Signal Content

If we ask what a signal (or the state of a machine) 'means', we are attributing an intentional state to the machine, to the originator of the signal.

If we examine how an actual example of a homeostatic mechanism behaves, we do not find intentional states - just physical circumstances.  We cannot (Kripke) 'decode' a thermostat (i.e. an actual physical device) unequivocally in terms of temperature maintenance.

We might talk about 'correct' and 'incorrect' signals, without attributing intentionality to the machine or the signal, but we would still have to make this judgement in terms of the overall function of the machine - and therefore in terms of the intentions of its creator or designer.

We can say 'at this point, line 4 should go high'.  If it does not, then this is a malfunction - either a mechanical failure, a design failure, or a poor mechanical implementation of a design.  (Are these all different?).

If we say 'line 4 high indicates that the lift door is open', and we find line 4 high when the door is shut, can we say that line 4 gave an 'untrue' signal? We can say the signal was incorrect.  If we say that the machine says the lift door is open when it is shut, then we can say the machine lied, or was mistaken, but this goes along with attributing intention - we make a normative choice here about the status of the machine; we treat it as an interlocutor.

Articulating tacit rules

Attributing conformity with a tacit rule to a non-interlocutor is always corrigible (Kripke).  Articulating a tacit rule we 'discover' we have been following changes if from being tacit - and also changes the way we talk.

Imagining that before we articulated the rule we must, indeed, have been following it is to make exactly the corrigible attribution we might make to a non-interlocutor.  When we move from tacit to articulated, we also move from corrigible to incorrigible.

Can we say that we moved from less clear to more clear meanings?

Evolution

Nature is indifferent between a successful heuristic and a true theory; but we cannot be, or our justification for saying so could be that this comprised part of a successful heuristic.

It is incoherent - for instance - to try to support our belief that the future will, in some way, be like the past by saying that it is a component of a successful heuristic.  "It's been OK so far" isn't a ground for complacency just because we've got away with it up until now ...

Truth telling and its effects ...

Telling the truth is not free.  Some can afford it better than others.

Our reflections on the price we are prepared to pay cannot be articulated in the language game in which the moves under consideration are to be made.

They might be articulated in another context - perhaps a privileged context, such as an academic discussion.  Such a discussion will have a different price list.

It is only when there is no context in which we can render ourselves intelligible that we are finally damned; that we become completely incoherent, and unable even to say so.

Patterns

Children are often given mathematical puzzles that involve working out how to continue a sequence of numbers.  Wittgenstein focuses on this kind of situation when discussing rules and mathematics.

There can, of course, be no 'right' answer to how to continue any finite sequence, since there will be an infinte number of potential rules that its generation might instantiate.  Given this, it is odd that children are - generally explicitly, and always implicitly - told that there is a 'right' answer.  Valid, but peculiar, rules will be rejected (e.g. suggesting that the sequence '2, 4, 6 ...' instantiates listing even numbers in the odd hundreds and odd ones in the even hundreds).

This is an essential part of teaching mathematics - this weeding of peculiar rules.  In fact, a sound intuition about which rules would be peculiar is a component of 'mathemtical ability'.

A child that is being given this mathematical training is getting two signals from the teacher with each exercise: one is the list of numbers, and the other is a reassurance that there isn't something 'tricky' going on - that this list really is a sort of message, a hint that some simple and obvious solution is the 'correct' one.

We handle numbers using manageable rules - which comprise a subset of the formally possible rules.  All the mathematical 'observations' made by humanity so far comprise a finite set, and so could instantiate an infinitely large number of mutually incoherent sets of rules.  We have selected from these the humanly manageable ones which appear to be coherent.

We have no way - we could have no way - of demonstrating that the set we have selected is the only possible one.  Such a demonstration would depend upon a set of rules which would have to have mathematical meanings.

If we had two sets of rules which were internally consistent (and so were consistent with all our 'observations') but inconsistent with each other, we could only choose between them on the basis of an observation we had not previously made - but the generation of mathematical 'observations' is a rule-driven business ...

Maybe we cannot state a complete set of rules, because we modify their meanings as we do mathematics - because doing mathematics is partly a matter of resolving rule ambiguities.

And that's apart from any open question issues which would arise.

Private Languages

Maybe this is the Private Language Argument:

Suppose I told you 'I have a private language, which I can use when I am thinking things to myself'.  You might ask 'What do you say to yourself, in your private language?'

I could answer in two ways:

(1) I could give you  a translation, in our shared public language, of what I say to myself.
(2) I could say that my private language was not translatable into a public language.

I could not, of course, simply give you a 'raw' private utterance, with or without a translation.  This would be to render the private language public.

(1) Is what we presently have.  If I tell you what I am thinking, I am doing as much as this - but this does not demonstrate that I have a private language:  It only demonstrates that I can use our public one.

(If we could conclude from our use of a public language that we 'must' have some private 'language of thought', then what would we have to conclude from our use of this private language?  That we must have an even more private language of proto-thought?  What kind of explanation could this possibly provide?)

If I want to demonstrate the possibility of a private language by an argument produced in our public language, I would need to have a way of pointing at the private language in a way which showed (a) that it was a language and (b) that it was not a public language.  But I can't point to it if it's private, and I can only show that it's a language if it can be spoken.

With respect to (2), the idea of a language which is in principle untranslatable is  (obviously) incoherent for Davidsonian reasons.  How could we know it was a language?

Evolved Function

There are two ways of describing an evolutionary process, just as there are two ways of describing a computational proces: (a) from an intentional perspective and (b) from a mechanical perpsective.  It is mistake to think that when we understand one we understand the other; and it is a bad mistake to think we can reduce the intentional to the mechanical.

It is possible that we may be able to 'reduce' the mechnical to the intentional, in the sense that we cannot understand rule following (and so mechanism) except in terms of a system falling under the scope of some intentional concept.

We can only render the evolution of function (an intentional category) intelligible by referring to higher functions.  If longer legs aid survival by allowing higher escape speeds, we 'explain' longer legs in terms of the function of speed ,whose relevance is the function of escape, whose relevance is the function of survival.  If we didn't understand what survival was, or why it was 'desirable', this explanation would have no force.

We could tell a different - much more complex - story about about the macrobiology of evolution, involving reproductive chemistry, the coding options available within DNA sequencing, genotype/phenotype relationships, protein folding and structure, environmental factors etc.  We can't, at present, tell enough of this story to be able to completely describe many large scale functional behaviours in terms of detailed mechanisms.  Even when we can do this, however, we will not be able to 'render' function as mechanism except by stipulation.  And this would expose our accounts of function to mechanical failure - just as if we tried to define a logical operator in terms of what a particular electronic digital component did.

In my last post I said we should think of machines as elements in a conversation between their designers and users.  When we discover a 'natural machine' we don't recognise the normative aspects of attributing function, and so we arrive at the argument from design.

And maybe this is lucky - especially if it is our visceral normative attributions which prevent us, on the whole, from seeing each other as machines.  Excessive intentional attribution is probably less dangerous to us than deficient intentional attribution.  From a functional point of view.

Talking Machines

We should think of a machine as part of a conversation between its creators and its users.  It's like the mechanial parts of our talking activity - the noises, the physiology, the writing technology.  They might break down, they might take different forms, but their character is given by their intentional content.

A machine's character is also given by the intentions of its creators, and the 'sharing' or 'communicating' these with its users.  It is not given my any physical description.

This applies most clearly to 'ideal' machines, such as Turing machines.

A real computer is part of a complex conversation between 'programmers' and 'users' - or, more correctly, between different classes of users.

The 'limitations' of these machines can only be given, in any intelligible way, in terms of the 'limitations' of intentional ascriptions - of 'rule following'.  If we say that there must be some rule which underpins the possibility of intentionality, we are making a mistake.

OQ Arguments, and Crispin Wright's 'dilemma'

For any potential source of knowledge (the reliability of our senses, religious inspiration, scientific method), we cannot both (a) have it as an empirical fundamental and (b) show it to be reliable.  These two are mutually exclusive, because any demonstration that our knowledge of (b) is reliable has to be based on some other fundamental.

We also cannot say 'this must be accepted without demonstration' because we have to explain the 'must'.

The Scope of a Rule

To speak, we must (a) attribute rule following and (b) attribute belief in natural laws (the 'rules' which the world follows).

Some of these rules have an indefinte scope. This can make it sound as though their scope precedes and exceeds the scope of language - of what we can talk about. We can only imagine such a thing by considering the possibility of a world that we cannot (in principle) talk about, but which, nevertheless, follows the rules. Whatever the value of this as a metaphysical speculation, we can't test it.

A specific case is the rules we presume to underpin the possibility of language itself. We imagine that logical and mathatical laws have a wider scope than simply guiding how we speak. This belief is associated with the kinds of things we say about them - among which are statements like 'these laws have indefinite scope'. Such a statement is true with respect to any testable case, since such a case must be articulable. Also, to qualify it would suggest that there was some articulable case in which they did not hold.

But its generality and lack of qualification do not need to extend to 'things we cannot talk about'. And there can be no rule that defines the boundary between what we can talk about and what we cannot.

When we explore this boundary, we generate paradoxes. This does not mean that the exploration is invalid or absurd. It means we must be careful about the conclusions we draw from these explorations - and about what we mean when we talk about them.

All the objects in the metaphysical world beyond the reach of speculation are paradoxical.

Conclusions drawn from the indefinite scope of our rules of intelligibility will also be paradoxical, since they depend upon speculations about these objects.

Meaning sneaks into the incompleteness and inconsistency proofs at the level of the rules themselves.

Worlds without people ...

Suppose we imagine a world - perhaps this world at a different time - without any people in it, so that 'there is no language in that world' is true. What are our grounds for saying that this, or that anything at all, is true in this 'non-linguistic' world? Only that you and I, now, can agree about it - and can agree about what any such world would be like.

We cannot say that there might be some other world, that we cannot imagine or agree about, and yet that there is some definite thing that must be true about it. What kind of thing could a truth that no-one could ever agree about be?

Barry Stroud

It may be a contingent matter that there was no language at the time of the dinosaurs, but only we can say so. We cannot imagine the truth of some state of affairs separately from the possibility of expressing it except by imagining a conception of it that cannot be translated into our conception. If I think 'at the time of the dinosaurs, there was no language' I am imagining being transported to that time and having no-one to talk to. What else could I be imagining? That some invisible, even non-existent, person was similarly transported ...

The dinosaurs could not know that it was true that they had no language.

If I say to you 'There was no language at the time of the dinosaurs', what non-linguistic 'fact' about the time of the dinosaurs are we expressing that makes that time different from what we would say of it in any case? We say a lot of things about dinosaurs, but we don't say 'and there was no language then'. What would this add if we did? We don't wonder whether they could talk to one another, and so go to find an expert who can tell us.

And it is not whether there was language at the time of the dinosaurs that is important, it's that we can, now, talk about the dinosaurs. This is no different from talking about anything else - including things that happen now, contemporaries of our linguistic abilities.

If I say something about the time of the dinosaurs, I am not concerned about whether what I say was true then, but with whether it is true now: and the only conception I can give to whether it was true then is whether I would say now that it was true then ....

What is it, exactly, that was true then except what we agree about now? And If we don' t have that agreement, how do we identify the thing we might have agreed about?

Private Language Questions

This is the real power of the Private Language Argument:

The question 'Is it possible to ask questions?' can have only one answer if it is a question. We might imagine, however, that it is something we could wonder about, non-linguistically. It would be like wondering whether it was possible to talk.

What would this 'wondering' be like?

It could be like privately articulating the question 'Is it possible to ask questions?' We could wonder whether any previous public questioning and answering we had done had really 'worked' - whether it had comprised real questions and answers. It's hard to separate this question from the question whether we had really been speaking at all in the past - whether we had understood what anything we thought we were saying meant. This is incomprehensible, but it has a specific incomprehensibility here: we could not know what we were 'privately' doing by asking ourselves 'Is it possible to ask questions?' as this private act depends upon the validity of the public practice of speaking - which we wish to query, or about which we have become sceptical.

What the PL argument says is that there cannot be some private language in which we could comprehensibly articulate these queries - e.g. about the possibility of asking questions or about whether it was really possible to talk to people.

The PLA can be reformulated in Davidsonian terms:

If some private language were to count as a language, we would have to be able to interpret it, or translate it, into the public language in which we are conducting our enquiry. I think this is what Wittgenstein meant by saying that otherwise we would not know what rules we were following or whether we were following them correctly.

If we can only render the concept of a rule intelligible in terms of intentionality (which I've argued for earlier), and we can only reliably attribute intentional states to interlocutors (from Kripke's paradox), then this result is what we would expect.

Another perspective on this: how would my (or your) having a thought in our private language be represented, and how would our thought be attributed to us, in the public language of our enquiry? If I say of someone that they have articulated the possibility that it might rain in their private language, then I might as well be saying of them that they thought it would rain.

I could say of someone 'she thinks in Polish', and imagine her thinking the Polish sentence for 'I think it will rain'. I can imagine her inventing a language which can be translated into our language, and using that language to say something that would translate as 'I think it will rain'. I cannot imagine her thinking it will rain, but somehow representing this in a language I cannot translate. Such a language might as well be random - have no rules, but just mean whatever she took for the moment it to mean. She might think it would rain, attach this thought to a noise, a colour, an unpleasant memory, her confusion about linear programming algorithms, and any of these would do. Anything at all could 'stand for' her thought that it would rain, and so anything could be 'translated' into our language of enquiry as 'I think it will rain', and different things could be translated so at different times. There could be no rules in this translation schema, and so it could not even be reliably applied from one occasion to another.

If we translated:

'X, on occasion Y translates from our language of enquiry into "It will rain" in Sophie's private language'

into Sophie's private language, we could not know whether our translation was correct.

And if we could not know whether our translation into Sophie's language was correct, we could not know whether the translation of any such statement back into our language of enquiry was correct, and so we could not know what anything in Sophie's 'language' meant.

Linguistic Rules and Bush Tracks

It's a mistake to think that because we can discern linguistic rules a language user must, in some sense, know what these rules are. This is like saying a train knows where the track is.

A language user may have a quite restricted kind of linguistic competence, and still be able to function - and to appear to function well.

Imagine someone who knew, say three thousand sentences and was competent in selecting the occasions on which to produce them. It might take considerable study to be sure that this was a case of rote learned responses, rather than what we might think of as full linguistic competence.

What is more important, though, is that these rote learned sentences would sound as though they had been constructed according to grammatical rule - whereas in fact they had just been memorised.

This isn't to say that there may be no such thing as grammatical rules, or other linguistic rules. It is to say that these rules are a tool of the linguistic student, not of the speaker. We should know this from the opening examples from the Investigations - the builders. The rules appear when we analyse, interpret, or translate the builders linguistic behaviour. We do not have grounds for saying that the builders are intentionally following these rules of grammar that we have discovered.

Grammatical rules are rules of linguistic behaviour, not of meaning or truth. We discover them as we discover 'laws of nature', and so these discoveries are subject to the same Goodmanesque or Kripkean uncertainties.

Hume, Is->Ought, Open Questions ...

Hume's paragraph on deriving ought from is suppresses the methodological rule by which he selects 'is' statements.

This rule is, of course, normative - and so Hume cannot allow it to be deduced from factual statements. It can, however, be induced - as his method for selecting facts.

This leaves him with a dilemma if asked to respond to an explicit statement of the rule: He can either (a) deny the validity of the rule (which undermines his is-ought insight) or he can (b) accept it as implied by his fact selection process, if not by the 'facts' that he selects. In the second case, the selection method must either include or imply the rule 'no value statement can be a fact', rendering the argument circular.

If the rule depends on the fact selection process, it doesn't depend on the facts. Does this leave the argument intact (though weakened by circularity)? The scope of Hume's 'facts' is narrow: arguably only the direct evidence on which natural science depends. Most 'social facts' include value statements or value commitments, from which 'oughts' can be deduced (as in Searle's argument).

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).