Friday, 10 April 2015

Rules, Norms and Robinson Crusoe

In lieu of a proper post about understanding and rules (it's coming, it's coming), I've posted some rough notes on my other blog concerning rules, norms and the "Robinson Crusoe" debate. I'm not putting them on here as a "proper" post, since they're just me thinking out loud. Still, have a look and see what you think. And, obviously, I'd be grateful for any comments, either here or on the other blog.

Thursday, 2 April 2015

Why Wittgenstein Matters

I hope in due course to say something myself on this topic. In the meantime, however, here's an excellent talk given by my former tutor, Dr Ian Ground, to the Royal Institute of Philosophy last February.

(I am also, by the way, still working on my next post about understanding. Progress is hard and slow, so bear with me.)

Wednesday, 4 February 2015

New Wittgenstein Blog

Adrian Brockless has been running a monthly Wittgenstein reading group for a while now in London. I've not been to a meeting myself, but they've certainly looked interesting (the last topic was rule following, and they're about to discuss Peter Winch's paper on "an attitude towards a soul"). Adrian has just set up a blog to accompany the group's discussions. No posts on it as yet, but hopefully this will be well worth checking out as it develops.

Monday, 20 October 2014

Understanding, States and Correctness

This post considers the negative account of understanding in §§138-242. By “negative account” I mean Wittgenstein's criticism of understanding as a state which governs our use of language. It is not, however, a straightforward description of that criticism as it develops in the Investigations. Instead, I've reordered the arguments in an attempt to bring out their key features – for the more I consider them the more it seems to me that Wittgenstein makes the same (or similar) points over and over again. Or perhaps it would be more accurate to say that the points he makes about a specific type of state turn out to be applicable across the board (and this, of course, is not a coincidence).

My next post will outline Wittgenstein's positive account of understanding – ie, his description of the role the concept plays in our lives and the circumstances within which it operates. Hopefully that won't take me ten months to write.


Let's start with the interlocutor's remark in §138:

But we understand the meaning of a word when we hear or say it; we grasp the meaning at a stroke, and what we grasp in this way is surely something different from the 'use' which is extended in time!

Why does this represent a challenge to Wittgenstein's claim that the meaning of a word is it's use in the language (§43)? Well, consider the following exchange between Lee and Jones:
Lee: I bought a piccolo today.
Jones: What does 'piccolo' mean?
Lee: This is a piccolo:
[Lee takes a piccolo out of his bag and shows Jones.]

At once the penny drops. Jones understands the meaning of the word “piccolo” at a stroke; she has grasped not merely what Lee meant in his specific remark, but how the word is to be used in a vast (indeed infinite) range of possible applications. But if that is so then how can meaning be use? Jones didn't study the use of the word in order to grasp its meaning – indeed her understanding seems prior to the use she will now go on to make of it. We might put it more strongly: how could a study of use (ie, what has happened in a relatively tiny number of past occasions) possibly determine how a word will function in an infinity of potential future cases?

Such thoughts can be built upon in a number of ways (as we shall see), but four points stand out:

  1. The above account presents understanding as an entity of some kind. As such, it readily suggests that understanding is a state. Jones was in one state before she understood, and acquiring the “entity” moved her into a new state: the state of understanding.
  2. The State governs our use of words. We use them as we do because we're in a particular state.
  3. The state enables us to use a word correctly. It is because Jones has acquired the entity that she can now respond to “piccolo” correctly when someone else uses it, and will also use it correctly herself. So the state must provide a standard of correctness against which usage can be assessed. Inevitably, this raises the issue of rules, since using a word correctly simply is using it in accordance with the relevant rule. So what is grasped at the moment of understanding would seem to be the rule itself (in some form or other). The state of understanding requires the possession of, and correcting functioning of, a rule.
  4. Achieving understanding seems to require a mental act of comprehension; the “entity” didn't merely fall into Jones – she grasped it. She not only saw the entity, she understood its meaning. She was conscious of the fact that she could now go on to use the word correctly.
Taken together, these points outline a foundational theory of understanding. But of course we still need to fill in the details: what sort of state are we talking about? In what sense does it govern use? And how does it provide a standard of correctness? As we shall see, the heart of Wittgenstein's objection to this whole approach lies in an unbearable tension arising out of the answers to these last two questions.

Physical States

We'll consider physical states first, if only because they seem the least mysterious option on offer. Here we are talking about “a state of an apparatus” (§149) such that it produces a given output for a given input. In the case of humans, the apparatus in question is held to be the brain, but it's worth noting that the theory doesn't rule out the existence of understanding in inorganic structures such as computers. For us, however, the process runs as follows: when we hear a word, the brain receives it as an input (or stimulus) which causes various neurological reactions (or processing), resulting in an output (or response). Clearly this account offers an explanation as to how the state governs use: it is a causal process. But how does it budget for correctness? Well, the state represents the rule for the use of the word. The rule is encoded into the brain's neurological structure just as the rules encoded into an iPhone's circuits allow Siri to answer our questions. The correct set of rules, properly encoded, results in the correct response for a particular verbal stimulus. Obviously our current understanding of the brain is too coarse to enable us to identify the detailed workings of these rules, but part of the theory's force lies in the thought that they must be there, or else understanding would seem to lie outside the realm of natural laws. It would be not simply mysterious but akin to magic (cf §158).

The first problem to note with this account is that it means my own understanding is hidden from me. I have no access to the rule I need to follow in order to use a word correctly (cf §153). So at best I can only infer that I understand a word – or we might say that I interpret my current state to be one of understanding. Likewise, when I use a word I can only infer that I'm using it correctly; I interpret its meaning in such-and-such a way. But from what do I make these inferences? What are my interpretations based upon? Consider a normal case of interpretation: I'm presented with the sentence “The cit purred and licked her paws”. Here I can easily infer that “cit” means “cat” (perhaps it's a typo). But the situation we're now presented with is one in which all our words need to be interpreted. So I'm not just inferring the meaning of “cit”; I'm also inferring the meaning of “inferring”. (And “meaning”. And “word”. And “and”.) It is difficult to make any sense of the phrase “I infer that by 'infer' I mean infer”. Worse still, the meaning of any interpretation I come up with will itself need to be interpreted, and so will the interpretation of my interpretation. And so on.

In fact, this same objection holds against the basic idea that the rules of language are encoded in our brains. Given that we have a language which is not a code, it's a simple enough job to decode a sequence like “Gl yv li mlg gl yv”. Likewise, it is because we have language in uncoded form that we can built coded versions of its rules into iPhones and PCs. But if all language everywhere is a code then the process of decoding can never end – and that amounts to saying that there's no such thing as a code in the first place. And no such thing as language, either. The concepts of interpretation, inference and encoding only make sense within a framework where there are some things which are not interpreted, inferred or encoded. The former are parasitic upon the latter. If you try to make them the fundamental basis for language then language itself collapses in a heap.

It's important to realise that the basic problem lies with the need for a standard of correctness. The physical state hides the rules of language, but you cannot follow a rule if you don't know what it is. And while it is possible to act in accordance with a rule without knowing that it exists, this can only happen within a broader practice of rule-following. Because the physical state makes following a rule impossible, it also makes acting in accordance with a rule impossible. And that, at bottom, is why notions such as interpretation can get no purchase here; we are attempting to act in accordance with a rule when there is no such thing as following it. In such a situation, the very notion of a rule disappears.

What this comes down to is that brain processes do not amount to rules. They may well operate according to causal laws, but there is a categorical distinction between rules and laws of nature. You don't need to know the laws of gravity in order to fall at 9.8 metres per second per second. And there is no such thing as correctly (or incorrectly) following that law. A causal process might do what we hoped or disappoint us; it might operate as predicted or might surprise us. But it cannot make a mistake. Therefore a causal process cannot provide a standard of correctness for usage, and therefore it cannot provide a foundational account of understanding.

Function and Behaviour

But it might be objected that we're needlessly over-complicating things. There's a much simpler way to allow for correctness in relation to physical states and processes. All that's required is for the outputs to be properly calibrated – that is, given the same input, the individual states reliably produce the same output. From this point of view it doesn't even matter if the states in question are completely different; after all, mobile phones can be constructed in numerous different ways but that doesn't bother us. It's the function that counts.

Interestingly, this approach leaves us in some doubt as to the precise criteria for specifying understanding. Our starting point was that understanding is a state; find the state and you've found the thing itself. But switching the focus towards function muddies the waters. Now it seems that understanding is a state plus the output (ie, behaviour). And since the state can take any number of different forms it seems to be the output which is doing all the heavy lifting here. The requisite state is merely whichever one happens to be in place when correctly calibrated outputs are produced. Doesn't it thereby drop out of the picture so far as defining understanding is concerned? Can't we say that understanding is simply correct behaviour? Or is there something important about the fact that the behaviour is produced – ie, that it's linked to inputs via the state? Should we take an inclusive approach and say that understanding is something which emerges out of the process as a whole, so that the total “system” includes all the linguistic interactions between people?

However, the real problem with this account is not which part to designate as the essence of understanding. The whole approach fails because, again, it cannot provide a standard of correctness. To say that we can generate a standard of correctness by properly calibrating outputs simply presupposes a standard – ie, we're assuming that the meaning of “properly calibrated” has already been settled. But that's precisely what we're supposed to be explaining! It's no help to say an output is right if it accords with everyone else's, because we've yet to fix what counts as “being in accordance with” in any given case. Here it's tempting to say that they're in accordance if they're the same – but this takes us in a circle, because whether two things count as “the same” depends on whether they are both in accordance with a particular rule. So we're defining “accords with” in terms of “the same” and “the same” in terms of “accords with”!

This brings out the significance of Wittgenstein's remarks on sameness which are dotted throughout §§138-242 (see §208, §§215-216 and §§224-227). It is tempting to suppose that the sameness of any two objects is an intrinsic feature of the world – something we can simply “read off” by regarding the objects themselves. From this point of view, a rule would be a description of the feature in question, and whether or not it was a correct rule would depend on how accurately it described the feature. So, for example, the rule “This = 'red'” would successfully correlate the word “red” with red objects because such objects were intrinsically the same as . The sample merely described redness and the rest followed as a matter of course. But, as we're already seen when discussing ostensive definition, this doesn't work as a foundational account because without a pre-established context the definition achieves nothing at all (cf §§28-31). In this case, the context is that the box is a sample of a particular colour and that we already know what counts as “being the same colour”. In different circumstances the rule would have had an entirely different effect. Likewise, if someone buys two copies of the morning paper has he bought two things once or one thing twice? That is, are we to classify the two papers as “the same” or “different”? The answer, of course, depends on why we're asking; it cannot be divorced from the activity in which the question arises. What a rule achieves depends on the context in which it is applied; it is woven into the way we live, the way we interact with things (with colours, for example). It does not function as a description.

But the function/behaviour approach implicitly assumes that rules describe “sameness” as an intrinsic feature of the world, and therefore, in themselves, provide a context-independent standard of correctness. All you need is the rule; the rest takes care of itself. But if you do not specify in advance what is to count as “being the same” (ie, “being in accordance with”) then any response can be classified as “the same” – or “different” – according to some formulation or other. It's easy to overlook this point because the examples of “sameness” used are extremely typical, and so we readily imagine a context in which they would be correct. But it is therefore we who are providing the standard of correctness. And this brings us back to our earlier point: far from showing how the input/process/output model gives rise to correctness, it simply presupposes what it is claiming to explain.

Mental States

Since the mechanistic causality of physical states presents an insurmountable difficulty, might not mental states prove more accommodating? Certainly there is something appealing about this idea. The mental realm offers something less rigid, something stranger – it is (we suppose) a nebulous realm whose workings we don't quite understand and yet seem capable of near miraculous results. This connects with the thought that, at best, a mechanical system can merely manipulate dead signs; you need the mind to breath life into the process and turn it into meaningful language (cf, the Blue Book, p3-4). And, in a way, such a transformation simply represents the move from mere causal “happenings” to correct or incorrect use. For unless our use of words can be right or wrong they cannot be meaningful – indeed, they do not even count as words. They are just bleats or scratches on a page.

Of course, in proposing mental states as a way forward we must be careful to avoid the difficulties we've already encountered. So, for example, such states must not operate in a causal or quasi-causal manner; a mental mechanism is no more helpful than a physical one. Likewise, a hidden state (or process) will be no use to us; that would simply re-introduce the absurdity of language as constant interpretation. Yet we still need to account for the way in which the state governs behaviour and produces a standard of correctness against which such behaviour can be judged.

Taken together, these requirements point us towards the linked notions of guidance (as opposed to causation) and characteristic experiences (as opposed to hidden processes). This picture can be fleshed out in a number of ways, but roughly speaking it hinges on the idea that we are directly aware of our understanding; we feel it. And this feeling guides us in our use of words. It suggests one use rather than another.

As usual, the devil is in the details. For example, is the feeling we get understanding itself? – in which case, we only understand a word while we're experiencing that feeling, which seems absurd. Or is it merely an indication that we understand? – in which case our actual understanding again seems hidden from us. Moreover, is there even such a thing as a characteristic experience of understanding? Do you have a characteristic experience every single time you hear the word “dog” or “the”?

Wittgenstein spends a considerable amount of time undermining the plausibility of the whole account, but it seems to me that it falls on two basic points he makes regarding correctness. First of all, feeling you understand isn't the same as understanding. Being sure you're right isn't the same as being right. Thinking you're following a rule isn't the same as following it (§202). If it was then there would be no way of settling disputes over rules, and therefore no standard of correctness for their use. Secondly, insofar as the experience guides me, it again fails to provide a standard of correctness, because “if it can guide me right, it can also guide me wrong” (§213). Guidance is not enough; I still need a means of evaluating it. And, of course, that cannot take the form of further guidance, because then I would need a means of evaluating the guidance about the guidance.

We've ended up back at the familiar regress, and this should alert us to the fact that “guidance” here is playing a similar role to inference and interpretation in earlier accounts. Guidance suggests an application of a rule, and that is akin to being offered an interpretation. But (we might object) can't guidance tell us what to do? Obviously it can, but then it's either up to us whether we do as we're told or we have no choice in the matter. In the former case, we need a standard of correctness by which to decide if it is right to obey. In the latter case, there's no question of a standard because we're again in a rigidly determined system. Our discussion repeatedly brings us back to this dilemma: either the state governs use in a rigid way, in which case there cannot be right or wrong usage; or else it leaves the final decision up to us, in which case we have no firm basis for a decision since that's precisely what the state is supposed to be providing.


But doesn't this introduce the role of the mental in the wrong way? Perhaps all this talk of characteristic experiences is besides the point. After all, item (iv) on our list spoke of mental acts rather than a passive awareness of sensations – isn't that what we need to add in order to provide a standard of correctness? This is what the interlocutor is getting at in §186 when he says, “The right step is the one that is in accordance with the order – as it was meant.” And it's a suggestion that brings us back to the difference between dead signs and living language. When I say something I don't just respond to a stimulus like a bell ringing at the push of a button; I talk about things and I mean what I say. And this meaning is (it seems) a conscious act which, inter alia, specifies how you should respond to my words. If you do so correctly (ie, as I meant it) then you've understood. Understanding, therefore, is the mental act of grasping the intention behind the word.

Hence (it seems) it is the mental act of meaning which supplies a standard of correctness and transforms dead signs into living language. But what do we actually mean here by “mental act”? What sort of act are we talking about? The answer seems to be something like picturing to ourselves what we mean by our words. As Wittgenstein (discussing “red”) rather scathingly puts it, “It is as if, when I uttered the word, I cast a sidelong glance at my own colour impression, as it were, in order to say to myself: I know all right what I mean by the word” (§274).

The most obvious objection to this account is that it doesn't seem to fit the facts. In the flow of a normal conversation we're not aware of performing acts of meaning relating to individual words or even groups of words. Perhaps (we might say) the acts go by too quickly to be noticed, or maybe they're performed subconsciously. But this just brings us back (yet again) to the problem of hidden criteria. If meaning is provided by associating a word with a sample and I'm not aware of making this association then how do I know what I mean by my words? The sample is supposed to be the rule that I follow, but, as we've seen, I cannot follow a rule of which I am unaware.

Even setting aside the issue of hiddenness, however, could the mental act achieve what we require of it? When the interlocutor introduces the subject of intention in §186, Wittgenstein immediately brushes it aside:
But that is just what is in question: what, at any stage, does follow from that sentence. Or, again, what at any stage we are to call “being in accordance” with it.

And consider in this context §239:
How is he to know what colour he is to pick out when he hears “red”? – Quite simple: he is to take the colour whose image occurs to him when he hears the word. – But how is he to know which colour it is 'whose image occurs to him'? Is a further criterion needed for that?

In other words, simply associating a word with a mental sample (or rule) isn't enough – it has to be the correct sample. Yet again, the process we posited in order to inject correctness into language itself requires a prior standard in order to make it work.

Let's pause a moment and consider the act of intending a bit more closely. I said it was a kind of picturing, but what might that involve? Here are three suggestions:
  1. I picture the object I mean by the word.
  2. I picture the use to which the word should be put.
  3. I picture both the object and its use as complementary parts of a whole.
How might the act described in (i) work? It seems to be a case of projecting the pictured object as a sample. If the picture is grasped by the hearer, she will use it to identify the appropriate objects in the world. But this just takes us back to where we were regarding the calibration of outputs. Which objects count as the appropriate ones? We are again trying to use a rule as a description, and so its status as a standard of correctness is either missing or assumed. Options (ii) and (iii) seek to rectify things by picturing the rule's application (it should be borne in mind that the “picture” need not be a literal one). But this amounts to providing a second rule for the application of the first – so the question now arises as to the application of the new rule. The regress has reappeared. (We are, in fact, right back in the difficulty described in §§139-141 and §146. See Understanding Part 1: Pictures for more on this.) Moreover, these aren't just problems for the person hearing the words; they apply to the speaker as well. Neither party has a standard of correctness and so neither can tell the meaning of what's said. (§504: “But if someone says, 'How am I to know what he means – I see only his signs?', then I say, 'How is he to know what he means, he too has only his signs?'”)

Still, it might be objected that we're not doing justice to the mental act. Of course it's not just about inwardly looking at a sign while uttering a word. How could anyone think that that would be enough? It's about meaning that the picture (or sample or rule) ought to be applied in a particular way. So this “meaning” is something that happens alongside paying attention to the picture. But what is this something extra? Wittgenstein brings out the strangeness of the claim in §332: “Utter a sentence, and think it; utter it with understanding. – And now don't utter it, and just do what you accompanied it with when you uttered it with understanding!” And again in §510: “Try to do the following: say 'It's cold here', and mean 'It's warm here'. Can you do it? – And what are you doing as you do it? And is there only one way of doing it?” The point, of course, is that whatever we end up trying is utterly unlike anything we'd be prepared to call “an act of meaning”. The very notion of meaning or intending as a substantive act begins to appear hollow.

At this point we are flirting with arguments which are given their fullest expression in the sections on private language (§§243-315), so I'm reluctant to delve into too much detail in this post. But I hope I've said enough at least to indicate how the interlocutor's account of intention struggles to fulfil its mission.

Logical Compulsion

Maybe, however, we're still looking at the mind in the wrong way. We've been talking about psychological laws and processes, and these are inevitably analogous to physical ones – so it's only to be expected that the same problems would emerge in both accounts. But isn't the important thing about the mind the fact that it is the medium by which we apprehend the underlying logic of things? We need a non-causal process which nonetheless does more than simply guide us; doesn't the notion of logical compulsion provide us with just this feature? For the force of logic is imperative yet not causal. Consider a typical logical statement such as “-(P˄-P)”. You cannot have both “P” and “not-P” at the same time. We are (somehow) forced to acknowledge the statement's truth – we cannot help but see that it is correct, that it must be correct. And yet this is not a matter of causation; if anything, the compulsion runs deeper than that. It is not founded upon the contingent laws of nature but upon the necessary structure of the world. It provides the framework within which such contingent laws can exist.

Tracing through the implications of this for language can take many forms, but the basic idea is that somehow logic locks words into a descriptive relationship with the world. Let's return for a moment to the distinction between language and dead signs. Seeing how the latter can become the former means answering a simple question: how do signs work? How is it that scratches on a page or emitted sounds actually mean something? How is it that they describe the world? Well, in the case of the Tractatus, it's a matter of logical structure. Names (which correspond to simples) are combined in a way that matches the logical structure of what they depict. They mirror the essence of a possible state of affairs. It is vital here that what is mirrored is a logical structure. That is what gives the connection its peculiar depth. Logic, we might say, acts as a kind of force which binds the two together in a depicting/depicted relationship. It adds an imperative quality to what otherwise would merely be a contingent likeness. So given that a particular mark or sound is being used as a sign, you must see that it represents such-and-such a state of affairs. Of course, this account doesn't explain anything. As an answer to “How do signs depict?” it amounts to replying “They just do”. And the Tractatus, at least, is audaciously upfront about this: you cannot say how the sign depicts. It shows that it does so by exhibiting the requisite logical form. This might seem unsatisfactory, but it must be the case because (the argument goes) it is a condition of the possibility of language itself.

The problems with this account are frustratingly familiar. First, it presents us with a sign which is a description of a possible state of affairs and yet which has the imperative quality of a rule. As such, it implicitly rests upon the idea of intrinsic sameness, and so it presupposes its standard of correctness. It says, in effect, that because the sign and state of affairs share a structure we cannot help but see that they're the same. But to say they share a structure is just another way of saying they're the same, and “they're the same because they're the same” says nothing. We still need to know what counts as “sharing a structure”.

It might be objected here that we've overlooked the sublime nature of logic; it is precisely because we're dealing with a logical force rather than an empirical one that this problem doesn't arise. Logic just is the realm in which the notion of intrinsic sameness holds good. Aside from the fact that this makes logic look uncomfortably like magic, it runs headlong into our other perennial problem: if logic creates an unbreakable bond between the sign and what it signifies then how is a mistake possible? Yet again we have produced an account of language which is too rigid to allow for the notion of correctness. And, as before, if we attempt to soften things – by an appeal to the guiding inner voice of “intuition”, for example – then we're left without a foundation for our judgements. Each guiding voice needs another to underwrite its advice.

We can express this as a problem arising from the fact that we're treating the sign as both a description and a rule. If you treat a description as a rule then its connection with what it describes is too rigid to allow for mistakes. But if you treat a rule as a description then it loses its imperative force and always stands in need of something further to justify its function. Rules and descriptions are conceptually distinct; running them together produces only chaos.


We have traced the idea of understanding as a state through various permutations, and each time we've been confronted by a similar set of difficulties, all of which centre upon the need to account for correctness. This suggests that the problem does not stem from choosing the wrong type of state, or misrepresenting its workings. It is far more fundamental than that. So where have we gone wrong?

Well, in a sense we have been misled by the very question we're trying to answer, viz: what is understanding? It's one of those questions that, according to Wittgenstein, produces in us a mental cramp. “We feel that we can't point to anything in reply to them and yet ought to point to something” (Blue Book, p1). He then remarks “We are up against one of the greatest sources of philosophical bewilderment: a substantive makes us look for a thing that corresponds to it”. And that's precisely our position regarding understanding; a form of expression (“we grasp the meaning at a stroke”) tempted us into treating it as a kind of entity. Of course, we weren't sure what type of entity we were dealing with, so we began to produce various hypotheses concerning its nature: it must be a physical state, or a functional state, a mental state, and so on. But the one constant amongst all this was that we were investigating a thing, and that by itself locked us into a particular way of considering it. Understanding, whatever its precise form, was a discrete entity which could be considered in isolation from its surroundings. It was context-independent. And insofar as it exhibited influence over other things, such as behaviour, it would do so in what was essentially a rigid, mechanical relationship.

And because understanding is bound up with grasping and following rules, those rules were themselves treated as things – mechanistic structures processing inputs in the brain or the mind or a platonic realm of logical compulsion. At the same time, however, these rules were assumed to function as descriptions, so that the computation was a matter of comparing a rule with a context-independent reality (either the reality of things being “thus and so” or the reality of the output which commonly followed from the relevant input). This act of comparison could only get off the ground if it was assumed that “sameness” or “being in accord with” was an intrinsic feature of the world, so that a standard of correctness was automatically provided along with the rule itself. That is, given a particular rule, it could not possibly be seen as anything other than a picture of such-and-such, and so there could be no doubt as to whether it was being correctly applied on any specific occasion.

Unravelling this illusion begins at §28 with the remark that “an ostensive definition can be variously interpreted in any case”. The ramifications of this simple observation echo throughout the Investigations in a variety of associated contexts; see, for example, §§85-87, §139, §§162-164 and §186. But the underlying point remains constant: a rule (or sample or picture) does not come with its method of application built into it. Imagining that it does so amounts to sublimating the concept; it turns the rule into something occult and utterly mysterious. And once this illusion is dispelled the whole mythology which has been built on top of it collapses. A rule cannot function as a description and therefore cannot state a context-independent fact (indeed, there are no context-independent facts). A rule is not itself a thing, so understanding the meaning of a word cannot be a matter of possessing this thing. Therefore understanding cannot be a state, for that was just another way of saying that understanding meant possessing (in some form or other) a rule. This is why normative explanations cannot be reduced to causes or rigid processes; they represent categorically different types of explanation, and the attempt to run them together only yields confusion (it's an example of what Wittgenstein calls “the crossing of different pictures” in §191).

And that, in turn, is why the switch from causation to something less rigid (guidance, interpretation, inference, etc) gets us nowhere. It introduces an alien type of explanation into the causal account. At the same time, it strips these new notions from their customary framework where they function alongside other, more direct practices. But it is only within this wider framework that they make sense; guidance is grounded in practices which are not themselves guidance, and the same goes for interpretation and inference (cf §1: “Explanations come to an end somewhere”). When we illicitly import them into the causal account, we expect them to provide a foundation for understanding, and this is precisely what they cannot do. Cut free from their own foundations they produce a regress whereby meaning itself vanishes into thin air: each interpretation requires a further interpretation and any response can be brought into accord with any rule according to some interpretation or other. In this situation, language collapses and with it go even our most basic concepts.

All this amounts to a damning indictment of the idea that we can found our concepts in a context-free description of reality – that we can start with “brute facts” about the world and show how features such as “understanding”, “knowing” and “meaning” arise out of them. (Think of the Tractatus: the facts are just there, and the philosopher's job is to explain how language reflects them.) The temptation to take this approach is built into the initial question: What is understanding? It doesn't express puzzlement about this or that aspect of understanding; it is not a question which arises out of a concrete difficulty. So it is unlike (eg) “What are the physical processes which underpin understanding?”. It is a question without a context and therefore invites a context-free answer. It is an example of the “engine idling” (§132), and the response it tempts us to make doesn't just fail to provide a convincing answer; nor does it merely destroy the concept it is trying to explain; it makes concepts themselves impossible – including the concept of brute facts. It is a response which brings itself into disrepute.

The context-free question is, we might say, a paradigmatically philosophical question. If, therefore, there's no legitimate way of answering it, that amounts to saying there's no such thing as philosophy itself. For Wittgenstein, however, the correct response is to describe the various contexts in which the relevant concept operates – the complex role it plays in our lives. So the answer is neither context-free nor an attempt to produce a single description covering all possible circumstances. And it is this approach which forms the basis of his positive account of understanding.

Monday, 18 August 2014

Yet Another Bloody Update

The first thing to report is that I'm now properly back on line, so that should make working on the blog a bit more straightforward. Unfortunately, that doesn't help with my main problem: how to condense (or finish) the mass of notes and half-written blog posts I have on understanding and rule-following. I'm finding that a formidable task, to be honest; at the last count I had nine half-written posts amounting to around 27,000 words in total.

Well, I shall just have to knuckle down and get on with it. To quote Goethe, "Don't let the best be the enemy of the good".

Tuesday, 15 April 2014


Yet another apology for my lengthy silence. Things have become a bit difficult recently (in various ways), and it’s hindered my progress with the blog.

A second reason, however, has a more philosophical stamp to it. The more I considered it, the more dissatisfied I became with dividing up §§138-184 and §§185-242 under the respective headings of “Understanding” and “Rule-Following”. True, that’s a fairly standard division (Hacker and Fogelin both use it, though McGinn doesn’t), but I don’t think it makes much sense. For a start, the so-called Understanding section is at least as much about knowing, meaning and rules as it is about understanding. Perhaps more importantly, however, I think the rule-following section is quite clearly a continuation of – and culmination of – the discussion begun at §138. There, the idea of understanding as an “inner Something” (a picture, formula, rule, process or whatever) leads to the question of how it ensures correct usage. This in turn raises the suggestion that the picture (etc) “forces a particular application on us” (§140). In the same section, Wittgenstein wonders about the nature of this compulsion: is it psychological or logical? (I’m probably overstepping the mark here, but I’m tempted to say this ROUGHLY equates to “Empirical or Rational”.) I seems to me that (again, roughly speaking) §§141-184 investigate psychological compulsion, while §§185-242 deal with the logical side of things.

Moreover, it is in this latter section that the crucial role of circumstances (raised in §154) is given its fullest treatment in the passages concerning customs and techniques. Finally, the discussion terminates with some startling and difficult comments that look back to (but also build upon) §142.

That, at any rate, is my current thinking on the structure of §§138-242. I’m still mulling over the best way to break things down into easily digestible chunks (there’s way too much going on to be covered in one or two posts), but hopefully I’ll have something worth putting up before too long (and if that sounds annoyingly vague…).

Sunday, 15 December 2013

Understanding: Discussion Summary

§§138-184 form a complex, interlocking series of remarks. Sometimes they seem to repeat themselves and and sometimes they go off on curious tangents. Wittgenstein wrote in his preface that “The same or almost the same points were always being approached afresh from different directions” and this is certainly a prime example.

It would be a mistake, however, to think that the discussion is presented this way simply because Wittgenstein wasn't up to the task of moulding it into a more conventional form. Rather, the presentation is bound up with the idea of philosophy as therapy. It attempts to reflect the experience of being in the grip of a particular way of looking at things. Our thoughts keep coming back to the same familiar notions – or, rather, those same notions keep reappearing in subtly altered forms. They cannot be despatched at a stroke; it requires patient work on the part of both the author and the reader.

At the same time, there's no doubt that it's easy to get lost in the maze of remarks that Wittgenstein presents us with. So I thought it might be helpful to provide a rough overview (which I actually wrote for my own benefit) to act as a guide when things get difficult. Obviously these notes aren't intended as a substitute for the text itself. They do not stand alone. Although here and there I offer some supporting arguments (usually in square brackets), my main aim is to highlight the flow of Wittgenstein's discussion and the connections between the various parts. Hopefully more detailed posts will follow in due course.

I have tried to keep things brief but I must admit it's turned into a bit of a monster. So in case reading it on the blog itself is tiresome I've prepared a PDF version without all this introductory waffle.

The discussion can be divided into four parts:

  1. §§138-150. This starts with the seeming clash between understanding as something grasped in an instant and meaning as something explained through use. It considers (and rejects) the claim that understanding amounts to having an “inner Something”. The following candidates are rejected: a mental picture; a mental state; a disposition. It concludes that “meaning” (and hence “understanding”) is more akin to having an ability.
  2. §§151-155. This considers a concrete case of “grasping in an instant” (“now I know”) which seems to go against the link to ability in §150. Again, it considers and rejects the claim that understanding consists of an inner Something. Specifically, it argues against understanding as a characteristic experience or an inner (mental or physical) mechanism. It concludes that understanding is logically (grammatically) bound up with the circumstances in which it takes place.
  3. §§156-178. To clarify this point, the case of “reading” is considered. As with understanding, the notion of reading as an inner experience or process is rejected. This time, however, the process is seen from the point of view of “deriving” or “being guided”, and the experience of reading is considered as an experience of this process taking place. But it is suggested that the pertinent difference between reading and (eg) pretending lies in the different circumstances of the two cases.
  4. §§179-184. The way in which circumstances enter into the concept of understanding is clarified. The nature of “now I know” is identified as a kind of signal rather than a report or description of an inner experience or state.

Now let's attempt a more detailed summary.

1. §§138-150. First Run-Through

§138 points out that when we hear a familiar word we understand it in an instant. This seems to clash with idea of meaning as something established by use, which is spread out over time. A number of questions are generated as a result:
  • What exactly is grasped when we understand a word in an instant?
  • How does what is grasped relate to use? How can it fit or fail to fit with use?
  • How can we grasp the whole use of a word in an instant?
  • How can observed use help us in novel situations – ie, how is it that we understand a word even when it's used in a sentence we've never encountered before?

Note how the mere idea of “grasping in an instant” suggests that what we need to understand is something “inner”. Again and again during the discussion observations about the public criteria for using the word “understanding” will be countered by objections relating to the first-person experience of understanding. One of Wittgenstein's main concerns is to clarify the place of such experiences in relation to the concept of understanding. He will not claim that they're of no importance (much less that they don't exist), but that we tend to misunderstand their role. Achieving a proper understanding, however, doesn't involve discovering a correct theory; what we need is to remind ourselves of the complex role understanding plays in our lives.

§§139-142: Pictures

What is grasped when we understand a word in an instant? The first theory considered is that we get a mental picture. If we have a picture of a cube and we pick out an actual cube then what is grasped fits the use. If we pick out (eg) a triangular prism then it doesn't.

But who is to say that picking out a triangular prism isn't a correct application of the picture? A picture by itself cannot provide a standard for its correct use. The same picture might be applied in two different ways, and we would say that it had a different meaning on each occasion. So the criterion for understanding is still the use rather than the picture itself.

[Note the link between this point, the discussion of ostensive definition (§§28-36), the discussion of deriving (§§163-164), and the discussion of “+2” in the rule-following argument (§§185-190). In fact, the discussion of ostensive definition foreshadows many of the points Wittgenstein makes about understanding. I'll group them together as the “anything goes” argument. I'm not saying they're the same argument each time, but they do seem connected. Further consideration might be rewarding.]

We're tempted to say that the picture forces an application upon us, but what this boils down to is that when confronted with a picture we often expect it to be used in a certain way – other possible applications do not occur to us. So “the picture fits the use” means “the use was the one we expected”.

Does all this mean there's no such thing as an application occurring to someone in an instant? No. It will often make sense to say such a thing. But we need to clarify the role of a statement such as “the application came before my mind”. This will happen shortly.

§§143-150: States, Dispositions, Abilities

As a prelude, however, Wittgenstein switches to a description of teaching someone the decimal series of numbers. It outlines certain broad, public criteria for saying that the pupil understands. It also emphasises (a) that at any stage the pupil's ability to learn may break down, and (b) that the type of instruction given will depend on the type of mistake the pupil makes. The point of this is to undermine the idea that getting the pupil to understand involves giving her a specific inner Something (a picture or formula, etc) the possession of which will be the source of correct performance. Rather, what the pupil is “given” depends on what she needs in order to perform correctly – and it's possible that there is nothing we can give her (ie, no course of instruction) that will achieve this.

Note also that the pupil's coming to understand is drawn out over time and there is no precise moment when we might say “now she knows”. There is no such thing here as “grasping in an instant”.

At §146 we get an important objection: applying understanding is not understanding itself. Understanding is the source of correct performance. Or, to put it another way, correct performance is derived from the source.

This is a variation on the “inner Something” idea; behind it lurks the notion that the “source” is a mental state – one in which a formula (or other method of application) occurs to the person who understands.

Wittgenstein makes two objections:

  1. A formula has the same problem as a picture. It does not come with its method of application built-in.
  2. Understanding a word is not best categorised as a mental state. Mental states (such as feeling anxious or euphoric) have specific duration and varying degrees of intensity. But I understand a word whether or not I'm thinking about it. My understanding isn't interrupted when I'm distracted by something, and so on.

Objection (b) prompts a revised version of the claim (§149): understanding is a state of an apparatus of the mind (or brain). (Wittgenstein sometimes calls this a disposition.) So, just as a pocket calculator can give us the answer to a sum because of its structure (ie, independent of this or that instance of calculating the answer), so I can be said to know a word because the structure for correct performance is in place even if I'm not currently using it.

Wittgenstein counters by pointing out that this gives us two separate criteria for saying someone understands: (i) ascertaining the structure, and (ii) observing performance. He doesn't here elaborate on the implications of this situation; they will emerge as the discussion continues (I'll call this the “two criteria” objection).

Instead (§150), we get a summary of sorts: the grammar of the word “know” is related to “able to”, but also to “understand”. I think this is emphasising a logical (grammatical) connection between meaning/understanding and performance. I also think it needs considering in conjunction with §155. But it all needs careful unpacking.

2. §§151-155: “Now I know!”

Wittgenstein now switches back to instantaneous grasping when he considers the significance of phrases such as “now I know” or “now I understand”. And this marks the start of his attempt to clarify the idea of an application coming before one's mind, which he mentioned in §141.

[Why did he first discuss learning the series of decimal numbers (§§143-150)? I think because that process is part of the general scene-setting (the circumstances) without which “Eureka moments” don't make sense. (Before you can even try to continue a number series you have to learn to count.) Compare this with ostensive definition. Viewed in isolation ostensive definition seemed to do something both fundamental and mysterious: provide an unmistakable super-bond between word and object. But its function was only seen aright when we reminded ourselves that actually it took place within a broader linguistic context. I think Wittgenstein is saying something similar about instantaneous grasping: if you overlook the broad context in which it actually takes place then it can seem to provide the essence of understanding and perform a truly mysterious function. The mystery is dissolved precisely by reminding ourselves of that broader context.]

Wittgenstein first points out that there are various things that might go on in someone who suddenly understands. This mitigates against the idea that understanding is a characteristic experience. But worse still, any of those experiences might occur and the pupil might still be unable to continue the series. So the experiences are not the “essence” of understanding; they are concomitant processes.

And now, just as when the notion of understanding as a mental state fell to pieces, we're tempted to posit a hidden process that lies behind what we actually experience. (Note how we've moved here from description to theory.) But if the process is hidden then how do I know that I actually understand? “How can the process of understanding have been hidden, given that I said 'Now I understand' because I did understand?” (§153). [This links back to the “two criteria” situation in §149.] If understanding is a state of an apparatus then how can I say I understand when (a) I don't know what that state actually is, and (b) whatever it is, I don't know whether it obtains?

Against this it is objected (§154) that there must be a state (or process). For if (eg) a formula occurring to me is not enough to provide understanding then something else must be necessary, and what could that something else be if not a hidden process or state?

Here we have reached an impasse. We seem forced to accept a theory despite the fact that it makes things worse (cf §112). This is a radical breakdown, and Wittgenstein's response amounts to a rejection of the whole approach that brought it about. Certainly something else is needed, he says, but not a state or process or disposition or structure or any type of inner thing. Instead we need to remind ourselves of the circumstances that warrant someone's saying “now I know” when the formula occurs to her.

3. §§156-178: Reading

To make this clearer, he introduces the analogous topic of reading. As with understanding, reading is a concept that tempts us to think its essential characteristic must be something inner. On the one hand, there is surely something computational about reading – we derive our words from the text – and this derivation is a process that takes place in the mind, or perhaps in the brain. On the other hand, reading is surely also a distinctive experience; just compare actual reading with pretending to read! These two characteristics might seem at odds with one another, but aren't they really different aspects of the same phenomenon? That is to say, isn't the experience of reading precisely the experience of the computational process taking place?

Here, then, we're once again confronted by the two notions that have dogged us throughout our investigation: process and first-person experience. And in the context of reading they seem to stand out even more compellingly than before. Accordingly, Wittgenstein's treatment of them is both richer and more probing in this part of the discussion.

§§156-158: Experiences and Mechanisms

As with §§143-145, we start with a (brief) description of the circumstances surrounding reading: learning to read, various criteria for saying someone is reading, the difference between a beginner and a fluent reader, etc.

This quickly prompts both the idea that reading “is a distinctive conscious activity” (§156e) and that some kind of mechanism must be at work (§156g). Against both, Wittgenstein observes that it makes no sense to talk about the first word a beginner reads (note the connection with §145b). The point here is that if reading is either a particular experience or the state of a mechanism then it ought to make sense to ask “what was the first word he read?” Concentrating on the notion of mechanism, Wittgenstein draws a highly significant distinction between our concepts in relation to machines and the way we apply them to living beings – even where the living being is used as a “reading machine”.

Against this it is objected (§158) that the different treatment merely stems from our comparative ignorance of the workings of the brain. Wittgenstein's response is to underline the claim's theoretical nature (a point already made at §146g), and to further suggest that it is a priori: whatever the evidence (or lack of it), things must come down to a mechanistic explanation.

[What is he getting at here? I think §158 is an enigmatic, troubling section. It seems to come alarmingly close to suggesting that our belief in causality as a universal principle is a kind of metaphysical superstition (cf Zettel, §609). It certainly requires careful, detailed analysis. For the moment, however, I'll offer a provisional, “middle of the road” gloss: “Why do you say things must come down to a mechanistic explanation? What evidence do you have for this? What we know is that 'understanding' is not used like a name for a mechanistic process – and certainly not a hypothetical one.” That's by no means the last word on the subject, but it'll have to do for now.]

§§159-161: The Conscious Act of Reading

The discussion now switches back to the first-person experience of reading. Isn't that the essential thing which distinguishes actual reading from merely pretending or the free-association of sounds with marks on a page?

Wittgenstein counters with the imaginary case where a drug produces a feeling of reciting from memory in someone who is in fact reading a passage he's never seen before. A variation is offered in which the person feels he is reading when he is actually associating words with signs in a completely unfamiliar alphabet. In the first case we would say he was reading despite his feelings to the contrary. In the second case, classification would depend on his reaction to the signs. If his words bore no clear relation to them (eg, he read “^#*” as “blue” on one occasion but as “left” on another) we'd say he wasn't reading. But if the same words were always associated with the same signs then we'd perhaps be more inclined to say he was. That is, it wouldn't be clear whether he was reading or not – it would be up to us.

[The point, of course, is that his experiences are not the decisive factor in either case. But nor is it exclusively down to what he does (for he might speak exactly the same words both times). Rather, it is down to what he does given the particular circumstances in each case. Wittgenstein underlines this with the experiment in §161. What is the difference between counting to twelve and reading the numbers from a watch dial? Again (I think) the implied answer is: the circumstances.]

§§162-164: Derivation

Here we switch back to understanding as a process – specifically the process of deriving. It's an idea that has sort of been “in the air”, but not directly confronted, ever since §146b (the same is true, by the way, of rule-following, which is closely connected to derivation; see §143a, §147 and §162).

Wittgenstein describes a particular case in which we'd be inclined to class reading as an example of deriving sounds from a text. But immediately comes the objection that we don't have enough here to be sure this is derivation; we taught him the alphabet, he read the words – what right have we to say that the link between the two was deriving?

[Note: (a) this already treats derivation as an “inner” or “hidden” thing; and (b) we're at once tempted to look inside ourselves and search for it there. This goes some way towards explaining why the discussion constantly switches between third- and first-person perspectives.]

So Wittgenstein alters his description to make it an even clearer case of deriving – that is, he changes the circumstances. But here it's objected (§163) that even if this is derivation, we can't assert it simply because the pupil looks at the chart and writes the “correct” letters. For whatever he writes might be classed as derivation according to some rule or other, and thus be “correct” (the “anything goes” argument – cf, §139). And now the very concept of derivation starts to look empty.

In §164 Wittgenstein offers the moral of the story: our search for the essence of deriving has led us into darkness, for there is no such thing. Instead, we have a complex family of circumstances in which the word “deriving” is warranted.

[A brief elaboration: when we treated deriving as a hidden essence it became logically distinct from its consequences. As a result, the very essence we thought we needed to find became empty. Compare this to the case of mental pictures (§139) and the “two criteria” objection to dispositions in §149. Also compare it to the famous “beetle in a box” example in §293. But our grammar doesn't treat deriving as if it was a thing in a box; instead it conceptually links it to various performances in various circumstances. These circumstances sometimes include what went through someone's mind (“I recited to myself 'Richard of York gave battle in vain' and derived the answer from that”). But sometimes they don't.]

§§165-178: “Experiencing the Because”

Yet again we revert back to a first-person argument. The discussion considers a cluster of related claims:
  • we know from our own experience that reading is a particular process (§165);
  • the words come in a distinctive way (§165);
  • they somehow cause our utterance (§169);
  • we feel the connecting mechanism between the word and our utterance (§169).

So finally the third- and first-person arguments come together. Reading is both a mechanism and a characteristic experience; it is a characteristic experience of a mechanism at work .

At each stage, however, Wittgenstein (a) exposes this account as a picture that we adopt rather than a straightforward description of the the facts, and (b) undermines the temptation to adopt this picture.

  • §165. If reading is a characteristic experience then it doesn't matter what sounds result as they can all be linked to the text according to some rule or other. [This brings together the “two criteria” argument (§149, §153, §§163-164), and the “anything goes” argument (§139, §§163-164).]
  • §165. Moreover, it's not enough that the written words makes the spoken ones “occur” to me, or “remind” me of it. That could happen yet what I utter might still be incorrect. Mere association is not enough.
  • §166. The claim that the words come in a distinctive way is a fiction. Consider normal cases of reading: we don't even think about how the words come or if there's something distinctive about it. We see the words and we make the sounds. What else do we know? Of course, we notice a difference when we (eg) associate sounds with squiggles but it mis-describes reading to therefore conclude that the words come in a special way. With reading it's automatic, with squiggles it isn't. That's the difference. And it's not an experiential difference; it's a circumstantial one.
  • §167. There is no single characteristic experience of reading. This undermines the idea that reading is a particular process that we experience. [To put it another way: even if there is a particular process, we cannot infer its existence from the experience of reading, for that experience is extremely varied.]
  • §169. It makes no sense to say that we experience the causing. Causation is established by experiment, tests, etc; it is not something that can be felt. [That would be like saying “I'm feeling inflation” when I'm shocked by how much the price of bread has risen.] This is a categorical (grammatical) distinction. Indeed, we do not say that the text is the cause of our reading – rather, it is the reason we utter the words that we do. That is, we appeal to a standard of correctness, not to causation.

§§170-178 provide a kind of summary. We assume that the difference between reading letters and associating sounds with squiggles represents (respectively) the presence and absence of influence. But being influenced (or guided) is no more a particular experience than reading is. It forms a wide family of cases, and the important factor is not the presence of a particular experience but the circumstances pertaining to any given case.

None of this bothers us during actual use, but when we reflect on these things mere circumstances can seem insufficient. We're tempted to posit a particular source of influence (in other words, we've started theorising). Maybe it's a strange feature of the words themselves – as if they exercised a kind of “thought control” – or a process (perhaps physical) operating behind the scenes. And now we take our varied experiences to be experiences of this elusive form of influence. We look at them “through the medium of the concept 'because'” (§177). Of course, it is right to say that we're influenced, but not because of any particular experience or process. Rather, it is correct to apply the word “influenced” in all these varied circumstances. [We have supposed an essence, but what we needed was to recognise a family resemblance concept.]

4. §§179-184. Back to Understanding

Wittgenstein now applies to understanding the insights gained from considering the case of reading. Not surprisingly, this involves reiterating the point made in §§154-155: “The words 'Now I know how to go on' were correctly used when the formula occurred to him: namely under certain circumstances. For example, if he had learnt algebra, had used such formulae before” (§179).

At the same time, he's aware of the temptation to take this point the wrong way. We might, for example, suppose that “now I know” (or “I understand”) is a kind of shorthand description of the circumstances – as if we somehow deduced that we understood from the fact that the situation was one in which “now I know” would make sense. This harks back to the interlocutor's point at §147: “When I say I understand the rule of the series, I'm surely not saying so on the basis of the experience of having applied the algebraic formula in such-and-such a way!” This is correct, but the interlocutor's mistake is to assume it shows that the circumstances are irrelevant. Rather, it shows that they do not connect with the language-game in the way he supposes. We do not appeal to them as a criterion of application; they are the context within which our criteria make sense. They “set the stage for our language-game” (§179).

A second temptation is to think that the circumstances form part of a causal explanation. Here the supposition is that if the right circumstances are in place (general education and other background features + the formula occurring to the pupil + a characteristic experience of understanding) then the pupil must continue the series correctly (§183). That this is mistaken can be shown from the fact that even where the phrase “now I know” is warranted it is still defeasible – the pupil might still be unable to continue the series correctly. In such a case we would normally say that the pupil's statement was wrong: he didn't in fact know. But it was still understandable. [Compare this to a case where you show a formula to someone with no mathematical training whatsoever and he says “now I know how to go on”. In those circumstances his claim is not so much wrong as completely bizarre.]

These considerations throw light on the role of “now I know”. It is best not thought of as a description of a mental state at all (§180). Rather it is a signal that the pupil is confident (perhaps certain) that he can go on correctly. But, of course, being certain you can give the right answer and actually doing it are not the same thing. We frequently find ourselves ruefully saying “I was so sure that answer was right!”

Next Steps: Rule-Following

I started this summary with a list of questions, but there's one I've not really considered so far: how can we grasp the whole use of a word in an instant? This is raised right back in §139 and is occasionally glanced at during the discussion (eg, §147: “I surely know that I mean such-and-such a series, no matter how far I've actually developed it”). Knowing the whole use of a word seems a criterion of understanding, just as knowing what a chess pawn is means knowing how it can move in any given position. But how can we know the whole use of a word? For however we've used a word up till now, what happens when we come to apply it in a completely new situation? How is our past experience supposed to help us?

Surely when I grasped the rule for using the word “cat” I didn't know that it could be used in the sentence “The cat sat on Jupiter's second-largest moon”? And yet in some sense I clearly did know that, for I was able to form the sentence without any trouble at all. It didn't come as a surprise to me that “cat” could be used in that context. The rule, it seems, guides us effortlessly through countless permutations that weren't envisaged when we learnt it.

How does it achieve this? Or, to put it more generally, what is the connection between the rule and its application? What keeps them in sync? That is the issue which forms the heart of the discussion in §§185-242.