Well, I think the notion of an essence cannot be made clear without being wrong — Banno

One examines the language and deduces the rules. — Banno

What's the connection between a rule and a definition? Is there one or none?

I have a feeling that Wittgenstein had a different view of language and philosophy. I deduce that he was of the opinion that there had to be some necessary connection between words and their referents (the standard non-Wittgensteinian take on meaning). Why else would he think meaning is use was such a big deal? Of course meaning is use. :chin:

Again, the slogan is misleading.

Wittgenstein showed us how to better answer philosophical questions by forgetting about essences, definitions and meanings and instead looking at what is being done in using words. In §201 he is reinforcing this way of doing philosophy by showing the limitations of considering just the rules of a language game. One must go beyond the rules and look at what is being done.

Well, I think the notion of an essence cannot be made clear without being wrong — Banno

Even Kripke's use?

We need to distingush two forms of ambiguity

Intensional Ambiguity of Extensions: A given extension, e.g. a sequence [s(1),s(2),s(3),...], corresponds to an infinite number of functions. This is an epistemic form of ambiguity studied by Theoretical Machine Learning and Statistical Learning Theory.

Extensional Ambiguity of Intensions: A given description of a function, say f(x) = 1/x, corresponds to an infinite number of possible extensions, e.g not only [1,1/2,1/3], but also [1,1/2,0,312,9998].
Although we immediately recognise the latter as being false, such pathological interpretations cannot be exhaustively ruled out by any finite description of f(x) . This is a semantic form of ambiguity that Quine and Wittgenstein were concerned with, that machine learning and statistical learning theory typically ignores.

Following Quine in Truth By Convention (1936), it is impossible to exhaustively define a function extensionally in terms of a graph-plot of the function's values, since any graph plot is finite, leaving many semantic holes. Therefore the meaning of a function cannot be explicitly stated by convention, and the same is true for the meaning of logic.

The upshot is that conventions cannot explicitly describe or prescribe how users use mathematics and language in general, which implies that linguistic conventions are largely a post-hoc expression of how people decide to use language in practice, rather than the converse.

There is a very real and evident problem with the way that Wittgenstein describes obeying a rule, and that is that this way of looking at rule following pays no real respect to the internal mechanism of the mind of the person who is said to have obeyed the rule.

So for example, if we give two distinct people from two distinct parts of the world, the same division problem, they might use completely different mental techniques to come up with the same correct answer. Since they both have the same correct answer, we'd say that they both followed the same rule. But if we timed the activity, we might find one quicker than the other. And if we enquire as to the procedure, or give them a difficult 'long division' problem, so that we can observe their mental activity being expressed on paper, we'd see that they each followed a different mental procedure. Therefore there is a real issue of very distinct mental processes each leading to the same conclusion, and the observation of obeying the same rule, because each produces the correct answer, when the processes being followed are actually distinct.

Wittgenstein showed us how to better answer philosophical questions by forgetting about essences, definitions and meanings and instead looking at what is being done in using words. In §201 he is reinforcing this way of doing philosophy by showing the limitations of considering just the rules of a language game. One must go beyond the rules and look at what is being done. — Banno

I'm beginning to see dimly what you're driving at. — Dr. Watson

The sequence 2, 4, 8,... can be made to fit with an arbitrary number of patterns i.e. a word's usage pattern can be made to match any rule whatsoever. — Agent Smith

This takes Witt's realization as the discovery of a paradox which importantly impacts our ability to have any certainty at all (Kripke will call this the skeptical paradox). However, that a "rule" may not be able to be pre-determined nor causal, only means that rules do not do what we want them to: to make the judgments of our actions, or others', certain beforehand--to already know the best thing to do. Rules just do not play the part in meaning and justification that we want (think we need) them to. Our judgments are made afterward (as @Banno points out), based on the criteria for doing such a thing; and so the discussion of obeying a rule is not an explanation (of how action or expression works--by rules) nor "resolved" as @Hermeticus characterizes it; it is an example (of what matters in being said to have "obeyed a rule"), as there were examples of calls and slabs, and chess: to show us the mechanism for them each individually, and the limitation of them to be a general analogy.

I suppose what I mean to inquire is whether there's any difference at all between essence (of a word) and rule (how a word is supposed to be used)? — Agent Smith

Wittgenstein puts it (#371) that the essence of a thing is shown in how we would judge it to be such a thing. That what is essential to us is what interests us, how we value it, what differentiates it, refines it, etc. Here, the essence of "obeying a rule" is brought out in what criteria make up our judgment of how rules are obeyed.

Therefore [ because two people can come to the conclusion in different ways ] there is a real issue of very distinct mental processes each leading to the same conclusion, and the observation of obeying the same rule, because each produces the correct answer, when the processes being followed are actually distinct. — Metaphysician Undercover

But isn't this an observation about following a rule? and not about obeying a rule? We need not have "followed" a rule to be said to have obeyed it. "Why did you drive under the speed limit?" "I followed the rule." or "What speed limit? I'm just driving here." But is it our lack of rationality that causes the fear here? or that there remains a lack of certainty, even if "rules" are involved?

:zip: Wriggle finger. — Cratylus

Despite my many attempts to grasp Wittgenstein's point, I have to confess nec caput nec pedes.

Wittgenstein's right on the money when he claims that words are essenceless across domains, but I fear people misunderstand this to mean that words are sans an essence within domains. Domains herein loosely corresponds to language games.

Imagine there's a rule on how to use a particular word. I apply the rule (as I apprehend it). However, my rule is not the same as your rule and yet the first few instances the two of us have used that word are compatible with both our rules. That we're using two very different rules is hidden for this reason.

It seems the rule following paradox has something to do with private languages (beetle-in-the-box)

But isn't this an observation about following a rule? and not about obeying a rule? We need not have "followed" a rule to be said to have obeyed it. "Why did you drive under the speed limit?" "I followed the rule." or "What speed limit? I'm just driving here." But is it our lack of rationality that causes the fear here? or that there remains a lack of certainty, even if "rules" are involved? — Antony Nickles

I really can't see the distinction you are trying to make here. What would it mean to obey a rule without following it? Notice "follow" implies a temporal posteriority, as does "obey". I really don't see how one could obey a rule without following it.

In fact, in reading your post, I do not understand your use of "rule" at all. You appear to remove the necessity of temporal priority of "rule" in relation to "obeying a rule" by denying causality from "rule", but then you say that our judgements are "based on the criteria for doing such a thing". If a rule is not the criteria for making such a judgement, therefore causal in making such a judgement, then what is a rule?

In other words, what meaning could "obeyed a rule" possibly have, if the thing referred to with "rule" is not causal in judgement? Either the person acting must be caused by the rule to act in a way consistent with the rule, or the person observing must be caused by the rule to judge the one acting, as obeying the rule. If we are going to assume that there is such a thing as a rule, so that "obeying a rule" says something meaningful, I see no way to remove the causality of the rule from such a judgement.

It makes no difference to the issue of the causality of the rule in judgement, if the person acting judges oneself to be obeying a rule, or the person observing judges the actor to be obeying a rule. In each case, a person must interpret the rule, and interpret the act, and make the judgement as to whether the act "obeys" the rule. The fact that the person acting makes the judgement prior to the act being made, while the person observing makes the judgement posterior to the act, has little or no significance in relation to the rule being causal in the judgement.

In the parlance of computer science, criteria constituting what it means to obey a given rule falls under Denotational Semantics and in the case of a function refers to it's tabular definition.

For example, part of the tabular definition of the total function f(x) = 2x can be specified as
{(0,0), (1,2), (2,4)}. In general, we can provide partial definitions of f in terms of partial functions.

A central question that denotational semantics is supposed to answer, is given that we only have the time to write down partial functions, what does it mean to assert that f(x) has a complete tabular definition as a total function?

In contrast, how a rule is followed, which is in this case concerns how f(x) is computed, is addressed by Operational Semantics. For computer science, this refers to the infinite number of possible pathways for computing the value of a function in accordance with it's specification (as described in terms of denotational semantics)

Lastly, axiomatic semantics specifies the imperative implementation of a function as a computer program running on a finite state machine (recalling that the denotation of a function doesn't possess the notion of a state).

For natural languages, an individual's mental interpretation of their public language, which eludes ostensive definition, is analogous to the operational and axiomatic semantic aspects of formal languages which elude denotational definition.


The paradox of logic that Wittgenstein was colloquially referring to, that was initially raised by Lewis Carroll and formally expanded upon by Quine in his attack on the Analytic-Synthetic distinction, involve the fact that there isn't a way to derive the complete denotational semantics of any given function, either by fiat or by appealing to some other form of semantics, due to the essential incompleteness of any type of semantic specification. To put it colloquially, it isn't possible to give an exhaustive account of what it means to obey a given rule, because a tabular definition of the said rule can never be finished, implying that the intended meaning of a rule is publicly under-determined.

To use the example above, how can we nail-down the complete tabular definition of the total function
f(x) = 2x ? At most we can write a finite portion, and then intimate the rest with dots:-

f(x) := {(0,0), (1, 2), (2 ,4 ), ...}

but how can the gesticulated meaning of the dots "..." in this context be interpreted to refer to an implicit yet unambiguous definition? One might try appealing to supposedly finite denotational semantics in the form of recursion :-

f(0) = 0,
f(x) = f(x-1) + 2

But then we need to complete a table specifying how to map the variable x to f(x) for every possible value, which is impossible, so we have gained nothing. (Consider the fact that every computer program implementation of 'f' will overflow at some value for x, that varies in accordance with the operational and axiomatic semantics of the CPU, OS and compiled executable that varies in each and every use case).

Domain Theory is the theory appealed to by computer scientists for completing denotational semantics in such a way as to pretend that the 'private' axiomatic and operational semantics of a function are independent of it's 'public' denotational semantics. The theory fails to acknowledge the essential incompleteness of denotational semantics and merely hides the fact by implicitly defining the total function f(x) = 2x to be the fixed point of a functional F( g, x) , e.g

F(g,x) :: (Int -> Int) -> Int -> Int
F(g,0) = 0
F(g,x) = g(x-1) + 2 If g(x-1) is defined, else
F(g,x) = undefined

Applying F to the totally undefined function called 'bottom' and iterating repeatedly, leads to the increasing sequence of partial denotations

F (bottom, x) = {(0,0), otherwise undefined }
F( F(bottom, x), x) = {(0,0), (1,2), otherwise undefined }
F( F(bottom, x), x) = {(0,0), (1,2), (2,4), otherwise undefined}
...

The illusion of f(x) = 2x as a definite total function with complete denotational extension is generated by appealing to the definition of f as the fixed point f := F( f, x) , which can then used as a definition for the earlier expression

f(x) : = {(0,0), (1,2), (2,4), ...}

At the fixed point f, the above functional F ignores it's function parameter entirely and so the definition of f in this case is more or less identical to the earlier recursive definition of f above. Hence all this definition does is reinterpret the denotational ambiguity of f in terms of the denotational ambiguity of functionals.

The lesson here, is that the meaning of any word or rule isn't definable in closed form, ergo

i) The meaning of mathematics isn't reducible to logical axioms, and neither is the meaning of logic.

ii)The meaning of language is under-determined by, and cannot be grounded in, any explicitly stated convention, whether publically or privately given, as Quine and in high probability Wittgenstein, concluded,.

To put it colloquially, it isn't possible to give an exhaustive account of what it means to obey a given rule, because a tabular definition of the said rule can never be finished, implying that the intended meaning of a rule is publicly under-determined. — sime

The word "exhaustive" has a meaning here that most mathematicians would ignore. "f(x)=2x for all x that are positive integers" pretty much says it all.

F(g,x) = g(x-1) + 2 If g(x-1) is defined, else — sime

You call F(g,x) a "functional". This is not the commonly accepted use of the expression among math people. A functional operates on a function and produces a (real or complex) number. Not another function. Definitions in CS may differ from those in math.

I guess the arguments on this thread are too subtle for me. And I admit I don't read the long posts carefully. What I see is starts of patterns that follow from some algorithm, like f(x)=2x for x positive integers. Then one asks, Are there other algorithms that produce the same existing pattern? (2,4,6,...) :(2,4,6,8,...) vs (2,4,6,7,...) e.g. The answer is yes. This is such an obvious conclusion. I've never encountered this sort of conundrum in my research. Usually one tries to show a well-defined pattern arising from some sort of process has a certain property, like convergence or divergence.

But math is so incredibly diverse I'm sure there are those in the profession that ponder such possibilities.

@fdrake is more up on modern math. Comments?

Brouwer's philosophy of Intuitionism, in which ' x1,x2,... ' is interpreted as referring to partially defined finite sequence of unstated finite length, rather than as referring to an exactly defined sequence of actually infinite length. In other words, x1,x2,... is interpreted as referring to a potentially infinite sequence whose length is unbounded a priori, but whose length is eventually finitely bounded a posteriori at some unknown future date. — sime

What passages in the writings of Brouwer (or in writings about him) do you believe are fairly rendered that way?

Despite my many attempts to grasp Wittgenstein's point, I have to confess nec caput nec pedes. — Agent Smith

Well, as with @Metaphysician Undercover requiring that following and obeying be subject to the same necessity--not seeing that we may follow, for instance, our heart instead, or cross a line (in disobedience, but, necessarily, against it specifically)--maybe heads (e.g., following rules) and tails (e.g., deciding ends) are not the point and it is your desire to make something (find some knowledge) which is under investigation, and so confusion is the starting point, not a reason to give up. Instead of projecting, put yourself in the position of asking the questions he does, feel the reason for the others' statements that he quotes, etc. I remain open to answer any questions about what I wrote, or clarify.

Imagine there's a rule on how to use a particular word. — Agent Smith

Not providing an example makes rules sound ubiquitous, but, again, I would argue that the passage is investigating how we follow rules (what we want from that and the disappointment of it), and not making a claim that rules fundamentally make up our use of language or our actions. Does using a particular word usually involve "rules"? I can say you aren't using the right word ("you're really eager, not anxious"), but that is general, as is not using any word appropriately (and correctness can be for no reason, or just a boundary, or subject to debate even apart from whether a rule needs interpretation), but if we want a rule about a single word, maybe: don't shout fire in a crowded theater, though this seems a rule on the border for a word and an act? As would be rules about apologies, excuses, etc., and so then maybe it is essential to have an example here.

I apply the rule (as I apprehend it). However, my rule is not the same as your rule and yet the first few instances the two of us have used that word are compatible with both our rules. That we're using two very different rules is hidden for this reason. — Agent Smith

If we choose to follow the rule, we do the same thing. If we interpret the rule a different way, we act differently. If we disobey the rule, we interpret it the same, but defy it. And we may also not be following the rule, yet still act coincidently. Witt points out that following a rule is not like focusing on a line to see what to do next (#223), so the idea that we act the same way but somehow apprehend the rule differently is an illusion (an imagining to insinuate skepticism in order to create the idea of a specialness to us). Again, the paradox is something only if rules are to be everything, else it is a paradox showing the powerlessness of rules to provide the certainty we want (or the skeptical quagmire we create to allow us to be the center of the universe).

That went over my head I' afraid.

That out of the way, it seems I did catch Wittgenstein's drift which is, different rules may overlap until a point that is, this point itself determined by factors that yield divergent results (word usages) for either one or all rules at play.

The agreement among different people on how to use a word (rule) is then purely coincidental (the pattern just happens to match in the first few/hundred instances; a fluke).

What implications does this have on philosophy? Agreements, if any, are illusory. what about disagreements? Discovery of illusory agreements. Or, bewitchment by language.

That went over my head I' afraid. — Agent Smith

Well that's disappointing, but I remain willing to elaborate. Bottom line my point is that rules are an example he uses, not an explanation of how everything works (except rules). We want rules to be the answer because it satisfies the uneasiness we feel that our world is arbitrary--as you say, we call it "illusory" or "coincidental". The 101st time things don't go well does not, however, mean that everything is quicksand; only that sometimes we have to step in and reflect and consider and carry ourselves forward into the future of our lives, shared up to that point (not agreed).

I've interpreted Wittgenstein in my own way, perhaps in a language game of my own invention. If Wittgenstein is right, no language game is right or wrong i.e. anything goes, oui? After all, essence, the key ingredient for judgments right/wrong is missing.

What's the difference between share and agree? Could I share a word with someone without some agreement as to what it means with that someone?

If Wittgenstein is right, no language game is right or wrong i.e. anything goes, oui? After all, essence, the key ingredient for judgments right/wrong is missing. — Agent Smith

Witt is not examining how we judge a whole language game (say, the practice of cannibalism) so much as an act or expression within such a practice (a "concept" is his term); he looks at playing chess, pointing, calling, thinking, seeing an aspect, and... following/obeying a rule. So I would say he is not judging whether, say apologizing, is right or wrong, but whether your apology is right/wrong. Additionally, as Nietzsche broke ground on, we don't judge an apology as right or wrong (or true or false Austin will also say), but whether it is done correctly or incorrectly, is appropriate or inappropriate, felicitous or infelicitous. And the way we make those judgments is based on whether an apology hits the marks necessary to consider it an apology at all, whether it comes off well enough to be judged (after the fact) to be successful, etc. And Witt labels those marks as criteria, not (predetermined) rules (though some criteria are rules we have set for judging--say, for figure skating). They are the measures of a concept, to which he will make claims that he calls the "grammar" of a concept. And those "logical" requirements are the expression of what is essential to an apology being an apology (and not an insult or a back-handed threat)--so, no, essence is not missing. It just doesn't do what it was supposed to before (say, with Plato) in being universal, abstract, certain, etc. (not arbitrary, concidental), and he is not saying rules satisfy that role either. We do not have the certainty that if we follow a rule we will always be right, or if we obey a rule, we will never be wrong.

What's the difference between share and agree? Could I share a word with someone without some agreement as to what it means with that someone? — Agent Smith

The concepts we have are part of our lives together--we never got together and "agreed" on what an apology would be. We have shared our lives, our customs, alongside each other. And wrapped up in those practices are what matters to each concept, what counts towards it, how we judge it, how we fail or it falls apart, and the excuses, responsibilities, implications involved, etc. In looking--as we are told to--we gain a wider view of the unspoken criteria for our shared lives which we usually never consider.