Are you saying that you aren't able to respond when someone asks you to "fetch a red apple"? If not, then you know the rule to that language game to the extent that anyone else knows it (the extent to which it can be known). A rule is like a signpost. It is not meant to be exhaustive, it is only meant to be sufficient, that sufficiency being determined by use. — Isaac

It's easy to say "if you respond when someone says 'fetch the apple' then you know the rule", but this claim needs to be justified. First, we would have to say that there is a correct response, one which is according to the rule, because simple response is insufficient to demonstrate the existence of a rule. Further, it does not suffice to simply assume that any given response is the "correct" response. The judgement that any response is correct requires consulting the rule. Therefore, to justify "then you know the rule" requires that there is a rule available for consultation. Where do we find this rule, for consultation?

Eventually you have to arrive at simply what we have been instructed to understand by our existing within a social network, the near constant 'reverting to the mean' effect of each person trying to copy the other, which is what Sam has been trying to explain. — Isaac

The problem is, that you here, just like Sam26, refer to some vague "social network", as if the rules are just supposed to magically appear within this context, like the mouse which jumps from the dusty rags. I'm looking in detail, as Wittgenstein suggests, trying to determine the "particular rule", to validate this claim that there are rules within this social network.

Following on from his remarks at 55, using the example of a colour, he asks whether we could proceed without a paradigm/sample if instead we were to "bear in mind" the colour that a word represents. He suggests that such memories could provide us with the "indestructible" element sought at 55. However, the problem with this consideration is one which will famously return later: — Luke

Do you have a suggestion of how we ought to reconcile the following statements of 56 and 57:

56 ...This shews that we do not always resort to what memory tells us as the verdict of the highest court of appeal.

57 ...For suppose you cannot remember the colour any more?—When we forget which colour
this is the name of, it loses its meaning for us; that is, we are no longer able to play a particular language-game with it. And the situation then is comparable with that in which we have lost a paradigm which was an instrument of our language,
58. "I want to restrict the term 'name* to what cannot occur in the combination 'X exists'.

The statement of 56 seems clear, memory does not always have the final word in making such decisions. However, at 57 he seems to say that if we forget, then the meaning is gone. So in this sense, memory would be the "highest court" because it determines whether something has meaning or not. Also, it suggests that meaning is not indestructible as was earlier suggested, because when the memory is gone, so is the meaning.

I think the question is whether the Planck time is properly described as a discrete "unit of time" or as a limitation on our ability to mark and measure time, which in itself is truly continuous. Needless to say, I lean toward the latter. — aletheist

As I understand the issue, the physical evidence indicates that there is likely a real limit the spatial-temporal existence at this level. If this is the case then there is a separation, a lack of correspondence, between the conceptualization of a continuous space and time, and real physical existence. I believe that Peirce suggested replacing "infinite" with "infinitesimal", in our conceptions, as a way to deal with this problem.

I consider the real numbers to be an adequate model of any continuum for we can measure an arbitrarily small or large quantity without any problem. — TheMadFool

It is impossible that any numbers can model a true continuum because all conceptions of numbers are based in a conception of units, such that a number signifies either a unit or a multitude of units, and this is incompatible with a true continuum. This issue is very similar to the issue with true infinity. The "infinity" is defined as something which numbers cannot count, it is impossible by definition, that numbers can count infinity. Likewise, it is impossible by definition that numbers can represent a true continuum.

My understanding is that conceptualizing a true continuum requires the acceptance of infinitesimals. — aletheist

Infinitesimals does not resolve the problem because infinitesimals are units. So to model a continuum as infinitesimals is to model it as composed of discrete units.

It captures most or all of the features of that usage but without any contradictions at all. — MindForged

Obviously, this is what I disagree with. The mathematical conception of "infinite" clearly contradicts the colloquial definition of "infinite", I've demonstrated this over and over again, so you know what I mean and I will not demonstrate it here again. You simply assert that it does not contradict, while the evidence is clear, that it does.

Neither of the two distinct conceptions have contradiction inherent within. There is no contradiction inherent within the colloquial concept. Where the problem lies is in applying the concepts to what we consider as the real world. Each conception, the colloquial and the mathematical, has its own set of problems involved with application.

This is a claim there is no reason to accept. Aristotle believed actual infinities were impossible but they are not. — MindForged

That's not a proper representation of what Aristotle argued. He used the argument to separate "eternal" from "infinite" because Ideas, Forms, were described as eternal, and "infinite" was an idea. So he proceeded to demonstrate that "eternal" and "infinite" were incompatible. What he demonstrated is that anything eternal is necessarily actual, while anything infinite has the nature of potential. The latter, that the infinite belongs in the class of potential, must be read as a definition, a description, derived from observation. All instances of "infinite" are conceptual, ideas, and ideas are classed in the category of potential. From this premise, along with other premises, the conclusion that anything that is eternal is necessarily actual is derived.

This idea which you state, that actual infinities are possible, is produced from the conflation of Aristotle's two distinct aspects of reality, actual and potential, and consequentially time and space, in the modern conception of "energy". So the claim "actual infinities are possible" (which would be contradictory if we adhered to Aristotle's distinction between actual and possible), only demonstrates a failure to maintain Aristotle's principles. it doesn't mean that "actual infinities are possible" has any coherent meaning.

. For there to be a potential infinity, there has to be a predefined set of values that can be occupied otherwise the domain is precluded from study as it can change arbitrarily by an arbitrary amount. — MindForged

The whole point of "potential", under Aristotle's philosophy is that it cannot be studied as such. What we know, study, and understand, are all forms and forms are by definition actualities. "Matter" being classed as "potential", just like "ideas", is that part of reality which is impossible for us to understand. Potential is defined that way, it defies the law of excluded middle. There is an aspect of reality which is impossible for us human beings to understand because it violates the laws of logic, and this is "potential". Therefore, by its very definition, it is precluded from the study which you refer to. To assign a set of values, in order to study that domain is simple contradiction.

That domain is actually infinite. — MindForged

See, you have taken the category which is defined by "that which cannot be studied", "potential", which consists of matter, ideas, and the infinite, and you've applied some values (which is contradictory), and now you claim that this thing "infinite" is no longer in that category, it's in the category of actual. All you have done is changed the subject.

This is just an outright misrepresentation. Mathematicians did not simply redefine infinity to mean something contrary to its colloquial usage to disingenuously prove things about it. And hold this thought, I'll come back to it later when you say something inconsistent with the above. — MindForged

Yes it does, read the above.

"Defining features" are, ironically by definition, established by the definition in use. Otherwise we would never had words whose meaning varies across context and circumstance due to the resemblance in those varying contexts. — MindForged

This is clearly false, and indicates a misunderstanding of how logic and understanding proceeds. We identify a thing (law of identity), this thing as identified, becomes our subject, and we proceed to understand it through predication. The "defining features", how the subject is defined, ensures that the subject represents the object. This is known as correspondence, truth. It is evident therefore, that "defining features" is determined by correspondence between the logical subject and the object which is said to correspond to that subject, and not by "the definition in use". If it were the "definition in use" which defined the subject everything would be random with no correspondence to reality It is clear that "the definition in use" must be consistent with the known correspondence, truth. When the definition in use is not consistent to provide a correspondence with the identified object, we can correct the definition, saying that you are using an incorrect definition.

Contrary to your geometry misunderstanding, the reason parallel lines can meet is that the reason they cannot meet in Euclidean Geometry is because of how space is understood there (as planar). In Riemannian geometry, Euclidean space is understood simply as a space with a curvature of 0. But if space is curved then the provably such lines do intersect, such as on the surface of a sphere (i.e. lines of longitude). The actual Euclidean definition of what a parallel line is does not say the lines will not intersect. It's that if you have some infinite Line J, and a point P not on that line, no lines passing through P intersect with L. This does not hold if the space is different. It's only when one misstates the Parallel Postulate that it sounds contradictory to have intersecting parallel lines. You are forgetting that these notions are defined by the geometry, not separate from them. — MindForged

So this is very wrong because you have reified space, as if "space" were the subject, and there is a corresponding object which has been identified as "space". There is no such object being described here in geometry. The objects are all mathematical, conceptual, such as a "line". My point was that if there are two distinct concepts of "line", then there are two distinct objects referred to by that name "line" corresponding to the defining features which constitute the two distinct subjects under that name, "line". Therefore "line" ought not be used for identification of both of these objects.

Alternatively, as Peirce argued, there are no instants (NOWs) in any continuous interval of time, and there are no points in any continuous segment of a line. Time does not consist of instants and space does not consist of points; instead, those are arbitrary discontinuities that we mark within continuous space-time for our own purposes, such as measurement. The real numbers thus serve as an adequate model of a continuum for almost all uses within mathematics, but they do not constitute a true continuum. — aletheist

I agree with this representation, but the problem which TheMadFool points to is that there appears to be real points of discontinuity within time, which are represented by the concept of Planck time. So if time is represented as a true continuity, as you say Peirce suggests, how do we account for these fundamental units which cannot be further divided. What type of point would mark the beginning and end of these units of time?

But there is no problem here to resolve. The rule only appears vague when we are looking for something 'hidden' behind it. Absent of that, it is not vague and ill-defined at all. Do you have any great trouble speaking to people in ordinary language? — Isaac

I don't see the logic here. I have only a few problems speaking to people, but I don't see any rules here. I am not looking for something hidden behind the rules, I am looking for the rules themselves. But I see no rules, so I don't know what you're talking about. How could I be looking for something hidden behind the rules, when I do not even see any rules?

The chart of relations in 48 shows which types of action (in response to which utterances) are considered by the players to be in accordance with the rules. — Isaac

The chart does not show this though, that's the point. The chart is just an arrangement of coloured squares. Along with the chart there are instructions as to how to use the symbols, "R", "G" "W" "B". It is the instructions, whether the actions are according to the instructions, which determines whether the actions are according to the rules. So I have to associate "the rule" with the instructions, not with the chart.

Likewise, when the person calls "slab", the instructions are for the apprentice to bring a certain type of stone. It is the instructions which constitute "the rule", not the word "slab". The word "slab" is a representation of the instructions, for the apprentice, and the instructions are the rule. The word "slab" is a representation of the rule, which is what the apprentice is supposed to do upon hearing the word. What the apprentice is supposed to do, may also be described, and this description is another representation of the rule, which may be issued as instructions.

We can't. That's the point. — Isaac

I don't believe this. I think this is exactly what Wittgenstein is looking for the "particular rule". That is why he says we have to look up close, in detail, and he asks about the different cases in which a game is said to be played according to a particular rule. You might say that we cannot determine a particular rule, but that's not what Wittgenstein is saying. Consider, that the mouse comes out of the rags. Someone says, it's spontaneous generation. That's just like you, saying above, that the rules are there, in language, they must be or else we couldn't communicate. So we take it for granted, that the rules just come out of language, like the mouse just comes out of the rag, spontaneous generation. Now Wittgenstein says let's look into this language thing in detail, and see if we can determine exactly how these rules are coming into existence, just like examining the rags up close to see where the mouse is coming from. First off, we need to get a very clear idea of what a rule is, so we know exactly what we are looking for in there.

Wittgenstein believes that philosophy has been misguided by our grammar. I cannot put it much more basically than to say that it has been misguided into thinking that because we can "say" something in one context, we can analyse it without context. His work here is trying to show the effect of context on the meaning of words. It is not trying to map those effects, or explain them, or find some unifying theory behind them. — Isaac

I think it is a mistaken reduction to say that what Wittgenstein is trying to show in this book, is one single thing. What he is showing changes almost as quickly as the numbers change, and that's why reading this is such a long arduous process.

That's not a philosophical conception, that's as much the colloquial conception as anything else. — MindForged

Are you familiar with Platonic dialectics? We determine the meaning of a word by referring to how it is used in our society. This mean that the colloquial conception is the correct one. If mathematics is using a conception of "infinite" which is inconsistent with the colloquial conception, then this is an indication that they have not properly represented "infinite"?

Zeno's paradoxes are not resolvable under the colloquial understanding of infinity, but they are resolved by appeal to modern mathematics (calculus) which requires the hierarchy of infinities. — MindForged

Zeno's paradoxes were adequately resolved by Aristotle's distinction between actual and potential. The colloquial conception of "inifinite" is consistent with this distinction, though the colloquial understanding does not all the time include an understanding of this distinction, so Zeno's paradoxes may appear to one who holds the colloquial conception but does not understand Aristotle's resolution. The modern world of scientific discovery has long ago rejected Aristotelian physics, and with it the Aristotelian distinction between actual and potential. The concept of energy is clear evidence that this distinction has been lost to modern science. Because these principles, which resolve Zeno's paradoxes, were lost to modern science, Zeno's paradoxes reappeared as valid paradoxes.

The principles of modern mathematics do not resolve Zeno's paradoxes because the philosophers of mathematics have simply produced an illusory conception of "infinite", which is inconsistent with what we are referring to in colloquial use of the term. That's sophistry, and Platonic dialectics was developed as a means to root out and expose such sophistry. The sophists would define a word like "virtue" in a way which suited their purposes, and then profess to be teachers of this. However, Socrates exposed that what they were teaching as "virtue" was just their own little conception, which was completely inconsistent with the colloquial meaning of "virtue" (what the members of society in general regarded as virtue). Philosophers of mathematics have engaged in the same form of sophistry. They teach their own private conception of "infinite" which is completely inconsistent with what we generally mean by "infinite" (the colloquial meaning of the word), creating the illusion that this resolves Zeno's paradoxes.

Under your view absolutely nothing can ever replace a previous misconception because to change ones accepted theory of a concept entails just changing the subject. — MindForged

This is not true. What I am arguing is that if we change the defining features of a thing, then we are not talking about the same thing any more. Therefore we ought to give it a different name so as to avoid confusion. This is not a case of correcting a misconception, it is a case of introducing a new conception. We cannot say that one is a correction of a misconception, because they are distinct conceptions, having distinct defining features. The new conception ought to be named by a word which will not cause confusion with the old conception, or any sort of equivocation. For example, if the defining feature of parallel lines is that they will never meet, and someone says that they've come up with a new geometry in which parallel lines meet, then we ought not call these lines parallel, but use a term other than "parallel" in order to avoid confusion and the appearance of contradiction. They are distinct conceptions, not a correction of a misconception. Likewise, the new conception in mathematics, which is called "infinite" ought to bear another name like "transfinite" so as not to confuse the conception with what we commonly call "infinite".

You're essentially supposing all mathematicians are idiots who don't realize they have an unneeded or useless axiom despite the many criticisms of the formalism (including Cantor's work on infinity) of a century ago. — MindForged

That's ridiculous. I am saying no such thing, and I resent that because I have great respect for mathematicians, they are as far from "idiot" as you can get. But the mathematicians which I know do not create the axioms, as this is more of an activity of philosophical speculation. And I do believe that much of the philosophical speculations which provide the foundation for modern science and mathematics is misguided. And I would not call a misguided philosopher an idiot, because much philosophy is hit and miss, trial and error.

Philosophy does not have a different definition of infinity outside the colloquial ones which are inconsistent. A potential infinity is just that: potential, as in not an infinity. Check out any relevant excerpts from mereological and ontological work that relate to infinity and in virtually all of them infinity is understood based on the long-standing mathematical definition of it. The reasons for this should be obvious. — MindForged

I am not talking about potential vs. actual. I am talking about "infinity" as boundless (philosophical conception), and "infinity" as completed (mathematical conception). The two are incompatible.

All you're proving is as I said, that some colloquial definitions conflict with others. Who cares? If those definitions lead to insoluble paradoxes and cannot be applied where they ought to (in mathematics) then they need replacing. With the proper understanding of infinity and a developed calculus, we solves Zeno's paradoxes when philosophers could not because they did not have a workable definition of infinity outside the vaguely defined one. Mathematics does not wholecloth redefine infinity, it still has most of the properties it intuitively ought to have (continuously extendable, for instance), but has the unique benefit of being perfectly and probably consistent. — MindForged

Choosing one conception and rejecting the other does not resolve the incompatibility. Nor does it resolve the paradoxes involved with the one conception, by choosing the other conception. That's simply an act of ignorance.

if you could demonstrate that "infinity" (the philosophical concept) as boundless, and incomplete, is an incoherent, unintelligible conception, then we'd have reason to reject it in favour of the other, mathematical conception. Until then it remains a valid concept which is incompatible with the mathematical concept of "infinitely".

On the other hand, I reject the mathematical conception because I believe it was created solely for the purpose of giving the illusion that the issues involved with the philosophical concept of "infinite", as boundless and incomplete, could be resolved in this way, by replacing the conception. Despite your claims about how calculus and science rely on this conception of "infinite", I believe it serves no purpose other than to create the illusion that the problems involved with the philosophical concept of "infinity" have been resolved. In reality, mathematics could get along fine without this conception of "infinity". It would just be different, having different axioms. And, since this conception of infinity is just a distraction for mathematics, mathematics would probably be better without it.

Sometimes we say "infinite" and mean Aristotle's potential infinity , sometimes we mean a completed infinity (as in the cardinality of an infinite set) and other times we just mean some arbitrarily large number that we leave unspecified. Philosophy always makes recourse to.mathematics in understanding infinity, I don't know why you think otherwise. — MindForged

The fact that philosophy has a different definition of infinite which is inconsistent with your mathematical definition of "completed infinity" is clear evidence that philosophy does not make recourse to mathematics for its understanding of reality.

The actual understanding of infinity is the one that came from mathematics courtesy of Cantor and Dedekind. Philosophers almost uniformly appeal to this rigorous understanding they gave us because... — MindForged

All this demonstrates is that you are very selective in the philosophy which you read. Cantor's representation of "infinite" was confronted by Russell, and hence replaced by Zermelo-Fraenkel. But any thorough reading on the subject will reveal that the issue is far from settled.

Nevertheless there was, and still is, serious philosophical opposition to actually infinite sets and to ZF's treatment of the continuum, and this has spawned the programs of constructivism, intuitionism, finitism and ultrafinitism, all of whose advocates have philosophical objections to actual infinities. Even though there is much to be said in favor of replacing a murky concept with a clearer, technical concept, there is always the worry that the replacement is a change of subject that has not really solved the problems it was designed for. — Internet Encyclopedia of Philosophy

Notice specifically, "..there is always the worry that the replacement is a change of subject that has not really solved the problems it was designed for". This is my argument. By redefining "infinite" mathematics is not even dealing with what we generally refer to as "infinite'. It has created a completely new concept of "infinite". It has put aside the true concept of "infinite" which derives its meaning from continuity, in favour of an illusory one, a completed one, in order to create the illusion that it has resolved the problems of infinity. In reality the new concept of "infinite" has just distracted us from the true infinite. "Completed" is not a word which one could use to describe "infinite" in the way that we commonly use the word. Of course, your claim is that the "completed infinity" is the true concept of infinity. I disagree.

How do you not see the contradiction between "I have no arguments against the mathematical results of infinity" and "I think it's quite obvious that the mathematicians have it wrong"? It's one or the other, either you're not arguing against it and thus you cannot say it's wrong, or else you're saying it's wrong and thus have some argument against it. — MindForged

It's as I said, I have no arguments against the conclusions drawn by mathematicians from their concept of "infinite", what you call the "results". I do not even know these conclusions, or results, and I have no interest in them. I am arguing against their premise, their concept of "infinite". This is not contradictory, just a simple statement of fact, I am not arguing against the results (conclusions), I am arguing against the premise (their concept of "infinite"). And, I have no interest in these results.

Indeed, Peirce independently invented quantification; and he disagreed with Cantor and Dedekind about the real numbers comprising a continuum, because he viewed numbers of any kind as intrinsically discrete. He was primarily driven by a philosophical interest in true continuity, rather than a mathematical interest in infinity. — aletheist

Right, tell that to MindForged, who seems to think that mathematicians have resolved the philosophical problem of "infinity". In reality, mathematicians have redefined "infinity" to suit their own purposes, neglecting the real problem of infinity, which is associated with continuity. And this might lead some naïve philosophers to think that mathematicians have resolved the problem of infinity. All they've really done is created a new problem, a divided concept of "infinity".

You made the following statement, and I tried to answer it:

I don't understand how it is that you do not agree with this, namely, that any language-game, which by definition is social, necessarily has rules (implicit and/or explicit). — Sam26

So let me try again. I disagree with this because I am not sure as to whether an individual must know a language-game in order to learn a rule, or whether one must learn rules in order to play a language-game. Wittgenstein has not yet made clear this relationship between language-games and rules. And, if the former is true, then a language-game does not necessarily have rules because we would need to learn a language-game before we could learn any rules. So I hope this explains to you, and you now understand how it is that I do not agree with you that a language-game necessarily has rules.

In mathematics, pairing and counting are not activities at all; they are concepts, and there is no requirement that they ever actually be completed, or even be capable of actually being completed. One more time: Mathematics has to do with the hypothetical, not the actual. — aletheist

Right, so in mathematics you can count the infinite numbers without counting the infinite numbers.
That's exactly why I say it's a false premise. If you want to ignore that contradiction, and accept this hypothetical as a true premise that's your prerogative. I think this hypothetical is clearly and obviously false though, so I reject it as false, and I would not employ it as a premise, like mathematicians do.

What you refuse to acknowledge, is that I reject it on the basis that it is logically impossible by way of contradiction, because "infinite" means cannot be counted. You might claim that it's not logically impossible because "infinite" means something other than this in mathematics, but I think that's wrong, as an illusory definition of 'infinite".

That's my opinion and I will continue to defend it until someone demonstrates to me that I am wrong, that as something capable of being be paired or being counted is a better representation of what "infinite" truly represents. Simply pointing out that my opinion is inconsistent with the opinions of many mathematicians, does not demonstrate that my opinion is wrong.

You have offered no argument for this claim, you have merely asserted it over and over — aletheist

Can you not read? Or do you have an extremely short memory? Let me reiterate. "Pairing", like "counting" is a human activity which is not successful unless it is completed. "Infinite" is defined in such a way that human activities such as counting and pairing cannot be completed an infinite number of times. Therefore it is logically impossible to count, or pair, an infinite number.

Of course I've spelled this out for you numerous times already, and you've simply ignored it, claiming some unreasonable distinction between logically impossible and actually impossible. So I expect you to continue with this unreasonable ploy.

now you have completely undermined your own position by freely acknowledging that you are not using the relevant terms in accordance with how they are carefully defined within mathematics, such that there is no logical impossibility whatsoever. — aletheist

This in no way undermines my position. I've acknowledged this from the beginning, defining "infinite" in a different way allows for infinite pairing. My position is that this mathematical definition of "infinite" is mistaken because it doesn't properly represent what "infinite" refers to in common usage, and in philosophy. The mathematical notion of "infinite" is illusory.

Here's where you seem to go wrong. Where did anyone say this? Language would never get anywhere if this was the case. In fact, I've said the opposite, "[t]he rule (known or unknown) is in the bringing of the correct stone in response to the call." Moreover, what do you think learning a rule is all about? When one learns to act in accord with a command, one is learning to follow a rule. It doesn't require that you know a language, or that you know what a rule is. Animals can even participate in rule-governed activities. Think of learning to follow simple commands. The learning of language, and the learning of following rules are things that happen at the same time, viz., if you learn a language, then you are learning to follow rules (implicit or explicit, known or unknown). — Sam26

I know that you've said the exact opposite to this, that's the point. Wittgenstien, has not yet established the relationship between language and rules, to make the conclusion which you have made. So I suggested the very opposite to your conclusion, as still a possibility from what Wittgenstein has so far exposed. Whether a person needs to understand language to learn a rule, or whether a person needs to understand rules to learn a language has not yet been determined. So as much as you might assert that a person cannot learn language without learning rules, these assertions are irrelevant to the text we're reading.

Are you saying that it seems to you that this is how a rule exists, or that you understand Wittgenstein to be implying this is the case? Because if the latter, I get the exact opposite impression and I'm not sure what line of interpretation has lead you to that conclusion. — Isaac

Yes, I am saying that this is what Wittgenstein is saying about rules. Look at #53 and the three ways which StreetlightX has elucidated. This is how "the rule" appears to us, as something general, vague and ill-defined, as "various possibilities". Here's the statement which concludes 53:

"If we call such a table the expression of a rule of the language-game, it can be said that what we call a rule of a language-game may have very different roles in the game."

The table is "the expression of a rule", so I interpret this as the table is the means of representing the rule. Remember back to 50, the object plays a role in the language-game, which is the means of representation of the name. Now the table is the means of representing a rule. in 50, what is represented by the object is "the name", in this case what is represented by the object (the table) is "a rule". Consider that we have switched "the name" for "the rule", such that the object, which is the table, is a means of representation of "the rule".

Now the name itself, is used to signify a type of object "slab" for example signifies a type of object. And, the rule itself, is used to signify a type of action. Both are signifying "a type", so in each case what is signified is something general, as types are. However, we can point to the name, as something particular, "slab" etc., now let's move along and point to the particular rule.

I don't think it does appear as if it were one rule. Wittgenstein is pointing out three different roles rules can play in games. He's simply saying that rules do not play the same role in every game. This applies to any rule, it's not that one rule plays three different roles, it's that any rule could play any number of roles, there is no generalisable statement we could make about the roles rules play in games beyond a description of the roles rules play in games. That is what our close-to examination has shown. — Isaac

It has to be one and the same rule which is referred to at 53. The rule dictates a correspondence between the sign and the square. But how the rule does this is what appears to us in the form of various possibilities. This is what stymies our attempts to isolate the particular rule. We see the chart and we see the action of the people following the rule, but if we go to describe how the rule acts, we can describe it in a variety of different ways, despite the fact that it is one and the same rule which may be acting in a variety of different ways. How can we isolate the particular rule when it appears to exist in a variety of different ways at the same time?

Again, I'm unsure where you've got this impression from. If Wittgenstein was concerned to determine the form of the specific rule then he's going about doing so in a very obscure manner. He'd surely lay out as many language games as he could think of, and go through them one by one to arrive at some kind of Universal Rule Book. But we already have the first draft of such a book, the dictionary. — Isaac

But he makes a very succinct reference to playing games according to a "particular rule" at the beginning of 54. Yes, he is approaching the idea of a particular rule in a very obscure way, but that is because this is the way that the rules of language-games appear to us, in very obscure ways. The problem I see is that people like Sam26 simply assume that rules exist, they conclude that must exist in order for language to be successful, so they assume "rules" in some extremely general way as inherent within the social fabric. But Wittgenstein is saying at 52, that we must get over this biased, or prejudiced way of looking at this subject, and look right up close, in detail, and find these rules, and describe them, instead of just assuming that they must be there.

It's like assuming that the mouse came into existence from spontaneous generation. "Spontaneous generation out of grey rags and dust" is the bias, like "laws intrinsic within social fabric" is the bias. Now we need to look right up close, analyzing the details of that fabric, to find these rules. But there's an opposition to this up close, detailed analysis. Why? You seem to say that there is no such thing as the particular rule, while Sam26 seems to say that particular rules are necessary. Sam26 is saying that spontaneous generation is the necessary conclusion, while you are saying that there is no such thing.

Wittgenstein is claiming that the generalisable rule doesn't exist, so I don't see how it could be what is referred to as the particular rule. You may have to explain this a bit more clearly for me. — Isaac

I suppose I do not know what you mean by 'generalizable rule". So I don't know what you mean when you say that Wittgenstein is claiming that the generalisable rule does not exist. I haven't yet seen him claim that any sort of rule does not exist.

It would have to be something that is impossible for anyone even to conceive (like a square circle), — aletheist

That's exactly my argument, it's logically impossible, impossible to conceive of, just like a square circle is logically impossible. The point is that people claim to be able to conceive of square circles, just like they claim to be able to conceive of pairing infinite numbers. People claim all sorts of weird things, like a polygon with infinite sides. They do this by violating, or changing the definitions of the terms.

This is the exactly what I was talking about . The issue is you're misrepresenting what is being said. It should be patently obvious mathematicians do not define mapping (pairing) and infinity so as to make them jointly inapplicable. Just saying "I'm speaking English" isn't even beginning to honestly address this obvious fact. — MindForged

That's the point. In English we know that pairing infinite numbers is impossible, just like we know that counting infinite numbers is impossible. The way that we use and define "infinite" and the way that we use and define "pairing", ensures that this is impossible. if mathematicians want to define these two terms in a different way, so that it is possible to pair an infinite number, that's their prerogative. I am not here to police mathematicians. However, we ought to be clear that this "mathematical" language is inconsistent with common English, and also inconsistent with how "infinite" is represented in philosophy.

If your terms are not related to mathematics then you have absolutely no argument against the mathematical results relating to infinity. You're simply talking about something else. — MindForged

You may have noticed that I have no arguments against the mathematical results relating to infinity, although others like Devans99 do. I really don't care about the mathematical results relating to infinity, because what "infinity" means to a mathematician is something completely different from what "infinity" means to me, a philosopher. And, I think it's quite obvious that the mathematicians have it wrong, (they've created an illusory "infinity"), so I'm really not interested in the conclusions which they might derive from their false premises.

And yet we can; and yet we do, map series of infinite numbers, one against the other. — Banno

In case you've never noticed this, claiming that you've dome something, and actually doing it, are two different things.

Tell me the exact formal definition of a mathematical mapping and infinity within the context of form mathematics and prove the contradiction. — MindForged

If you have a problem with my terms (they are English), then address my posts and tell me where the problems are. If my terms are not related to mathematics, then don't worry about them, they pose no threat to this field which you hold sacred.

One more time: The fact that no one can actually pair all of the integers with corresponding even numbers has no bearing whatsoever on its logical possibility. — aletheist

And here's my "one more time". It is only a "fact" by definition, therefore the impossibility is logical. The only reason why no one can actually pair the integers is because they are stated to be infinite, and by this definition, it is impossible to do such. Therefore it is logically impossible to do such.

What you have shown is that you refuse to understand mathematics. — Banno

What I have shown is that I cannot understand mathematics because the language of mathematics contradicts my native language, English. This renders mathematics as incoherent and unintelligible to me. I know that you don't care about this. So be it.

don't understand how it is that you do not agree with this, namely, that any language-game, which by definition is social, necessarily has rules (implicit and/or explicit). — Sam26

We went through this at the beginning of the book. In Wittgenstein's use of "game" it is neither implicit nor explicit that rules are necessary for games. It may be the case, as StreetlightX suggests above, that "rules" implies "game", but "game" does not necessarily imply "rules". But I don't think that we've even progressed far enough in the close-up examination of rules to even be able to make this conclusion, that rules cannot exist outside the context of a game.

Any language-game by definition is a rule-following activity, if not, then there would be no consistency of actions that would make it work. — Sam26

Allow me to paraphrase the problem. If it requires that one knows a language in order for that person to learn a rule (I.e. if we can only learn a rule through language) then it is impossible that all language-games are rule following activities. At this point in the book, we haven't had the close-up examination required to determine the nature of "a rule", Therefore we cannot say whether it is the case that we must know rules in order to play a language-game, as you claim, or is it the case that we must be playing language-games in order to learn a rule.

So in essence you're right, from your premise. Wittgenstein has failed to show us the generalisable rule behind what we see in the examples, but that's because his examples are meant to show that there isn't one. — Isaac

I think the issue is that a rule only seems to exist in a general form. Look at the three examples which StreetlightX very aptly laid out. What appears as if it were one rule, may manifest in these three different ways. So Wittgenstein asks, what does it mean to play a game according to a "particular rule". He wants to look beyond this general sort of form, to determine the form of the specific (particular) rule.

So what you call "the generalizable rule", is what is referred to as the "particular rule". His examples so far show three possible ways that the particular rule could exist. We haven't determined which one is the actual particular rule, so we haven't yet found the existence of the particular rule. However, I wouldn't say yet, that he intends to show that there isn't one, because we need to read further ahead.

But not every game is like this. When I say "Nothung has a sharp blade" (§44), there is no need that Nothung actually be around, and in one piece, for this sentence to have meaning; but something like "is it the same length as Nothung?" would require there to be Nothung around to measure it against (notwithstanding a question like 'is it the same length as Nothung was?).* — StreetlightX

I think that there is really no such difference between these two examples. The only real difference is that one is in the form of a question. "Is it the same length as Nothung" really has the same sort of meaning as "Does Nothung have a sharp blade", when Nothung has been destroyed. In each case, "sharp blade" and "length" refer to properties of an object which once existed but does not exist anymore. There may be a distinction of primary and secondary properties here, but I don't think that's what Wittgenstein intends. So talking about the properties of non-existent objects, whether it be "length", or "sharp blade", in each case has some sense, but he hasn't really examined in what kind of way it has sense.

What I mean by "...the actions determine what's correct or incorrect" is that the actions within the social context is the means by which we distinguish what's correct or incorrect. No action, as I understand it, is intrinsically correct or incorrect, except as it is seen within the game, or as seen within a social context. No more than an arrow is intrinsically pointing, it points within the context of the actions associated with the arrow. — Sam26

This would require the assumption that the game, or "social context" has inherent within it, rules, by the means of which, such a judgement of correct or incorrect could be made. But we have not yet found, in the close-up examination of the details, the existence of any such rules. We haven't even gotten beyond the problem which is that philosophers are commonly opposed to making such a close-up examination. Why do you simply assume that there are rules inherent within "social context"?

why?

Actually, I take that back. Mapping an infinity of one sort against anther is a common mathematical practice. So you are wrong, or talking about something else. — Banno

Pairing, and mapping, are all activities just like counting is an activity. You cannot count an infinite number because this is contradictory to the definition of "infinite". You cannot pair an infinite number for the very same reason. You cannot measure an infinite number, nor can you map an infinite number, for the very same reason that these are activities which require completion to be successful.

You might assert that you have mapped an infinite number, and even show me your map. But since I cannot show you the infinite number (because this is contradictory), I cannot show you that your map does not correspond with the infinite number.

All I can do is demonstrate logically that it is impossible to map an infinite number because this is contradictory. Do you recognize the truth of "it is contradictory to claim that you could measure an infinity"? Do you recognize the truth of "to map something requires that it be measured in some way"? What makes you think that mathematicians have done what is logically impossible, mapped infinity? How naïve are you? Suppose I told you, that if you keep going in this direction, counting, you will eventually reach infinity. Would you believe that I have mapped infinity?

They will just insist there is a contradiction. When you ask then to formally show the contradiction, they will just say it's weird, or that it's not possible to actually map two infinite sets or something like that. — MindForged

What a short memory you have. I actually demonstrated the contradiction to you in numerous different ways, because each time I demonstrated it you would change the goal posts in an effort to avoid my demonstration. That's why I had to demonstrate it to you in so many different ways, you kept trying to wiggle out from under the crushing force of blatant contradiction.

As explained in my last several posts, pairing infinite numbers is contradictory due to the definitions of "pairing" and "infinite". If you want to get back under the crushing force of contradiction, and try to wiggle out again, then be my guest, and try to demonstrate to me how this is not contradictory. But since your memory seems to be very short, let me remind you that you were not able to wiggle out last time.

It illustrates that actual impossibility does not entail logical impossibility. — aletheist

But that's irrelevant because I was only arguing logical impossibility all along, which as I explained is the only real form impossibility.

No; the whole point here is that pairing the members of infinite sets cannot actually be completed, yet it is still logically possible. — aletheist

It is the definition of "infinite" which necessitates that pairing infinite sets is impossible. How on earth do you assert that it is still logically possible without changing the definition of "infinite"? That's MindForged's tactic, to produce a different definition of "infinite", but that leaves "infinite" as utter nonsense. As Devan's99 has demonstrated over and over again, the definition of "infinite" employed by set theory is illogical.

If pigs had large and powerful wings, then pigs could fly. The truth of this hypothetical proposition is not affected by the fact that pigs do not actually have large and powerful wings. If one were to pair all of the integers with the even numbers, then one would never run out of even numbers while still having integers left. Again, the truth of this hypothetical proposition is not affected by the fact that one cannot actually pair all of the integers with even numbers. — aletheist

How is this relevant?

A square circle is logically impossible because the definition of a square and the definition of a circle are mutually exclusive — aletheist

Right, let's hold that thought.

There is no such incompatibility between the definition of an integer and the definition of an even number; in fact, the alleged paradox is rooted in those very definitions, which place no finite limitation on either set. — aletheist

The incompatibility is not between "integer" and "even number", it is between "pairing" and "infinite". "Pairing" is a task which requires completion. If the task is incomplete, they are not actually paired, and the "pairing" attempt is a failure. "Infinite" denies the possibility of completion, therefore the "pairing", is of logical necessity, in the case of the infinite, a failure. Therefore either the integers are infinite in which case pairing is impossible, or the integers are not infinite, in which case pairing is possible.

Where does that sense reside? It resides in the complexity of the language-game, grammar, rules, and actions (correct and incorrect) within social contexts, all of these work together to establish meaning. — Sam26

—An example of something corresponding to the name, and without which it would have no meaning, is a paradigm that is used in connexion with the name in the language-game. — Wittgenstein 55

As per my discussion with Luke, the paradigm referred to here cannot be an object, because the word still has meaning when the object is destroyed. How do you think it is, that something so general, and abstract, as "grammar, rules, and actions (correct and incorrect) within social contexts", is "a paradigm", as "paradigm" is used above? Remember, at this point in the text Wittgenstein is considering a specific problem which is an attitude of opposition by philosophers, toward looking up close, at details, and our example is the "particular rule".

That is actual impossibility, not logical impossibility. — aletheist

When definitions deny the possibility of something due to contradiction, this is a logical impossibility, like a square circle is a logical impossibility. That one could make a bijection of infinite numbers is logically impossible because the definition of "infinite" (what it means to be infinite), contradicts the definition of "bijection" (what "bijection" means). Therefore "infinite bijection" is excluded as a possibility because it is contradictory, i.e. logically impossible.

I don't know what you would be referring to with a distinction between "actual impossibility" and "logical impossibility", because all impossibilities are logical impossibilities. The only way that we have of demonstrating, or knowing that, something is impossible is through logic. Therefore actual impossibilities are logical impossibilities, because all impossibilities are logical impossibilities. An actual impossibility might be one based in sound logic, while a not-actual impossibility might be one based in unsound logic.

Unless I'm mistaken, you seem to be suggesting that 53 (and I suppose 54) are somehow answering the question raised at 51. — Isaac

I wouldn't say that Streetlight is suggesting that this question is being answered. "Dealing with the question" does not necessarily imply answering it. Witty says that we must look at it in detail, from "close to". Putting something under a microscope is a way of dealing with it, but it doesn't necessarily answer the questions we have about it. What is often the case is that we have to compare the up close view with the far back view, and often reciprocate back and forth with our attention, to establish a relationship between the aspects observed from up close and the aspects observed from far back.

Consider the (2) example, where the chart is as Streetlight calls a "fall-back". If we were looking only up close, we might not even observe the existence of the chart, because we might not see any of the instances of referring to it. And such instances, if observed, might be obscured and taken as irrelevant because they are few and far between and the chart may not be recognizable as such. Therefore we wouldn't even know that such a chart existed until we compare the far back look and see a pattern of reference. Then we could turn back to an even closer close-up to focus directly on the chart, and see exactly what the chart consists of.

Here's something I think we ought to take note of. At 51 he says we must look at the detail, the up-close. Then at 52 he says:

"But first we must learn to understand what it is that opposes such an examination of details in philosophy."

So the expose which follows in the next section may be an explanation of this opposition to examining detail, rather than an actual examination of the detail. Think of the "particular rule", as the thing to be examined in detail. Are we examining this thing in detail, or are we examining the opposition to examining it?

Why do you want to lock down the use of “convinced” in this way? What purpose does that serve? — DingoJones

Because that's the way the word is generally used. So if you used it in another way the interpreter might misunderstand you. And if that "other" way of using it is intentional, it could qualify as deception. Would you tell me that you were convinced of X even though you knew of reasons to doubt X? If so, I would say that you used "convinced" deceptively. If Terrapin Station said "I am convinced that all of reality is physical", and I found out that Terrapin was actually considering possible reasons why this is not true, I would conclude that Terrapin used "convinced" in deception. Instead, Terrapin said "everything seems to be physical empirically". The use of "seems" instead of "convinced" tells me that Terrapin is in some way open to other possibilities. Therefore the possibility of that form of deception is excluded by a careful use of words. That's the purpose.

...that's what a bijection does... — jorndoe

The numbers are infinite. The bijection is necessarily incomplete. Therefore, in this case the bijection "does not", and that contradict "does".

Being convinced does not mean you think there could not possibly be any reason to doubt. — Janus

Yes it does mean that. If you thought that there was any reason to doubt it. you would not be convinced. Being convinced means that you are not aware of any reason to doubt it. That this is "possibly" not the case can not enter into your mind or else you would not be convinced.

Convictions, unless they are untreatable faith-based convictions or based on tautology, are always open to possible future doubt even if no present reasons to doubt seem to exist. Think of science, and you will understand. — Janus

What may or may not happen in the future (that one may doubt in the future what one is convinced of now) is irrelevant, what we are talking about is being convinced now. If I am convinced now, I have no doubts. and I also "think there could not possibly be any reason to doubt". If I thought that there could possibly be a reason to doubt, I would not be convinced.

A proposition is not contradictory merely by virtue of stating something that is actually impossible, only if it states something that is logically impossible--which is certainly not the case here. — aletheist

As I said, it is impossible "by definition". This means that it is logically impossible, contradictory. "Infinite" is commonly defined in such a way that it is impossible, by definition, to pair up infinite things, because the task would never be complete. It is only by changing the definition of "infinite" to something else, that this might become possible. But then "infinite" loses it's meaning, so what's the point? You simply change the definition of "infinite" to create the illusion that the logically impossible is actually possible.

Just thought I'd mention the difference.

Also, you'd be one of those people asking Socrates: what is the point of asking what clouds are . — frank

Socrates was asking because he wanted to see if the people talking knew what they were talking about. Most often they did not. So that's the point to Socrates asking what clouds are, to demonstrate that the people talking about clouds didn't really know what they were talking about, because they couldn't even say what clouds are.

So you're talking about the average temperature and the average CO2 concentration over a 60,000,000 year period? What's the point in that?

Both are expressions of belief, aren't they? — Janus

No, in my customary usage they are not both expressions of belief. "It seems to be the case" implies doubt and therefore not a belief that it is the case, as doubt is opposed to belief. One cannot doubt that it is the case, and believe that it is the case, at the same time. "I am convinced that it is the case" expresses belief that it is the case.

My translation (Anscombe) reads "definite rule" rather than "particular rule".

I already know about it. The fact remains that we cannot actually pair them because there is an infinite number of them. So the proposition states something which is, by definition, impossible (i.e.it is contradictory). Therefore we ought to reject it as a falsity, as is customary for propositions which are recognized as self-contradictory. Whether or not it's basic high school mathematics is irrelevant. If it's a falsity it ought to be rejected.

How many integers are there? Infinitely many. How many even numbers are there? Infinitely many. If we paired up each integer with an even number, when would we run out of even numbers, but still have integers left? — aletheist

The problem being that we cannot pair them up because there is an infinite number of either one of them. So your conditional "if we paired up each integer with an even number", is a statement of an impossibility, and therefore must be dismissed as a false premise.

If we paired up every number with its square, when would we run out of one or the other? Never. — aletheist

Again, an impossible conditional. "If we did X" when X is impossible.

Logic convinces us that p.

Therefore, p seems to be the case, no? — Terrapin Station

No I wouldn't say that. When logic convinces me that p is the case I would assert that p is the case, not that p seems to be the case. I would only say that p seems to be the case if I wasn't convinced that p is the case.

"Seems to be the case" is simply another way of saying, "I believe this to be the case." — Terrapin Station

So you recognize no difference between "I am convinced that p is the case" and "p seems to be the case"? No wonder I do not agree with your ontology.

Notwithstanding that, I'm not seeing how what you're saying is not covered by "the player's responses". I don't have the text with me so you may need to correct me if I'm wrong, but I'm pretty sure it's plural and so would be talking about the responses of all the players as a whole. — Isaac

Yes it's plural, and that's exactly what I said earlier, "the players' responses" indicates a multitude of players. In my experience of observing games, and playing games, when one person steps outside the bounds of the rules (makes a mistake), that individual is corrected by the others playing the game. This is the obvious example of how an observer of the game would know when a player makes a mistake, that player is corrected by the other players. So, the issue, I wonder why Witty does not use this obvious example, and instead opts for an obscure example, which in my opinion (for the reasons explained) is not a good example. Why does Witty not state the obvious, the observer knows when a player makes a mistake because that player is corrected by the others?

Again I don't think Wittgenstein is in any way ruling this out, he's just also including the possibility of the player knowing themselves. Consider running offside in football. The player may not have intended to break the rule, but they only need look around to see that they have. — Isaac

But running offside in football is not making a mistake, it's a perfectly acceptable play, the players do it all the time and they know when they are doing it. They also know that it puts an end to the play, that's the rule. If the player kept running, as if the play was not ended, that might be a mistake, the player didn't know he stepped offside. However, there is also the possibility that the player knew, and then continuing on with the play is an attempt to cheat.

The issue here is what qualifies as making a mistake. that's what Witty asks at 51 "what is the criterion by which this is a mistake?". This is what Luke was getting hung up on at 53, what constitutes "the rule", because the same rule maybe expressed in different ways, and therefore it may play different roles in the game. So which expression of the rule, or role, do we turn to in determining when there is a mistake? We cannot say that there has been a mistake unless we know what the rule is. You think stepping offside in football is a mistake, according to your expression of the rule, I think that this is not a mistake, but continuing to play after stepping offside is a mistake according to my expression of the rule..

But none of this is relevant to the point and I don't want to get sidetracked. I'd rather just say yes, Wittgenstein chose a bad example. It doesn't change anything about the point he's making.

The point I see that you are leading toward is clearly relevant to the private language argument and if we're to keep this process on track (I'm guessing that's the aim, yes?) then we'd better leave that discussion for when we get there. — Isaac

I agree that we're going off track, but the point to stress is that the question hasn't really been answered yet, by Wittgenstein, what constitutes making a mistake.

Do you mean like "physicalism is false"? I assert that this is the case, though it doesn't seem to be the case. If this is what you mean, then logic often convinces us that what seems to be the case is not actually the case. So we assert on the merits of logic, that what seems to be the case is actually not the case, and what seems to be not the case is actually the case.

I don't get you. Are you trying to change the subject?

First, how do you know it is "usually not the case" that when a player makes a mistake they do not themselves recognise it. This sounds like an empirical conclusion. Do you have any studies to back this up with? — Isaac

I've played many games, and I've made mistakes in relation to the rules. I've also observed others making such mistakes. When I make a mistake, it is because I did not know, or did not understand the rule. So when the mistake is made I do not know or do not understand the rule. Therefore I do not recognize that a mistake was made, and so I do not make any indications in my actions that a mistake was made (as in W's example), because I do not know that a mistake was. What is required is that another player point out to me that a mistake was made. For some reason, Witty does not exemplify this.

If it is possible that a player genuinely not know they've made a mistake, and this is the case most of the time, then it follows that this would be the case for most of the players at any given time. — Isaac

Isn't that what "making a mistake" is though, not knowing that you were making a mistake? If you knew that it was a mistake, you would not proceed in that action which constitutes "the mistake". So it is impossible that the person making the mistake knows, at the time of making the mistake, that it is a mistake, because this contradicts the nature of "mistake". So, after making the mistake, if the person is to recognize that a mistake has been made, something must occur to bring to that person's attention, the fact that a mistake has been made.

Now consider the observer, trying to learn the rules from observation of the play. The observer is going to recognize that a mistake has been made by the same means that the player making the mistake recognizes that a mistake was made. So the observer doesn't determine that a mistake was made, from the actions of the player who makes the mistake, but from whatever else it is that occurs which would make the player making the mistake recognize that a mistake was made.

If there's a rule that most of the players at any given time are not aware of, then where is that rule kept? — Isaac

Right, I think that this is an issue which will arise. What if there's a rule that only one of the player is aware of, a private rule? As I tried to outline in my earlier posts in this discussion with you, we can avoid that problem by making a clean separation between prescriptive and descriptive rules. If, all the rules of the game are learned through observation as we've been discussing, the rules are purely descriptive. There is no such thing as a prescriptive rule, what one ought to do in this type of game, the one that is learned purely through observation. Therefore, such a "rule" which is known only to one or a few players, is not a rule at all, by this definition of 'rule" because it cannot be observed as a rule. The activities of this one, or very few players, which you describe as players who are aware of a rule that the others are not aware of, are actually exceptions to the rules, and are therefore mistakes.

No, I don't think that's what everyone does. If it seems to be the case, or appears to be the case, I do not assert that it is the case.