Counting

It's easy to imagine a counting system which functions for some particular set of circumstances, but doesn't work 'in general' (e.g. the 'one, two, many' system occasionally attributed to some 'primitive' people by anthropologists).

Does this mean that we can ask the question whether our counting system might be like this, and that we might discover (in some future) that it didn't really work in some particular, presently unimaginable (undescribable) circumstance?

The best answer to this question would show that the question could only be asked if our counting system did not suffer from this defect. This would avoid the problems associated with giving a general account of counting (Peano/Gödel), or with 'defining' counting in terms of the system we presently use.

Gödel's method, which shows that Peano's calculus can be represented arithmetically, suggests a route to an answer: if our question can be represented arithmetically, and would only make sense if this was the case, then we might regard it as answered.

There are loads of bear traps, though - 'making sense' being an obvious one. But if we think of 'making sense' in terms of practical interpretation (the playability of the game), then there is still a possibility of a transcendental demonstration.

Would we be saying that only a person who counts the way we do could understand the question the way we do? Along with a Davidsonian attitude to conceptual schemes, this might work (so long as we avoid trivial solutions, e.g. that part of the understanding included a certain kind of generalisability of our counting method).

We would need to avoid any tacit reference to an 'ideal' counting system - one which is at the top of some abstract hierarchy in which, say, arabic numerals are above roman or greek numbering systems; or in which 'more' abstract rank above 'less' abstract systems.

(I think I can only deal with (a) games which are playable, and (b) potential games which are not - and there is no final method for discriminating between these. But, of course, the playability of a game, e.g. a question asking game, can have general consequences.)

Gödel's proofs depend upon demonstrations of indescribable worlds - no 'practical' arithmetic would encounter his numbers. There are many things we might think we could do with numbers (e.g. rank them, perform certain algebraic operations on them) which are not possible with Gödel's examples. They aren't quite as minimally articulated as we might think a 'number which is indescribable' would be - we have some elements of their descriptions.

(Some objections to Cantor's proofs arose from their consequence that there were indescribable numbers?)

Is it OK to say that we can't ask any questions of an indescribable collection? Does it matter if it only contains numbers? Can we make this restriction, given the representational power of numbers?

Perhaps the possibility of speculation depends upon arithmetic being constrained in a way which (a) cannot be fully articulated, and (b) avoids Gödel's consequences.

We can allow such a constraint as (a) because it is the kind of constraint implied by the possibility of a theory of truth for our language. There must be some constraint, but we cannot fully articulate it.