The notes below end up suggesting a more nuanced view of what is going wrong in metaphysical debates about the existence of ordinary things and numbers, according to which Carnap has correctly diagnosed that a kind of impossible jumping out of a framework is being attempted. But on the more nuanced view, this kind of jumping out is possible in some cases. (Carnap's view is that it never is, except as a potentially misleading way of switching to talk about the practical question of whether to adopt a linguistic framework.) That it is possible in some cases better explains why we so much as attempt it in the case of ordinary things and numbers.
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'If someone wishes to speak in his language about a new kind of entities, he has to introduce a system of new ways of speaking, subject to new rules; we shall call this procedure the construction of a linguistic framework for the new entities in question.' - What is the status of this proposition? Is it meant to be a tautology? What does it take to be a 'new kind' of entity in the relevant sense? For in some reasonable sense it seems clear that we can begin to talk about a kind of entity which we have not previously been talking about without introducing a new linguistic framework. The framework can have, so to to speak, advance provisions for that in some cases.
'To recognize something as a real thing or event means to succeed in incorporating it into the system of things at a particular space-time position so that it fits together with the other things as real, according to the rules of the framework.' - Shades of pragmatism, or coherentism. Is 'recognize as F' a success verb here? Can't we have an idea of something which does fit well with our other current ideas but which is nevertheless the idea of something which doesn't actually exist?
Also, why a particular position? That seems false. Surely we can come to recognize the existence of something without knowing where it is. Also, the language Carnap uses here is quite unclear because while he is actually talking about changes on our end, i.e. in our linguistic representations and thoughts, he makes it sound like we are operating with the things themselves, incorporating them into a system.
- 'From these questions we must distinguish the external question of the reality of the thing world itself.' - This is a strong case for Carnap (who is talking here about the world of ordinary, observable physical objects), but one might wonder if it fails to generalize. Isn't there something special about our talk about the thing world? Carnap himself admits that the members of the thing world are 'the simplest kind of entities dealt with in the everyday language: the spatio-temporally ordered system of observable things and events'.
Couldn't it be this - the special foundational role played by talk of ordinary things - that makes it nonsense to ask whether they exist? (Or makes the question unsettlable?) Couldn't you have other cases where you've set up a framework which seems to licence certain "internal" existence statements, but where you can quite intelligibly and productively ask whether the things posited really exist at all? (And where this is not plausibly construed, as Carnap would want to construe it, as a practical question about whether to accept certain linguistic forms?)
It may be instructive to attempt to construct a clear, if artificial, example of this. (Here is a first thought, though there may well be much better examples available: a legal linguistic framework may treat the existence of a court as a basic assumption, without which the framework could not be applied. This doesn't mean we can't drop the legalese - step outside the legal linguistic framework - and ask about the existence of the court.)
Remember, linguistic frameworks can be embedded in larger linguistic frameworks. And in that way, we may be able to call the existence of the entities posited in some framework into question outside that framework, by remaining in a larger containing framework which allows us to treat the question in what Carnap would allow is a 'scientific, non-metaphysical' way. So that in the larger framework, the existence of the kind of entity in question is a question which may be answered empirically or a priori, in a non-trivial and non-metaphysical way, while in the embedded framework, the existence of the kind of entity in question is a basic assumption. I.e. something without which we can't really get off the ground with the embedded framework at all.
'The acceptance of the thing language leads on the basis of observations made, also to the acceptance, belief, and assertion of certain statements. But the thesis of the reality of the thing world cannot be among these statements, because it cannot be formulated in the thing language or, it seems, in any other theoretical language.' - Again, it seems like the thing language is a special case here, which makes Carnap overgeneralize. In the cases of less basic, less foundational frameworks, the latter disjunct 'or, it seems, in any other theoretical language' may be weak indeed.
Now Carnap turns to numbers, and again his case there is strong.
So, what I am saying is no threat to the core of Carnap's way of understanding what is wrong with metaphysical questions about the reality of ordinary things or the reality of numbers. Rather, it may lead to a nuancing of this and greater plausibility for it.
The problem isn't that you can never get outside a framework and ask about the reality of the things posited in the framework. On the contrary, you often can. And that helps explain why the attempt in the case of things and numbers is made at all.
So this more nuanced view has greater explanatory power. On Carnap's simpler view, according to which there is no such thing, ever, as getting outside a framework to ask about existence (literally, not as a disguised practical question), it is less clear why we would ever try.
Furthermore, the very idea of this going outside a framework, i.e. the very idea of what Carnap would call an 'external existence question', becomes clearer on the view I am suggesting. Rather than this mysterious thing which cannot in any possible case be done, it becomes something which can happen, and which we have examples of. On that basis, we may then argue that certain cases which bother philosophers are such that there is no properly analogous going-outside-the-framework to be done.