A Mathematical Interpretation of Wittgenstein's Rule Following Paradox

An argument against using pattern sequences in IQ tests.

Quoting jgill

An argument against using pattern sequences in IQ tests.

I'm reading this elementary textbook on math. The only way one can determine the persistence of a pattern i.e. know that a pattern you discovered will continue (is the correct one) is if you can explain it.

What means this?

Reply to Agent Smith Are patterns exactly equal to rules?

Notice the difference between simply listing the array: 2, 4, 8 ...

and requiring such array to: start from 2, multiply the previous number by 2 to get the next number in order.

Well, you might as well argue that we understand each word from observing patterns. Such as the way we follow the rule of the word "multiply" by doing what gives us results that accord with what we observe: we observe that multiplying 1 by 2 is 1+1; 2 by 3 is 2 + 2 + 2, etc, so when we follow my more specific rule above, we just follow the pattern to get 2+2 = 4; 4+4 = 8... So the rule is still just patterns packed together, each allowing some discrepancy of understanding.

Btw, I think at some point in Philosophical Investigations Wittgenstein actually discussed the margin where the rule learner turns from observing patterns to understanding rules - his conclusion if I remember correctly is just as obscure. I'll take a closer look at what he was trying to elaborate on.

Quoting D2OTSSUMMERBUG

Are patterns exactly equal to rules?

More or less. If we figure out a pattern, we have a rule that helps you find the next term in a sequence. If we work out a word's usage pattern, we have in our hands a rule that determines where it should be used.

.

Wittgenstein out of context is problematic. There are a bunch of paragraphs leading up to this statement, that are very helpful in understanding what Wittgenstein is on about. Most importantly the paragraph just before that sentence, which gives a practical example:

Ludwig Wittgenstein:
200. It is, of course, imaginable that two people belonging to a
tribe unacquainted with games should sit at a chess-board and go
through the moves of a game of chess; and even with all the appropriate
mental accompaniments. And if n>e were to see it we should say they
were playing chess. But now imagine a game of chess translated
according to certain rules into a series of actions which we do not
ordinarily associate with a game—say into yells and stamping of feet.
And now suppose those two people to yell and stamp instead of playing the form of chess that we are used to; and this in such a way
that their procedure is translatable by suitable rules into a game of
chess. Should we still be inclined to say they were playing a game?
What right would one have to say so?

Ludwig Wittgenstein:201. This was our paradox: no course of action could be determined by a rule, because any course of action can be made out to accord with the rule.

Do note that Wittgenstein employs the term "was" not "is". This is by no means a coincidence or mistake. The paradox was but is no more, Wittgenstein resolves it in his very next sentence.

Ludwig Wittgenstein:
The answer was: if everything can be made out
to accord with the rule, then it can also be made out to conflict with it.
And so there would be neither accord nor conflict here.
It can be seen that there is a misunderstanding here from the mere fact
that in the course of our argument we give one interpretation after
another; as if each one contented us at least for a moment, until we
thought of yet another standing behind it. What this shews is that
there is a way of grasping a rule which is not an interpretation, but which
is exhibited in what we call "obeying the rule" and "going against it"
in actual cases.

The entire point of Wittgensteins argumentation was that "interpreting a rule" and "obeying a rule" are two completely different things.

Ludwig Wittgenstein:
202. And hence also 'obeying a rule' is a practice. And to think one
is obeying a rule is not to obey a rule. Hence it is not possible to obey
a rule 'privately': otherwise thinking one was obeying a rule would be
the same thing as obeying it.

Now, let's take your numerical example:
From a 2, 4, 8 sequence we could interpret all kinds of pattern (rule) that this sequence follows - but as it has been established, our interpretation of the rule has nothing to do wether we're obeying it or not.

But what exactly is our rule then? Where does it come from? How can we confirm it?
The rule, in the case of such a sequence, is determined by the author. They are the ones that write down the numbers partaining to a pattern. The only way for us to know wether our interpretation of such a sequence is truthful to it's rules is to ask the one who made the rules if we are correct or not.

In other simplified words:
The only way to know if you're playing chess or if you're only doing something that looks like playing chess is consulting the rulebook of chess.

Consider the propositions

"The sequence x1, x2, ... is determined by the function f(x)"

"The function f(x) is determined by the sequence x1, x2, ... "

There are two permissible interpretations of the word 'determined' in the above propositions:

A) As an imperative when for instance normatively insisting that " f(x) means the sequence x1,x2,..

B) As a descriptive hypothesis when for instance alleging that a given sequence x1,x2,... obeys f(x)

Classically, the sign 'x1,x2...' is interpreted as an abbreviation for a particular sequence of infinite extension for which there isn't time to write the whole sequence down. Under this identification, interpretations of the form A leads to the identification of f(x) as also denoting a particular domain and image of infinite extension. Thinking of functions extensionally in this way leads to scepticism whenever it is asked if some unbounded sequence S obeys a given f(x), given that only a finite prefix x1,x2,..xn of S can be observed. In conclusion, hypothesis of the sort (B) aren't verifiable when thinking this way.

On the hand, in the Russian school of constructive mathematics, functions aren't directly interpreted as representing entities of infinite extension, but as being finitely describable computable maps whose domain is unbounded. But this can lead to the same impression of such functions as having actually infinite and precise extensions if one thinks of such functions as denoting ideal and physically infallible computation. Thus the same platonistic skepticism about rule-following arises as in the classical interpretation.

The alternative to adopt 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. Likewise, the domains and images of functions are also interpreted as being potentially infinite rather than as being actually infinite. Relative to this philosophy, rule-following scepticism is avoided due to the fact that the meaning of potentially infinite sequences and functions are both understood to be semantically under-determined a priori.

The problem of induction (Hume-Wittgenstein):

There's no necessity to a law of nature - things might've been a certain way, are that way, but there's absolutely no reason why it should be that way in times to come.

Crucial difference: Hume's insight implies a law of nature's violated. Wittgenstein's rule following paradox doesn't mean a law (rule) is broken; au contraire, Wittgenstein is saying is that all observations (word usage) are compatible with any conceivable law.

:chin:

Quoting Hermeticus

The entire point of Wittgensteins argumentation was that "interpreting a rule" and "obeying a rule" are two completely different things.

I believe this is a good interpretation, and the difference here amounts to the difference between a descriptive rule and a prescriptive rule. When we produce a descriptive rule, we come up with 'this is the way things behave'. That's a conclusion of inductive reasoning. There is nothing within the rule itself, to compel that a thing will behave like that in the next instance, so what causes the thing to behave like that, is a completely different issue.

If we say that being caused to behave according to a rule is what "obeying a rule" means, then we see the difference between "interpreting a rule" (as in understanding the described behaviour), and "obeying a rule", (as in causing oneself to act according to the described behaviour). On the other hand though, we can say that "obeying a rule" is to act in a way which can be judged as being consistent with the described rule.

The difference between these two interpretations of "obeying a rule" is the difference between judging the cause, and judging the effect. Wittgenstein opts for the latter, making "obeying a rule" something which is observed after the fact, rather than something decided prior to the act, in the sense of interpreting a prescriptive rule, and acting accordingly. So the prescriptive rule is not relevant to Wittgenstein's position on rule following, and we must be careful when reading him not to misunderstand.

Quoting Agent Smith

au contraire, Wittgenstein is saying is that all observations (word usage) are compatible with any conceivable law.

Not quite. Wittgenstein only criticised logical conceptions of meaning, especially in relation to the view that the meaning of a proposition is static and a priori decidable . He didn't criticise individuals for their idiosyncratic interpretations of rules and language, which generally don't invoke theoretical interpretations of meaning.

There is a world of difference between speculating that an event E must logically follow from the a priori definition of a law L, versus recognising for oneself post-hoc, that E follows from L.

For example, often when you judge for yourself that two colours are the same, (which you usually do without any external guidance), your recognition wasn't contingent upon you invoking a priori definitions of the colours involved and calculating a truth value.

This is the reason why Wittgenstein wasn't a verificationist - meaning doesn't normally involve processes of verification - ergo Wittgenstein wasn't against the idea of private meaning.

Quoting sime

Not quite.

In other words,

some observations are not compatible with any law. That means...

Reply to Agent Smith

Right, very interesting. It implies that the patterns that you point out are actually commensurate with your specific thought process going into the problem. Not that you've necessarily discovered a pattern that is actually there, but that your brain discovered only that pattern that makes sense to your brain. Either because of how we perceive symmetry, or some other pattern processing function. But, then again, a good test as to whether that was the case, would be to experiment with the usefulness of the specific pattern in question. Which is where I think things like sacred geometry come from out of history. Very cool topic, Smith.

-G

Quoting Garrett Travers

Very cool topic, Smith.

:up: You're :cool: too, sir/madam as the case may be.

Reply to Agent Smith

Sir, will do just fine. :cool:

Quoting Metaphysician Undercover

the difference here amounts to the difference between a descriptive rule and a prescriptive rule.

The differentiation between "descriptive rule" and "prescriptive rule" is fantastic. :ok:

We could also say the difference boils down to epistemological demand as well, which is why one method raises no questions, while another only raises questions.

A descriptive rule faces the typical scrutiny of any epistemological consideration along the lines of subject/object matters. That's the interpretation part. Descriptions require relations, a context in which they can be established. We could say a truly complete description of anything would require us to describe the entire world as context alongside with it.

A prescriptive rule on the other hand doesn't make any demand to knowledge at all. It is a dictation of how things must be. The context - as far as I can think - is always already implied by engaging with the rule itself (context of life, math, games, society, law, language etc)

Quoting Metaphysician Undercover

The difference between these two interpretations of "obeying a rule" is the difference between judging the cause, and judging the effect. Wittgenstein opts for the latter, making "obeying a rule" something which is observed after the fact, rather than something decided prior to the act, in the sense of interpreting a prescriptive rule, and acting accordingly. So the prescriptive rule is not relevant to Wittgenstein's position on rule following, and we must be careful when reading him not to misunderstand.

I actually think that's exactly what Wittgenstein himself is trying to get across. To not misunderstand one for the other.

Ludwig Wittgenstein:It can be seen that there is a misunderstanding here from the mere fact
that in the course of our argument we give one interpretation after
another;(descriptive rule) but that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" (prescriptive rule)

Quoting Garrett Travers

Sir, will do just fine. :cool:

:up: I knew you were a knight! Sir!

Quoting Hermeticus

Now, let's take your numerical example:
From a 2, 4, 8 sequence we could interpret all kinds of pattern (rule) that this sequence follows - but as it has been established, our interpretation of the rule has nothing to do wether we're obeying it or not.

But what exactly is our rule then? Where does it come from? How can we confirm it?
The rule, in the case of such a sequence, is determined by the author.

I find this confusing.

If there genuinely is a pattern with 2,4,8,... then that pattern will describe the number chain or series to infinitum or otherwise it's a wrong pattern or the series of numbers is basically without a pattern, patternless. Here to talk about rules it would be better to talk about algorithms in the general sense. And either you have an algorithm that correctly tells you how the series 2,4,8,... goes or either you have the wrong algorithm or the series is non-algorithmic.

Nothing to do with the author, the subject. Our understanding or incorrect understanding about the series doesn't brake this logic.

Quoting ssu

If there genuinely is a pattern with 2,4,8,... then that pattern will describe the number chain or series to infinitum or otherwise it's a wrong pattern or the series of numbers is basically without a pattern, patternless. Here to talk about rules it would be better to talk about algorithms in the general sense. And either you have an algorithm that correctly tells you how the series 2,4,8,... goes or either you have the wrong algorithm or the series is non-algorithmic.

yes, in the case of a potentially infinite sequence of numbers, it is meaningless to consider any particular function, let alone algorithm, as being descriptive of the sequence unless and until the sequence comes to an end. Until then, one cannot even decide whether or not the sequence is computable. Nevertheless it is meaningful in the meantime to speak of falsified hypotheses in relation to the sequence.

However, the problem goes further than that, because on Kripke's interpretation, the skepticism is calling into question the very meaning of "algorithm", and hence the distinction between algorithmic versus non-algorithmic processes, which computing and constructive mathematics take for granted. Such philosophies treat the definition of an algorithm to be isomorphic with the input-output pairs generated by it's execution, which presupposes the existence of ideal calculators. But as we know practically, physical implementations of algorithms have finite capacity and finite reliability, making the intensional definitions of functions misleading with respect to their implemented behaviour. Kripke is asking how it can be decided that a sequence corresponds to a given total function, given the irreparable inability to define what the 'correct' outputs of the function are for most of it's inputs.

Quoting sime

yes, in the case of a potentially infinite sequence of numbers

That what is generally depicted with 2,4,8,...
Not that it ends sometimes, which would be 2,4,8,.....,a.

Quoting sime

it is meaningless to consider any particular function, let alone algorithm, as being descriptive of the sequence unless and until the sequence comes to an end.

?

if the sequence is for example N, then the correct algorithm is "list all natural numbers". And natural numbers don't come to an end.

Either there is the correct description or their isn't. Those are your choices, they aren't meaningless.

Quoting ssu

if the sequence is for example N, then the correct algorithm is "list all natural numbers". And natural numbers don't come to an end.

To understand the paradox using your example, you have to distinguish the intensional definition of a function, such as one reproducing the natural numbers

i.e. f(n) = n for all n in Nat

from a potentially infinite list of elements such as

S = { 1, 2 , 3, ... }

that looks like it might be a prefix of Nat.

By 'potentially infinite' I mean that S is finite but of unknown size and whose elements are only partially defined a priori (in the above case, only the first three numbers). it's remaining elements are denoted by the dots "..." and are a priori unknown and decided when S is instantiated.

Obviously, until S is fully instantiated it cannot be decided as to whether S is a prefix of N or some other function. The paradox concerns the fact that one cannot know in advance what the prefixes of N are.

One can try to define the prefixes of Nat intensionally as a function, namely

P(n) = {0,1,....n} for all n

But in order to know what all of the prefixes are as implied by this definition, we would need to define what it means to treat n as a free-variable that can be substituted for any natural number. But if this definition of a free-variable is also intensionally defined, we will have gotten nowhere. So the only way to decide what the prefixes are, is to resort to writing them out extensionally for some random number of terms, which we can represent as the potentially infinite set

{ {0}, {0,1}, {0,1,2} , ...}

This set will be instantiated with a random number of prefixes, relative to which it will be decided, in a spur-of-the-moment bespoke fashion, as to whether or not S is a prefix of Nat.

Quoting Agent Smith

I'm reading this elementary textbook on math. The only way one can determine the persistence of a pattern i.e. know that a pattern you discovered will continue (is the correct one) is if you can explain it.

Prove it continues. Mathematical induction is one way. Providing an algorithm is another. For example, I am working on a theorem now that has the product

[math]\prod\limits_{k=1}^{n}{\left( 1+\frac{k}{{{n}^{2}}} \right)}[/math]

And its fairly simple to prove that, for all positive integers n,


[math]\prod\limits_{k=1}^{n}{\left( 1+\frac{k}{{{n}^{2}}} \right)}\le e[/math]

Reply to jgill

But here you predetermined the pattern of the sequence. Suppose I saw the ten first members of the pattern of the outcomes of your products. All smaller then e. Are we sure that only your prescription for generating the numbers (2, etc.) is unique?

Quoting Agent Smith

This was our paradox: no course of action could be determined by a rule, because any course of action can be made out to accord with the rule.
— Ludwig Wittgenstein

Reply to jgill
Given some finite set of numbers, how many formulae are there that fit?

Given some finite set of actions, how many rules are there that fit?

In both cases, an indeterminate number.

Wittgenstein's response is found further into the same remark:

...there is a way of grasping a rule which is not an interpretation, but which, from case to case of application, is exhibited in what we call “following the rule” and “going against it”.

One shows one has understood a rule not by stating it, but by following it.

Following or going against a rule is using the rule. So he is making the point that stating the rule is not grasping the rule; using the rule is grasping the rule.

Hence, don't look to the meaning, look to the use.

Quoting Cornwell1

Suppose I saw the ten first members of the pattern of the outcomes of your products. All smaller then e. Are we sure that only your prescription for generating the numbers (2, etc.) is unique?

Good point. Fortunately that's not an issue normally in a mathematical discussion. Of course there may be other "rules" generating that finite sequence. For the most part in math research one generates a sequence of numbers according to a given process, and then tries to ascertain whether an observed outcome is true for all numbers so generated. I'm happy it doesn't go the other way around!

Quoting jgill

I'm happy it doesn't go the other way around!

Haha! You could make it your opus magnus. "Okay, for n=56545434566, is the product smaller than e... yes! Next n..." :joke:

Quoting Hermeticus

I actually think that's exactly what Wittgenstein himself is trying to get across. To not misunderstand one for the other.

It can be seen that there is a misunderstanding here from the mere fact
that in the course of our argument we give one interpretation after
another;(descriptive rule) but that there is a way of grasping a rule which is not an interpretation, but which is exhibited in what we call "obeying the rule" (prescriptive rule)
— Ludwig Wittgenstein

I don't exactly agree. For Wittgenstein, "obeying a rule" is to be observed and judged to be acting in a way which is consistent with the rule, hence his use of "exhibited". The need for a prescriptive rule really disappears for him. For a person to obey a prescriptive rule, in the sense of 'I should respect the rule and do what I ought to do', this requires that the person interpret the rule, then move to act according to one's interpretation of it. Notice that the interpretive part is what he is trying to avoid. So for him, "obeying a rule" is to be described as acting in a way consistent with the rule. And the means by which the person comes to act that way becomes sort of irrelevant. The person might just be copying the actions of others, or whatever.

Reply to Banno

The way a word is being used (following a rule) let's us know what its definition (comprehension of the rule) is in a particular language game. The definition (rule) changes with the language game one is playing e.g. the word "god" means different things in theism proper, deism, and pantheism, these being distinct language games, each with its own unique rule (definition) for the word "god". In that sense, we could say that the word "god" lacks an essence, a common thread that runs through all of the aforementioned domains.

Questions:

1. True that, sticking to the example above viz. "god", words lack an essence that's cross-domain (the pantheistic god is different from the deistic god and both have no essential connection with the theistic god). However, within a given (one) language game (say deism), how does one pin down meaning? Doesn't the word "god" in deism have an essence? 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)?

2. Indeed, as regards a word, there's a difference between stating the rule and following the rule governing that word. Comprehension of a rule is best demonstrated by a person following the rule rather than just being able to state it. Step 1 (A rule) [math]\rightarrow[/math] Step 2 (Comprehension of the rule) [math]\rightarrow[/math] Step 3 (Following the rule). There's a lot going in step 2.


Reply to jgill

The way I understand it is that when we're asked to explain a pattern, a necessary step if one is to make the case that one has homed in on the right pattern, we must be able to demonstrate why the pattern, well, makes sense. Consider the pattern in the sum of the series 1, 3, 5, 7, 9...

1 + 3 = 4 (perfect square)
1 + 3 + 5 = 9 (perfect square)
1 + 3 + 5 + 7 = 16 (perfect square)
.
.
.
The sum of the odd integer series above is always a perfect square.

Now how do I explain it?

I use tiles/polyominos to show that when I use a monomino and a tromino, I can construct a square. Add a pentomino and another square can be constructed, so on and so forth.

Quoting Agent Smith

Now how do I explain it?

[math]\sum\limits_{k=1}^{n}{\left( 2k-1 \right)}=2\sum\limits_{k=1}^{n}{k}-n=2\frac{n(n+1)}{2}-n={{n}^{2}}[/math]

Nighty nite, my friend

Reply to jgill G'nite.

Reply to Agent Smith

Language games are neither discrete nor inviolable. Deism, theism and pantheism are not constituted by different uses of "God".

To your question, is there a difference between a rule and an essence... Well, I think the notion of an essence cannot be made clear without being wrong. If oyu think otherwise, have a go for us.SO rules and essences are not he same sort of thing.

IN your second point you seem to think that the rule proceeds the use. More usually, it would go the other way. Unless one is Tolkien, does not construct a dictionary and then use the language. One examines the language and deduces the rules. The rules are usually post hoc.

More interestingly, use overrides the rules, in cases such as nice derangements of epitaphs.

Reply to Banno

Quoting Banno

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

Quoting Banno

One examines the language and deduces the rules.

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:

Reply to Agent Smith 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.

Quoting Banno

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

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.

Quoting Banno

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.

[quote=Dr. Watson]I'm beginning to see dimly what you're driving at.[/quote]

Reply to Agent Smith Quoting Agent Smith

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.

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.

Quoting Agent Smith

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)?

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.

Quoting Metaphysician Undercover

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.

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?

Reply to Antony Nickles

[quote=Cratylus]:zip: Wriggle finger.[/quote]

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)

Quoting Antony Nickles

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?

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,.

Quoting sime

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.

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.

Quoting sime

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

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?

Quoting sime

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.

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

Reply to Agent Smith Quoting Agent Smith

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

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.

Quoting Agent Smith

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

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.

Quoting Agent Smith

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.

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).

Reply to Antony Nickles 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.

Reply to Agent Smith Quoting Agent Smith

That went over my head I' afraid.

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).

Reply to Antony Nickles 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?

Reply to Agent Smith Quoting Agent Smith

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.

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.

Quoting Agent Smith

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?

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.