this, but these judgements can only be considered when they are expressed - when statements of them form part of the conversation. At this stage, the 'internal' becomes just whatever it was - whatever private precursor or accompaniment we happened to have.

If we imagine this 'internal' separated from the external statement, what is it that we are imagining? Just our own beetles. Not irrelevant, but not an 'internal representation' either.

Whatever is going on 'inside my head' (whatever beetle scratches there ...) it only gets out, it only becomes available to philosophical theory, as the intentional component of my participation in this conversation. And by that stage, all the important bridges have been crossed. It has become my 'beetle', and not [my beetle].

In this context, as well, the question of 'first person authority' becomes an issue of linguistic competence - the 'capacity' to correctly report on (or otherwise reflect) our intentional states when we speak. We attribute this 'authority' when we attribute interlocutor status - "You have no idea what you really think" is, here, more or less equivalent to "You do not know how to speak". This is not, of course, a playable move - it can't be part of a shared conversation.

Recursive roots

Here is a (wildly) speculative consequence of what I'm saying:

There's a relationship between inductive mathematical proofs and recursion - recursive functions can have their values determined by a definite finite algorithm, and by demonstrating that a function is recursive we are demonstrating that such an algorithm exists (whether or not it is practicable).

Inductive proofs depend upon the induction axiom in formalisations of arithmetic. By showing that zero has a certain property, and that the successor of any number with this property must also have it, we show that all numbers have it. This is an axiom of arithmetic (and it needs to be formaly stated in a way which avoids talk of 'properties'). The 'numerical' root of induction - zero - is also defined axiomatically as the cardinality of the empty set. Many proofs for the whole of arithmetic must depend upon this inductive principle, and its stipulated root.

It is this formalised arithmetic - which establishes induction and zero as axiomatic - which can represent axiomatic systems and theorem generation in general and so its own axiomatisation a` la Gödel.

If I have the right picture of a natural language, then the recursive roots of empricial demonstration will comprise the statements which must be true for the language - and so for argument - to be possible. These roots are not arbitrary, though, since they appear directly as statements about the possibility of language and argument (which cannot be rendered 'formally' - i.e. stripped of semantic content).

Could we write down a specification for a 'naturalised' arithmetic which took the possibility of arithmetic and of computation as its recursive roots, rather than a specific set of (semi-arbitrary?) generative rules?

Some might say: but these rules just are arithmetic. No conception of arithmetic is possible without them.

Maybe this is a place where we are deceived by our intuitions. Can we ask questions like 'Is arithmetic fundamentally about counting, about succession?' or 'Is arithmetic fundamentally about computation, about argument?'. And we have reasons for thinking a certain kind of computation (algorithmic, recursive) goes along with the possibiltiy of counting, of enumeration.

Suppose, however, that we started with a different number: not the cardinality of the empty set, but the Gödel representation of the statement that there is some statement x such that x is a theorem of arithmetic. On the face of it, this does not require completeness, but it does require consistency (since otherwise ~x would also be a theorem). This number must exist in any arithmetic (I think?), and must allow the generation of other numbers (or it wouldn't be arithmetic?) in a way similar to the way zero and succession work in Peano arithmetic.

The value of x (and so also its representation) is a function of the axiomatic system and rules selected, and the coding system used to Gödelise them (as Chaitin's Omega number is a function of the actual structure of the machine it describes). This is true of the theorems of arithmetic generally, of course, but these depend upon a definition of zero that is not open to further interpretation, that 'has no semantic content'.

(Maybe we should think of the value, representation, and 'meaning' of zero as being generated by the rules and coding system as well. But zero is also a word in our natural langauge...)

Anyhow: If there is some number which is the code for a statement that there is some theorem of arithmetic (in the syntax of the axiomatisation that is being encoded), then is that number (or its isomorphs in differently coded and axiomatised, but 'semantically equivalent' systems ... can this make any sense?) a candidate for a general recursive root for a 'natural' arithmetic?

Could its 'existence' be more 'necessary' than the unique cardinality of the empty set?