At 143 Wittgenstein writes that in copying the series of natural numbers, there is a normal and an abnormal learner's reaction. We can assume, given that most people are able to count to ten, that the normal reaction is one of eventually writing down the series correctly. Wittgenstein notes that the possibility of communication will depend on the student "going on to write it down by himself". He then says (my emphasis):

And here we may imagine, for example, that he does copy the figures by himself, but not in the right order [...]

Or again, he makes ‘mistakes’ in the order. [...]

Or he makes a systematic mistake; — PI 143

This is not intended to be some sort of theory of developmental learning.

At 144, Wittgenstein asks what he means by his statement that "the pupil's ability to learn may come to an end here". He notes that he does not report this from his own experience "(Even if I had such experience)", and he asks us what is he doing with this remark?

After all, I’d like you to say: “Yes, it’s true, one could imagine that too, that might happen too!” — But was I trying to draw someone’s attention to the fact that he is able to imagine that? — PI 144

He wants us ("you"; the readers) to say “Yes, it’s true, one could imagine that too". To imagine what? To imagine that the pupil's ability to learn may come to an end here. The second line of the above quote relates to the first, as he then asks whether it was his objective to draw our attention to the fact that we could imagine that. His proceeding comments indicate that his objective was not merely to have us imagine it. On my reading, his objective was to put an alternative picture (or "sequence of pictures") before us to accept, where our acceptance "consists in being inclined to regard a given case differently". The objective of his statement that "the pupil's ability to learn may come to an end" is to change our way of looking at things.

However, if your account is correct, then this raises several questions. For example, where Wittgenstein asks in the second line of the above quote:

But was I trying to draw someone’s attention to the fact that he is able to imagine that?

You claim that this question translates into:

But was I trying to draw the pupil's attention to the fact that the pupil is able to imagine that the pupil's ability to learn may come to an end here?

Isn't that a very odd (or oddly phrased) question? Why would Wittgenstein ask it?

Furthermore, what alternative picture does Wittgenstein put before the student (other than the "picture" of the series of numbers which are written down and placed before him)? What "way of looking at things" is required in order for the student to copy the numbers on the page in front of him?

This is not intended to be some sort of theory of developmental learning. — Luke

Oh, then what is it? He distinguishes random from systematic mistakes in a theoretical way, and says that there is no sharp distinction between the two. How is this anything other than theory?

The second line of the above quote relates to the first, as he then asks whether it was his objective to draw our attention to the fact that we could imagine that. — Luke

Sorry, to have to reiterate, but he doesn't say "we", he says "he", referring to the theoretical student.

Isn't that a very odd (or oddly phrased) question? Why would Wittgenstein ask it? — Luke

I think it's poorly phrased, yes, that whole little section 144 is, that's why it's so hard to understand, but if understood properly it's not odd at all. He's explaining why we come to the end of the person's capacity to learn, if we have to correct his systematic mistakes by teaching him our methods of procedure as an offshoot or variant of his own.

There's two options given for correcting a systematic mistake. One is to wean him off a bad habit, the other is to teach him our way as an offshoot or variant of his way. If we correct his systematic mistakes in the way of correcting a bad habit, then we change his way of looking at things, he sees his old way as a bad habit which must be broken. This demonstrates that he is open, and accepting of having his way of looking at things changed. He is therefore capable of further learning.

If we can only correct his mistake by having him learn our method as an offshoot of his own, we do not change his way of looking at things. He does not see his way as a bad habit. This is because he is not open and accepting to having his way changed, and so his capacity to learn from us is limited by this. If he proceeds in doing it our way, it is what I called above, a pretense (and Wittgenstein will get into pretending later). It's as if he is saying I'll do it your way, just to please you, and get past this step, but I do not agree with you, and you will never get me to see things your way.

There is a dichotomy between we and he at 144. We have a normal way of looking at things, and do not make systematic mistakes in counting numbers. He, the theoretical student has an abnormal way of looking at things, and needs his systematic mistakes corrected. Correcting a person's propensity for a systematic mistake requires changing one's way of looking at things.

Furthermore, what alternative picture does Wittgenstein put before the student (other than the "picture" of the series of numbers which are written down and placed before him)? What "way of looking at things" is required in order for the student to copy the numbers on the page in front of him? — Luke

Having the appropriate "picture" associated with the appropriate words is how Wittgenstein has been describing "understanding". So, for the person to properly understand the "formation rule" involved in counting the numbers, it is required that the person has the appropriate "picture". If the person is making systematic errors, it is necessary to change that individual's way of looking at things.

Sorry, to have to reiterate, but he doesn't say "we", he says "he", referring to the theoretical student — Metaphysician Undercover

You're reading too much into "he". Wittgenstein often uses the third-person male pronoun ('he', 'him') as a general reference to any person, which was common practice at the time. For example:

69. How would we explain to someone what a game is? I think that we’d describe games to him

31. ...We may say: it only makes sense for someone to ask what something is called if he already knows how to make use of the name.

32. ...Augustine describes the learning of human language as if the child came into a foreign country and did not understand the language of the country; that is, as if he already had a language, only not this one.

The "someone" in this case is a reader of the text.

What "way of looking at things" is required in order for the student to copy the numbers on the page in front of him? — Luke

143. ... he copies the series 0, 1, 2, 3, 4, 5,... like this: 1, 0, 3, 2, 5, 4

Let's turn this around. Has he copied a series of numbers? What is that series? How does one have to look at it in order to continue?

Skip the first number, 0, and write down the next number 1 followed by the skipped number 0, then skip the next number 2 and write down the next number 3 followed by the skipped number 2, then skip the next number 4 and write down the next number 5 followed by the skipped number 4. The series continues 7, 6, 9,8 (unless I made a mistake).

Skip the first number, 0, and write down the next number 1 followed by the skipped number 0, then skip the next number 2 and write down the next number 3 followed by the skipped number 2, then skip the next number 4 and write down the next number 5 followed by the skipped number 4. The series continues 7, 6, 9,8 (unless I made a mistake). — Fooloso4

What he was actually doing was pairing the numbers into twos iteratively then inverting the elements.

{0,1,2,3,4,5}
becomes
{ (0,1), (2,3), (4,5) }
becomes
{ (1,0), (3,2), (5,4) }
becomes
{1,0,3,2,5,4}


A different rule that expresses the same series.

Yes, I think it is easier to see if looked at that way.

Yes, I think it is easier to see if looked at that way. — Fooloso4

I think I might disagree with you though. There isn't always a unique justification for someone using language in any given way, there can be plurality of understandings consistent with it.

Your way is probably how a child might see the error, my way was probably how W. constructed the series.

How does one have to look at it in order to continue? — Fooloso4

Maybe he can't continue.

I think I might disagree with you though. There isn't always a unique justification for someone using language in any given way, there can be plurality of understandings consistent with it. — fdrake

I don't know what you are disagreeing with. I though of one way the student might be looking at it. You suggested another. I think the example you gave in better and would be the first I would employ if I had to teach this series (unless someone suggests another that is easier to see); but of course, both are dependent on first understanding the series that the student appears not to have understood.

I agree that there can be plurality of understandings consistent with it. I take this to be one of the main points Wittgenstein is making here. It is consistent with a quote I cited some pages back:

What a Copernicus or a Darwin really achieved was not the discovery of a true theory, but of a fertile new point of view. (CV 18)

I agree that there can be plurality of understandings consistent with it. I take this to be one of the main points Wittgenstein is making here. It is consistent with a quote I cited some pages back: — Fooloso4

Ah, sorry, seems we agree on the interpretation of W. then. :)

Maybe he can't continue. — Luke

Perhaps with this student it is, but I don't think Wittgenstein intends for this to be the end of the matter. I take the larger point to be that by changing the way we look at a problem the problem can be resolved.

Perhaps with this student it is, but I don't think Wittgenstein intends for this to be the end of the matter. I take the larger point to be that by changing the way we look at a problem the problem can be resolved. — Fooloso4

I see little textual support for this.

That it is not the end of the matter? If so, what would be the point of his even bringing it up?

That by changing the way we look at a problem it can be resolved?:

122. A main source of our failure to understand is that we don’t have an overview of the use of our words. a Our grammar is deficient in surveyability. A surveyable representation produces precisely that kind of understanding which consists in ‘seeing connections’. Hence the importance of finding and inventing intermediate links.

The concept of a surveyable representation is of fundamental significance for us. It characterizes the way we represent things, how we look at matters. (Is this a ‘Weltanschauung’?)

[emphasis added]

308. How does the philosophical problem about mental processes and states and about behaviourism arise? —– The first step is the one that altogether escapes notice. We talk of processes and states, and leave their nature undecided. Sometime perhaps we’ll know more about them - we think. But that’s just what commits us to a particular way of looking at the matter. For we have a certain conception of what it means to learn to know a process better. (The decisive movement in the conjuring trick has been made, and it was the very one that seemed to us quite innocent.) a And now the analogy which was to make us understand our thoughts falls to pieces. So we have to deny the yet uncomprehended process in the yet unexplored medium. And now it looks as if we had denied mental processes. And naturally we don’t want to deny them. [emphasis added]

309. What is your aim in philosophy? To show the fly the way out of the fly-bottle.

That by changing the way we look at a problem it can be resolved? — Fooloso4

I was commenting on PI 144 when you quoted and responded to me. You appear to be referring to something else.

Sorry for the misunderstanding.

Although I quoted you I was not offering a direct answer your question:

What "way of looking at things" is required in order for the student to copy the numbers on the page in front of him? — Luke

Rather I was trying to consider how the student might have looked at the series of numbers he wrote on the assumption that if we can understand how he looked at it we might be able to provide another way for him to look at it that would conform to the normal series. To that end I pointed out that the series he wrote is probably not random, that there is a logic to it. Perhaps this was obvious to others. When it comes to my mathematical abilities Wittgenstein would probably have concluded: our pupil's ability to learn has come to an end.

Rather I was trying to consider how the student might have looked at the series of numbers he wrote on the assumption that if we can understand how he looked at it we might be able to provide another way for him to look at it — Fooloso4

What I dispute (consistent with Baker and Hacker's exegesis) is that Wittgenstein is referring to the pupil's way of looking at things at all. Instead he is referring to our (the reader's) way of looking at things. I see little textual support for your claim as it relates to PI 144.

You're reading too much into "he". — Luke

It's not a matter of what I'm reading into "he", it's a matter of determining the proper referent of "he". If "he" refers to the pupil in Wittgenstein's example, who requires having our method taught as an offshoot, or variant of his method, and may therefore have his capacity to learn come to an end here, instead of what you claim, that "he" refers to the reader of the text, this is a substantial interpretational difference.

Furthermore, what alternative picture does Wittgenstein put before the student (other than the "picture" of the series of numbers which are written down and placed before him)? What "way of looking at things" is required in order for the student to copy the numbers on the page in front of him? — Luke

Remember, the "picture" here is in the mind, a mental picture. that is how Wittgenstein has described understanding words, as associating them with mental pictures. Teaching the pupil would constitute changing the pupil's mental picture. If the pupil's capacity to learn has come to an end, then our capacity to change his mental picture, (his way of looking at things), has come to an end. It really doesn't make sense to assume that Wittgenstein is talking about changing the reader's way of looking at things.

The problem here is that Wittgenstein has given us an example which cannot be related to anything real, therefore we cannot make sense of the example. The fact is that we learn to count through verbal training (repeat after me), not through the use of written symbols. So he has given us problems which are completely unrealistic, and cannot be comprehended, because they could not occur in they reality of education. In reality, the student learns the order of numbers through hearing them, so the mental image is an aural image, and not a "picture" at all. The order is a temporal order, (two comes after one) and is acquired by the learner through a process of memorizing. We do not learn orders by observing right to left, or left to right on a paper, (though we might learn them with a progression of flash cards, but speaking is more efficient and natural). We learn orders through memorizing a temporal progression of sounds, not a temporal progression of visual images. So Wittgenstein's example, of learning an order through written symbols does not make any sense to us.

it's a matter of determining the proper referent of "he" — Metaphysician Undercover

I've wasted too much time on this and given too much credence to your preposterous reading already. I was just optimistic that you might for once be able to see something so obvious.

After all, I’d like you to say: “Yes, it’s true, one could imagine that too, that might happen too!” — But was I trying to draw someone’s attention to the fact that he is able to imagine that?

How could Wittgenstein be saying in the first line he'd like the reader to "imagine that", but then in the second line be asking about the pupil's ability to "imagine that"? It's ridiculous to think that he switches from reader to pupil in between these two lines. He clearly wants us to say that we could imagine that, and then he asks whether he was trying to draw attention to our ability to imagine that.

I think you are right. I had not read the paragraph carefully enough and had skipped your exchange with MU for reasons I won't go into.

I think the way of looking at things refers to the problem of understanding. At 143 he asks:

How does he come to understand this system?

Again at 146 he asks:

Has he understood the system if he continues the series to the hundredth place?”

Looking back, I see that this is consistent with what your quote from Baker and Hacker said, and what prompted your question about the student's way of looking at things. With all the noise the signal gets lost.

I think you are right. I had not read the paragraph carefully enough — Fooloso4

I suspected that might be the case. No worries and thanks.

I'll be away with reduced access to internet over the next week. Will hopefully catch-up when I'm back. Keep up the good stuff!

OK, so let's assume it's as you say, it's only the reader being referred to with "he" here at 144. The reader says "I can imagine that too", and this means that the reader imagines this as well as something else, and so, the reader's way of looking at things has been changed. What changed the reader's way of looking at things is understanding the phrase "And here too our pupil's capacity to learn may come to an end."

What is required then, is that we, as the readers, imagine that the student's capacity to learn has come to an end. Has anyone here, other than me offered an explanation of (i.e. an imaginary scenario) within which the pupil's capacity to learn has come to an end? I have produced this imaginary scenario, which Wittgenstein asks for, this other way of looking at things, in which the student's capacity to learn has come to an end. Regardless of what "he" refers to, I have explained how the student's capacity to learn has come to an end. Do you understand this? Do you have an imaginary scenario within which the student's capacity to learn has come to an end?

145: The imaginary pupil is supposed to have learned to write the series 0 to 9 in the way which we call correct, consistently, numerous times. Wittgenstein now proposes that he teach the pupil the recurrence of this series in "the tens". At some point, we can decide that the student has the capacity to continue the series independently. We can say that he has "mastered the system". But how far must he be able to produce the series, before we can draw such a conclusion? "Clearly you cannot state a limit here".

As an aside, which may or may not be relevant, it may be useful to note that there is a significant difference between learning to count the numbers verbally and learning to write the series of numerals. In writing, we learn the digits, 0 to 9, and all the following numbers can be represented infinitely, from repeating these digits in distinct patterns. In counting verbally, we need to learn new names as we go "twelve", "thirteen", "twenty", thirty", "hundred", thousand", "million", etc.. So, whereas the writer of the symbols may obtain the capacity to continue the series indefinitely, independently, the speaker of the numbers cannot continue independently because one must always learn new words continuously, as one gets to the higher and higher numbers. There is no formula for creating the words for the numbers, which would allow one to continue indefinitely, as there is for creating the written symbols, because we must learn distinct conventions as we go.

There may be a hidden reason why Wittgenstein's example consists of learning written numerals rather than consisting of learning the numbers we count verbally. He may be trying to reveal this problem to us. In real educational situations, we learn to count verbally first, because aural training is much more efficient. When we proceed in our education, to the point of writing the numerals, as in Wittgenstein's example, we already have an understanding of how to count. So we do not really proceed simply from this process of learning how to write the symbols, to mastering the system, because there is another important ingredient which is learning how to count, which we learn through the aural process. There is an important synthesis, as we pass from learning spoken orders to learning written symbols, which is somewhat neglected here.

This may become more evident later when he discusses "reading" . Reading is completely sound oriented. The symbol represents a sound, whether imaginary or aural, and orderings are learned through verbal demonstrations. We are able to discern that M comes after L in the alphabet, by saying this part of the alphabet within our minds.

At this point, we need to recognize and respect the difference between learning an order (memorizing it), and applying a principle to create an order (application). One can learn, memorize, the numerals from 0 to 9, and reproduce them, over and over again correctly, without having grasped the principle of application. Knowing the principle of application is what enables the person to proceed in recreating the same order in higher numbers. Wittgenstein is now proceeding to emphasize this distinction, the difference between learning something through memory, therefore being able to repeat it from memory, and knowing how to apply a principle, whereby the person might take what has been learned from memory and proceed through application of a principle, to use what has been memorized, in completely new situations. Only when the person is capable of proceeding in application, has the person "understood the system". Now he will proceed to investigate, what signifies to us, that the person has understood the system.