Understanding Part 1: Pictures §138-142

1. The mythology of understanding as something “inner”

As I discussed in my previous post, a rapid series of modulations brings us to the topic of understanding when, at the end of §138, the interlocutor says:

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!

This is probably the strongest objection so far to Wittgenstein's account of meaning. Its power stems from the way it ties into various extremely tempting ideas about the process of communication. Above all, it suggests that understanding is some kind of thing that we acquire when we learn the meaning of a word, and which is subsequently represented to us whenever we hear that word.

The appeal of this idea is bolstered by various simple reflections. When we come to know the meaning of a word we gain understanding – so we must've gained something! Likewise, if we don't know the meaning of a word then we lack understanding – so “understanding” must be whatever it is we lack. And although we might exhibit understanding in performance (eg, by using a word correctly), that is not understanding itself; it is merely evidence from which others can infer that we do indeed possess understanding.

Moreover, when we hear words we are immediately aware of our understanding. The words strike us in a quite vivid and particular way. We don't simply hear sounds or see ink-marks on a page (or shapes on a computer screen) – we hear (or read) language. We experience the meanings that it conveys. To see that this is so, just compare the case of reciting a passage we don't understand (having learnt it parrot-fashion, perhaps) with that of reciting one whose words are familiar to us. Surely it is undeniable that what happens inside us in the two cases is completely different?

Such considerations present a challenge for Wittgenstein. When we understand a word we understand its meaning. But what we grasp in the instant of understanding doesn't seem to be anything like a use. (And, in any case, is use something that could be grasped in an instant?) But if what we grasp isn't use then meaning can't be use.

2. Pictures and their application

Wittgenstein begins his assault on this appealing conception by considering the notion that understanding is a picture that comes before our mind when we hear a word (§139). It's a natural enough place to start given that his early philosophy propounded what's called “the picture theory of language”. Moreover, that theory was itself a refinement of a venerable philosophical position dating back at least as far as Locke's Essay Concerning Human Understanding. In my last post I gave a rough sketch of the account of communication that the picture theory tends to suggest. It might be helpful to repeat that here:

Person A has the thought, in the form of a picture, that things are thus-and-so. She wants to communicate her thought to B so she “converts” it into a proposition: she speaks, writes, uses sign-language or Morse code, etc. The words she uses relate to the elements of her picture and are arranged in a similar fashion. Next, B perceives A's proposition and he re-converts it into a thought. That is to say, the words he hears (reads, etc) produce a picture in his mind. That picture is his understanding of what A said. If he has the same picture he has understood correctly; if he has a different picture then he has misunderstood (and if he has no picture at all then he has not understood at all). The correctness of his picture can be inferred from his behaviour. Of course, we can't be sure B's picture matches A's but so long as he fetches the right object (etc) then it is reasonable to assume that it does.

In attacking the idea of understanding as a picture, it is this last part of the account that Wittgenstein focuses upon: the link between the picture and subsequent behaviour. He points out (§139) that, whatever might happen in our minds, our behaviour is still a criterion for understanding. If someone consistently uses a word incorrectly then we say that he or she doesn't understand its meaning. So understanding is clearly bound up with use in some sense. But if understanding is a mental picture then what is the link between that picture and use? “Can what we grasp at a stroke agree with a use, fit or fail to fit it?” (§139.) Wittgenstein suggests an answer to his own question: let's assume that when I hear the word “cube” I get a picture of a cube in my mind. If I point to a cube my use fits the picture, but if I point to a triangular prism instead then my use has failed to fit the picture.

At first blush this seems straightforward, but Wittgenstein argues that it will not do. The problem is: who is to say that pointing to the triangular prism actually is an incorrect use of the picture?

His objection is couched in terms of the picture's “method of projection”. I assume that's a glance back to Wittgenstein's time as an engineering student and refers to techniques used in technical drawing. I'm afraid I have roughly zero understanding of such things (I looked up geometric projection on Wikipedia and it didn't help) but there are other ways of making the same point. Basically it comes to this: a picture by itself stands in need of a method of application. If no method is stipulated then it's impossible to say whether a picture has been used rightly or wrongly.

So, returning to the example of “cube”, the word presents me with a picture:

I then look about me and spot a triangular prism. I realise that in both cases they have equal ends and parallel rectilinear figures and that their sides are parallelograms. I therefore point to the triangular prism because, in that sense, it is the same as my picture. In other words, I have used my picture as an example of a prism (for a cube is a prism too). That might seem an unusual way of applying the picture, but who is to say that it's wrong?

To see how far use can deviate from expectation, consider the following example: I present someone with a dog and an iPhone. I then give him a photo of the dog and ask him to choose which of the two items it most resembles. He chooses the iPhone because, like the photo, you can hold it in your hand, put it in your pocket, etc. Of course most of us would automatically choose the dog, but that is only because we are already familiar with the activity of picking things out from photos. It is something we’ve done countless times and so it probably wouldn’t even occur to us to use the photo in a different way. But there is nothing that says choosing the dog must be the correct response in all circumstances. Indeed to someone with a different upbringing from ours the dog might seem an absurd choice.

The picture, don't forget, is not being cast as an aid to understanding; it is supposed to be the thing itself. But it's hard to see how it can play that role when it provides no standard of correctness. If this observation seems familiar, that's because it is closely analogous to the point made about ostensive definition in relation to meaning. There, it was supposed that the sample by itself could establish a link between word and object, that it was completely unambiguous and therefore unmistakable. But it turned out that it only functioned as part of an established practice of describing the rule for the use of a word. And it's a very similar story in the case of the picture (which is, after all, a kind of mental sample). We have the sample, but what we lack is the application. (I should also mention that as well as looking back to ostensive definition this point also anticipates aspects of the discussion of rule-following. See, for example, §201: “if every course of action can be brought into accord with the rule, then it can also be brought into conflict with it. And so there would be neither accord nor conflict here”. This overlap should not be surprising; the concept of understanding is internally connected to the concept of meaning. Each helps define the other. And they are both closely bound up with the concept of rule-following.)

3. Compulsion

The reminder that a picture can be used in various different ways also serves to highlight (and undermine) another temptation regarding understanding as an “inner” phenomenon: the assumption that the relation between representation and what is represented somehow takes care of itself. We don't need to stipulate a method of application because the picture does it for us. It's especially easy to think like this when we focus on the immediacy and fluency with which we usually understand language. We don't have to struggle to grasp the meaning – in fact, it's impossible for us not to understand. It's as if the picture somehow carries its meaning within it, like a kind of spirit, and exposure to the picture transfers this spirit to us, so that we cannot help but see the picture as an image of this object. The words (or rather the images they produce) force an application on us (§140).

In the Tractatus this seems to be regarded as a kind of logical compulsion. So long as picture and fact share the same logical structure there cannot be doubt about what's represented. Of course, this idea is exploded by the simple observations in §139 (it is, in fact, another example of a rule misrepresented as a necessary feature of the world). But there are other ways of presenting the idea of compulsion. As Wittgenstein says in §140, “we might also be inclined to express ourselves like this: we're at most under a psychological, not a logical, compulsion”. He then adds a typically cryptic coda: “And now, indeed, it looks as if we knew of two kinds of case.” What's he getting at here?

I think this is a warning against defending a preconception by taking refuge in theoretical explanations. We assumed we were under compulsion; the idea that it was a logical one has proved empty, so now we say it must be “some other kind”. And, happily, we hit upon the notion of “psychological compulsion” as a plausible-sounding alternative. (Already we can see the bewitching idea of a mental mechanism looming on the horizon.) But notice how vague all this is! How much do we actually know about psychological compulsion? Is it clear that this is an example of it? And how do we propose to find out? Will we be conducting field experiments or can we decide things from our armchairs? Isn't the idea of psychological compulsion just a guess – and a guess that invokes a mysterious realm of mental structures, subconscious computation, and so forth?

There's a second, perhaps more fundamental, issue. We've been attempting to clarify the concept of understanding, but now it looks as if we're sliding into a quasi-causal explanation. But that's like trying to find out what a watch is by examining the structure of its cogs and springs. Such an investigation may be useful in various ways, but it won't tell you anything about the role watches play in people's lives. For that you need to describe their use, not their internal mechanisms. The notion of compulsion tempts us away from a conceptual investigation towards an ersatz scientific one which is doomed to failure from the outset.

Well, Wittgenstein may be alluding to such thoughts in §140, but he doesn't elaborate on them at this point. Instead, he sidesteps the quagmire of compulsion and brings things down to earth: “our 'belief that the picture forced a particular application upon us' consisted in the fact that only the one case and no other occurred to us.” This, of course, is a move from the theoretical back to the descriptive. Why no other case occurred to us is not Wittgenstein's concern. But the fact that they didn't made it seem like the picture could only be applied in one way even though (as we now see) that's not true at all.

It is this observation that subsequently allows him to answer the question in §139: how can what's grasped in an instant fit or fail to fit a use? “[T]hey can clash in so far as the picture makes us expect a different use; because people in general apply this picture like this” (§141). What seemed to be a mysterious (perhaps impossible) relation between two entirely different phenomena turns out to be remarkably simple and ordinary.

4. Building in the application

Maybe, however, compulsion isn't the only answer. Perhaps we could build the application into the picture itself, thereby allowing it to perform its allotted function. Wittgenstein considers this in §141. “How,” he asks, “am I to imagine this?” The answer he suggests is a picture of two cubes with lines of projection between them. Something a bit like this perhaps:

Actually, I don't think that quite works. The cube on the left presumably represents the mental image and the one on the right represents actual cubes in the world (and the lines attempt to show the relation between the two). But we no longer have the mental image that the left-cube pictures! So we're going to need two separate images: first, the picture of the cube by itself and, second, the picture of the two cubes showing how the cube in the first picture is to be applied. But this raises a further problem: how do we know the relationship between these two images? How do we know that the left-cube in the second image represents the cube in the first image? Aren't we going to need a third image depicting the relation between the two we already have? And won't that in turn require a fourth and fifth image linking the third image to the first and second ones? And so on.

In a roundabout way this comes to the same objection that Wittgenstein makes about his own proposed image: whatever its content it will still just be another picture, and will therefore stand in need of an application. It's not simply that the initial picture did not provide its application (as if that were a kind of oversight); it could not do so. Application cannot be provided in that way.

Is Wittgenstein claiming, then, that there's no such thing as an application coming before one's mind? Of course not, but (he reminds us in §141) understanding what that amounts to involves looking at the use of the phrase “the application came before his mind”. It does not involve a doomed attempt to posit a hypothetical picture with miraculous powers. Instead of analysing the phrase himself, however, Wittgenstein leaves it up to us investigate. I think it's worth having a go because the conclusions it suggests (to me, at any rate) tie in with many of the points he will shortly be making about other accounts of understanding. Here's what I came up with:

a. Suppose I've been teaching someone to use mathematical formulae. Now I show her the series 0, 1, 1, 2, 3, 5, 8 and ask her to continue it. She ponders for a while and then cries “I've got it!” I ask her what's occurred to her and she writes down the formula “Fn = Fn-1 + Fn-2”. Then she correctly continues the series. That would count, I think, as an example of the application coming before her mind. Something occurred to her and it helped her continue the series.

b. However, it would also count if, instead of the formula, she writes down “Each new number is the sum of the previous two”. So what comes before her mind needn't be one specific thing.

c. But suppose instead when I show her the series she simply says “Oh, that's easy!” and correctly continues it: “...13, 21, 34, 55...”. And when I ask what occurred to her she says “Nothing – it was obvious.” (Perhaps she's worked through the series numerous times before.) Here no application came before her mind; she just knew what to do. In other words, correct performance by itself is not a sufficient criterion. Did she understand my order to continue the series? Yes, but that shows you can understand something without an application coming before your mind.

d. Finally, suppose that she writes down the formula as in the first example, but then has no idea how to continue the series. It later transpires that she'd previously seen the formula written below the series and that's why it occurred to her. Here again the application has not come before her mind even though what occurs to her – the formula – is exactly the same as in the first example. So correct performance might not be sufficient, but it's clearly important.

But what, then, is the difference between (a) and (d)? From the point of view of the criteria for the phrase “the application came before her mind”, the difference is not what's in her mind but her past experiences: her education, training and so on. The formula only counts as the application in the right circumstances. We might sum up all these reflections in the following way: however you slice it, understanding involves more (and sometimes less) than an application coming before one's mind.

5. Conclusion

A picture cannot fulfil the role of understanding by itself. This is not to deny that at least sometimes a word can bring an image before our minds, nor that the image might help us to use the word correctly. But it cannot show that we have used the word correctly; it supplies no method of application and hence no standard of correctness – and the criteria for applying the term “understanding” are closely bound up with correct usage.

However, the appeal of supposing understanding to be an inner “Something” is not so easily dispelled. Thus, for example, we might think that (questions of application aside) the pictorial account of understanding is implausible simply because it doesn't seem to match what actually happens. When we hear someone speak we don't get a torrent of pictures running through our minds one after the other. Yet instead of making us abandon the idea, this objection typically leads to attempts at refining it. We might wonder, for example, if the pictures are somehow unnoticed. Do they go by too fast? Are they perhaps unconscious? And need they be conventional images at all? Mightn't they take a form that we wouldn't easily recognise as pictures – formulae, for example? Now comes the thought that what we're really talking about is something that allows us to compute the correct output for a given input. And if that's the case, then isn't “understanding” a question of having the right mental structure – one that facilitates correct responses? And need this structure actually be mental? Couldn't it be a state of the brain? And so on.

It is these types of “refinement” that Wittgenstein considers in the remaining sections on understanding. They all, I think, flounder on the point already made: “understanding” is bound up with application, and correctness of application can only be given by an established practice. Nevertheless, it is important for Wittgenstein to deal with these new variations because otherwise the suspicion will remain that he has been over-hasty. The prejudice in favour of the inner “Something” is deep-rooted; digging it out requires tracing it through all its manifestations. So that's the next post.

From Logical Form to Understanding: the Transition from §137 to §138

§138 begins a sustained investigation into the concept of understanding. I'll begin unpicking its various strands (which is a daunting prospect) in my next post, but here I want to examine the way Wittgenstein introduces the topic. Basically §138 perplexed me when I re-read it recently and it took a fair amount of effort to tease out the trajectory of its thought. That might just have been dimness on my part (in which case I apologise in advance for what's to follow), but I suspect that for many readers of the Investigations – especially those unfamiliar with the Tractatus – it can be easy to miss what Wittgenstein is doing in this section.

Typically, the shift between propositional form (§§134-137) and understanding (§§138-184) is treated as one of those abrupt changes of subject the author warns us about in his preface. Fogelin, for example, states that in §138 “Wittgenstein turns from his attack upon Tractarian themes” (Fogelin, Wittgenstein, second edition, p144 – I like the word “Tractarian”, by the way, and intend to make full use of it from now on). Hacker and Baker, by contrast, are a bit more nuanced. In Wittgenstein: Understanding and Meaning we are told that “§§1-142 of the Investigations are Janus-faced, looking back to the errors of the Tractatus […] and forward to the very different account now being unfolded” (Volume I, p357). And then, a little later, “§§138–142 mark a change” (ibid). So, it would seem, the breach-proper comes at §143 and §§138-142 form a sort of pivot or gateway into the full expression of Wittgenstein's new philosophical approach. Nonetheless, even here there's little or no sense that the discussion of understanding arises out of what comes before it, nor are we offered an account of how or why the one leads to the other. (Hacker and Baker might cover this in the exegesis, but – forgive me – I've never read it. There are limits, you know.)

But that's curious, because when you actually read the text the transition feels pretty seamless. Usually when Wittgenstein makes a jump it's not too hard to spot (eg, §134 or §243), but in this case §138 is clearly presented as a continuation of the discussion in §137. Indeed, the topic of understanding is raised before we've hardly realised it. At the same time, however, it must be admitted that there's something a bit odd about the transition. It reads like a continuation and yet, upon closer inspection, it's not at all clear how it fits with what's gone before. Is Wittgenstein resorting to a kind of authorial sleight-of-hand? Why would he bother – especially given his near-pathological aversion to fudging things? Why not just make the jump and have done with it? Time for a closer look.

§§136-137 consider the claim that the concept of truth and falsity (truth-functionality) fits the concept “proposition”. That is to say, the two concepts necessarily go together. Wittgenstein exposes this idea as a grammatical remark misleadingly disguised as a description (see Propositional Form). At first blush, the start of §138 merely seems part of this debate; with typical doggedness the interlocutor suggests a further example of “fitting” in order to shore up his position:

“But can’t the meaning of a word that I understand fit the sense of a sentence that I understand? Or the meaning of one word fit the meaning of another?”

When you think about it, though, this is strange. What does the idea of a word “fitting” the sense of a sentence have to do with truth-functionality “fitting” propositions? How does the one support the other? It seems a glaring non sequitur. Of course, they both use the notion of “fitting” in relation to language – but, frankly, so what? If that’s all it amounts to then the interlocutor might just as well have said “But surely there’s such a thing as words fitting (or failing to fit) on a page?” We need something stronger than that.

To make matters worse, Wittgenstein's response is in some ways just as puzzling: “Of course, if the meaning is the use we make of the word, it makes no sense to speak of such fitting” (§138). This seems to suggest that if meaning is use then there's no such thing as a word fitting the sense of a sentence or of one word fitting the meaning of another. But this is surely wrong! Not only do we often talk about words fitting in this way, but when we do we are usually concerned precisely with the question of use.

So, for example, someone with little English who wanted a drink might wonder whether it was correct to say “Pass me the jug” or “Pass me the water”. In such a case we would tell them that it didn't really matter; in this context the words “jug” and “water” both fitted the sense of the sentence. The two expressions achieve the same thing; they are used in the same way. (Imagine a culture where, instead of “jug” or “water”, they had a word meaning “water-in-jug” that was always used on such occasions. So if you said “Pass me the jug” they'd empty out the water before handing it over, and if you said “Pass me the water” they'd pour the water into your hands. In such a situation neither “jug” nor “water” would fit the sense of the sentence – and again that is entirely due to the conventions governing use.) Here “fitting” is akin to aptness; it's a question of which words are effective, which words get the job done.

The final puzzle in this puzzling section comes with the interlocutor's response to Wittgenstein's criticism, which suddenly shifts the focus away from “sense” and on to “understanding”. With palpable exasperation the interlocutor says:

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!

This is a pivotal moment in the Investigations and I'll be discussing it in detail in my next post. But for our present purposes the important issue is the connection between sense and understanding. It may or may not be true to say that the phenomenon of grasping meaning “at a stroke” sits uneasily with the claim that meaning is use, but how on earth is this bound up with the idea of a word fitting the sense of a sentence?

So now we have three questions:

1. How does the idea of a word fitting the sense of a sentence support the idea of truth-functionality fitting the concept of a proposition?
2. Why does Wittgenstein respond by seemingly ruling out the idea of such fitting on the grounds that meaning is use?
3. What is the connection between the sense of a sentence and understanding?

The answer to all three, I think, lies in what the interlocutor means by “the sense of a sentence”. In common usage, explaining the sense of a sentence typically involves paraphrasing it into a form that's easier to understand. In such a context, therefore, “sense” is roughly synonymous with “meaning”. But it shouldn't be forgotten that §§134-137 deal explicitly with the Tractarian notion of propositional form, so when the interlocutor raises the subject of “sense” in §138 he is still looking at things from the point of view of the Tractatus. That is, he is using “sense” according to the definition given at TLP 2.221: “What a picture represents is its sense”.

This brings us up against the picture theory of language. According to the Tractatus, the essential thing about language is that it pictures logically possible states of affairs. And it does this by mirroring the logical structure of what it represents. So “The cat is on the mat” presents us with objects arranged in a particular way, just like a drawing might. The assertion it makes may or may not be true – to find out we’d have to compare the sentence-picture with reality. But even if it’s false it still has a sense because what it pictures is possible: cats can be on mats. Compare this, however, to “The cat is on selfishness”. That proposition is nonsense precisely because it attempts to combine phenomena in an illicit way; selfishness can’t be sat on any more than it can be eaten or set on fire. So what the proposition attempts to represent cannot even be pictured and therefore lacks sense.

Now, how can we tell which words can legitimately be combined with which? That is down to the a priori structure of the world, for it is this structure which determines the combinational possibilities of objects. And these combinational possibilities must, in turn, be reflected in the logical syntax governing language. It is as if a sentence beginning “The cat is on…” leaves a gap which can only be filled by words of the right type. Or, to put it another way, only the right type of word will fit. This, I think, is the type of “fitting” that the interlocutor has in mind at the start of §138. It belongs to a conception of language as a structure made up of different shaped holes into which only the right kind of “word-peg” can be slotted. And this type of “fitting” has nothing to do with aptness or use; it is modelled on the physical sense of the word, as when a plug fits into a socket or a hand fits into a glove. Its ultimate legitimacy comes from the world rather than linguistic convention.

Now consider the very next line of the Tractatus, TLP 2.222:

The agreement or disagreement of its sense with reality constitutes its truth or falsity.

A proposition with a sense must be a picture of a possible state of affairs. As such, that possible state of affairs must either obtain or fail to obtain. In other words, a proposition must be either true or false. So, according to the Tractatus, the very notion of sense necessarily brings with it the notion of truth and falsity. You cannot have one without the other.

Now I think we can begin to see the connection between §137 and §138. So far as the interlocutor is concerned, the requirement for propositions to be true or false arises out of the definition of “sense”. Criticism of the former inevitably implies criticism of the latter. It is as if the interlocutor – worried by the attack on truth-functionality – moves back a step to protect the prior link in the chain of reasoning. For if that holds good then (it seems) the subsequent links must also be preserved.

To put it more generally, both claims are part of the same broad conception of language, ie: language as a kind of calculus whose rules reflect the necessary structure of existence and rigidly determine the way words operate. The various elements of the conception are closely interwoven (that is part of the beauty of the Tractatus); they hold each other in place and, consequently, if any one element is attacked the others automatically rise up in its defence (this, I think, sheds light on Wittgenstein's remark in the Blue Book (p44) that “no philosophical problem can be solved until all philosophical problems are solved”).

The answer to question (2) is also now clear. It is not “fitting” per se that Wittgenstein objects to so much as the interlocutor's specific interpretation of the term and the philosophical theory that lies behind it. (We might say that the interlocutor's remark at the beginning of §138 is somewhat duplicitous – he is trading on the ordinary meaning of “fitting” and “sense” to make his claim seem straightforward and uncontroversial.) The Tractarian account of “fitting” does indeed make no sense if meaning is use. For as we have seen, from the viewpoint of use, “fitting” is a matter of the role the word plays rather than its ability to mirror a supposed a priori world-structure. There is such a thing as a word fitting the sense of a sentence, only not in the way the interlocutor claims.

Are we any closer to answering question (3)? I think we are, but only if we read between the lines. The interlocutor's exclamation at the end of §138 (quoted above) marks the point where we move from discussing ideas explicitly set out in the Tractatus and on to what we might call the “uncredited assumptions” that lurk behind those ideas and help make them seem plausible. And one of those assumptions concerns the link between sense and understanding.

To clarify this we need to return to the picture theory of language. As we've seen, according to that theory a proposition pictures a possible state of affairs which is its sense. But, of course, a proposition doesn't just come out of nowhere – it expresses a thought, and a thought is itself a picture (TLP 3: “A logical picture of facts is a thought”). Therefore “In a proposition a thought finds an expression that can be perceived by the senses” (TLP 3.1). Now, what is it to understand a proposition? 4.024 tells us: “To understand a proposition means to know what is the case if it is true”.

From these brief remarks it is possible to construct an account of communication that goes like this: A has the thought that things are thus-and-so. And this thought is in some sense a picture – that is, whatever form it takes it must combine objects in a way that mirrors a possible state of affairs. A wants to communicate her thought to B so she “converts” it into a proposition. She speaks, writes, draws, uses sign-language or Morse code, etc. The elements of the proposition are arranged in a way that mirrors their combination in A's thought (and thus also mirror a possible state of affairs in the world). Now B perceives A's proposition and he (so to speak) re-converts it into a thought in his mind. And that thought is, of course, a picture. The sense of the picture is what it represents, and it is precisely this that B needs to grasp in order to understand the proposition. Grasping the sense of the picture (seeing that it represents this possible state of affairs) and understanding the proposition are one and the same thing.

This account will be familiar to anyone who's studied Empirical philosophy (especially Locke). The connection it makes between sense and understanding couldn't be more direct: grasping the sense is understanding the proposition. Understanding, therefore, is a thing (specifically, a picture) that we acquire instantaneously when we perceive the proposition. And that is surely right, isn't it? For when we hear or read words we understand them at once; we don't have to wait until we've studied their use before we know what they mean. So if the sense of a sentence isn't to be cashed-out along Tractarian lines how do we account for instantaneous understanding?

That thought, it seems to me, completes the chain of argument that leads from the discussion of propositional form in §137 to the topic of understanding at the end of §138. Wittgenstein's criticism of “fitting” in relation to truth-functionality causes the interlocutor to defend the notion of sense from whence the necessity of truth-functionality originated. And criticism of the notion of sense in turn provokes an appeal to the nature of understanding, because the immediacy of understanding can only be understood (it seems) if something like the Tractarian account of sense holds good. By now we have indeed moved away from the explicit theories of the Tractatus (which is notoriously silent about how thought, understanding etc actually work), and so in a way Fogelin, Hacker et al are right: §138 represents a break. But it shouldn't be thought that we've moved on to completely unrelated issues. We have moved from the explicit theories of the Tractatus to the implicit assumptions that lay behind them.