Maybe the exact statement which (indirectly) generates the contradiction is:
"The rules have greater scope than what we can describe"
which must come out as:
"The rules have greater scope than the language which gives them sense".
Thursday, January 27, 2011
Monday, January 24, 2011
Rule based proofs
Suppose we have a set of rules for constructing proofs for a system, and the rules could (in principle) be used to construct a proof of the validity of the system. And suppose that we have a proof-computational analogue of validity, as Gödel has. But we add one more thing: the system we are working with is a 'language', and our concept of a rule is grounded in the intelligibility of this language - so that we give an account of 'rule' in terms of some actual rules, and we we give an account of these recursively (or transcendentally) as I've indicated elsewhere in this blog. (We show that the failure of one of these to be a rule would be inconsistent with the intelligibility of the language.)
Would we then be trying to ground the intelligility of the language (it's capacity to test validity) in the intelligibility of the language (upon which the rules of demonstration ultimately rest)? And, if we were, why would this circularity produce a contradiction, rather than just be vacuous or invalid?
Maybe because these are the only two options - vacuous circularity or inconsistency. Although that's a bit vague and long-range.
Arithmetic is a good subject for this kind of investigation because of its capacity to model an indefinite range of rules and representations. It's because it can do this that it generates the incompleteness and inconsistency theorems.
The cycle I'm thinking of looks something like this:
(1) It is possible to state a rule, because we couldn't speak otherwise - it is a consequence of intelligibility
(2) Some of the rules we must be able to state are rules of logic and of arithmetic
(3) We can use these rules to specify a general theory of validity
(4) A demonstration of validity is, essentially, a demonstration of intelligibilty...
This is like saying that a rule we use applies outside of any possible scope of use. That it is 'universal', where this includes things we can't talk about as well as things we can.
Would we then be trying to ground the intelligility of the language (it's capacity to test validity) in the intelligibility of the language (upon which the rules of demonstration ultimately rest)? And, if we were, why would this circularity produce a contradiction, rather than just be vacuous or invalid?
Maybe because these are the only two options - vacuous circularity or inconsistency. Although that's a bit vague and long-range.
Arithmetic is a good subject for this kind of investigation because of its capacity to model an indefinite range of rules and representations. It's because it can do this that it generates the incompleteness and inconsistency theorems.
The cycle I'm thinking of looks something like this:
(1) It is possible to state a rule, because we couldn't speak otherwise - it is a consequence of intelligibility
(2) Some of the rules we must be able to state are rules of logic and of arithmetic
(3) We can use these rules to specify a general theory of validity
(4) A demonstration of validity is, essentially, a demonstration of intelligibilty...
What happens as we go around this circle?
Maybe we discover that either (a) we stop meaning anything (we execute the recitations of a machine) or (b) the meanings change as we iterate.
(a) would be a result of being able to give a complete behavioural account of what was going on (which we could only do in a 'meta-language'). Once we've done this, we can't constrain the meaning of the behaviour we have described.
(b) means the rules change as well. To change the meaning of a rule is to change its consequences - what can be deduced from its truth. This means that we don't get a 'closed' system of proofs.
(This is all consistent with (i) the current conversation being the top of the meta-linguistic hierarchy and (ii) unspecifiable constraints on meaning, allowing developmental change - especially as we change meta-linguistic perspective.)
How could we say that the 'meanings' of the rules (Peano plus logic?)that Gödel employed changed?
Or is it the meaning of 'validity' which changes?
His computations might be 'meta-computations', since he - in some sense - demonstrates that they are 'in principle' executable, but does not execute them.This is like saying that a rule we use applies outside of any possible scope of use. That it is 'universal', where this includes things we can't talk about as well as things we can.
Wednesday, January 05, 2011
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.
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.
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