Rules and Arithmetic

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".

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...

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.

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.

If we wonder how we can talk, do we also wonder whether ... ?

If I wonder whether it is possible to talk, it cannot be because I'm wondering whether 'it is possible to talk' is true.

I must be wondering in some other way - some way which requires interpretation. I must be in a potentially ambigous intentional state, whereas if I say that I wonder what day it is I cannot - by accident, for instance - be wondering something else.

If I say 'I wonder whether it is possible to talk?' you might be puzzled: I can't be believe we can't talk without rendering my statement uninterpretable. If my statment means anything, then I can't be seriously wondering.

Some other rules we can't follow ...

If we can only give an account of rule following in terms of intentionality, then we need to be able to associate a particular rule with an intentional state. Lets say that the 'rule of addition' is what someone is following when we agree with them that they are adding.

We can render a number as a rule by representing it as 'the number that ... ' etc. In order to be a proper example of a number we need to be able to do certain things with it - e.g. determine whether another (arbitrary) number is greater or smaller than it, determine the outcomes of using it in certain arithmetical contexts etc.

It seems likely that the paradoxical numbers pointed to by Cantor, Borel, Gödel and Chaitin aren't fully numerical in this sense. There are arithmetical operations we can't do on them. (Are they like 'incomplete concepts'?).

A number like this can't be fully represented in terms of intentionaly tractable rules, and if we can only avoid Kripkean chaos by rendering rules in terms of intentional states - attributable to interlocutors in a (necessarily playable) language game - then these numbers begin to soften in the mist, as the idea of an 'intentionaly intractable' rule begins to look analytically incoherent.

Maybe Gödel's paradoxes only arise when we assume that we can construct these rules. He has put together a set of rules that defines validity for the system within which they are (presumably) validly constructed. Maybe we can only imagine doing this if we can imagine a rule that is independent of an intentional state - that is prior to, rather than depending upon, the possibility of attributing such a state.

To be able to do computations based on Peano's axioms and the relevant logical transformations is to be able to unambiguously follow certain rules (i.e. is to be able to avoid Kripke's chaos). We cannot show that we have avoided ambiguity by applying some further rules which, mysteriously, avoid Kripke's problem ...

We have to be careful of specifying rules which we cannot fully state, and which cannot (therefore) be defined in terms of an attributable intentional state. We should be careful of saying 'there is some rule such that following it would have the consequence X'; and, of course, of saying 'there is some number such that ...', particularly when we know this rule, or this number cannot be stated. This is the case when a rule is meta-specified as the rule which must be followed in order to validly state rules (and to attribute rule following).

Language

Imagine a hierarchy:

(1) Mechanincal, natural.

At this level, we describe everything in terms of physical interactions. It is a world without semantic content.

There will appear to be only one way of describing the world mechanically, and we might think of this as 'reality'.

(2) Signalling

At this level, we divide the world into systems and sub-systems. These have formal characteristics which might be instantiated in a number of different physical ways. Each unit is attached to one or more other units by a 'channel' which has a 'band width'. The bandwidth is the number of different signals the channel can carry. Channels are asynchronous - they carry signals in one direction only. For synchronous signalling, we need more than one channel (even if both might be implemeted using the same physical substrate).

To be a channel, at least one possible signal must be 'useful'. By this, I mean something like: we can specifiy the conditions under which it would be sent, and what effect it has on its recipient.

The formal characteristics of the system as the rules that describe the behaviour of the units and the structure of the signals.

There will always be more than one way of dividing up level 1 in this way - it will be a matter of taste or convenience whether we describe some things in terms as internal structure of individual units or in terms of sets of interacting sub-units.

(3) Semantic

At this level, we give the signals and the behaviours 'meanings'. These are always provisional (Kripke). Perhaps we might say that the a meaning interpretation is a 'model' of the system, but perhaps this would be confusing.

In any case, this is the level at this we introduce intentional content. We can say 'Unit A has told Unit B that the temperature has reached 0 degrees', or 'Unit A has instructed Unit B to switch on the heater'.

(4) Validity attributing

A signal can be interpreted as an adjudication on the correctness or incorrectness of (a) a semantic interpretation or (b) its intentional content. Things get pretty wobbly at this stage. What is really going on here?

Particularly difficult issues arise if we interpret a signal as adjudicating on the correctness of our interpretation of it.

Is this a model people use? It looks like it to me. It can be partly 'unpacked' in some cases, and this contributes to the illusion.

But the whole thing is an 'interpretation' ...

Talking about semantic content

Does this make sense:

Giving semantic content is like giving a dictionary definition. If you already know how to speak the language, it gives you a way of extending your ability at the margins.

Its similarity to rendering formal syntax is misleading: of course we can write down some rules - this is not a useful model, though, because of the OQ arguments. We can only write down some rules if we can already write down something.

And dictionary definitions are only really useful if we also have some tacit 'hooks'. If we have only synonyms, then the only argument for replacing the synonym with the defined word will be convenience - we can say something more quickly, or in a less clumsy way.

We have to be able to show as well as tell.

We might, like the ordinary language philosophers and the linguists, articulate the way we talk. As we do this, we change the way we talk - words have their meanings 'tidied up', and so changed. Sometimes they are changed in more dramatic ways, that we take time to articualte (as scientific terms change meaning - space, mass, distance).

Discovering how we talk, how we can talk, and what the world is like are not three different enterprises.

What's wrong with the picture theory of language is what's wrong with semantic content, and also what's wrong the 'languages of thought'. This isn't to say that these models, these metaphors, have no empirical value - they may be important heuristics for certain kinds of studies. They just can't answer any interesting epistemological or ontological questions.

Chomsky, Davidson, & learning ...

This may be a caricature.

Once I learn to ride a bicycle, I can ride it through many landscapes. Once I can sing in tune, I can sing many tunes. There may be rules about following maps or reading music, but not about riding bicycles or singing in tune.

We can analyse map-reading metalinguistically. We can generate isomorphs of the map, or of a musical score. These things require us to be able to talk about them, agree about them, in certain ways. We can't do the same for the 'methods' of singing in tune or bicycle riding, both of which are needed for music and bicycle navigation.

While there are some rules of grammar and syntax, and some rules of meaning there are also just some things that we find that we agree about in practice - that just don't give us enough trouble for us to reflect on them. When we try to reflect on them ('how does reference work?') we find ourselves in a Wittgensteinian predicament. Some aspects of these things seem too obvious to remark on and yet too obscure to explain.

A language may well need generative meaning rules, but this is not all it needs. There must also be things we do not yet know that we will agree about - things we might call 'discoveries' when we encounter them. Discovering that we agree about something may look like new knowledge about the world or about ourselves. In many cases, it may be a matter of taste which - it may come down to metaphysical prejudice.

We can do something like following rules that we can not write down - we know this from the open question arguments. We also know that we learn to speak, and that we can learn to do things without being able to articulate the rules which govern them.

Where does this leave Davidson's argument for a constructive account of meaning? At least we know that if we tried to pursue his programme, the meanings of things we said about meaning would change ...

Constructivist meaning

Suppose we had a set of Davidsonian meaning rules. Could the sentences constructed using these rules be arbitrarily long, or complex? We could not set a limit based on cognitive capacity, because this cannot be represented within the rules. How could we write down a limit that wasn't arbitrary?

Would we otherwise be committed to saying, for some 'long' sentences, that they had a meaning but it was not practical to work out what it was?

We might say that a complex computer programme, represented in a high-level language, has a 'meaning' in this way. As soon as we begin to summarise or analyse the code - in terms of it's functions, or into different functional 'blocks' - we are hostage to interpretive ambiguities.

Is this also true about a large piece of language? A book, or this blog?

Does the whole book have a 'meaning' which could be analysed in terms of a Davidsonian theory?

Interpretation

We can imagine a language without the tools to reflect on its own activity - one which functions, but which cannot be used to formulate validity rules or make intelligible attributions of truthfulness.

I think that there are commercial exchanges which take place in a languages like this. We shouldn't insist on interpreting them in terms their use of tokens we also use - they will certainly sound like nonsense if we do this. This is how we should undertstand Moore's paradox - as challenging the conventional meanings of the tokens it employs. Finding that a translation schema for an unknown language produced Moorean statements in critical contexts would invalidate the schema.

We can also imagine a language which does have the relevant capacities - a language in which it begins to be possible to do philosophy.

And we can imagine a language in which it is possible to speculate about how to interpret other languages - or other things as languages.

In our language, a comprehensive validation is one which shows that the falshood of a statement is inconsistent with the possibility of using the language. A comprehensive validation of an interpretation, then, would be a demonstration that the falsehood of the interpretation was inconsistent with the possibility using the language.

We know, from Kripke, that interpretations are always provisional. This doesn't mean that some specific interpretation cannot be ruled out, though. Can we imagine that someone who is quad/adding is actually playing chess? And if we cannot, is this a logical or a cognitive limitation? Is such an interpretation incoherent or just impossibly complex to apply?

Formally, the second seems more likely, but it isn't easy to work out what this means.

In any case, we don't need to worry about that - the interpretive judgements we are making are those that can be made intelligibly within the game we are presently playing.

Is there a problem with giving an account of what it is that we are interpreting, before we interpret it? Imagine a space with a dimension for every freedom of the human body, which could be used to describe any position that we might adopt. A function in this space could describe a movement. What would the function, or functions, associated with 'he waved goodbye' look like? How would we (a) distinguish them from others and (b) recognise them from the mathematics? This is like Wittgenstein on smiles which are only a millimetre too wide ...

We are good at finding the things in the world that we need to be able to recognise in order to be able to talk about it. We know this because we are able to talk about it. It is not a consequence of this that we must be able to say how this knowledge is acquired - contra Davidson. That something is possible is not the same as that it is explicable, or we would be able to explain the possibility of explanation, which is incoherent.

Models

A model provides a grammar and a vocabulary. When it 'works' we think there is some additional similarity, some isomorphism, which underlies this. All there is is a judgement - and we cannot speculate on whether such judgements are, in general, reliable.

Talking about talking

We can imagine a language which doesn't have certain self-reflective capacities - e.g. which doesn't have an explicit truth predicate. This is because we would still be able to translate this language into the one we are presently speaking.

(We can't, of course, translate - or, therefore, imagine - a language which did have a truth predicate, but whose truth predicate worked in a significantly different way from 'ours'.)

Some language-like games might not be able to have a truth predicate. The introduction of a truth predicate (which would have to, again, be isomorphic with 'our' truth predicate) would render them unintelligible. Bargaining games are like this - when a salesman swears that everything in the brochure is true, we know he cannot be using 'is true' in the same way as, say, a logician or a natural scientist. On the other hand, the bargaining game would be unplayable if it was complemented by a logical or natural-science-style truth predicate, because this game depends upon a certain amount of conventional dishonesty. The extent of this dishonesty is explored by the participants in the game, but not explicitly explored - it is explored practically, by finding out what moves 'work' and what moves don't. Salesman do not make good philosophers, but this does not render the bargaining game intrinsically corrupt.

What would render it corrupt would be dishonest practice on the part of any players. I'm going to give a definition of this here which I'm not sure I can fully support, but which I think is roughly right:

Dishonest practice is practice which exploits the expectations of other players in a way which sacrifices the playability of the game to the instrumental aims of the practitioner.

Since the bargaining game cannot include a truth predicate, and since it is hard to construct a normatively self-reflective game without a truth predicate, it is possible that the bargaining game cannot be used to explictly adjudiate on honest practice. To test any practice against the definition, we would need to make a judgement about the objects and consequences of moves within the game - and particualrly about whethe a move rendered the game unplayable. This last, I think, would definitely require self-consious reflection on the possibility of truth-telling.

This raises a tricky moral issue: Can we, from outside, using a language with the capacity to articulate truth related issues, adjudicate on the language of the bargainers, which does not? We would be presuming to be able to translate this language, while knowing that we could not discuss the quality of this translation with the bargainers - we could not ask them whether the translation was correct.

We could, of course, learn to play the game. If we had an appropriate facility with it, we could then reflect on this facility using our native meta-language. This looks tricky: the character of our engagement would be different just because we had access to the meta-language. A physicist discussing mass with a weights and measures inspector does not participate in the weights-and-measures game in the same way as another inspector would.

This may be harmless. We certainly wouldn't want to say that the physicist and the weights and measures inspector didn't understand one another - we wouldn't want to say that they weren't able to play this game together. We would, however, find that their game became more difficult if it developed in a certain self-reflective way. There would be some truths about mass that they could only share by changing the game.

Natural Laws

We can't just say that it is a law of nature that nature follows laws.

What kind of law can this law be?

We can only talk about a predictable nature - anything else would be incomprehensible. Natural laws are the grammar of this talk.

Conviction, Certainty and Argument

The world is careless of our phenomenological convictions - it produces them without warranting them. We might imagine that we have some 'evolutionary guarantee' that the internal products of our sensory apparatus contribute to our survival, but this comforting mythology can't be grounded, itself, in our phenomenological convictions; and Darwinian evolution selects on the basis of publicly available behaviours and characteristics - not on the normative character of intentional states. It is indifferent between lucky misconception and 'true knowledge'.

We might argue that, over time, evolution would tend to weed out ignorance and misconception. There isn't any evidence for this beyond our own, present, existence. This existence, though, is the outcome of just one of many rolls of the evolutionary dice; and we know already that it is very probably temporary. Attributing intentional states to past 'evolutionary successes' in support of a phenomenological evolutionary epistemology would be circular and gratuitous.

It is absurd to say that our whole language game is reliable because it has 'survived' an evolutionary process which can only be given an intelligible account of if the game is reliable ...

Martian intentionality

And we couldn't unambiguously attribute intentional states to them - such as knowing how to tell the truth in Human.

Goodman, chess machines, and physical standards

It's as though we decided to mean by 'green', 'parrot coloured'. Now a previously grue parrot would remain green, but the meaning of green would change.

Chess playing computers

While we might learn a lot about chess from the behaviour of a computer, and might learn how to play from such a computer, we cannot define chess in terms of the computer's behaviour. The rules of chess determine whether the computer is playing correctly - they are not tested against the behaviour of the computer.

This is also true of people: we might believe that people act rationally, but we cannot define rationality in terms of the behaviour of any individual or group - however apparently exemplary.

This extends to their internal 'behaviour' - no neurological account of brain processes can capture the normative aspects of rationality, just as no actual 'or' gate could count as the effective standard for the logical operator.

Under certain circumstances, we might have to choose between deciding that the standard meter had grown or that our prior measurements all needed to be proportionally corrected. If we standardise the meaning of 'or' on the behaviour of an 'or' gate, the meaning of 'chess' on the behaviour of a computer, or the meaning of 'rationality' on either a neurological process or the behaviour of an individual or group, we do not know what adjustment this might lead us to have to make - we do not know what changes in meaning we might have to accomodate.

The standard metre can only change in one way, resulting in a uniform scalar adjustement to length. The standard 'or' gate could change the meaning of 'or' in a way which rendered the world unintelligible.

The standard chess computer might redefine chess in terms of any option available to it - whether or not it was functioning correctly.

Rules and Laws

So I think this is where I am:

Language requires attribution of intentional states, and is 'a priori' possible or we wouldn't be able to say so.

Certain irreducible intentional states - 'thoughts' (Frege?) - cannot be expressed unless it is possible to attribute generic properties. Predication requires universals.

To know how to talk is, among other things, to know how to attribute these generic properties.

Also - 'X knows how to talk' attributes some specific generic properties, including the ability to follow rules, to X.

To be able to attribute universal properties is part of being able to talk, it is part of being able to follow the rules of talk.

Asking whether the world 'really' contains universals (including whether it 'really' follows general laws); or asking whether certain people 'really' follow rules is like asking whether it is possible to talk about the world, or whether these people do, in fact, talk. It's a Moorean error - a question which can only have one answer if it is to make any sense at all.

To talk about a world is to talk about a world of this kind.

Rules and Intentionality

Various people (Crispin Wright, Ruth Millikan & others) have drawn attention to a potential role for intentionality in resolving Kripke's paradox, since intentional states seem to have the scope and reliability we would wish to attribute to competent rule-following.

I think it would be a mistake to take this (as Millikan and Wright do) as a cue for producing a better account of intentionality, if by 'account' we mean 'reductive account'. For reasons that I've probably laboured elsewhere in this blog (more open questions ...), I think any reductive account is almost certain to generate a version of Kripke's paradox.

An account of intentionality in terms of physical behaviour (e.g. of a machine) is obviously ruled out by this kind of objection, since questions about the reliability of the machine would always make sense - and these questions would be questions about whether the machine appropriately fulfilled its function; whether it appropriately 'modelled' the intentional state it was meant to underpin.

It is difficult to think of how a reductive account could be given that could avoid this machine model objection.

In concert with accounts of truth and meaning (to which intentionality is obviously related), a recursive account can, however, be given. Since we have to be able to attribute intentional states in order to be able to speak, and since we can speak, we can start with the intentional states we must be able to attribute in order to be able to have this conversation (we don't need a complete list - just an articulable selection).

Following from this, we can give an account of rule following in terms of these intentional states, or from others that depend upon them. To follow a rule is to act in accordance with an intentional attribution.

Intentionality and rules

If I am in a certain intentional state - say believing something particular to be true (that today is Friday), then I will in general behave and speak in a ways that is consistent with this. The ways that might be consistent cannot be spelled out in advance, and not only because they are complex - also because unexpected situations will arise.

Who will adjudicate? I and my interlocutors, playing the shared language game within which the statement of my intentional state is held to be true. The intelligibility, the playability, of these adjudications will be one part of what makes the whole game possible; and if the become idiosyncratically unintelligible then so does the original statement of my intentional state - we will find that we did not know, after all, what we meant by attributing this state to me.

This process of adjudication and exploration is what the history of mathematics reveals to us about addition. Someone who has been quadding so far, but thinks they were adding, is not a competent participant in this conversation.