[FOM] expressive power of natural languages

[Mark Lance]

This is a follow-up on the line of thought introduced by Monroe.  The first question is what one means by "a natural language".  Some mathematicians and some formal linguists try to see a natural language as a fixed structure.  But language as a phenomenon - as a social practice - as Monroe notes - involves constant revision.  In particular, it is  absolutely central to the power and social function of natural language that one can introduce new concepts.  This can be as mundane as naming a new baby - thereby introducing a new name into circulation - to postulating a theoretical entity to explain some elaborate natural phenomenon.  In between - or maybe to the side - we have such wildly impredicative phenomena as characterization of social movements, assignments and criticisms of social identities, descriptions of linguistic habits and practices themselves, etc.  If this is right, if this is all essential to language, the question of whether natural language has the expressive power of some formal language is literally obvious.  The "formal language" - that is all the talk we engage in when we do mathematics - *is* natural language.  We say "let x be a structure such that …".  How could any of that noise/marks on paper/etc mean anything if it weren't introduced in natural language.  

Now on the one hand this might seem merely a verbal quibble: What the original question meant to ask was whether natural language prior to the introduction of formal methods is a language of second order.  But there is a substantive issue.  It is crucial not to obscure what makes thinking about the power of natural language so hard: and this is precisely the fact that it incorporates everything, in the sense of holding open the potentiality for any expansion whatsoever.  And by supposing that one can wall off some well-defined sub-practice - "natural language without formal methods" - and treat it (a) as a determinate totality about which questions like "what is it's expressive power?" can be asked, and (b) suppose that we have not thereby turned it into something that is utterly unlike natural language, we may well be obscuring precisely the most interesting and important feature of natural language.