§51-52

Were I to divide the PI into chapters, §51 would mark the beginning of a new one (which goes on till about §66). What distinguishes this section is that Witty will run through a whole series of different iterations of language-use, in order to pick out wide variations of such uses. If §50 began to establish that there can be different roles for words in language-games, and that those roles could be changed, §51-§66 will cash this insight out across a whole range of language-games. The idea in these parts is not so much to look for some underlying similarity between these uses (Witty will return to the issue of invariance later), so much as precisely to look for variance, and to acknowledge that such variance exists.

(These sections correspond very nicely to what @John Doe mentioned as Witty's general modus operandi, which he put as: ""It's experience...wait no, that's too broad, it's ways of seeing and acting, what no..." is what the book is aiming to get us to do as readers".)

So §51 is basically a series of rhetorical questions set out to lay the ground: in saying that the word "R" corresponds to a colored square, what exactly is going on here?: "what does this correspondence consist in? In what sense can one say that certain colours of squares correspond to these signs?" - one ought to read these question as: is there only one sense in which this correspondence can obtain? Or - and this is what I think Witty is driving at - are there are various ways in which such a correspondence is set up? Witty runs through two examples, the first of which he puts into question, and the second of which he leaves open to consideration, but their specifics are not important. What is important is that there is variation in what it could be for "R" to correspond to something in the first place.

Witty ends the rhetorical questioning with something like a methodological imperative: if we "want to see more clearly... we must look at what really happens in detail, as it were from close up." This more or less characterises the strategy in the upcoming sections. One resonance to hear in all this is something like an 'anti-theoretical' stance: something like - don't come up with an 'a priori' theory of correspondence - look and see what happens instead, and note how wide the variety of things are that count as 'correspondence'.

§52 is a cute little dig at philosophy, which, on Witty's account, doesn't engage in the 'close up' strategy he will employ here. He leaves the question open: "why not?".

I feel as though I've fallen behind, but things are getting serious (and more difficult) now. I've also spent a little longer on §50 to try and get clear in my own thinking.

§49. What does it mean "to say that we cannot define (that is, describe) these elements, but only name them?" Referring to his example of the 3x3 colour matrix at §48, W proposes the "limiting case" of a (one square) 1x1 matrix having a definition or description that is "simply the name of the coloured square". He states that "a sign "R" or "B", etc. may be sometimes a word and sometimes a proposition. But where it 'is a word or a proposition' depends on the situation in which it is uttered or written." W states that the word "R" might be a description or a proposition if it is being used within a language-game to refer to a coloured square. Alternatively, the same word "R" might be a word or a name if its use is being taught to others or to oneself, and where it is, therefore, only being prepared for use within a language-game. As Wittgenstein states:"...naming and describing do not stand on the same level: naming is a preparation for description. Naming is so far not a move in the language-game..."

§50. "What does it mean to say that we can attribute neither being nor non-being to elements? One might say: if everything that we call "being" and "non-being" consists in the existence and non-existence of connexions between elements, it makes no sense to speak of an element's being (non-being)..."

That is, if being (and non-being) consists in (or is defined as) the connections between elements, then it makes no sense to speak of the being (or non-being) of the elements themselves. This is clarified by his next statement:

"...just as when everything that we call "destruction" lies in the separation of elements, it makes no sense to speak of the destruction of an element..."

That is, if destruction consists in (or is defined as) the separation of elements, then it makes no sense to speak of the destruction of an element.

"One would, however, like to say: existence cannot be attributed to an element, for if it did not exist, one could not even name it and so one could say nothing at all of it."

It's hard to see the problem here, since it sounds perfectly sensible to say that if a particular element did not exist then we could not name it or talk about it. But this will become clearer with Wittgenstein's remarks on the standard metre which immediately follow, which he calls "an analogous case".

I won't rehash the standard metre discussion, except to re-quote Fogelin who interprets Wittgenstein here as saying that it makes no sense "to use something as a standard and simultaneously judge its accordance with that standard."

Wittgenstein then asks us to imagine a similar case to the standard metre in which a sample of "standard sepia" is kept hermetically sealed in Paris. W states that it will likewise "make no sense to say of this sample either that it is of this colour or that it is not".

Wittgenstein's reference to this being "an analogous case" makes clear that when he earlier said that existence cannot be attributed to an element, what he implied was that neither existence nor non-existence can be attributed to an element.

In the reference work 'Wittgenstein's Philosophical Investigations' edited by Dr Arif Ahmed, Dale Jacquette identifies a principle at work which connects the opening lines of §50 (regarding the existence of elements) with the standard metre and standard sepia examples. Mr Jacquette calls it the "polarity or complementarity principle" such that whenever "the predication or its negation or complement serve no purpose in a genuine language-game, the predication and its negation or complement are judged to have violated a rule of philosophical grammar". In the case of the standard metre, since it makes no sense to say that the standard metre is not one metre long, then it likewise makes no sense to say that it is one metre long.

Wittgenstein goes on to say that a standard or sample, such as the standard metre, "is not something that is represented, but is a means of representation". A 'means of representation' is a way of representing something. The standard metre is an example of this since it introduces, or prepares the way for, our use of the metre (as a length) in our language-games. However, once introduced, our use of the metre as a length is 'something that is represented'; something that is used within our language-games. The parallels of this distinction to the distinction drawn between names and descriptions at §49 are now apparent: preparation for use in the language-game vs. use in the language-game.

Hence: "And to say "If it did not exist, it could have no name" is to say as much and as little as: if this thing did not exist, we could not use it in our language-game.—What looks as if it had to exist, is part of the language. It is a paradigm in our language-game; something with which comparison is made. And this may be an important observation; but it is none the less an observation concerning our language-game—our method of representation."

I actually want to come back to §50 for a bit and 'intervene' in the debate that was going on between @Luke and @Banno a little earlier. My post on it before was trying to hew closely to the text, but I want to try something a little more free-form to really tease out the significance of the section. Because there's definitely something weird going on with it, and I wanna get at exactly what. Anyway, I want to start with this trilemma by Banno:

is the referent of "One Metre" a length, or is it a stick, or is it a process?

I say it is a length. — Banno

I think this is right, but something is missing. Or it is not the whole story, rather. Surely, the Paris meter is a length... of one meter. But is it only a length? Or is it also, in addition to a length, something else as well? Well, Wittgenstein would say: it is the means of measuring a meter's length. But here I wonder: can it not be both? And if I were to say this, would I be disagreeing with Wittgenstein? Here's my thesis: I would not be disagreeing with Wittgenstein, because Witty is approaching the question of the meter rule from a very particular angle, and outside that angle, it's perfectly possible to agree with Banno that the meter is a length.

So, how to pull this have-my-cake-and-eat-it-too act off? Like this: it can be both, but not at the same time. To wit: note the peculiarity of Witty's discussion of the meter rule, which takes place in the context of names and descriptions, simples and composites. The whole discussion is basically a conditional: if the Paris meter is the simple by which we measure meter lengths, then we cannot say of it that it is a meter nor not a meter long. If its role is that of being the standard by which meters get their measure, then the question of it's being a meter cannot be sensibly posed. But, as per Witty, roles are anything but fixed, and are themselves context-bound.

So, as @Ciaran, rightly pointed out, when I'm measuring my shed with my tape measure, I simply don't give a damn about the Paris meter. It doesn't even have a role in my particular activity of measuring the shed. The Paris meter might as well be just another stick. And if someone, out of the blue were to ask: How long is the Paris meter? One could well reply: a meter long, give or take some minor variation in wear and tear.

But say I start to question if my tape really is a meter long. Maybe I bought the tape from a dodgy store. Then I invoke the Paris meter and I ask: is my tape of the same length? But now my friend, a committed Cartesian, comes along and opines: but maybe the Paris meter is wrong, and you can't even be sure of that! What can we say to our friend? It's at this junction, when the Paris meter is playing the role of a standard, that Witty's insight becomes relevant: the 'right' reply to our Cartesian friend is something like: don't be daft, that's not a sensible position to hold.

Another way to put all this is: the Paris meter is just another stupid stick. It 'is' neither a length, nor a process, nor really anything in particular. But, if it has, or is given, a role in a language-game, that role determines what we can and cannot say of it. And in its role as a standard, we can neither say it is or is not a meter long. Outside that role, we can of course say, with no trepidation, that of course it's a meter long. Witty's discussion is explicitly one in which the Paris meter does occupy that role. To put it yet another way: Witty's pronouncement on the Paris meter is not a general-purpose statement, and it would be a mistake to treat it as such. It refers to it only in its capacity (role) as a standard.

To say all this is to keep in mind the 'relativity' of 'words' and 'sentences' in §49, where the same thing can be a word or a sentence "depend[ing] on the situation in which it is uttered or written"; with the caveat that, depending on which role it has, different things may be said of it. And moreover, that something cannot be both at the same time. One could in fact call this a 'complementarity principle', a la Bohr on particles and waves - only here we're talking names and descriptions, simples and composites, lengths and standards.

Does this parse things out nicely?

Is this an adequate reading? — StreetlightX

Yes, I think so.

ETA: although I think the standard metre is a special case as its purpose is only to set the convention.

It makes no sense to assert that the standard metre is one metre long, because this proposition implies that the standard metre might not be one metre long, which is absurd. Therefore, the standard metre is the one thing of which we can say neither that it is, nor that it is not, one metre long.

This relates to my 'verification' comments to Banno.

There is no referent of "One Metre" at 50 because Wittgenstein here attempts to remove it from any context in any language-game. It is removed from the context of use. The naming, as Luke describes, is the preparation for language-games. We can't say that the referent is "a length" because this is just an idea in people's heads, and this type of occult explanation is what Wittgenstein is trying to avoid. I think it is best to think of the standard metre as an object which represents the word "metre". Therefore, just like in his descriptions of ostensive definition, if someone were to point to the standard metre and say "one metre", we still have the same question of what type of thing is the person pointing to when saying "one metre".

My opinion is that this will prove to be circular unless words are given occultish mystical status, because a word is itself a physical object. So we have one physical object, the standard, representing another physical object, the word metre, without any actual means of establishing this relationship.

I'm not attempting to interpret Wittgenstein here, nor will I make much effort in sticking closely to the text, I'm trying to give an account of why I don't think there's really anything mysterious about the meter stick 'out in the wild', so to speak.

I understand the problem with the meter stick thusly. We have a game of measuring, moves in this game consist in (metaphorically) holding something next to the meter stick and measuring its length. The issue we're discussing arises, then, when we interpret the meter stick as an thing to be measured using itself; the problem being, how can it make sense to say that the meter stick is a meter long when we're using comparison to the meter stick to measure? The alleged problem with this is that the length of the meter stick will always be a the length of the meter stick, so it's not appropriate to say we measure any length using it in the game of measuring using the meter stick as a standard.

I agree with that up to a point, if we can grant that measuring practices in general behave like the game I have described above, it wouldn't make sense to say that the meter stick was a meter long or that it was not a meter long. However, this doesn't do the full richness of the 'game of measuring' the Paris meter stick is involved in justice, and with appropriate recognition of nuance the paradox loses its bite.

The key thing, it seems to me, is bound up with what it means to compare an item in our game of measuring with the meter stick. If we grant that a comparison takes place between distinct elements, then it is inappropriate to measure the meter stick with itself. If we constrain 'comparison' to mean 'must be done between distinct units in the language game, one of which is the meter stick itself', then applying the meter stick to itself is not a comparison in the sense of comparison at work in the language game. But I believe it is a comparison in the broader language game of length measurement, standardisation and unit ascription.

Call the sense of comparison at work innately in our meter stick language game above C1.

The comparison which occurs between the meter stick and itself is not a comparison in sense C1, but it is nevertheless a comparison of length/extension. In the game of length/extension comparison, we can place an object next to another (metaphorically) and see if they are the same length, one ground rule of the game has to be that an object has the same length as itself, because we need to be able to see if lengths are the same or not regardless of what unit of length they are expressed in. It is required that a length is the same as itself as a constitutive rule of of the broader length comparison language game, because the game operates on its items as token length representations rather than as actual objects for measurement. Precisely, then, comparisons in this broader game are comparisons of lengths relative to lengths and not lengths relative to the meter stick. Call this comparison sense C2.

So in terms of C1 comparisons, the meter stick can't be held up to itself - this is not an appropriate move in the language game, but in terms of C2 comparisons it absolutely is; C2 comparisons operate on lengths.

The confusion arises when we take a C1 comparison and substitute in a C2 comparison without noticing that the scope of the discussion has changed.

The specific length of the meter stick doesn't really matter, the same confusion would arise if we were talking about any standard of length; this implies that the role the meter stick plays is just an example of a general pattern. We can play a game with lengths called standardisation, whereby we elect a privileged object which induces a length scale on all other objects, whereby other objects are associated a number that measures their magnitude. The same can be set up analogously for token length representations. So, at work in this language game of standardisation is a notion of ascription which isn't innate to the original language game or the length comparison language game. This ascription sets up a scale which interacts with C1 and C2 comparisons.

Scaled C1 comparisons are a modification of the first game where we additionally associate a number with an object besides the meter stick which is a multiplier of the meter stick's length. Scaled C2 comparisons are when we associate a number with a length itself.

We can then see that the meter stick functions in three ways which we usually treat as equivalent, it allows C1 length comparisons, C1 length comparisons are moves in the broader game of C2 length comparisons. It modifies C1 and C2 length comparisons by ascribing scales to extensions, which allow us to name lengths through the scale.

So the meter stick names a privileged length for scaling, it provides comparisons between distinct objects, it provides comparisons between possibly non-distinct lengths, and the understanding we bring to the functioning of the meter stick elides all of these subtleties... Only for Wittgenstein to present the nexus of their interaction as bloody confusing if you think about it. Which it is.

The confusion arises when we take a C1 comparison and substitute in a C2 comparison without noticing that the scope of the discussion has changed. — fdrake

I think we agree. This is part of what I meant when I said that Witty's pronouncement on the Paris meter is not a general-purpose statement, but one that only applies to it in its role as a standard (what I think you're referring to as a C1 comparison). I also think you're right that Witty's presentation of the issue is confusing because he doesn't make this narrow application clear, and it can come across as a general purpose statement about the Paris meter as such. But a close reading will dispel any such reading I reckon. Particularly the fact that Witty says that the pronouncement does

"not to ascribe any remarkable property to it [the Paris meter], but only to mark its peculiar role in the game of measuring with a metre-rule."

And further on that it:

"is none the less an observation about our language-game - our mode of representation".

i.e. it is not a statement "about" the Paris meter qua metal rod sitting in a basement somewhere, but only the 'role' that it takes on in a particular language-game (such that, given a different language-game, where it might have a different role - or none at all - as with your C2 - the statement simply would not apply, and we could well say of it that it is a meter long).

I also think you're right that Witty's presentation of the issue is confusing because he doesn't make this narrow application clear, and it can come across as a general purpose statement about the Paris meter as such. But a close reading will dispel any such reading I think. — StreetlightX

Right, and I also think it's important to note the following section of §50, which possibly shows that the application is to much more than just the standard metre:

In this language-game it is not something that is represented, but is a means of representation.—And just this goes for an element in language-game (48) when we name it by uttering the word "R": this gives this object a role in our language-game; it is now a means of representation.

W's extension of the standard metre example to the word/name "R" here could indicate that the preparation/use distinction applies to all names in our language... I think?

W's extension of the standard metre example to the word/name "R" here could indicate that the preparation/use distinction applies to all names in our language... I think? — Luke

Yeah, to all things that have the same kind of role that both names and the Paris meter occupy in their respective games.

We still have the unresolved problem of ostensive definition though. We have no clear way of knowing what "kind" of role is being represented by the name. The Paris metre represents the word "metre", but there is still a need to signify which aspect of it represents "metre"? And this requires a further language-game as already demonstrated. So if naming is the "preparation" for a language-game, this still requires an underlying language-game which makes naming intelligible. Therefore naming cannot actually be the preparation for language-games in general, though naming might be the preparation for specific types of language-games.

It appears like Wittgenstein at this point in the book is moving from language-games in general, toward a specific type of language-game.

§51. Wittgenstein reminds us of his description of language game (48) where the words "R", "B", etc. correspond to the colours of the squares. He asks "what does this correspondence consist in; in what sense can one say that certain colours of squares correspond to these signs?" Wittgenstein notes that the account given in (48) "merely set up a connexion" between the signs ("R", "B", etc.) and the colours. He states that it was presupposed that the use of signs in the language-game would be taught via "pointing to paradigms". "Very well", Wittgenstein says, but of what does the correspondence consist in with the "technique of using the language" (i.e. in use, not in preparation)? He queries whether the person describing a square always uses the appropriate sign. W asks: what if an error is made? Furthermore, "what is the criterion by which this is a mistake?" Or, does the correspondence between sign and colour consist in some mental connection made by the people using the sign, such that when using "the sign "R" a red square always comes before their minds"? Wittgenstein suggests that to discover the answer in this case, we "must focus on the details of what goes on; must look at them from close to."

§52. Anticipating resistance to his therapy, Wittgenstein states that "we must learn to understand what it is that opposes such an examination of details in philosophy."

Hi . At first, I thought I was in agreement with your post, but something doesn't sit right, so I hope you can clarify.

The key thing, it seems to me, is bound up with what it means to compare an item in our game of measuring with the meter stick. If we grant that a comparison takes place between distinct elements, then it is inappropriate to measure the meter stick with itself. If we constrain 'comparison' to mean 'must be done between distinct units in the language game, one of which is the meter stick itself', then applying the meter stick to itself is not a comparison in the sense of comparison at work in the language game. But I believe it is a comparison in the broader language game of length measurement, standardisation and unit ascription.

Is the C1 comparison simply comparing the metre stick to itself (somehow)?

Also, I don't understand what the 'broader' type of comparison (referred to in your last sentence above) is supposed to include. Is this 'broader' comparison also included in C1?

The comparison which occurs between the meter stick and itself is not a comparison in sense C1, but it is nevertheless a comparison of length/extension.

What is "a comparison in sense C1"? What type of comparison is being made in the C1 sense if it is not a comparison of length/extension?

It is required that a length is the same as itself as a constitutive rule of of the broader length comparison language game, because the game operates on its items as token length representations rather than as actual objects for measurement. Precisely, then, comparisons in this broader game are comparisons of lengths relative to lengths and not lengths relative to the meter stick. Call this comparison sense C2.

Are you saying that 'token length representations' are abstract units of measurement? Therefore, C2 comparisons are not made relative to the metre stick, but to the metre unit?

So in terms of C1 comparisons, the meter stick can't be held up to itself - this is not an appropriate move in the language game, but in terms of C2 comparisons it absolutely is; C2 comparisons operate on lengths.

Are you saying that C2 comparisons involve the use of the metre unit, and so here comparisons can be made between the metre unit and the metre stick?

What is "a comparison in sense C1"? What type of comparison is being made in the C1 sense if it is not a comparison of length/extension? — Luke

I'll try and make it mathematically precise.

By a C1 comparison I meant specifically 'comparing something with the meter stick', so it is a length comparison with the meter stick and only with the meter stick. I was thinking of it like placing something beside a meter stick in order to measure using it, this is one sense in which the meter stick can be used to measure. The way we'd use it to measure wood to be cut and so on.

Looking at the logic of the thing, C1 comparisons take the form of a binary relation M, which looks like xM(meter stick), where x is anything which can be measured except the meter stick (by construction). If we were then asked 'how does (meter stick)M(meter stick) function?', we can't say as a C1 comparison since x is now also the meter stick. The domain of x is all objects (stuff we'd measure IRL) except the meter stick. This means the meter stick cannot be used to establish its own length through C1 comparisons.

A broader sense of length comparison, C2 in my post, takes the form of a binary relation N where we have xNy where x and y are possibly the same. Comparing the length of two items tout court. The domain of x and y are all lengths. The picture I have in my head here are comparisons of magnitudes which represent distances, as if comparing the distance from 0->1 and 1>2 on the number line. Notice that this comparison doesn't need units to make sense.

C2 is broader insofar as it allows length->length comparisons and there is no privileged object which must occur exactly once in every comparison (like the meter stick in C1).

We can do these comparisons without actually associating numbers with the various lengths, notice that in order to set up the meter stick as a standardisation of length we have to be able to say that it is exactly 1 of itself long; this is because all objects and their lengths, then, are a multiple of the meter. So when we 'bring the meter stick to measure itself', we're comparing lengths irrespective of the standardisation (in the sense that we can forget the standardisation is there as it only sets the stage/scales the axes of the space of comparison), when we compare something in sense C1 it will always be done with respect to the meter stick.

Also notice that the ability to say 'the meter is 1 of itself long' is actually a little modification of the previous two senses, insofar as we have established the meter as a standardisation of length-length comparisons as well as object-object comparisons. This means that the meter plays the role of a dimension in length-length (C2) comparisons and the role of a... measuring stick... in C1.

The confusion here, then, is rooted in substituting the meter as a unit of length (dimension) into a comparison which only makes sense (by construction) while using the meter stick as a measuring object.

The Paris metre represents the word "metre", but there is still a need to signify which aspect of it represents "metre"? — Metaphysician Undercover

Don't we already know that?

Are you saying that 'token length representations' are abstract units of measurement? Therefore, C2 comparisons are not made relative to the metre stick, but to the metre unit? — Luke

Are you saying that C2 comparisons involve the use of the metre unit, and so here comparisons can be made between the metre unit and the metre stick? — Luke

I think so Luke, hopefully all my extra words helped.

Thanks for the clarification, . I think I agree (but maybe I still don't get it?). In relation to the reading of the text, I think you pretty much nailed it in one of the opening paragraphs of your original post:

The issue we're discussing arises, then, when we interpret the meter stick as an thing to be measured using itself; the problem being, how can it make sense to say that the meter stick is a meter long when we're using comparison to the meter stick to measure? The alleged problem with this is that the length of the meter stick will always be a the length of the meter stick, so it's not appropriate to say we measure any length using it in the game of measuring using the meter stick as a standard.

I'm viewing my attempt to neuter the paradox from as beginning from noticing that it makes good sense to say that a meter is 1 meter long from a certain perspective. If we look in the game of 'unit standardisations of length', we can find yards and meters and lightyears and so on, if we look at the conversion rates of these we'll always find that they're in direct proportion with each other, and each unit is in direct proportion with itself with proportionality constant 1.

Once we've set the stage for the units of a dimension, we can largely forget that the units are there. 3 of a unit is always more than 2 of a unit and so on. The specificity of the meter doesn't actually matter for length comparisons, it's rather a preparation for length comparisons which facilitates those length comparisons through the ascription of numerical magnitudes.

The comparison of numerical magnitudes itself is something that can occur independent of the ascription of any scale. So we take something where a scale is implicit (Wittgenstein's measuring game) and then destroy that implicit dependence through a seemingly innocuous question - the question actually invites us to violate the established rules in a subtle way.

Once we've set the stage for the units of a dimension, we can largely forget that the units are there. — fdrake

Speaking for a moment outside of just the PI, this 'forgetting' of the origin has always seemed to me to be bound up with some of the most interesting philosophical questions out there - when the contingency of origin turns into intra-systemic necessity, which then begins to function wholly autonomously from that origin. This is the genesis story of sense, of number, of the discreet, of systematicity as a whole. I even want to say of the infinite. But these are just side remarks. But I think the Paris meter discussion is one of the places in Wittgenstein where he touches upon this.

Just in case this breaks your mind as much as it breaks mine in this context, angles are actually dimensionless quantities - even though we can subdivide a rotation into regular parts arbitrarily, all of those parts remain proportional to radians. Which are dimensionless. That they're dimensionless is required by trigonometry; if you try to do trigonometry with degrees or what have you a conversion to radians is implicitly done in order to allow you to take sines and cosines and so on, otherwise doing something like sin(1 meter) makes the quantity dimensionally inconsistent (sine( 1 meter) = 1 m + (1/3)m^3 ...).

So it appears there are standardisations which don't come along with unit ascriptions, too.

Just to clear my thoughts: is right to say angles are inherently proportional? Do they (always) express a ratio? Having trouble thinking this through.

That might be the crux of it. It's possible to understand angles as transformed ratios of lengths; but we could similarly understand lengths as transformed times (like lightyears). I don't know if it makes sense to see angles being derived from of lengths - we'd still have the ability to quantify rotation even without triangles.

we'd still have the ability to quantify rotation even without triangles. — fdrake

In which case movement - difference - would still be primary, no?

I don't imagine there's a way to think about angles without requiring thinking about relative positions (differences of positions) of (probably the same) shapes in the plane, no. Whenever you draw an angle there's a starting point and an end point, or a comparison of objects in which one serves as a base.

Though, you can codify rotations as distinct entities, they can be represented as matrices (which are transformations of points). I imagine this step of abstraction is similar to the one going from '1 meter' to 'length 1' (like C1 to C2 in my previous post). In this way you can forget the 'starting point' by making it an arbitrary application of rotation. IE rotating something 90 degrees is still the same rotation even if you do it on a horizontal or vertical line, even though it does not produce the same shape.

I imagine this step of abstraction is similar to the one going from '1 meter' to 'length 1' (like C1 to C2 in my previous post). — fdrake

Very cool. So much interesting stuff happens at this intersection. Part of me wants to say that it's the source of all paradox. But I'll stop this train here - too off-topic.

I'm viewing my attempt to neuter the paradox... — fdrake

You've used this term more than once. What do you consider to be the paradox? I don't think Wittgenstein views it or intends it as a paradox.

I see being unable to say that a meter stick is a meter long, nor that it's not a meter long, based upon a hypothetical situation that otherwise makes sense, is a paradox. Perhaps it's better to say that it's extremely counter intuitive.

I don't think Wittgenstein really believes that we can't say a meter stick is 1 meter long, I think he's using the example to illustrate what can happen when we pay insufficient attention to the prerequisites for our language use; even maybe how asking a question in the wrong context; or a poorly formulated question; leads to batshit insanity.

I think he's using the example to illustrate what can happen when we pay insufficient attention to the prerequisites for our language use; even maybe how asking a question in the wrong context; or a poorly formulated question; leads to batshit insanity. — fdrake

This seems like a good moral to draw!

Don't we already know that? — Luke

How would we know that, by referring to some paradigm?

".

Actually I'm having great difficulty understanding Wittgenstein's use of "paradigm". At first I thought it was similar to "grammar", but the use of this word is becoming more and more prominent. What does he mean by "pointing to paradigms"? Can anyone help me out with this?