A speaker doesn't require any understanding in their use of words? Where does Wittgenstein demonstrate this? — Luke

I didn't say that. I said that people can use a word despite having misunderstood how it is used by others. Having understood the use of a word by another does not exist as a constraint on speaking that word. So for example, we can mimic and imitate others without understanding the meaning (the others' use). This is demonstrated by Wittgenstein in this section, 138-141. The process which leads to speaking words (application), is distinct from the process which is understanding the spoken word, such that understanding the spoken word is not required for speaking the word.

Now clearly we accept two different kinds of criteria for this:
on the one hand the picture (of whatever kind) that at some time or
other comes before his mind; on the other, the application which—in
the course of time—he makes of what he imagines. (And can't it be
clearly seen here that it is absolutely inessential for the picture to exist
in his imagination rather than as a drawing or model in front of him;
or again as something that he himself constructs as a model?) — 141

Notice the very last phrase here, "or again as something that he himself constructs as a model". The model for application may be constructed by the person, completely independent of anyone else's usage.

How is it unintelligible given your claim that "you can use a word however you please, and this use provides meaning for that word"? I used those words however I pleased, therefore I must have provided meaning for those words. So what makes it unintelligible? — Luke

If you do not understand what I am doing, then to you what I am doing is unintelligible. Using words is a case of doing something. It is very common that people do not understand meaning (the meaning is unintelligible to them). For example, it is very difficult to understand what Wittgenstein is doing in many parts of the PI, so for many people much of the text is unintelligible. But we're getting off topic here.

If you do not understand what I am doing, then to you what I am doing is unintelligible. Using words is a case of doing something. It is very common that people do not understand meaning (the meaning is unintelligible to them). — Metaphysician Undercover

You claimed that "you can use a word however you please, and this use provides meaning for that word". But is it actually meaningful if nobody understands? In order for people to "not understand meaning", there must be meaning there to be misunderstood. I used the words "elephant of cheese red line upon whiskey very distance" how I pleased and you don't appear to have understood. But how do you know whether there was any meaning there?

You claimed that "you can use a word however you please, and this use provides meaning for that word". But is it actually meaningful if nobody understands? — Luke

That's a good question. I'd say yes. If I swing the hammer, and miss the nail, is the action still meaningful? I think it is, despite the failure. To try, yet fail, is still meaningful, as 'trial and error' proves. Perhaps you think it's not. That's a matter of opinion. If someone speaks, and no one understands, is that action still meaningful? I think it is, despite the failure. Perhaps you think it's not.

I used the words "elephant of cheese red line upon whiskey very distance" how I pleased and you don't appear to have understood. But how do you know whether there was any meaning there? — Luke

The premise is "meaning is use". You say here, that you "used" those words. Therefore there is meaning here, despite the fact that I did not understand. The premise forces that conclusion. It appears to me, that you do not agree with Wittgenstein's premise. Would you prefer "meaning is understanding"? But wouldn't that make meaning a mental thing? The problem is that we cannot have both, 'meaning is not mental', and 'meaning requires understanding', because understanding is mental. So if we are to understand 'meaning' as non-mental, we need to rid ourselves of the notion that meaning requires understanding.

Once again though, Witty looks to 'test' his account to check if there is, in fact, a two-stage process at work, which he ultimately wants to deny. As he puts it, it looks as though there were first the picture, then its application — StreetlightX

I question whether this is something that he ultimately wants to deny. Given that a picture can have more than one application, it seems important that a picture and its application are separable.

To conjure up a 'picture of a cube' is to already have an application of it in mind. It may not be the only application there is ("it was also possible for me to use it differently"), but this doesn't imply that there are two stages from meaning to application. Rather, the application is always-already inherent to the meaning. — StreetlightX

I agree that there aren't two stages from meaning to application, but Wittgenstein is instead talking about the distinction between a picture and its application. A picture (either mental or material) is essentially "lifeless" and does not force or contain its own application or meaning.

141. At 139 Wittgenstein demonstrated that a mental picture evoked by the hearing/saying of a word does not force a particular use/meaning of that word, because different methods of projection (i.e. different applications of the mental picture) can produce different uses/meanings of the word.

Wittgenstein now considers whether hearing a word might be able to produce not only a mental picture but also its own application/method of projection. How would this work? Wittgenstein suggests a schema showing lines of projection from one cube to another. However, this schema encounters the same problems as the mental picture, viz. it is just another picture. As Wittgenstein notes: "does this really get me any further? Can’t I now imagine different applications of this schema too?"

Nonetheless, is it possible for an application to come before one's mind when they hear a word? Wittgenstein says we need to get clear about this expression:

Suppose I explain various methods of projection to someone, so that he may go on to apply them; let’s ask ourselves in what case we’d say that the method I mean comes before his mind.

The only case in which we'd say that one of several particular methods of projection had come before his mind would be where the student goes on to apply it; by an actual application.

On the one hand there is the picture, and on the other hand there is the application of that picture. Wittgenstein notes that it doesn't matter whether the picture is mental or material.

Returning to the question of 'fit', Wittgenstein now asks whether there can be a 'clash' between picture and application. Only in so far as the picture makes us expect a different use, he says. A picture might suggest a particular use because that is how it is normally applied.

Therefore there is meaning here, despite the fact that I did not understand. — Metaphysician Undercover

I intentionally used that string of words to be meaningless. I used those words how I pleased but my use did not provide meaning to those words, so your claim is false. It is your claim that "you can use a word however you please, and this use provides meaning for that word", not Wittgenstein's. I don't disagree with Wittgenstein; you simply fail to understand him.

142. At the end of 141, Wittgenstein tells us that a picture can suggest a particular use because that is how it is normally applied. At 142, he states:

It is only in normal cases that the use of a word is clearly laid out in advance for us; we know, are in no doubt, what we have to say in this or that case. The more abnormal the case, the more doubtful it becomes what we are to say.

This seems to indicate that language is a technique or skill that fluent speakers master, and that there is a certain degree of regularity to our language games. Moreover, there is a regularity to our form of life. Although Wittgenstein does not use the phrase here, I take PI 142 to be one of the best examples of form of life in the text, which he goes on to describe:

And if things were quite different from what they actually are —– if there were, for instance, no characteristic expression of pain, of fear, of joy; if rule became exception, and exception
rule; or if both became phenomena of roughly equal frequency —– our normal language-games would thereby lose their point. — The procedure of putting a lump of cheese on a balance and fixing the price by the turn of the scale would lose its point if it frequently happened that such lumps suddenly grew or shrank with no obvious cause.

Daniele Moyal Sharrock suggests (here) that 'form of life' refers to very general facts of living, which can be identified with the certainties referred to in Wittgenstein's On Certainty. Form of life can also account for Wittgenstein's comment at boxed section §139(b) with the picture of an old man walking up a hill: a Martian might describe the old man as sliding downhill, but there is no need "to explain why we don't describe it so".

I intentionally used that string of words to be meaningless. I used those words how I pleased but my use did not provide meaning to those words, so your claim is false. — Luke

Well no, just because you claim it, doesn't make it so. That's the difficult aspect of "intention". What you intend does not necessarily come from what you do, that's a failure. So your intent to use the words to be meaningless was a failure, because meaning is use, and that is impossible by way of contradiction, if we maintain that principle You just tried to do the impossible, and failed.

The fact is, that you used the words as an example, and although you intended the words to be meaningless, as an example of a meaningless use of words, your action of using the words this way was meaningful, as an example. One cannot escape the reality that intentional acts are meaningful, simply by claiming I intended to do something meaningless, therefore what I did was meaningless.

Your claim of success in this attempt at a meaningless use of words, just demonstrates that you are using "meaning" in a way other than Wittgenstein does. Maybe you misunderstand "meaning", maybe Wittgenstein misunderstands "meaning", maybe we all do. One thing is quite clear, we can use the word however we please. Whether this use has meaning is another question. Wittgenstein obviously thinks that it does. Even the most vague sentence has a perfect order [98]. It is the fact that you ordered those words to be in the array that they are, which gives the sentence meaning, regardless of whether they are understood.

You might say "I intended to make an unintentional act", and speak the truth in saying this, if you truly believed that you could do this. And I could believe that you had this intention, to make an unintentional act. But if you intended to make that act, clearly it was not unintentional.

One cannot escape the reality that intentional acts are meaningful — Metaphysician Undercover

Irrelevant. Your claim was about the meaning of words, not the meaning of acts.

Oh dear, what are you thinking? Using words is a type of act, is it not? There are many meaningful acts which human beings engage in, using words is one of them. There is a family of "meaning". Remember, the point though, in relation to language "meaning is use". Do you think that the meaning of the words is distinct from the meaning of the act, using the words? If it's different, then meaning is not use, as the meaning of the words is distinct from the use of the words. If it is the same, and meaning is use, then we are talking about the meaning of acts, acts of using words.

Do you think that the meaning of the words is distinct from the meaning of the act, using the words? — Metaphysician Undercover

Your original claim was:

"you can use a word however you please, and this use provides meaning for that word."

Now you are pretending that your claim was instead something like:

"you can act however you please, and this act provides meaning for that....act?"

I don't believe this was your original claim. Furthermore, if you (as a reader) found my nonsense statement to be unintelligible and I (as the speaker) can find no meaning in it, then to whom is it meaningful? As you say, "just because you claim it, doesn't make it so."

I don't see your point. Are you arguing that using words is not an act, or are you arguing against Wittgenstein's principle that meaning is use. Either way, I don't see that you've made an argument.

You used the words, therefore they have meaning. Why do you ask "to whom is it meaningful?" unless you think that meaning is something in a mind? It is not, and therefore it doesn't have to be meaningful to anyone, and yet it is still meaningful.

Wittgenstein will proceed now to apply the distinction made at 142, between normal use and abnormal use, to the process of learning what he calls a "formation rule". His example is the rule required to produce the series of natural numbers [143]. A "mistake" constitutes abnormal use, and he here distinguishes a "systematic" mistake from a "random" mistake, and notes that there is no sharp distinction between the two. At 163 () he moves to completely remove the possibility of a random mistake (random abnormal use of words), by showing that to produce a truly random use requires following a rule. At that point, the possibility of random abnormal use is ruled out.

We are inclined to think that a systematic mistake is one which can be corrected. If the student makes mistakes, what is required to correct him, is to change his way of looking at things [144]. at 145 the question is asked, when can we say that he has mastered the system, that he knows the rule? So he proceeds further, 146-151, to question what does "knowing" the rule involve, and he will make a distinction between "understanding" which is proposed as the source of correct usage at 146, and the act of applying the rule [147].

Here's a couple points of criticism which I have concerning this section. First, his example of learning the series of natural numbers is not expressed to represent the way that we normally learn these numbers. He expresses it as copying written symbols, when in reality, we learn to count first, verbally, by mimicking spoken words. Also, at 145 he presents this education as first learning the digits 0 to 9, instead of the way that we normally learn to count, as 1 to 10. I believe that it is crucial to recognize these points if one is actually interested in understanding how we learn to count.

In the first place, the 'repeat after me' method which we normally use, is a method of memorizing an order of sounds, and repeating that order. As a method of memorizing there is a temporal extension to this process of learning, and there is no 'eureka moment' of, 'now I understand'. The second point is very important, because it involves the actual learning of the principle. Learning to count involves learning one object, then two objects, then three objects, so that there is a process of adding an object each time. We do not start with zero, because "zero objects" is difficult and incomprehensible to the young mind. In any case, this method of recognizing one object, two objects, three objects, etc., is the method of learning the application. We recognize that two is one more than one, and three is one more than two, etc.. In this process, learning the principle of application, it is possible that there is a eureka moment of recognition.

So there is two distinct processes involved here. First, there is the learning of the names, one, two, three, four, etc., and this is learning how to count. Then there is learning the application, which involves recognizing the number of objects. So Wittgenstein is correct to suggest a distinction between knowing the rule (being able to mimic, count), and being able to apply the rule, but since his analysis of the learning process is a bit off, his attempt to express this distinction is vague and confused

143. Wittgenstein describes a language game of teaching the natural numbers, wherein a series of numbers are written down and a student is required to copy them. Wittgenstein notes that there is already at this stage "a normal and an abnormal learner's reaction". He suggests we might guide the student's hand in writing out the series, but that the "possibility of communication" (i.e. the "pupil's ability to learn" or the possibility of teaching the student further) will depend on the student "going on to write it down by himself". Wittgenstein asks us to imagine that the student copies the figures but in the wrong order (in most instances), writing "sometimes one, sometimes another, at random"; and "at that point communication stops". The difference between this and the case of making (random) mistakes, Wittgenstein says, "will of course be one of frequency".

The student may also make a "systematic mistake"; regularly making the same mistake at the same place. "Here we shall almost be tempted to say that he has understood us wrongly". However, there is not a sharp distinction between (what we call) "random" and "systematic" mistakes. Wittgenstein notes that it may be possible to wean the student from a systematic mistake; otherwise, it may be possible to teach the student the normal way of copying "as an offshoot, a variant of his". However, the "possibility of communication" could end here again (if this fails).

A picture (either mental or material) is essentially "lifeless" and does not force or contain its own application or meaning. — Luke

A lot rides on this and I think there may be a genuine question of how to understand this. One part of me wants to agree that a picture doesn't contain its own application - this is after all the thrust of the current sections. But on the other hand, I also want to say that in all instances of what Witty would call 'everyday use', a picture, by virtue of it always being used in some manner or another (read: applied in some manner or another), is always-already a 'living' one, and that 'lifelessness' is always a derivative phenomenon, which happens when words are taken out of their everyday use. In other words, picture and application 'come apart' only in instances of breakdown; in 'normal' cases, there is no separation. A single phenomenon breaks into two, and not - two phenomena are brought together in this case or that case (this would be a case of 'fitting', which is exactly what Witty is arguing against).

But on the other hand, I also want to say that in all instances of what Witty would call 'everyday use', a picture, by virtue of it always being used in some manner or another (read: applied in some manner or another), is always-already a 'living' one, and that 'lifelessness' is always a derivative phenomenon, which happens when words are taken out of their everyday use. — StreetlightX

I don't disagree, but in the distinction between a picture and its application, the use is the application (of the picture). Recall that the same picture might have various actual applications. Rehashing the metaphor, we might say that this actual use is what gives the picture its "life". Whereas I see Wittgenstein as constantly attacking the mistaken idea that a picture by itself - especially, a mental Something - can be a source of meaning.

§143

§143 begins the section on learning and understanding. For a great deal of what is to come, Witty's overarching question is something like: how does, or how can, someone ‘continue on’ after being shown only a few limited examples? A useful distinction to keep in mind is that between what Witty considers learning proper, and what we might call rote learning. Learning by rote simply is a kind of mere learning by imitation, as it were, following the letter of what has been shown; for Witty, genuine learning must go beyond this: to have properly learnt something is to be able to go on in new ways, to encounter something never yet encountered yet still know what to do with it.

Unsurprisingly then, §143 sets the scene for the questions to come: having written down a series from 0 to 9, how does a pupil go on? How do we, or ought we, characterise deviancies and conformities? In fact, Witty takes a step back and considers the mere effort of copying - what happens when a pupil can’t even do that? Witty here speaks of the point at which ‘communication stops’, and at which ‘our pupil’s ability to learn comes to an end’. My feeling here is that Witty is here drawing on his notion of ‘explanations com[ing] to an end somewhere’ (§1), such that were even the mere act of copying too hard to follow, more than teaching is at stake here (I’m tempted to say: the student does not share our form-of-life - maybe our student isn’t even quite human? A lion?).

The line about there not being a clear-cut diction between a random and systemic mistake had me puzzled, but this reading from Oskari Kuusela helped: “The distinction between not following a rule (making frequent random mistakes as opposed to merely occasional mistakes) and following a variant rule (making a systematic mistakes) is not sharp. Thus, while we may readily say of a pupil who makes constant random mistakes that she is not following a rule, the verdict is less straightforward in the case of a systemic mistake”.

The line about there not being a clear-cut diction between a random and systemic mistake had me puzzled, but this reading from Oskari Kuusela helped: “The distinction between not following a rule (making frequent random mistakes as opposed to merely occasional mistakes) and following a variant rule (making a systematic mistakes) is not sharp. Thus, while we may readily say of a pupil who makes constant random mistakes that she is not following a rule, the verdict is less straightforward in the case of a systemic mistake”. — StreetlightX

The random/systematic distinction here I think more neatly maps onto mistakes being patternless and mistakes having a pattern.

My little foster brother often counts like {75,76,77,78,79,90} or {75,76,77,80}, in the first case he's incrementing by 10 extra, he does this more often when the number preceding the multiple of 10 and the multiple of 10 share a digit name, like 'seventy-nine' 'ninety'. The second mistake is similar, 78 shares a digit-name with 80. He also sometimes just skips to the next multiple of 10 irrelevant of context, and sometimes skips numbers towards the upper end of a multiple of 10 (like, he struggles with 76,77,78,79 themselves sometimes).

So, for any given case, you can't tell whether he's 'just skipped it' or he's skipped it due to one of the systematic patterns. This isn't just an epistemological limitation, there's nothing in his use of words (including gestures, and intonation...) that determines one or the other. Sure, you can always guess, but there's nothing like 'evidence' of which mistake he's made, sometimes at least.

He's also too young to have developed insight, or an 'inner watchman' for his thought processes, so you can't ask him why he's struggling and, even if you could, he cannot notice these patterns himself, so he can provide no answers.

So, for any given case, you can't tell whether he's 'just skipped it' or he's skipped it due to one of the systematic patterns. — fdrake

Ah, this is super useful. I was thinking - and Wittgenstein's phrasing encourages this - of two different mistakes, one which might be called systemic and one random; but thinking of the 'same' mistakes as being indiscernible between the one and the other helps make sense of the passage alot.

The line about there not being a clear-cut diction between a random and systemic mistake had me puzzled, but this reading from Oskari Kuusela helped: “The distinction between not following a rule (making frequent random mistakes as opposed to merely occasional mistakes) and following a variant rule (making a systematic mistakes) is not sharp. Thus, while we may readily say of a pupil who makes constant random mistakes that she is not following a rule, the verdict is less straightforward in the case of a systemic mistake”. — StreetlightX

I think what he is driving at is the ability to learn. Notice that a systematic mistake demonstrates the possibility of being corrected, and the random mistake does not. If the person were making truly random mistakes then that individual would be showing no effort, no attempt to learn. Learning takes effort, so effort is evidence of the capacity to learn, and systematic mistakes demonstrate effort while random mistakes would demonstrate no effort.

He will later (163) question whether there even is such a thing as a random mistake (read my post above), because to make a random order requires following a rule. This relates back to 98, even the simplest, vaguest order is a perfect order when there is no ideal by which to judge order. So there is a sort of paradox involved, because the person who refuses to learn, makes no effort to learn the norm and insists on doing things in a random way, is already following some sort of rule, to do it in a random way, or in a way other than the norm. The norm replaces the ideal here as the principle for judging order, like in nominalism. "Now, however, let us suppose that after some efforts on the teacher's part he continues the series correctly, that is, as we do it." -145

and the random mistake does not — Metaphysician Undercover

Nonsense. You just intervene differently in the case of a clear systematic mistake and a random one.

You can tell someone why, or guess how, they made the mistake if there is a systematic error. You do this by exploiting whatever contextual and behavioural cues you can.

If there isn't a systematic error, you can still correct the mistake by telling them what the answer is, or what they ought to do.

I'm not sure whether this is helpful (mainly because I disagree with it), but Baker and Hacker offer the following reading:

...understanding, misunderstanding and not understanding are distinguished by the difference between reacting correctly to training, making systematic mistakes, and making random mistakes.

Nonsense. You just intervene differently in the case of a clear systematic mistake and a random one.

You can tell someone why, or guess how, they made the mistake if there is a systematic error. You do this by exploiting whatever contextual and behavioural cues you can. — fdrake

I was just trying to explain Wittgenstein's point:

Perhaps it is possible to wean him from the systematic mistake (as
from a bad habit). Or perhaps one accepts his way of copying and
tries to teach him ours as an oflfshoot, a variant of his.—And here too
our pupil's capacity to learn may come to an end. — 143

The point being that we are inclined to think that the person making the systematic mistake can be corrected, as Wittgenstein goes on to explain at 144, we correct him by changing "his way of looking at things".

If there isn't a systematic error, you can still correct the mistake by telling them what the answer is, or what they ought to do. — fdrake

If he is making so-called random mistakes we cannot correct him by telling him the answer, because we need to instill the proper method in him. "Random mistakes" implies that he is using no method whatsoever, he has no "way of looking at things", his actions are random. So simply telling him one answer, 4 comes after 3 for example, will not give him the method. You might continue, and tell him 5 comes after 4, and 6 after 5, but that's what he was already given in the first place, and he was incapable of comprehending the method, and this is evident because he reproduces the numbers randomly. We want the pupil to learn the method, so he can carry on, and if he is just making random mistakes when asked to repeat what he has learned, then he has demonstrated no capacity for understanding any method. If the mistakes demonstrate something systematic, like in your example, then we figure the person is using some sort of method and can be corrected.

144. Wittgenstein reflects on the purpose of his preceding section 143, and particularly its final line: "the pupil's ability may come to an end". He notes that he did not report this from his own experience, so what was he doing with that remark? He states that he wants the reader (you) to imagine its possibility, and to be inclined to compare and accept this "picture". Ultimately, he says, he wants the reader to "regard a given case differently"; that is, he wants to change the reader's "way of looking at things".

Baker and Hacker provide the following explanation for the type of alternative picture Wittgenstein may be referring to here:

W. is drawing our attention to a logical possibility, reminding us of a particular contingency, in order to reorient our way of looking at things. At what things? At the phenomena associated with understanding and meaning. He aims, in particular, at getting us to conceive of understanding quite differently from the way we are tempted to construe the concept: namely, as akin to an ability rather than as a mental state or process. If we compare understanding with abilities and think of manifestations of understanding as exercises of abilities rather than as causal consequences of inner states, we shall look quite differently at the phenomenon of sudden understanding, and also cease to conceive of understanding as a reservoir from which applications of understanding flow.

... that we do have certain elementary abilities (to imitate, react in standard ways, recognize shapes and colours, continue activities in a common pattern, etc.) is a general brute fact of human nature (cf. PI p. 56/48n.), which is crucial for our having the kind of language we have.

Ultimately, he says, he wants the reader to "regard a given case differently"; that is, he wants to change the reader's "way of looking at things". — Luke

I don't agree with that at all. He's talking in the third person, referring to "him", "he is capable of imagining that?", "put the picture before him", "his way of looking at things".

He is clearly talking about the pupil who's capacity to learn has come to an end, not the reader. He is saying that the person whose capacity to learn has come to an end is incapable of recognizing that his own capacity to learn has come to an end. "But was I trying to draw someone's attention to the fact that he is capable of imagining that?" In this sentence, "he" refers to the pupil of 143, not to the person whose attention Wittgenstein might be trying to draw. He already said he would like "you" to imagine that, in the prior sentence, now he is asking whether the pupil ("he") is capable of imagining that, what he wants you to imagine.

Consider that he has distinguished between teaching him the correct way (what is accepted by us, "our way"), and teaching him the correct way as an offshoot of his own way. "His way" remains the principal way for him, and his way is not the correct way. However, we manipulate him, and "his way", until he produces an acceptable facsimile of our way. This is a sort of pretense. He pretends to be doing it our way, when really he's doing it his way and making a representation of our way. His capacity to learn "our way" comes to an end here because the teacher has no way to distinguish the pretense from the real, to bring him around to doing it the correct way, and the pupil has no way of imaging the other way, and the fact that his way is not the correct way.

In the section where Wittgenstein describes reading, he will elaborate on such pretense, pretending to read.

We have two distinct sets of circumstances in which the pupil's capacity to learn the formation rule comes to an end. One is when he is making random mistakes, and the other is when he learns what appears to be the correct way, as an offshoot or variant of his own way (pretends). In the latter, he is incapable of imagining the correct way because he will not release his own way of imagining things. We cannot change his way of looking at things. because he has found a way to incorporate, or subsume, our way of looking at things within his own way.

He is clearly talking about the pupil who's capacity to learn has come to an end, not the reader. — Metaphysician Undercover

I know you're beyond help (or a troll), but in the interests of futility:

At 144, he asks: "What do I mean when I say “the pupil’s ability to learn may come to an end here”? ...what am I doing with that remark?" He then goes on to explain the purpose of that remark.

Why would Wittgenstein explain his remarks in the text to a fictional person in the text? Do you think that the pupil at 143 is reading the Philosophical Investigations? Why would he refer to the pupil as "someone"? These questions are rhetorical.

PI 144. ... (Indian mathematicians: “Look at this!”)

In Zettel Wittgenstein says:

461. ... (I once read somewhere that a geometrical figure, with the words "Look at this", serves as a proof for certain Indian mathematicians. This looking too effects an alteration in one's way of seeing.)

When W. says: “the pupil’s ability to learn may come to an end here”, I think he intends for us to look at it in two ways. First, the student can go no further because the way he is looking at it is not the way that one must look at it in order to conform to the practice. Second, we can go no further in teaching the student unless it occurs to us that he may be looking at it differently, and so, we must change the way we are looking at the student's inability to learn.

Of course the larger issue here is not to teach a series of numbers. Philosophical problems may arise not because we are not thinking rigorously enough but because we are not looking at the problem is a perspicuous way.

Working in philosophy … is [working] .. On one’s own way of seeing things. (CV 16)

At 144, he asks: "What do I mean when I say “the pupil’s ability to learn may come to an end here”? ...what am I doing with that remark?" — Luke

Right, he asks that question and answers it with this:

:

Well, I should like you to say: "Yes, it's true, you can imagine that too, that might happen too!"—But was I trying to draw someone's attention to the fact that he is capable of imagining that?

Notice, he begins by speaking to "you", "you can imagine that too". Then he proceeds in this way. "But was I trying to draw someone's [a reader's] attention to the fact that he [the fictional pupil] is capable of imagining that?"[No, he's not.] And then he proceeds to talk about that fictional pupil. "I wanted to put that picture before him, and his acceptance of the picture consists in his now being inclined to regard a given case differently: that is, to compare it with this rather than that set of pictures. I have changed his way of looking at things."

You, the reader are capable of imagining what Wittgenstein asks, the fictional pupil is not. Wittgenstein says "I wanted to put that picture before him...", and "his acceptance of the picture" would incline him to look at the case differently, thus Wittgenstein would "have changed his way of looking at things. However, Wittgenstein could not change the fictional pupil's way of looking at things, the best that could be done according to what is stipulated by the example, is "to teach him ours as an offshoot, a variant of his" way of doing things. So the pupil's ability to learn may come to an end here. His way of looking at things has not been changed, and it may well be that it cannot be changed.