I think it is arguable that nearly all humans find counting and the basic arithmetical operations intuitive, so it's not arbitrary, Mathematicians have specialized skills that enable them to find things intuitive that the layperson cannot even comprehend because they don't have the requisite training or ability.

It looks like we are going to continue to disagree, but that's OK with me. I believe I would change my mind if given good reason to, but I haven't seen anything approaching such a reason thus far.

Imagine there is a wedding, and there are 68 guests from one side of the family and 57 from the other side. — Janus

Of course, applications of "+1" include practical applications. The point is that the rule must be applied to each case; it does not reach out to the future and the possible and apply itself in advance.

There is a forward problem of mapping rules — Apustimelogist

Yes, that's part of W's point. We can apply the rule to imaginary or possible cases, but we have to formulate them first. We cannot apply a rule to infinity. Hence mathematical induction.

It is natural simply because we can intuitively get the logic — Janus

.
There's truth in this. In some ways, "getting" a logical point is like "getting" a joke, If someone doesn't "get" modus ponens or a joke, we don't formulate more arguments. We try to help them "see" the connections.

But the fact that we mostly agree is not inevitable, not guaranteed. It is a "brute fact", which is the foundation of logic (and other rules). Bedrock is reached.

Or, to put it another way, if these agreements fail, we become bewildered and attribute the problem, not to the rule, but to the person who cannot follow it.

An afterthought.

Kriipke's sceptic does not escape from all this. Posing the problem takes for granted that we can recognize ("get") the difference between addition and quaddition. So posing the problem is based on, and does not bring into question, the agreement..

Could there be an arch-sceptic who cannot see the difference? Perhaps. But such a person could not join in our debate.

Yes, that's part of W's point. We can apply the rule to imaginary or possible cases, but we have to formulate them first. We cannot apply a rule to infinity. Hence mathematical induction. — Ludwig V

Yes, obviously induction is one of the big parts of this, but I wasn't intentionally referring to that. I was referring to the idea of starting with some repertoire of rules and using it to generate some behavior (e.g. the behaviors people acceptably think of as addition). Very true, the induction problem applies even to this issue which just emphasizes Wittgenstein's points, and I have been an advocating for my interpretation him this whole time, even if inarticulately.

I think it is arguable that nearly all humans find counting and the basic arithmetical operations intuitive, so it's not arbitrary, Mathematicians have specialized skills that enable them to find things intuitive that the layperson cannot even comprehend because they don't have the requisite training or ability.

It looks like we are going to continue to disagree, but that's OK with me. I believe I would change my mind if given good reason to, but I haven't seen anything approaching such a reason thus far. — Janus

I genuinely think we agree on more than you think but i think you have a different understanding or interpretation of the issue that is put forward.

The question of intuition is arbitrary because this is about the notion of objective rules or meanings. Why does intuition matter for objectivity? A putatively objective scientific theory should be true regardless of intuition. The truth of thermodynamics doesnt depend on my cats ability to find it intuitive. Intuition doesn't stop behaviour being describable in a certain way, and if you want to appeal to intuition then I will have to ask you to define what you mean further, which you haven't tried to do so far because I think you will know that will be very difficult (even if you could, I think its always possible to provide some quus-like alternative, or continue the regress of definitions or perhaps point to counterexamples like Moliere did in terms of how your counting example cannot be identical to addition semantically); however, without such definitions, how can I know you mean what you mean and rule out alternatives. It points to how vacuous the explicit semantics of these things become as opposed to implicitly based demonstrations of our ability to follow rules (but then its hard to explicitly characterize when and why these rules are broken). You have appealed to implicit ability as a defence several times which is why I think we agree more than you think. But the problem isnt about skepticism towards whether we can perform certain behaviors, its about objective semantic characterizations. Appealing to your intuitive ability to perform a behavior that you cannot even define properly is not an explicit semantic characterization!

Kripke's mistake (assuming I am recalling his position correctly), was phrasing the skepticism as a circular question to a mathematician where he asked to defend the validity of his judgements, as in

"How do you know that your present usage of "plus" is in accordance with your previous usage of "plus" ?"

That question is easily viewed as nonsensical, since it is easily interpreted as asking a person to question their own sanity. Similarly bad phrasing, leading to pointlessly circular discussion is found throughout the philosophy literature on private language arguments.

Different conceptions of logic and semantics cope variously with the question of meaning skepticism. For instance, Classical Logic with set-theoretic semantics, as in Model Theory, lends to the idea of semantics being static, a priori, unambiguous, infinite and transcendent of the finitely observed behavior of a formal system that is said to correspond to the semantics. Such "picture theories" of meaning, that place semantics in an exalted position above the cut and thrust of computation and IO, naturally provoke skepticism as to the relevance, utility and even existence of semantics, as evidenced by the existence of formalists of the last century. Similarly, I think Kripke's (misconceived) interpretation of Wittgenstein was partly born out of this obsoleted semantic tradition that he was part of, but couldn't see beyond due to the lack of a formalized alternative approach.

Intuitionism copes better with semantic under-determination, because it assumes less meaning to begin with; it interprets infinity as referring to unspecified finite extension of indefinite length, implying that all data is finite and that all symbolic meanings have a finite shelf-life. So it doesn't consider mathematics or logic to consist of an actually infinite number of semantic facts that finite linguistic practices must miraculously account for. Consequently, intuitionism permits a tighter identification of logic with a suitably non-standard version of set-theory, narrowing the opportunity for semantic skepticism.

A more modern alternative is to place syntax and semantics on an exactly equal footing, by considering them to refer to opposing sides of interaction of a dialogue between two or more agents, where what is considered to be syntax and what is considered to be semantics is a matter of perspective, depending on who is asking questions and who is answering them. Girard's Ludics is a formalisation of this pragmatic idea of meaning as interaction, and is of relevance to the rapidly emerging discipline of interactively-typed languages and interactive AI, in which no individual party has full control or understanding of the language they are using, whereupon the meaning of a type or symbol is identified with it's observational history of past-interactions.

Can you demonstrate how quus is dealt with by the approaches you have said?

Girard's Ludics is a formalisation of this pragmatic idea of meaning as interaction — sime

Well, on the face of it, this sounds not disimilar to Wittgenstein's meaning as use.

But what would you look for in an extraterrestrial signal if you were assessing for rationality? You'd probably want to see intention, right? What tells you that an action was intentional?

Some would say we want to see some signs of judgement. For instance if we would take a sequence of constants as a sign of intelligence, that would tell us that the aliens consciously chose those numbers. Choice entails normativity. They picked this number over that one.

All of this is wrapped up in rule following, which is normativity at its most basic. To follow a rule means to choose the right action over the wrong ones.

If it turns out that there's no detectable rule following in the world, normativity starts to unravel and meaning along with it. Is that how you were assessing the stakes here? — frank

I want to post Kripke's summation of his own argument. On page 107-109:

Let me, then, summarize the 'private language argument' as it is presented in this essay. (I) We all suppose that our language expresses concepts - 'pain', 'plus', 'red' - in such a way that, once I 'grasp' the concept, all future applications of it are determined (in the sense of being uniquely justified by the
concept grasped). In fact, it seems that no matter what is in my mind at a given time, I am free in the future to interpret it in different ways - for example, I could follow the sceptic and interpret 'plus' as 'quus'. In particular, this point applies if I direct my attention to a sensation and name it; nothing I have done determines future applications (in the justificatory sense above). Wittgenstein's scepticism about the determination of future usage by the past contents of my mind is analogous to Hume's scepticism about the determination of the future by the past (causally and inferentially). (2) The paradox can be resolved only by a 'sceptical solution of these doubts', in Hume's classic sense. This means that we must give up the attempt to find any fact about me in virtue of which I mean 'plus' rather than 'quus', and must then go on in a certain way. Instead we must consider how we actually use: (i) the categorical assertion that an individual is following a given rule (that he means addition by 'plus'); (ii) the conditional assertion that "if an individual follows such-and-such a rule, he must do so-and-so on a given occasion" (e.g., "if he means addition by '+', his answer to '6S+ 57' should be '125"'). That is to say, we must look at the circumstances under which these assertions are introduced into discourse, and their role and utility in our lives. (3) As long as we consider a single individual in isolation, all we can say is this: An individual often does have the experience of being confident that he has 'got' a certain rule (sometimes that he has grasped it 'in a flash'). It is an empirical fact that, after that experience, individuals often are disposed to give responses in concrete cases with complete confidence that proceeding this way is 'what was intended'. We cannot, however, get any further in explaining on this basis the use of the conditionals in (ii) above. Of course, dispositionally speaking, the subject is indeed determined to respond in a certain way, say, to a given addition problem. Such a disposition, together with the appropriate 'feeling of confidence', could be present, however, even if he were not really following a rule at all, or even if he
were doing the 'wrong' thing. The justificatory element of our use of conditionals such as (ii) is unexplained. (4) If we take into account the fact that the individual is in a community, the picture changes and the role of (i) and (ii) above becomes apparent. When the community accepts a particular conditional (ii), it accepts its contraposed form: the failure of an individual to come up with the particular responses the community regards as right leads the community to suppose that he is not following the rule. On the other hand, if an individual passes enough tests, the community (endorsing assertions of the form (i)) accepts him as a rule follower, thus enabling him to engage in certain types of interactions with them that depend on their reliance on his responses. Note that this solution explains how the assertions in (i) and (ii) are introduced into language; it does not give conditions for these statements to be true. (5) The success of the practices in (J) depends on the brute empirical fact that we agree with each other in our responses. Given the sceptical argument in (I), this success cannot be explained by 'the fact that we all grasp the same concepts'. (6) Just as Hume thought he had demonstrated that the causal relation between two events is unintelligible unless they are subsumed under a regularity, so Wittgenstein thought that the considerations in (2) and (3) above showed that all talk of an individual following rules has reference to him as a member of a community, as in (J). In particular, for the conditionals of type (ii) to make sense, the community must be able to judge whether an individual is
indeed following a given rule in particular applications, i.e. whether his responses agree with their own. In the case of avowals of sensations, the way the community makes this judgement is by observing the individual's behavior and surrounding circumstances.

Because it makes sense of your questions :D -- when I first read your questions I realized I just needed to do some of the homework. So far I've been arguing only that there is a skeptical problem or skeptical question that I see from your OP, and haven't gone so far as to offer a solution or response or even to draw out implications.

And I'm glad I did some of the homework. Kripke's mind is wild to ride along with. Look at all these incredible connections he's able to draw out, and look at how he's able to distinguish so many possible beliefs at once while maintaining a single thread of thought! It's impressive.

I think what I'd say is that there are ways of detecting if someone is following a rule, it's only that these ways are not a state of affairs in the world. Rather it's an acceptance by a community. At least this is the solution I see Kripkenstein offering. The conditions of assertability aren't in truth-conditions, but there are still conditions of assertability. You just have to learn what they are.

What Kripkenstein's skeptic points out is that our common belief that "1 + 1 = 2" doesn't have truth-conditions, but rather conditions of assertability, and in comparison with Hume's skepticism we learn the conditions of assertability through repetition and acceptance by a community of rule-followers: the force of habit reinforced by communal acceptance.

So not quite an undermining of all normativity, but possibly a re-adjustment on philosophical interpretations of meaning.

I think what I'd say is that there are ways of detecting if someone is following a rule, it's only that these ways are not a state of affairs in the world. Rather it's an acceptance by a community. At least this is the solution I see Kripkenstein offering. The conditions of assertability aren't in truth-conditions, but there are still conditions of assertability. You just have to learn what they are. — Moliere

I think what he's saying in the passage you quoted is that we have to look to the language community to discover why we ever talked about rule following in the first place.

On the other hand, if an individual passes enough tests, the community (endorsing assertions of the form (i)) accepts him as a rule follower, thus enabling him to engage in certain types of interactions with them that depend on their reliance on his responses.Note that this solution explains how the assertions in (i) and (ii) are introduced into language; it does not give conditions for these statements to be true.

So we still don't have any basis for determining that S followed a particular rule. We just treat certain circumstances as if she did.

So not quite an undermining of all normativity, but possibly a re-adjustment on philosophical interpretations of meaning. — Moliere

I agree. My bringing rationality into it was just a side effect of studying the link between meaning and normativity. You end up falling into discussing rationality with that topic.

Kripke's mind is wild to ride along with. — Moliere

Very true.

So we still don't have any basis for determining that S followed a particular rule. We just treat certain circumstances as if she did. — frank

True.

If I'm understanding the argument: in place of truth-conditions Kripke resolves the sceptical problem with the sceptical solution that the community provides assertability-conditions. There's no fact which justifies the assertability-conditions, though.

Finally, the point just made in the last paragraph, that Wittgenstein's theory is one of assertability conditions, deserves emphasis. Wittgenstein's theory should not be confused with a theory that, for any m and n, the value of the function we mean by 'plus', is (by definition) the value that (nearly) all the linguistic community would give as the answer. Such a theory would be a theory of the truth conditions ofsuch assertions as "By 'plus' we mean such-andsuch a function," or "By 'plus' we mean a function, which, when applied to 68 and 57 as arguments,. yields 125 as value."

...

Wittgenstein thinks that these observations about sufficent conditions for justified assertion are enough to illuminate the role and utility in our lives of assertion about meaning and determination of new answers. What follows from these assertability conditions is not that the answer everyone gives to an addition problem is, by definition, the correct one, but rather the platitude that, if everyone agrees upon a certain answer, then no one will feel justified in calling the answer wrong.

I have a distinct sense that some level of consensus is developing. :smile:

And I'm glad I did some of the homework. — Moliere

And I'm sorry I didn't. He seems to come out so close to W that there doesn't seem much mileage in asking whether his view is W's or not. I didn't know that.

So we still don't have any basis for determining that S followed a particular rule. We just treat certain circumstances as if she did. — frank

I'm not sure that we don't have to re-think what "S followed a particular rule" means. Even if S's application of a rule agrees with ours, it is always possible that the next application may differ. We even find this happening empirically, when some circumstance reveals that a friend has a very different understanding of a rule we both thought we agreed on.

If I'm understanding the argument: in place of truth-conditions Kripke resolves the sceptical problem with the sceptical solution that the community provides assertability-conditions. There's no fact which justifies the assertability-conditions, though. — Moliere

I don't believe Kripke is offering a resolution. He's just explaining why we think we're justified in picking out rule-following. I think he leaves us free to reshape our conceptions of meaning in anyway we might want to. :grin:

The way that makes sense to me is to read the essay as presenting Kripkenstein's views, rather than Kripke's. Is that how you're reading it? (which, to be fair, that's how he starts out the essay -- saying it's an essay not on his view as much as an impression of his while reading Witti)

The target audience seems to be philosophy professors because he advises bringing the challenge up for consideration by students. Kripke isn't a philosopher who preaches, so he doesn't really deal in dogma, like say, Nietzsche does. Is that what you mean?

Fair point about the target audience. I'm asking about the speaker of the essay. If Kripke isn't offering a resolution, then I'm asking: what about these resolution-like looking paragraphs at the end? Who is offering them? That's what I mean. Who is the speaker?

Oh, yes that's Kripke. I guess I just don't see it as resolution per se. He's just explaining why we expect rule following. Hume's problem of induction is primarily about why we expect contiguity past to future. This expectation isn't empirical, not rational, so why? One potential answer is habit. Another answer is Kant: we expect contiguity because it's coming from us in the first place.

So again, why do we expect rule following? It can't be empirical because there are no facts to observe. A rational answer would only mean something to a rationalist like Leibniz. So why do we expect that there is rule following and that this accounts for meaning? Could be habit.

You see, in both cases, the fundamental issue isn't resolved. Answering "habit" doesn't create rule-following facts for us. As with the problem of induction, we still have the gaping hole where we expected empirical data to support our assertions. Obviously, since Hume's problem attracted Kant's approach, we might expect that Kripke's problem would do something similar. Meaning isn't based on objective rule following, so maybe there's something innate about it. Maybe this innateness is a touchstone that meets each episode of communication, including this one.

I've been trying to think of a good response @Janus but have been unable, so perhaps this will do better. I believe this expression may be close to what you've been getting at?

You see, in both cases, the fundamental issue isn't resolved. Answering "habit" doesn't create rule-following facts for us. As with the problem of induction, we still have the gaping hole where we expected empirical data to support our assertions. Obviously, since Hume's problem attracted Kant's approach, we might expect that Kripke's problem would do something similar. Meaning isn't based on objective rule following, so maybe there's something innate about it. Maybe this innateness is a touchstone that meets each episode of communication, including this one. — frank

"innate" with respect to meaning is something I wouldn't deny as true, but only as unsatisfactory. It may be the case that innateness of meaning is the touchstone that allows you and I to communicate. When it comes to poetry, especially, that's where I gravitate towards -- asking for more words to explain words.

However we'd like to know more about something than "this is just what it means". This is getting back to a question I don't know how to answer: what do I want from a theory of meaning? To disappoint, I don't know what I want from a theory of meaning. Somehow I just ended up here with these questions, probably because I like to ask after seemingly silly things ;)

I think I'm tempted to simply accept the conclusion: there are no rule-following facts. Same with Hume and causation, though I really do admire Kant's attempt to overcome Hume's skepticism towards causation.

I think I'm tempted to simply accept the conclusion: there are no rule-following facts. Same with Hume and causation, though I really do admire Kant's attempt to overcome Hume's skepticism towards causation. — Moliere

I think the question is reaching for something beyond the limits of language, so I think you've got the right idea. :grin:

Nice.

"How do you know that your present usage of "plus" is in accordance with your previous usage of "plus" ?" — sime

There remains a problem for teachers, asked to mark off that little Jenny has learned how to add. We test Jenny on 2+3, 7+9, and so on, for some finite number of examples; and yet we give Jenny the epithet "Able to add numbers of any length". This is not justified by any number of examples, but perhaps it is by her showing mastery of the iterative process involved; she can cope with each of the limited number of cases - carrying, adding zero, and so on, and so there is no reason to think that she could not add together numbers of arbitrary length.

The proof of the pudding here is in the doing.

Kripke's challenge isn't about finding proof of something. It's not an epistemic question. It's metaphysics.

But as for Jenny, what you want to do is sneak in that magical phrase "for all practical purposes." For all practical purposes, Jenny has followed the rules.

Peace out, guys! Thanks for the discussion.

Of course, applications of "+1" include practical applications. The point is that the rule must be applied to each case; it does not reach out to the future and the possible and apply itself in advance. — Ludwig V

My only point was that the logic of +1 and its concatenations is the conceptual basis of counting and arithmetic, and that its ability to serve practicalities, while alternative stipulated rules cannot show its non-arbitrary nature.

But the fact that we mostly agree is not inevitable, not guaranteed. It is a "brute fact", which is the foundation of logic (and other rules). Bedrock is reached. — Ludwig V

I can't think of any examples of failures of consensus concerning basic arithmetic.

The question of intuition is arbitrary because this is about the notion of objective rules or meanings. Why does intuition matter for objectivity? A putatively objective scientific theory should be true regardless of intuition. The truth of thermodynamics doesnt depend on my cats ability to find it intuitive. — Apustimelogist

The truth of scientific theories is not intuitively self-evident in any way analogous to the truth of basic arithmetical results. So, scientific theories are never proven. That the math involved in thermodynamics is sound may be self-evident, but that doesn't guarantee that it has anything to do with some putatively objective reality.

It's not an epistemic question. It's metaphysics. — frank

Well, neither is quite right. It's a question about meaning. What do we claim when we say "Jenny can add"? And more generally, what do we claim when we say that someone follows a rule?

And the sceptical answer is that there is no fact of the matter. This is Kripke's great argument against realism.

The truth of scientific theories is not intuitively self-evident in any way analogous to the truth of basic arithmetical results — Janus

So what, some truths are intuitive and some are unintuitive. Their intuitiveness has nothing to do with objective truth. Intuition is a product of your subjective inclinations. Saying that something is more truthful because it is intuitive is like saying your subjective inclinations has something to do with objective truth.

So, scientific theories are never proven. That the math involved in thermodynamics is sound may be self-evident, but that doesn't guarantee that it has anything to do with some putatively objective reality — Janus

And just because a rule is unintuitive doesn't refute it being objectively true.

So what, some truths are intuitive and some are unintuitive. Their intuitiveness has nothing to do with objective truth. Intuition is a product of your subjective inclinations. — Apustimelogist

I would say that the only intuitively self-evident truths are logical or mathematical, and I don't see that as being merely a subjective matter.

And just because a rule is unintuitive doesn't refute it being objectively true. — Apustimelogist

I don't know what you mean when you talk about a rule being objectively true.

indeed, it seems that
Kripkes proof shows rules are not objectively true.

I don't know what you mean when you talk about a rule being objectively true. — Janus

That there is a fact of the matter about what rule is being followed.

I would say that the only intuitively self-evident truths are logical or mathematical, and I don't see that as being merely a subjective matter. — Janus

It is a subjective matter because you are appealing to your intuition subjectively and you cannot rule out the other possible rules you can use.

That there is a fact of the matter about what rule is being followed. — Apustimelogist

Oh, OK, I would say that is uncontroversial.

It is a subjective matter because you are appealing to your intuition subjectively and you cannot rule out the other possible rules you can use. — Apustimelogist

Judging from the ordinary understanding of basic arithmetic and logic I would say their results are self-evident to anyone who cares to think about it.

Kripkes proof shows rules are not objectively true. — Banno

Again. I'm not clear on what it would mean for a rule to be objectively true or false. So, do you mean that Kripke has shown that the idea of objective truth just doesn't apply to rules?

Yes, but if you care to think about it, the behavior is consistent with other rules. Your selection of a single rule is based on intuition not on some evidence that contradicts the alternatives. There is nothing stopping someone from saying that they are following the other rules, but supposedly you would just disregard their testimony straight off.

Arbitrary rules like quaddition do not yield reliably workable results, or at least I haven't seen anyone showing that they can. The logic of addition is that you can keep adding forever, and that logic is based on the fact that there is no reason you cannot keep adding forever. Quaddition is just an arbitrary countermand of that unlimited iteration. What's the point, when it cannot even yield workable results? And as for people claiming they are following "other rules", there might be some plausibility to that if the other rules yielded the same results.

Arbitrary rules like quaddition do not yield reliably workable results, or at least I haven't seen anyone showing that they can — Janus

Its logically valid so I don't see the issue. Also, don't forget that quus is only one example of many other possible rules so actually you have been using some other strange rule since you started learning math and you have been using it fine. In fact you have been using many rules at the same time. Its all totally workable. Again, the point is underdetermination so its not about whether one rule is workable or not, any time you use addition it has an underdetermined characterization, and your ability to use it and practise it has little to do with that.