Kripke's skeptical challenge
This challenge comes from Saul Kripkes Wittgenstein on Rules and Private Language (1982). Note that Kripke advises against taking it as an attempt to correctly interpret Wittgenstein (which is a convoluted statement considering the nature of the challenge), but rather it's a problem that occurred to him while reading Wittgenstein. This post is the challenge in my words:
We start with noting that there is a number so large, you've never dealt with it before, but in our challenge, we'll just pick 57. You've never dealt with anything over that. You and I are sitting with a skeptic.
I ask you to add 68+57.
You confidently say "125."
The skeptic asks, "How did you get that answer?"
You say "I used the rules of addition as I have so often before, and I am consistent in my rule following."
The skeptic says, "But wait. You haven't been doing addition. It was quaddition. When you said plus, you meant quus, and: x quus y = x+y for sums less than 57, but over that, the answer is always 5. So you haven't been consistent. If you were consistent, you would have said "5.""
Of course you conclude that the skeptic is high and you berate him. He, in turn, asks you to prove him wrong. Show some fact about your previous usage of "plus" that demonstrates that it wasn't "quus."
Up next: the implications of the challenge and possible solutions.
We start with noting that there is a number so large, you've never dealt with it before, but in our challenge, we'll just pick 57. You've never dealt with anything over that. You and I are sitting with a skeptic.
I ask you to add 68+57.
You confidently say "125."
The skeptic asks, "How did you get that answer?"
You say "I used the rules of addition as I have so often before, and I am consistent in my rule following."
The skeptic says, "But wait. You haven't been doing addition. It was quaddition. When you said plus, you meant quus, and: x quus y = x+y for sums less than 57, but over that, the answer is always 5. So you haven't been consistent. If you were consistent, you would have said "5.""
Of course you conclude that the skeptic is high and you berate him. He, in turn, asks you to prove him wrong. Show some fact about your previous usage of "plus" that demonstrates that it wasn't "quus."
Up next: the implications of the challenge and possible solutions.
Comments (256)
Sorry. There's something I'm missing. If I apply the definition of addition to 68 and 57, I get 125, not 5. What you are describing, "quus," is a different operation which is not consistent with that definition.
You haven't been doing addition. It was quaddition.
I think this is where I'm supposed to berate you.
Right. You say: "No! I've been doing addition, not quaddition. Stop embarrassing yourself, you baboon!"
Then I ask you for a fact about your previous behavior that shows that the rule you were following was addition rather than quaddition.
Is this something about word-games and their context?
In another thread I was saying thus, and I think it might have some relevance about context and the meaning of terms (like plus and quus):
For physicists, "nothing" has a different connotation than the classic philosophical notions of nothing. It just needs zero energy to be considered "nothing" in physics I guess. And of course, that is unsatisfying in a philosophical sense that the theoretical principles and laws and fields that underlie this "nothing" still need to be accounted for.
So yeah, various terms can be thought of differently (have different definitions and uses) in different language communities.
Quus guy's logic is using it differently than plus guy's.
The only other answer is Quus guy simply misinterprets the language-game of a particular mathematics-using community.
Does my behavior include my invisible, to you (and perhaps to me), mental processes? If it does, I say "I already have given you that fact."
Yes, definitely. The challenge ends up being about the meaning of any word.
Quoting schopenhauer1
:up:
Cool.
I mean, who is to say the tribes that have a word for "one", "two" "three" "anything more than three" is wrong? If used in a way that everyone gets by, there you go.
In the challenge, it's granted that you know everything there is to know about your mental processes.
I think the problem is that following the rules of addition are exactly the same as following the rules of quaddition up to the number 57. What in your mental processes would have been different so as to prove that you weren't quadding rather than adding?
I'll have to come back to this.
To me, it seems like the same idea really, but a real life example of how math is radically different. The rule is you can add to three but any more, it's just a "a bunch of stuff" (you mine as well say 3+X). The focus here should not be the content but the fact that there is a different rule on how addition works in that language community.
In other words, it's almost a "nominalism" versus "essentialism" argument. Early Wittgenstein versus later Wittgenstein might be another phrasing. Logical positivists versus post-modernists. And on and on.
Ah... Now, maybe, I understand your point. I'd forgotten that I'd never encountered 57 before. Let me think... Ok, for natural numbers, the definition of "addition" can be traced back to counting. Are you saying that I can count to 56, but for any larger number I'm doing something different?
It's specifically about your assessments of past behavior. You assume you know the rules you were following. Kripke's skeptic suggests that there is no fact of the matter. The fiction of "quadding" is just meant to illustrate this.
I got this wrong. Kripke's challenge is not about epistemology. It's metaphysics. That's the point of the emphasis on facts.
Wow.
Rats. Now I'm back to not getting it again.
On the web, I found a discussion of this issue. Here's a link:
https://iep.utm.edu/kripkes-wittgenstein/#H1
It doesn't make things any clearer to me. I give up.
Your challenges still helped me flesh it out, so thank you.
Anytime you need somebody to be confused, I'll be happy to help.
:lol:
Hey, general relativity came out of little games.
Why? Because if we weren't following any specific rule in the past, then it follows that we aren't now, in spite of my confidence that I know now what I mean by "plus."
I'm not quite following why this is true. Why does meaning have to be rule following? Why can't it pop into thin air in the present?
There's something I'm missing
Surely the only thing you need to prove historically that you weren't quadding is to show any instance where you've added two numbers > 57, right?
If I've done proofs via induction using addition, doesn't this show that I've taken addition all the way to the infinite in the past?
That or I smugly pull out a crumpled sheet of paper from my pocket with the Peano Axioms written on them. I inform the skeptic that, as a good positivist, I only preform arithmetic by starting from this sheet and working up from there. "Show me how it is possible to derive quusing from these axioms and I will accept your proposition."
Still, I get the point. Defining systems only in terms of past use seems to miss something.
Yes, but in the thought experiment, you've never done that. The idea is that in real life there's a number you've never added up to before. For the sake of presenting the challenge, we just pick 57.
Quoting Count Timothy von Icarus
I think so, but in the challenge, you've never added numbers up to above 57.
Quoting Count Timothy von Icarus
He grants that math has specified rules, but is there a fact that shows you're following those rules every time to add? Do you really take the sheet out?
Quoting Count Timothy von Icarus
He sees it as an outcome of the private language argument. This is the PDF text if you become fascinated enough to read it. :grin:
Wittgenstein on Rules and Private Language
Then I would put it on THEM to prove to me instead of doing it myself. If they refused, I would ignore them from then on for wasting my time. :D
Kripke poses the challenge:
The answer is simple: the rules of arithmetic. We either follow them correctly or we do not. When Kripke substitutes 'quus' for some cases of '+' he in not substituting one expression of a rule for another. Quus has no place in the rules of arithmetic. Kripke or his skeptic is not interpreting or misinterpreting the rules of arithmetic, he is disregarding them.
If our ability to follow rules correctly and consistently is not dependent upon the application of a privately held conceptual understanding of the rule (the justified mental fact), but can be explained in terms of training and conformity to standard practice, then what remains of the skeptical problem?
The challenge is to point to some fact that shows which rule you were following in the past. Remember, the challenge is not about epistemology. It's not about how we know what rule you were following. It's conclusion, and the one Kripke doesn't see Wittgenstein ever ruling out is this: there was no rule following. If you disagree, he's asking you to prove it.
This is not an exegesis of Wittgenstein. It's not an attempt to correctly capture what he thought out of the elusive text he wrote. This thread is about considering a Kripkean challenge.
New rule: replace an a with an i.
That's wonderful. There's probably a calculator program already on there, though. See if you can find it. :cheer:
These analytic truths are arbitrary, so there is no correct usage outside your agreed upon rules.
You chose 57, but 59 would have been better because the number after 59 is in fact 1:00.
If we're dealing in synthetic truths, we see the same thing. The rules governing planetary travel show a predictable course and the coordinates can be predicted so that it would appear which number would follow next, until something interferes with the travel. Would we then say we're not following the word game because the next in sequence wasn't predictable from the last in that one instance?
Ah right. Well, in that case, if I'm disregarding the obvious silliness of the whole thing, then I would think about the sorts of problems I was trying to solve with addition, and think about it those sorts of problems would be solvable by quaddition.
Perhaps I used addition once to count how many apples me and my brother picked together. I picked 5 and he picked 10, so together we picked 15. I'd then think, would quaddition give me the correct answer? In this case, yes, but in the case where one of us picked 57 or more? No, clearly not.
Quaddition doesn't generally solve the sorts of problems I've thus far been using addition to solve, so no, I haven't been doing that.
A summary and survey of the literature.
Quoting Fooloso4
Quoting frank
Kripke's skepticism is based on his assumption that there must be some fact independent of and other than the fact of the practice of addition.
:lol:
Quoting Hanover
For Kripke's challenge, we want a fact that shows intentional rule following. This entails justification and correctness. We usually wouldn't look for that kind of rule following in a planet because we imagine they just blindly do what they're going to do and we identify a structure in it. We then use that historic structure to predict where it's headed (which is what technical analysis of a market is, btw.)
An example of a fact that might work is dispositionality: which says it's a fact about the world that you have a predisposition to answer "125" instead of "5". That kind of thing.
He grants that there's such a thing as the practice of addition. He's asking for a fact that shows you've actually adhered to this practice as opposed to the practice of quaddition.
I'm finding it to be pretty mind blowing, but I can see how it would seem silly to some.
Quoting flannel jesus
In the challenge, addition and quaddition produce the same results up to 57, and that's as far as you've ever gone. If there is no fact about which one you were doing all this time, then it shows that if meaning arises from rule following, there is no meaning. That's the crazy part.
There is more than one sense in which we say someone is following a rule. If I if I ask a child what the rule of counting is more than likely she cannot state a rule but will simply demonstrate how it is done by counting.
If someone is going to tell me what's in my mind - and telling me I've been using quaddition instead of addition is doing just that - then they should have a good reason for believing that. I have a good argument for why I've been using addition. What's the counter ?
In the challenge, you've never dealt with numbers above 57. Addition and quaddition give the same answers up to that point. The question is: what fact would show us that you were adding and not quadding?
Quoting flannel jesus
If I could give you a fact about which rule you were following, then the challenge would fail. I think I need to flesh out the criteria a proposed fact has to meet in order to crash the challenge. I've just been busy lately. Need to collect the army of brain cells.
As long as we are dealing with quantities less than this imaginary number that has not been dealt with before, then there are a multitude of rules we might invent that we could say are being adhered to. It is only when we encounter this number that we can say say that what follows is or is not arithmetic, for the rules of arithmetic do not allow that two positive integers added together will be less than either one.
That's correct.
So if up until we get to this number, which as far as we know no one has ever encountered, there is no discernible difference between plus and quus and puus. The practice is the same. What then is the skeptical objection?
That there is no fact about which rule you were following.
Do they? What about 68 + 1? I mean 68 is the outcome of, say, 30 + 38. I need to do addition to be able to do quaddition; I don't need to be able to do quaddition to do addition.
So if I'm asked to "add 68" that wouldn't make sense und quaddition.
True: 68 = 57 + x
False: 68 = 57 quus x (that's always 5)
So how does addition flow into quaddition? What's the rule here? Which of the following is correct:
1 quus 68 = 5
1 quus 68 = 6
I can argue for both, but I don't know enough about quaddition to decide on my own. I'm way more familiar with addition. This may be the result of an unnoticed stroke, though. Who knows?
If what is being done is in accord with addition then it does not matter which rule one thinks they are following.
The fact that Kripke is able to make a distinction between addition and quaddition means that there is in fact a discernible difference. No arbitrary rule imposed under conditions that do not occur should lead to skeptical confusion.
It goes like this:
Quoting frank
:up:
"In the discussion below the challenge posed by the sceptic
takes two forms. First, he questions whether there is any fact
that I meant plus, not quus, that will answer his sceptical
challenge. Second, he questions whether I have any reason to
be so confident that now I should answer '125' rather than '5'.
The two forms of the challenge are related. I am confident that
I should answer '125' because I am confident that this answer
also accords with what I meant. Neither the accuracy of my
computation nor of my memory is under dispute. So it ought
to be agreed that ifl meant plus, then unless I wish to change
my usage, I am justified in answering (indeed compelled to
answer) '125', not '5'. An answer to the sceptic must satisfy
two conditions. First, it must give an account of what fact it is
(about my mental state) that constitutes my meaning plus, not
quus. But further, there is a condition that any putative
candidate for such a fact must satisfy. It must, in some sense,
show how I am justified in giving the answer '125' to '68+57'.
The 'directions' mentioned in the previous paragraph, that
determine what I should do in each instance, must somehow
be 'contained' in any candidate for the fact as to what I meant.
Otherwise, the sceptic has not been answered when he holds
that my present response is arbitrary. Exactly how this
condition operates will become much clearer below, after we
discuss Wittgenstein's paradox on an intuitive level, when we
consider various philosophical theories as to what the fact that
I meant plus might consist in. There will be many specific
objections to these theories."
So
1. We need a fact that explains why I'm compelled to answer 125.
2. We need a fact that contains the "directions."
It is reassuring to know that we have saved addition from Kripke's skeptic ... at least for the time being.
Here I think he is simply wrong. My mental state and whatever my meaning might be has no bearing on how to properly add numbers.
Quoting Fooloso4
Yeah, I've read this. I guess I shouldn't have posted. (I'm still confused about the meaningfulness of "68" from a quus-centric world-view, but that might just be marginally on topic.)
The answer is 42, I guess.
Ha ha! Yea. Except it wasn't addition that was in danger. :wink:
As always. :razz:
I believe counting is intuitive, so no need for rules. All the basic arithmetical operations can be shown with actual objects like stones or marbles. Once this is intuitively grasped the rest is just naming the numbers. The "rules" are just formulations of what is already easily made obvious. by showing.
Hmm. Might be better to say it is a ritual. Touch one shape, say "one", touch the next, say "two"...
We watch a child do this, and then count the cats, and the chairs, and the fingers, and as Hume pointed out no finite list of such examples logically implies that the child will get it right next time. So when do we say that the kid knows how to count?
Some laugh at the primitives who go "one, two, three, many..." but we do the same when contemplating grains of sand on a beach.
Thinking of counting as "intuitive" underplays the need to teach kids how to count. It's an initiation into a language game.
An initiation which would be impossible if the child did not intuitively get the logic of it. Once understood the logic can be extended indefinitely, and excluding brain damage, should not ever be lost.
What I was after was showing how Kripkenstein relates to Hume's skepticism towards induction.
Can you explain how you see Kripke's "skeptical challenge" relating to Hume's skepticism regarding induction? I have always understood the latter to be merely pointing out that induction is not deduction, that causation is not logically necessary.
As I said I don't think about it in terms of following rules, so your question is not relevant.
In any case even if it were a matter of following rules my knowing that the child has followed a set of rules is not the same as the child following a set of rules. I don't have to know something, or even be able to know something, in order that it be the case.
I see no reason to think that once a child understands the logic of addition that they would ever lose that understanding, barring, as I said previously, brain damage or senility.
Except that the topic is following rules.
Ok, leave it.
I thought the topic was "Kripke's skeptical challenge". If the challenge is based on an inconsistency that shows up when thinking of counting as rule-based, have you considered the possibility that not thinking of it as rule-based, but as intuitive, might dissolve the apparent issue?
Kripke presents this as the discovery of a problem; Cavell reads Wittgenstein as stating a truth. There is no fact that ensures the extension of a concept into the future or a new context. Unless it is a game or math, we do not follow a rule to reach a certain effect or conclusion. I got into this here
True, but the skeptical argument goes beyond that. When you communicated in the past, you weren't following any particular rule. Meaning does not arise from community rule following.
And I can agree with that too. Im not denying the skeptics argument.
:up:
Quoting Paul Moser and Kevin Flannery, Kripke and Wittgenstein: Intention without Paradox
The upshot is that Wittgenstein's understanding of intention "does not fall within the scope of the sceptical paradox."
Edit: Added link
About what? Everything?
About the coherence of his position, about the claim that there is no fact about S that constitutes S's meaning plus rather than quus, and about the claim that the challenge represents a new form of scepticism.
Yep, the generally agreed view is that the problem Kripke posits is not found in Wittgenstein, that Kripke should not be seen as engaged in exegesis.
Yes, I checked as well. I tried to quote more but the OCR is flawed.
Quoting Banno
But it would be interesting if Wittgenstein had already provided an answer to the challenge that Kripke derived from his work. Granted, I don't know the chronology of when each wrote what.
---
This is somewhat interesting but I don't think Kripke is himself engaged in a Hume thing.
Its interesting there's this impression that Kripke has misunderstood Wittgenstein or not even attempted to but I find Kripke's interpretation more or less aligns with what Wittgenstein seems to say throughout the book imo. Maybe I am misinterpreting it towards my philosophical inclinations though (but I don't actually believe that).
I just saw a hawk fly up and land. A crow began sounding an alarm. I've been seeing that since childhood. It's meaningful to me, not because anyone involved is consciously following rules, but because it's following a well worn pattern, and I have an innate need to make sense of events.
Could it be that this is the same thing that's happening with language use? Events transpire, people make noises, and I need to make sense of it. So you don't have to know what rule you were following when you spoke. There doesn't have to have been any rule at all. Your speech took place, and now we both receive it and go to work fitting it in with the rest of what we know. And then this becomes cyclical, so there's an expectation that when you speak, someone will try to make sense of it.
It's like a whirlpool.
Some:
I still think gave the most apt reply.
Hmm. What is a pattern, if not some sort of rule-following? OR perhaps, there are two ways of showing that you understand a pattern - by setting it out explicitly in words, and by continuing it.
So here's the problem. Consider "101010..."
Someone says "you are writing a one followed by a zero, and you intend us to understand this as continuing in perpetuity"
Someone else says "The complete pattern is "101010010101", a symmetrical placement of one's and zero's".
A third person says "The series continues as "101010202020303030..." and so on, up to "...909090" and then finishes".
Our evidence, "101010...", is compatible with all of these, and much more besides.
It's not the absence of rules that is puzzling, it's their abundance.
Yes, explicit rules are in a way post hoc.
It's not clear from this passage that the authors have ever heard of the private language argument.
Right. Kripke isn't saying there's no such thing as rules.
Quoting Banno
I guess the question is whether rule following is something we sort of stamp onto certain kinds of events?
Well, something along those lines happens when we say that little Jenny can count.
:up:
Then I ask you to prove tI've been doing quaddition, not addition.
Draw 57 tally marks. Ask the skeptic how many there are. If the answer is "57", draw 68 more. Have the skeptic count them all. That should be a good enough answer for him.
I wouldn't be able to. Someone else brought up objections as if the question is about what one can prove with regard to rule following. It's not about proof. It's that there is no fact (a situation existing in the world) that signifies which rule you followed.
It probably would have helped if I had explained the basics of meaning normativity first, then set out the challenge. But I'm stuck now trying to get my head around the ins and outs of that. It's a pretty thick topic.
The problem generalizes so it encompasses all language use. One imagines that you create meaningful speech by following certain rules. The Private Language argument suggests that this isn't what's really happening.
If you're interested, I'll eventually fill out details about meaning normativity, and then move further along in Kripke's work to describe his own thoughts about where we land after considering the challenge.
Quoting RogueAI
See this comment I made in this thread 25 days ago:
Quoting Janus
As far as I can see this solution dissolves the supposed problem. Much ado about nothing...
If someone could prove you were doing quaddition it would equally defeat the point of the thought experiment in the same way that proving someone can do addition would.
This just shows a misunderstanding of what is at stake imo. It is clearly obvious that in general, people do not have a problem in performing coherent behaviors that help us fulfill goals and desires in the way we want. The point is that describing this behavior and its "rules" is chronically underdetermined, chronically indeterminate. It is clear that we do not perform behaviors in a top down way as a consequence of explicit conceptualizations of rules. Rather, rules are post-hoc classifications and inferences we impose on our own behavior. Our behavior and our abilities arise in a completely implicit, automatic fashion; they are the product of complex neuronal processes that are completely hidden from us and can endow our thought and abilities with a Humean kind of arbitrariness which we often don't stop to take the time to notice. People in A.I. often talk about the problem of interpretability where by our machine learning programs chuck out solutions which are difficult for us to understand or we have no idea how it came to the solution. They can do things in ways that from our perspective seem very non-linear. I think the exact same happens in our own cognition and brains, which should not be surprising given the fact that a brain is just a big machine learning architecture. I think we often consider our own cognition human interpretable because we explain concepts in terms of other concepts which our brains have already chucked out but look at the Munchausen trilemma in philosophy: this approach doesn't go very far.
Well said, thanks. Meaning normativity opens up into ideas about rationality. If we reject meaning normativity theses, do we end up also rejecting our common sense ideas about rationality?
In counting we need symbols to begin to understand just how much we have of something. Hence learning the base-10 system, and having to memorize the order of numbers prior to being able to count to 57 -- we're already using a number system by the time we're counting, and so counting presupposes understanding the domain of numbers with some kind of symbolic system. Counting is tied to the natural numbers, where in quussing we're clearly in a different domain -- but all three rules, counting, adding, and quussing all look the same up to the number 57 because that's where the domains of interest are the same, and the operations are similar and so the outputs are the same within that small domain, and because we're using the same number system to represent the numbers. Not that changing bases would matter, I just mean we have a number system with bases, rather than a number system that consists of "one, two, three, and many" or something like that.
Quusing is clearly derivative of adding, and so it seems a bit silly -- but I'd say there's no fact to the matter between choosing between, say, counting on the natural number line or counting on the rational number line until you get to a point where there is a difference, like the square root of 2 and suddenly you see that you have a new kind of number to deal with. But for all that there is still a difference between these sets, it's just not in the rules of counting, adding, or quusing.
:up:
Isn't counting adding 1 to the previous number? Also, if I skip count by a number, aren't I adding that number each time?
I think "counting" is almost a primitive. It's such a simple operation or concept that we'd have a hard time defining it rigorously. But I'd put "counting" as more primitive than addition, because addition holds for more domains than counting -- such as fractional numbers that fall in-between the counting numbers.
Without defining the domain counting is strange. You can't count to the square root of 2 on the natural numbers, for instance. Counting will never get you to the real number line. And if we allow division, at least, it's pretty easy to operate on the natural numbers such that we need more numbers than what we count. One might say a difference between quusing and adding is that adding is a part of all arithmetic, and so we have access to division, where quusing is the same as addition up to a certain point but what makes it different are the rules and the domain.
Quaddition is clearly a philosophical toy, but modular arithmetic works similarly in that there is no number beyond a certain point within the mod space. Quaddition just defines, arbitrarily, what happens after you reach the end.
I don't think rules are imposed, they describe behaviors which are entrenched and replete with their own logic.
Quoting Moliere
Addition, subtraction, multiplication and division are all, as far as i can see, basically counting, and counting is basically naming different quantities. Think about the abacus.
Sorry, late reply. Ideally I would like to think of myself as having an anti-normative stance philosophically, generally anyway. I'm not really sure what rationality is and think it probably means various things in different contexts. I think it is probably something that follows conventions more than we think, is fallible. What is rational depends on what people's preferences are too, in the sense that you might not look at someone rationally if you cannot even see what end they are trying to meet, or perhaps even if youre just a staunch believer that acts need "useful" ends.
I think what I just said is trying to get at a kind of rigorous version of what rationality is but then again I don't think that really has much to do with common sense rationality in daily life. As I said I don't even know what rationality is really but its a very intuitive concept in daily life.
If what we do seems to be the product of these kind of blind processes as Wittgenstein seems to emphasize in Philosophical Investigations, then it will never really matter how we explicitly characterize something like this because life, society must go on anyway and constraints on what is "good" or "rational" will inevitably emerge in a self-organizing way, whether we have a proper understanding of them or not. I don't think there can be any strong objective notion of rationality though.
Quoting Janus
Yes, that's what I said, you must have interpreted impose differently. But my point is we construct those descriptions.
Cool. I'm discussing normativity elsewhere, so I probably won't be adding to this thread. :up:
Depends on what you mean by arbitrary. There is a reason we tend to label things in a certain way and its to do with how our labelling and descriptions are literally physically, mechanically caused by a complicated brain that has evolved to infer statistical structures in our sensory inputs and learn.
Maybe in that way, it is not arbitrary, because our brains have evolved to do a certain thing, they do it well and, and there tends to be similarities in what human brains do, whether as due to social influences or without those influences.
At the same time, does this mean our concepts could not have been otherwise, for what ever reason? I don't think so. Is our brain not an arbitrary structure which could have been different in some way and so learned concepts differently? yes. Those concrpts are obviosly motivated by what we observe, just I don't think it doesn't mean we can't interpret what we observe in different ways in principle
Even just the fact that people can come up with ideas like quus shows that there are arbitrary ways we can define, construct, draw the boundary on things. We might think of them as unintuitive but I think that kind of reasoning is as arbitrary as the concepts themselves.
Yes, and there can be no disembodied brain, or brain in isolation from environment. We are blind to the worldly process of construction, so it is not we who construct, but we who are constructed from moment to moment.
We are all more or less similar, and animals too, so there will be similarities and differences. Our concepts could have been otherwise, if we were. Could we have been otherwise? Of course, it is, logically speaking, possible; but that means no more than that imagining ourselves having been different involves no contradiction. How can we find out if it is really possible?
It seems obvious we can interpret what we observe in different ways; that is different people can. Or one person may be able to imagine other possibilities than those which are simply found to be the case.
We can come up with arbitrary, even ridiculous, ideas like quus, to be sure; language even enables us to speak of round squares and many other absurdities. Language can even make the mind seem as though it is disembodied, a free-floating locus of identity. We are the locus-eaters, reifiers of myriad concepts, generators of nuclear identities. Poetry is a great benefit.
Semantics really, isn't it?
There is no self!
Quoting Janus
Well you start to get into a slippery slope here because modality is something we make use of all the time whether in daily life, intellectual discussion, conceptualizations etc. This kind of skepticism, while very fair, is also I think is an argument against all your thinking, not just in this discussion.
On the otherhand, I could just ask you whether you think you could use the operation quus. Yeah, I'm appealing to the same kind of modal quandry but I would be surprised if you said you were unable to.
There's also examples on real life where people categorize concepts differently, like colours in different cultures. Of course, some Amazonian tribe will see the same colours as us, but they will categorize them differently, which is essentially the crux of this problem.
Quoting Janus
Yes, and what is in question is whether there is a fact of the matter about who is correct.
The abacus might be a bad example for me because it would emphasize what I've said: I can certainly count the beads on an abacus, but I don't know how to do the arithmetic operations with an abacus. I never learned how to use it.
Similarly we can count marks, or we might know the the arabic numerals, but we may not know how to solve an addition problem without some sort of knowledge of figuring sums. That ability might even be relative to the numeral symbols we use -- thinking here about the trick of stacking numbers on top of one another and adding them by column from the right. Seems like that'd be difficult to do with Roman numerals.
Not really, I think it is literally true that we are being created moment by momentuntil we are not. We do not create ourselves. We don't even know what causes the thoughts we have to arise in awareness.
Quoting Apustimelogist
I don't see a slippery slope, but rather a phenomenological fact that we make a conceptual distinction between what is merely logically possible and what might be actually, physically or metaphysically, possible. We don't know what the real impossibilities are, but we inevitably imagine, whether correctly or incorrectly, that there are real, not merely logical, limitations on possibility.
Quoting Apustimelogist
I think we mostly do assume that there is a fact of the matter, but of course we have no way of knowing that for sure or of knowing what a "fact of the matter" that was completely independent of human existence could even be.
Quoting Moliere
If you wanted to count a hundred objects you could put them in a pile, and move them one by one to another pile, making a mark for each move. Then if you wanted to add another pile of, say, thirty-seven objects you just move those onto the pile of one hundred objects, again marking each move. And then simply count all the objects or marks. I don't see why we should think that all the basic operations of addition, subtraction, division and multiplication cannot be treated this way. We really don't even need to make marks if we have names for all the numbers and we can remember the sum totals.
Being able to count "1" is significant, as is being able to recognize when you have 0 of something. Then the journey from 1 to 2 is the act of grouping -- absence, presence, and sameness. A nothing, a something, and a set. After you have a set then I think the successor function makes perfect sense -- keep doing the thing you already did, add 1 and go to the next spot. I'm not so sure before that.
Also: division is what allows us to start asking things like "What about the numbers in between 1 and 2?" -- before that we'll just be dealing with wholes. Then we start adding all kinds of numbers to what appeared to be nothing but counting and moving stones. But that we can divide sets into equal portions, or set up ratios between numbers, I'd say is distinctly not counting as much as comparing, because some of the numbers in between 1 and 2 cannot be represented with a ratio of stones. The square root of two cannot be represented by a ratio of stones in the numerator and denominator, so it can't be counted to by counting two ratios, but it's still a number between 1 and 2. We only get there through operating on the numbers, rather than counting. But it's still arithmetic because we're just dealing with constants and what they equal.
Basically I'd say that arithmetic is more complicated than counting.
Here is an accessible version: "Kripke and Wittgenstein: Intention Without Paradox," by Paul Moser and Kevin Flannery.
Of course I agree that arithmetic is more complicated than counting, all I've been saying is that it is basically counting. It is the symbolic language of mathematics that allows for the elaborations (complications) of basic principles.
And I would also argue that it all finds its basis, its genesis, in dealing with actual objects, Thinking in terms of fractions, for example, probably started with materials that could be divided.
Yes but for the purpose of this topic it doesn't really matter. Talking about the nature of the self is does not really have an impact on what I mean when I say we construct concepts, at least not in this context from the way I see it.
Quoting Janus
And my point ia you are doing this with pretty much every conversation you are having about philosophy. Philosophy is an armchair science so a huge amount of its arguments rely on this same kind of conceivability of what seems correct, what seems possible, logical, metaphysical or otherwise.
Quoting Janus
I don't think there can be a fact of the matter independent of human experience and even within experience, people find themselves unable to determine a solution to issues like this quus one. Its chronically underdetermined, there is no objective way to see it that can definitely rule out all of the others. Thats the vision that makes most sense to me anyway.
Quoting Janus
Again, this has nothing to do with what we can and cannot do. The whole point is this underdetermination occurs in spite of these abilities. You said earlier that you don't even really know the causes of your thoughts or how they arise. So you know the causes of your understanding of addition? Or quantity itself? Can you actually articulate non-circular definitions of these concepts. I'm not sure you can, they are totally intuitive and implicit. You can demonstrate to me how to add but you can't tell me the rule and the only way I can even learn off of your demonstration is that I have a brain intelligent enoigh to learn, mirror, make inferences but then again we have no personal idea why or how our brains do that. We don't know what makes it that an idea suddenly clicks and why. I can apply the same quus-type thought experiment to the concepts that you are using in this scenario. We can equally do this counting thing exactly in the way you wanted but the point is not being able to count or perform addition, its to have uniquely determined descriptions of what you are doing.
My point in making that distinction was that some concepts, like counting and addition come naturally, and other concepts like quaddition are arbitrary artificial constructs.
Quoting Apustimelogist
I don't see the phenomenological dimension of philosophy as "armchair speculation", but rather as reflection on what we actually do.
Quoting Apustimelogist
Well, there certainly cannot be an ascertainable fact of the matter, which is independent of human experience, I'll grant you that much. I see the quus issue as not merely under-determined, but trivial and of no significance, and I wonder why people waste their time worrying about such irrelevancies; but maybe I'm too stupid to see the issue, in which case perhaps someone can show me that I'm missing something.
Quoting Apustimelogist
The causes of our thoughts are presumably neuronal processes which have been caused by sensory interactions; my point was only that we are (in real time at least) "blind" to that whole process. I don't believe we are phenomenologically blind to activities like counting and addition and I think it is a plausible inference to the best explanation to say that these activities naturally evolved from dealing with real objects. I'm not claiming to be certain about that, just that it seems the most plausible explanation to me.
I don't really understand the connection as I have read in your comments so far tbh. Neither do I see a real significance in the distinction between "natural"and "artificial" concepts.
Quoting Janus
Well thats more or less what I mean.
Quoting Janus
Well most philosophical issues are arguably trivial and doesn't make much difference to what people do in the world. Most people haven't even heard of these issues so why do they matter. As I have already said, the quus issue has no relevance or consequence for people's ability to do things but I think if you are interested in notions of realism or whether we can have objective characterizations, problems like this are very interesting and central.
Quoting Janus
I think if you consider that quantitative abilities and counting might be primitive processes we cannot non-circularly decine then I would say actually, yes we are blind to these. We are able to count, we don't know why we can, just like someone extremely good at mental math wouldn't know why they are so good at problems other people find impossible... the answers just come to them very quickly. Addition and counting as the same and would involve other blind processes like the ability to immediately discriminate the things you are counting etc etc.
Yes they obviously are natural abilities and they evolved but again, this is completely missing the point. The point isn't about our ability to do things, this is uncontroversial. Its about descriptions and characterizations of the things we find ourselves doing.
I disagree that arithmetic is basically counting for the reasons I've stated: there are some numbers you cannot count to which you can get to within the arithmetic operations. This is an ancient problem, so I'm not sure how much the symbolic language matters. The symbols simplify and make things easier for us, but this is a problem that's not derived from the symbology: link on incommensurability (which should show why I keep harping on the square root of 2)
Probably, yes. But as the influential codger said:
[quote=Kant]There can be no doubt that all our cognition begins with experience...But even though all our cognitions starts with experience, that does not mean that all of it arises from experience[/quote]
Quoting Moliere
If you have four piles of four objects then you have sixteen objects, three piles of three objects then you have nine, two piles of two objects you have four. This obviously cannot work with two objects, so I'm not seeing the relevance to deciding whether addition, subtraction, multiplication and division are basically derivable from counting operations.
Without the symbolic language of numerals the irrational nature of the square root of two would not have been discovered.
This is the passage from Kant I am familair with"
In respect of time, therefore, no knowledge of ours is antecedent to experience, but begins with it. But, though all our knowledge begins with experience, it by no means follows that all arises out of experience.
So, it addresses knowledge, not cognition. How do we arrive at a priori knowledge? It is not given directly in sensory or somatosensory experience, but I think it is gained by reflecting on sensory and somatosensory experience, and I don't understand the Kant quote to contradict that or to be suggesting any other source for synthetic a priori knowledge.
If you can't derive addition from counting then how are you proving you are doing addition?
Natural concepts are those which inevitably evolve out of experience like space, time, number, difference, similarity, causation, constitution, form, material, change, grammar, logic and so on. Artificial concepts are those which are purely derived from stipulating arbitrary sets of rules. The latter are parasitic on the former.
Quoting Apustimelogist
Well, it's not what I mean. Armchair speculation I would class as metaphysics, not phenomenology.
Quoting Apustimelogist
I don't see the relevance at all, and no one seems to be able to explain clearly what it is, so...
Quoting Apustimelogist
We are not blind to considering how counting and the basic arithmetical operations can be instantiated using actual objects. This is not the case with quus.
Quoting Apustimelogist
You can derive addition from counting. Counting basically is addition.
But so what? Unless you can show that someone cannot use those "parasitic" concepts and that they don't or can't work, then what is to say it matters what is "natural" when that itself is dependent on contextual factors of how your brain is structured and the things you happen to learn. I am the type to think that just because everyone agrees on something, doesn't make it some how unique or objective. I think ultimately what is "natural" just boils down to something like an impelled preference and I don't see that as a valid way of arguing that something is somehow unique, correct or objective.
Quoting Janus
I don't really see how phenomenology is not another form of armchair speculation in a similar way.
Quoting Janus
The relevance for what? Its simply the issue of whether the descriptions you ascribe to behavior is uniquely determined as opposed to underdetemined or indeterminate.
Quoting Janus
I can demonstrate quus with objects just as well as I can with addition. Neither am I blind to doing that with counting. What I am blind to is a good description of what counting or quantities are. I know these things very intuitively, I am very good at doing them. Its difficult to give a explicit account in a way that I would personally find satisfying imo.
Quoting Janus
I dunno, it seems to me with what [quote="Moliere;838856][/quote] has been saying that what these concepts mean and how they relate to each other is not trivial in a way that questions whether counting actually does much at all in this context. You want to use the example of counting tonshow you can get to what we deem thr correct answer but I think demonstrating your ability to meet a goal is not the same as specifying a description or meaning of what you actually did.
Well, we see things very differently, and for that I would say, there is no antidote. You keep mentioning objectivity, which has nothing to do with what I've been arguing.
Quoting Apustimelogist
It's not mere speculation because experience is something we can reflect on and analyze. Metaphysics is not based on experience at all but on imaginative hypothesizing.
Quoting Apustimelogist
Some descriptions of some behaviors are more determinate than others, obviously.
Quoting Apustimelogist
I don't believe you can.
You are not presenting any arguments, just baseless objections, it seems, so the conversation is going nowhere.
Well what have you been arguing?
Quoting Janus
I think in many ways reflecting on experience is just that though. I feel like people can have radically different views of what experiences are, what feelings are, what they actually perceive, and how do people make something of their perceptions other than by intuition?
Quoting Janus
Hmm, thinking about it, I think it might be difficult if your intuitions are set on counting rather that quounting. But maybe a quonter would find no problem with it.
I have been talking specifically about synthetic a priori knowledge of what is intrinsic to embodied experience: spatiotemporality, differentiation and the other attributes I mentioned.
Quoting Apustimelogist
Maybe...I remain unconvinced.
Well I would say there is still a difference between know-how and know-that when worded like that.
But not the significance that know-how doesn't give a determinate know-that
Well, if they're not derivable from counting then your argument against quusing isn't really talking about the same kind of thing since you've outlined a procedure for deciding if someone is quusing by pointing out that we can count beyond the quuser. But if it's not counting then that doesn't really demonstrate that a person is adding or quusing. The operations are distinct, rather than reducible to counting.
I don't believe that know-how can always be translated into a determinate know-that. And any such translation will always be an abbreviation, a reductive conceptualization.
Quoting Moliere
I'm saying that squares are derivable from counting; my point was that the square root of two cannot be instantiated with physical objects (derived from counting) like the rational squares and square roots can. Think about the relationship between the words 'ratio' and 'rational'.
You could come up with a million absurd and arbitrary rules like quusing, and all I can say is "so what?". The logic of counting is inherent in cognition; even animals can do basic counting. And I see no reason not to think that basic arithmetic finds its genesis in counting. Give me a good reason not to think that and I will reconsider.
I'd say that basic arithmetic's genesis is in abstraction more than counting. But whether that's a good reason or not is up to you.
Mathematics is strange because there are no physical instantiations of it, really, and yet it's still true. It's always abstraction. With 0 you have to recognize something that isn't there. With 2 you have to look over the differences in physical objects to see what's the same between them. With 1 you have to individuate from the rest of the world: "this is an object distinct from the world as a whole. here we have a part"
I'd put mathematics on par with language as a whole rather than counting. Counting is an operation whereby we find the number. We don't even need things, as you've stated. You just go to the next number.
But what is the next number?
With modular arithmetic the number after 12 could be 1, or if we use military time the number after 24 is 1. Since we're in the domain of time this makes perfect physical sense. It's just a way of marking the day rather than the total time. Sometimes that's more important than a count "from the beginning of time".
Quaddition is certainly an arbitrary rule. (one might be tempted to say to the external world skeptic the same thing) It's a toy.
But the rub here is that addition is too -- it's just more useful than quaddition because of the world we happen to be referencing. But if we were referencing clocks then a different, modular arithmetic might be better suited.
So maybe a more plain-language way of putting the question @frank opened with (though I haven't read the text he's supplied, so I could be wrong): the skeptic might be asking how do you know the answer is not "the time is about 10:25" given that 125 divides into 12 10 times with a rough estimate of 25 minutes.
The challenge is about rule following, specifically about rule following activity that's now in the past. It's not an epistemic problem. It's not about what a person knows about which rule they followed. It's that there's no fact (a situation existing in the world) even in terms of mental states that satisfies Kripke's criteria for a rule-following-fact.
The idea of quadition was just to convey the problem. Kripke wasn't trying to do philosophy of math, although there have apparently been philosophers of math who were interested in it.
My thoughts on it (so far) is that it fits pretty well with my belief that we aren't as rational in practice as we tend to think we are. I think some people would assume that means I end up a behaviorist, but I'd say they're making the same mistake again. They think their post hoc rationalizations are the way the world really is. It's not.
Counting starts with concrete objects and then becomes possible in the abstract with the advent of numerical symbols.
Quaddition seems to arbitrarily countermand the natural logic of counting and addition; the logic that says there is neither hiatus nor terminus.
I'm not trying to do philosophy [s]as[/s]of math. I don't think I'd reduce rationality to rule-following either.
I think what @Janus's position amounts to is that there is a kind of fact, namely the familiar rules of arithmetic, which is the natural way to believe a person to be thinking about the question "how many?"
I'm taking the position that this is not an adequate reply, and attempting to give examples, like modular arithmatic, that are actually used where the procedure is the similar to the philosopher's toy of quaddition. Just because quusing is a philosopher's toy in comparison to addition that doesn't mean we have a fact to the matter about which rule is being followed -- there are other, more "practical" operations of arithmetic which can serve the same function as quaddition in the set-up. So the familiar reply to the skeptic -- to ask the skeptic to justify their skepticism -- can be overcome because there are practical (natural) examples that look identical to addition that are not philosopher's toys.
Quoting frank
Heh. I don't think I'm that deep. I see a question, but I don't see a resolution.
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?
I don't believe arithmetic to be merely rule following, but I think it is something we get intuitively on account of its being naturally implicit in cognition. Some animals can do rudimentary counting, which means they must be aware of number.
So, it begins with recognition of difference and similarity, then gestalted objects, then counting of objects, and this basis is elaborated in the functions of addition, subtraction, multiplication and division. Mathematical symbols and the formulation of arithmetical rules then open up the possibility of endless elaboration and complexification.
I hope that makes it clear how I see it. I'm happy for others to disagree, provided they disagree with things I actually think, and not some imagined position based on their misunderstanding.
But counting is just applying the rule "+1", so it doesn't escape Kripke's question.
But Kripke's question is a mistake. A rule doesn't state a fact; it gives an instruction. So the question here is what counts as following the instruction. The facts can't possibly determine that on their own. It requires acceptance of my response to the instruction. But my acceptance of my response is empty. Acceptance of a response must, in the end, come from other people.
If we want a complete description of behavior then I believe that a better term would be a neurobiological-ist which I think many people would find totally reasonable perspective!
There would only be a logic to countermand if there was a sensible definition of these things in the first place which specified the correct behavior without requiring prior understanding... and if rules like quaddition provided different outcomes to addition. Sure, only considering quaddition on its own doesn't provide the right answers but considrr that there are an infinite number of possible alternative characterizations which you can even use in any number of combinations.
It is therefore possible to use alternative concepts without any difference in behaviour. How is that countermanding logic? It cannot be. This is in the same vein as Quine's indeterminacy of translation also.
Again, the only recourse you have is "Naturalness" and given that I don't think you can give me a satisfying definition of counting or quantity, that to me is almost like begging the question without being able to tell me what you are even begging, so to speak. The only reason I know what you are saying is that I have an implicit undrrstanding of what you are talking about. Not necessarily an explicit one.
You've already said that you think this stuff is implicit so I think it must mean we agree more or less but you are failing to distinguish that there is the explicit notion of these rules and then the implicit "blind" notion. This is maybe why we are talking at cross purposes because you agree about the implicit thing, so do I. The whole debate however is about the explicit characterization.
How do you respond here to @Ludwig V's point?
Quoting Ludwig V
Here there's a few bases from which we could confuse one another: arithmetic as a practice, arithmetic as a part of our rational intuition, arithmetic as rule-following, arithmetic as it was in its genesis, and arithmetic as it is. It might depend on which we're thinking about in our assertions how we evaluate the skeptical position.
Quoting Janus
Hard to attain, at times. All we can do is re-state, try again, and all that. I read you as taking an intuitionist stance, as in mathematics is a part of our natural intuition that's even shared with other creatures, and so the skeptic has no basis because the skeptic is framing arithmetic in terms of rule-following when there's more to arithmetic than rule-following, such as the intuitive use of mathematics, whereas the skeptic's use is derivative of that (and so is an illegitimate basis of their skepticism, considering that the skeptic is undermining their own position in the process)
Let me know if that's close or not.
If they don't make any difference, how are they alternative?
On the other hand, it is perfectly possible for two or more of us to get along quite well for a long time with different interpretations of the same concept or rule. The differences will not show themselves until a differentiating case turns up. This could happen with quaddition or any other of the many possibilities. Then we have to argue it out. The law, of course, is the arena where this most often becomes an actual problem.
Quoting Apustimelogist
Quoting Apustimelogist
What is fundamental to understanding concepts is not their definition, but knowing how to apply the definition. That is a practice, which is taught. Learning to count and measure defines number and quantity.
Quoting Janus
There is a natural logic of these things. But we had to learn how to do it. It seems natural because it is a) useful and b) ingrained. "Second nature".
Quoting Moliere
There's certainly a difference between arithmetic in its genesis and arithmetic as it is. For the ancients, arithmetic was developed for severely practical reasons. The first texts on the subject are clearly meant to enable administrators to provision and organize the work force or the army (Ancient Egypt). The Greeks did not count (!) either 0 or 1 as numbers - it was the Arabs who included them. Arithmetic as it is includes all sorts of crazy numbers - irrational, complex, etc. Yet it is always the use of the numbers in calculations that drives the changes.
However, the idea of arithmetic as rule-following and the idea of arithmetic as a practice are closely related. If you ask me to justify my claim that 68+57 =125, I can do so. But if you ask me to justify my application of the rule "+1", I can only start to teach you to count. Counting is a practice, which is either done correctly or not, where correctly means what we agree on (bearing in mind that pragmatic outcomes provide a semi-independent check on purely subjective mutual agreement).
This is what Wittgenstein means by saying "justification comes to an end" or "This is what I do".
Quoting Ludwig V
As stipulated the rules of quaddition do provide different outcomes:
Quoting frank
Addition gives "125' and quaddition gives "5". Which one is correct? Imagine there is a wedding, and there are 68 guests from one side of the family and 57 from the other side. Addition tells you to provide food and seating for 125 guests, and quaddition tells you to provide food and seating for 5 guests. Now you tell me which one will turn out to have been correct.
Quoting Moliere
I do favour intuitionism in this. If the skeptic could provide different rules of counting and addition which do not consist in infinite iterability and yet can always come up with pragmatically workable solutions as in the simple wedding example above, then it might be time to start taking it seriously. How do you thinking structural engineering would fare if it started using quaddition instead of addition?
It is natural simply because we can intuitively get the logic once we have our attention drawn to, and become familiar with, its basics. We can apply the rules because they make cognitive sense, so we don't require another set of rules to tell us how to use the rules of counting and addition. We don't even really need to be able to explicitly state the rules, just as it is with grammar in the case of language. The fact that there are several different possible grammatical structures which are exemplified in different languages doesn't change this; the logic remains basically the same, it is only the order that changes.
Well you can use sets of concepts with different meanings to refer to the same thing, enables by the natural underdetermination.
There is a forward problem of mapping rules to behavior in which case, I can use any number of multitude of different concepts and combinations of concepts in order to produce the same behavior as you might get from addition.
There is also the inverse problem of mapping behavior to rules in which case, even under some single case of differentiation, there is always a multitude of alternatives that underdetermine what the successful rule actually is at any given time.
Quoting Ludwig V
Well that suggests you have a definition in the first place. Neither would I say that you can define these concepts by the behavior itself. But yes, part of my view all along is the distinction between explicit definitions which are chronically underdetermined and the implicit behavior which we have a mastery of but is difficult to give explicit descriptions.
Quoting Janus
My point here is the forward problem as described earlier. Even though quaddition has particular outcomes, someone can generate all of the behavior of addition and define it, have definitions, without using addition, even if they require a plethora of other concepts to make it work. And again, this all depends on people agreeing with all the necessary concepts which are required to make something like quaddition work. My understanding of all concepts is scaffolded on prior concepts and prior implicit understanding or abilities that have been learned by practise without definitions.
I don't think it is true that the same outcomes as addition could be achieved using some other set of rules or concepts "to make it work". I see no reason to think that. Can you stipulate a set of rules and/ or concepts that will always yield the same results as addition? If you cannot, then how could you know it would be possible?
Even if you could come up with something, that wouldn't change the fact that addition is intuitively gettable, while the alternative is just some arbitrary set of rules that happened to work, and which would be parasitic on the gettability of addition in any case.
I think you can. If you can make up arbitrary rules like quaddition then you can think up infinite many rules which give describe all the same processing ability.
Quoting Janus
To you maybe. It might be totally unintuitive to a different kind of being. Addition might be arbitrary or unintuitive to someone else just like how you might find the notion of some operator that subsumes division, addition etc etc unintuitive.
But do they yield answers that are pragmatically workable?
Quoting Apustimelogist
Counting is intuitive to humans and apparently some animals. I doubt there are sentient beings which would find it not to be intuitive if they had the ability to count. Of course, there are sentient beings who cannot count, but that would not be a lack of intuitive ability, but simply a lack of the necessary intelligence.
Intuition really doesn't matter because its arbitrary. What is intuitive to a human may not be intuitive to an animal or an artificial machine. What a mathematician finds intuitive might be different from a layman.
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.
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.
Quoting 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.
Quoting 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.
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.
Quoting 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.
Quoting 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!
"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?
Quoting sime
Well, on the face of it, this sounds not disimilar to Wittgenstein's meaning as use.
I want to post Kripke's summation of his own argument. On page 107-109:
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 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.
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.
Quoting 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.
Quoting Moliere
Very true.
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.
Quoting 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.
Quoting 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.
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 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?
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.
Quoting 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 the question is reaching for something beyond the limits of language, so I think you've got the right idea. :grin:
Quoting 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.
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.
Quoting Ludwig V
I can't think of any examples of failures of consensus concerning basic arithmetic.
Quoting 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.
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.
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.
Quoting Janus
And just because a rule is unintuitive doesn't refute it being objectively true.
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.
Quoting Apustimelogist
I don't know what you mean when you talk about a rule being objectively true.
Kripkes proof shows rules are not objectively true.
That there is a fact of the matter about what rule is being followed.
Quoting 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.
Oh, OK, I would say that is uncontroversial.
Quoting 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.
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.
Quoting 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.
What other "strange rule" have I been using? Basic arithmetical procedures are simply the infinite iterability of addition and subtraction, and the fact that things can be grouped together in terms of different quantities.
I agree that many rules have been extrapolated out of these basics, but the extrapolations are not arbitrary in the kind of way quaddition is. They just show what can be done with these basic conceptual tools.
So, I don't agree that these basic procedures are "under-determined".
We apparently see things very differently; so much so that I cannot even tell where you are coming from with this.
other rules like quus. there are probably a multitude of them which are consistent with all of the addition you have ever done so far in your life and you can't rule them out.
Quoting Janus
uhhh don't you mean quu-nfinite quu-terability?
Quoting Janus
why should it be that just because a description is general or extrapolatable means it is any more or less true than a description which is specific. Is the fact you are using addition any less true than the more general description of using an operator? is the more general description of being a mammal somehow more true than the more specific description of being a human?
How can they be consistent if they don't yield the same results.
Quoting Apustimelogist
No, I wasn't referring to gibberish.
Quoting Apustimelogist
All that seems irrelevant. I may be missing something to be sure, but if it is so, no one seems to be able to point it out. I've reached burn-out on this...
Quoting Janus
It's worth remembering that in geometry, it turned out that rules other than Euclid's (with all their intuitive plausbility) turned out to yield consistent systems, which, in the end, turned out to have "practical" applications.
Quoting Banno
I don't think that's a particularly interesting result. Rules are instructions, so they aren't either true or false. That is, the rules of chess are not true or false; but they do yield statements that are true or false, such as "Your king is in check".
Quoting Banno
Well, I would suggest that what is at stake is the refutation of a certain conception of what rules are - the idea that logic/mathematics is some kind of structure that determines the results of all possible applications in advance. Nothing can reach out to infinity. What we have is ways of dealing with situations as they come up which do not appear to have any limitations to their applicability. (That phrase could be misinterpreted. I mean just that "+1" can be recursively applied indefinitely. What we can't do is apply it indefinitely.)
Quoting Apustimelogist
Yes, but that doesn't mean that we cannot have ways of responding to, and dealing with, problems as they come up - if necessary, we can invent them - as we do when we discover irrational numbers, etc. or find reasons to change the status of 0 or 1. In the case of 0, we have to modify the rules of arithmetical calculation.
Because I can construct a rule just like quus which is consistent with all of the additions you have done so far in your life and would yield the same results as all sums you have ever done, analogous to how someone may have never done sums with larger than 57 will have been doing an operation totally consistent with both quus and plus. Now the question is whether there is an empirical difference that differentiates the rule you have been using so far as either addition or this other rule? The answer is no, because so far every answer for addition is the same as this new rule. And remember there will be a multitude of these rules; for all your past behavior, this will be underdetermined and this will continue to be the case for t+1, t+2... t+n ad infinitum for every new sum you do.
Quoting Janus
The point is that you have defined what you mean by the fact that addition is diffetent to quudition, but the words in this description are susceptible to the same kinds of skepticism, and further attempts to elucidate what you mean will bring a regress of these definitions on which skepticism can be applied.
Quoting Janus
well you seemed to be appealing to the extrapolatability or generalizability of addition as to why it is more true but I don't see why this is a fact in making a description more true or not. why should it be that a description that extends to more cases than another be somehow more true?
Quoting Ludwig V
Buy "you are following x rule" is factual. What do you think is the interesting result of this story then?
Quoting Ludwig V
I don't think this problem has anything to do with practical problems. The quus issue has no bearing on someones ability to perform math.
Does Kripke question the extent to which consciously following a rule even applies?
For example, having worked with digital logic a fair bit, I have all the powers of 2 up to 2^13 memorized and if I see 2048 + 2048 I simply recognize that the sum is 4096 without following any step by step decimal addition rules.
What is supposed to be the significance of arriving at sums via different cognitive processes?
Hume's questioning of the place of causation doesn't yield reliably workable results. Scepticism isn't as much about reliable workable results as truth.
Quaddition's workability isn't really at issue. I think the sceptic would say "no, that's not useable for engineering. But what's the fact you can point to that lets us know the engineer is using addition?" Quaddition is there as a conceptual contrast to addition to help in understanding the question "What's the fact I can point to that justifies my belief that I'm adding?"
To make a similar function to quaddition that'd be easier to accept in light of engineering: Instead of Quaddition we could posit Googol-ition -- where the rules of arithmetic are the same up to a googol. If you find an example of an engineer whose used a number that high, then you can raise the googol to the power of a googol, and posit the googol^googol-ition. What's being asked after is a fact which demonstrates that we're performing addition, and googol-ition is there to give a conceptual contrast (and highlight that there's no factual difference, or at least make that challenge).
And the sceptic believes there is no fact at all -- there's a rule being followed rather than a truth being stated.
Does that make the question make sense?
Not sure about Kripke but Wittgenstein definitely mentions stuff l like that in philosophical investigations. rules and explicit definitions are more like signposts than prescriptions on how to behave. in fact, i think a major point in PI is that meanings and definitions in language are effectively so impoverished that it should render language un-usable, but it doesn't. A repeated theme it seems to be this underlying inscrutable, implicit underlying behavior where there is room for the kind of indeterminacy, fuzzyness or perhaps plurality about how people accomplish things.
Excerpt from PI:
"232. Let us imagine a rule intimating to me which way I am to obey it; that is, as my eye travels along the line, a voice within me says: "This way!"What is the difference between this process of obeying a kind of inspiration and that of obeying a rule? For they are surely not the same. In the case of inspiration I await direction. I shall not be able to teach anyone else my 'technique' of following the line. Unless, indeed, I teach him some way of hearkening, some kind of receptivity. But then, of course, I cannot require him to follow the line in the same way as I do.
These are not my experiences of acting from inspiration and according to a rule; they are grammatical notes.
235. It would also be possible to imagine such a training in a sort of arithmetic. Children could calculate, each in his own wayas long as they listened to their inner voice and obeyed it. Calculating in this way would be like a sort of composing.
234. Would it not be possible for us, however, to calculate as we actually do (all agreeing, and so on), and still at every step to have a feeling of being guided by the rules as by a spell, feeling astonishment at the fact that we agreed? (We might give thanks to the Deity for our agreement.)
235. This merely shews what goes to make up what we call "obeying a rule" in everyday life.
236. Calculating prodigies who get the right answer but cannot say how. Are we to say that they do not calculate? (A family of cases.)
237. Imagine someone using a line as a rule in the following way: he holds a pair of compasses, and carries one of its points along the line that is the 'rule', while the other one draws the line that follows the rule. And while he moves along the ruling line he alters the opening of the compasses, apparently with great precision, looking at the rule the whole time as if it determined what he did. And watching him we see no kind of regularity in this opening and shutting of the compasses. We cannot learn his way of following the line from it. Here perhaps one really would say: "The original seems to intimate to him which way he is to go. But it is not a rule." "
Today we're talking in the meta-language about the object-language of yesterday, or right now we're talking in the meta-language about the object-language of addition. What, in the object-language, is the fact that we're adding at all? Would you say that this version of the question is easily viewed as nonsensical?
One of the things that I keep thinking on is how I tend to think of facts not as things but rather as true sentences. So in reading the essay, to make it make sense, I'd probably put it that -- rather than there is no fact to the matter -- there are no truth-conditions which make 68+57 equal 125. It's true because that's the answer we should obtain according to the conditions of assertability, but there are no truth-conditions that make it true.
In saying that much -- the question begins to make a kind of sense because mathematics is abstraction. So, in a way, there shouldn't be truth-conditions of addition. If there's a physical unit involved then there are possibly truth-conditions, but that's not the question. It's much more a question about meaning because of the abstraction. (at least, as I'm understanding it so far)
Thanks for taking the time. I'm reading PI right now, but haven't gotten that far.
Insightful stuff.
It can be really difficult to read to be fair. Its one of those books where possibly what the book says has not been as influential as what othwrs have said about the book.
I admire your memory! But isn't it exactly the same as we all (?) do when we memorize the standard multiplication tables and recall what 12x11 is. (It's just a convention that we stop at 12. The table for 13 is no different in principle from the table for 12.) Multiplication reduces to addition, but adding 12 11's by that procedure is long and boring. By memorizing the standard multiplication tables, we have a quicker way of dealing with some questions and of calculating bigger numbers. (Incidentally, how do you deal with 2 to the power of 35?)
Quoting Apustimelogist
I agree with that. Though Wittgenstein would ask what makes the sign-post point? Again, there's a practice of reading sign-posts, which we all somehow pick up/learn. Perhaps by recognizing a similarity between a pointing finger and the sign-post.
Quoting Apustimelogist
I agree with that. It's a pointless difficulty. Like most sceptical arguments. I like Hume's response - essentially that it is not possible to refute the argument but it has no power to persuade me to believe the conclusion. But that's not how the philosophical game is played - for better or worse.
Quoting Apustimelogist
There's a nest of complications buried in that. In one way, you are just raising the original question again. However, there is a fact of the matter involved - that I gave 125 as the answer to the question. Whether I was following the rule "+1" is another question. In one way, it depends on whether I had that rule in mind when I gave the answer. In another way, it depends on whether we agree with the answer - and that may depends on the wider context (consistency and practical outcomes).
Quoting Moliere
Forgive me, I don't really understand what "conditions of assertability" are as opposed to "truth-conditions". Are they facts? In which case, we may be no further forward.
Quoting Apustimelogist
That's an interesting question. In one way, the desired result is to defuse the question so that I don't get bothered by it - that is, don't need to take it seriously. Whether that's interesting or not depends on whether you are philosophically inclined or not.
But I am learning from this. One result is that I now know how to defuse Goodman's "grue". Another is that it seems that Kripke has made the private language argument superfluous. I need to think about that. A third - minimal - result is that Kripke has added to the stock of examples that pose Wittgenstein's problem. The fourth is that I notice that we have all appealed to the wider context, both of mathematics and of practical life to resolve it. Kripke's case is effective only if we adopt his very narrow view, The wider context makes nonsense of it. (I'm not saying that a narrow focus is always a bad thing, only that it sometimes gets us into unnecessary trouble.)
Wittgenstein says somewhere that he has got himself into trouble because he is thinking about the pure world, but what we need to do is return to the rough ground.
Quoting Apustimelogist
I agree with that. One of the difficulties is that the text is not difficult to understand (contrast Hegel or Derrida). The difficulty is to understand what the point is. That's where the commentators can help - and sometimes hinder, so don't read them uncritically.
This book is not written, as most philosophy books are, in the belief that laying out the arguments clearly ("clearly" is complicated, of course) is the most effective way of changing someone's mind. No doubt it is, sometimes. But W thought that philosophical problems were not really susceptible to that treatment. So he provide hints and leaves you to work out what he's getting at. Some of his followers do the same thing.
That's where the commentators can help - and sometimes hinder, so don't read them uncritically. You'll need a general introduction to start with. Sadly, I'm so out of date that I don't know which are the best ones. But there'll be reviews that will help you choose and you could a lot worse that read an encyclopedia entry, which would be shorter.
I think it'd depend upon how we're trying to judge if someone knows something or not. With arithmetic those conditions are spelled out in books and habit and embodied within a community of arithmetic speakers. I'm thinking that it has more to do with a community's process of acceptance than facts.
So the teacher has a handful of representative problems which if the student is able to do without aid we then accept them as part of the community of arithmetic speakers.
Same goes for accepting whether a person knows the meaning of such-and-such for particular topics, or whether they know a language: the meaning isn't a fact as much as what you have to do in order to be accepted within a community of languagers.
Yup, and then the issue regresses as to what makes someone recognize a similarity between pointing a finger and sign-posts. This is all what I meant when I said that meanings and definitions are so impoverished that language should not be usable, yet it is.
Quoting Ludwig V
I think for me, its about how such insights might reveal something about how brains and minds work.
Quoting Ludwig V
Well I am just saying that it is a factual statement regardless of whether there turns out to be or not be a fact of the matter. It straightforwardly makes sense as a factual statement.
Quoting Ludwig V
Or perhaps even what it means to have a rule in mind.
Quoting Ludwig V
Can you elaborate?
Yes, it just seemed relevant to me to point out that there are a variety of mathematically legitimate ways that can yield correct mathematical results and therefore it seems weird to me to focus so, on whether some particular rule was used in some specific case. So I brought it up in hopes of getting a better idea of what Kripke was trying to get at.
Quoting Ludwig V
With a calculator. :wink:
Well though the math thing you get at is related, its not exactly the same. The rule thing is about definition and description and is just meant to be a single example of a type of issue that is generic to everything. I guess the point is that semantic definitions and descriptions are not intrinsically embedded in the world; instead, we impose labels on the world at out own discretion and there are no fixed set of boundaries for those concepts or force us to impose concepts in a particular way.
Yes. There isn't a way of resolving that without going beyond that way of thinking. W's does that. His appeal to games, practices, forms of life etc. is an attempt to explain it. As a general thesis, it is quite unsatisfactory, (cf. God of the gaps), but as a tactic applied in specific situations, it works well (as in this case). There's an obvious catch that it may be misapplied. But that doesn't necessarily mean it is never appropriate.
Quoting wonderer1
I think the intention is to distinguish between a heuristic which may be useful in some circumstances, but not in all, and how we would settle the question whether the output of the heuristic is correct or not. The intriguing bit is why we accept one way of calculating as definitive (conclusive). Kripke's problem muddles up the two different ways of getting an answer.
Quoting Apustimelogist
That works in some ways. But the picture of the world out there, waiting to be "carved at the joints", is partial. The world reaches in and prods us, tickles us, attracts us and repels us. We do not start out as passive observers but as engaged actors in the world - which does not always behave in the way that we expect.
Quoting Apustimelogist
H'm. My posts are quite long as it is. I'm concerned I might outstay my welcome or run up against the TL:DR syndrome. Some focus would help.
Quoting Moliere
I get that distinction. Indeed, arguably an assessment whether the knower is in a position, or has the capacity, to know p is appropriate in assessing any claim to knowledge. And I can see that final truth will often be distinct from any such assessment. (The jury has a perfect right to find the prisoner guilty or not. Yet miscarriages of justice do happen - and proving that is different from proving whether the prisoner is guilty or not. (A miscarriage might have reached the right result.)) But I still feel that the distinction is quite complicated. After all, the truth would be the best assertability condition of all, wouldn't it? And the assertability conditions would themselves be facts, wouldn't they? Of course, they need not be the same facts as the truth conditions.
Quoting Ludwig V
For me, what the point of the impoverishment of language shows is that the way we use words and concepts does not trickle down prescriptively from definitions and meanings that possess some invariant, essential nature. Rather, definitions are idealizations that are constructed or inferred in a bottom up manner from the statistics and dynamics of experience. The kinds of fuzziness, ambiguities, context-dependence, indeterminacy that characterizes Wittgenstein's analyses can be explained by appealing to the nature of how brain processes perform inference, effectively extracting lower-dimensional, more coarse-grained, more generalized underlying patterns (concepts) from complicated observations. These are extremely complicated machines processing an extremely complicated world and so the processing they do does not necessarily reflect very simple, linear, straightforward transformations between observations and the resultant inferred concepts.
Essentially the missing link in Wittgenstein's observations is the fact that we have a brain, one whose processing is extremely complicated yet also totally hidden from us, generating our complicated thoughts and behavior from below on the fly, making it look like we are acting in these kinds of mysterious ways that seem somewhat messy and underdetermined by our concepts and so can only be described as "games, practises, forms of life".
"The world reaches in and prods us, tickles us, attracts us and repels us" as a product of the mechanistic message passing and hebbian timing-dependent learning between neurons that are physical enslaved by the patterns of activation at our sensory boundaries (e.g. retina, inner ear, receptors under the skin), impelling the perceptions forced upon us, complicated behavior we are capable of, the higher-order concepts that we construct, but also the metacognitive insight that such concepts could have been otherwise. The brain completes the picture.
Quoting Ludwig V
I dunno; I think looking at this way, as I seem to understand what you have said, plays down everything else that Wittgenstein seems t be getting at in philosophical investigations.
Quoting Ludwig V
Well you just put forward your four points without any reference to what you mean by those points. Basically, all these points are lacking a "how".
Quoting Apustimelogist
I hope so. It's the only way that we get reliable information - and, in great part, we do.
Quoting Apustimelogist
I'm sure there's a lot of quick and dirty solutions and heuristic dodges involved. Anything remotely like formal logic would be too slow to be useful.
Quoting Apustimelogist
I was only talking about relying on a memorized table, instead of doing the basic calculations. It's an example of a quick and dirty solution.
Quoting Ludwig V
This "paradox" is structurally the same as Kripke's. Here's the link to Wikipedia https://en.wikipedia.org/wiki/New_riddle_of_induction, which mentions, but does not discuss, Kripke and his solution. I think that Wittgenstein's discussion of rule-following applies to both of these puzzles. Does that help? To take it much further would probably require another thread, don't you think?
Quoting Ludwig V
This point is made elsewhere. The complication is that the private language argument does rely on some of the things he says about rule-following, particularly the importance of understanding what does and does not conform to the rules about ostensive definition. But numbers are not sensations, so the cases are not exactly the same.
Quoting Ludwig V
W likes lots of examples. In one way, Kripke's case is just another one, although W does mention the point at PI 201 "This was our paradox: no course of action could be determined by a rule, because every course of action can be brought into accord with the rule. The answer was: if every course of action can be brought into accord with the rule, then it can also be brought into conflict with it. And so there would be neither accord nor conflict here." I had forgotten this quotation. In time, I could no doubt find what he was referring back to. It gives a short answer to both Goodman and Kripke.
Quoting Ludwig V
Isn't that an accurate reflection of what we've been saying about practices?
I hope that's helpful.
I am not sure I would characterize them all as heuristics or "quick and dirty" solutions since they are just the same processes that underlie everything we do. Its just that the actual statistical structure of much of the world is much more complicated and non-linear than the simpler idealized concepts we like to dealing with in academics.
I think the problem with the answers that brains give though is they are finely contextualized by different personal histories, individual differences in brain structure, noise etc. What people learn and the information they store is probably different for everyone, but in places like academia we want to remove all ambiguity. The side effect of neat clean concepts is they lose all the fuzzy non-linearity which makes them exceptionally good at being used in real life.
:100:
Yeah, I'll admit it's complicated. Or at least vague. I don't know if the truth is the best assertability condition, though, because here we have truths that we arrive at because of the conditions of assertability -- at least this seems to make sense of Kripke's position as an interesting position. If it all came back to truth then what's the deal with pointing out that there's no fact to the matter?
Also I think Kripke takes us to this place in his essay, but then doesn't say much more. I'm still uncertain that I have the exact right interpretation of Kripke here, too -- this is just what comes to mind when I attempt to make sense of Kripke's arguments.
There's something queer for myself at least in holding that facts are true sentences, that mathematical sentences are true, and yet they are not true in virtue of truth-conditions. It would seem that under this interpretation that I'm committed to some way of coming to know true sentences aside from truth-conditions. Given that we're talking about meaning that seems to be where I'd have to go. And there's a historical precedent there in the analytic/synthetic distinction, but I wouldn't want to rely upon that distinction because I pretty much agree with Quine on it being fuzzy.
So, yes, to hold to my interpretation of Kripke's conclusion along with some of my other beliefs and defend them I'd have to do some work on what these conditions of assertability are.
Yes, that's true. I'm a bit inclined to say the W sees that "fuzzy non-linearity" as inherent in all concepts. So what do we say about logic? What makes it special? (I'm not asking because I know, or think I know, the answer.)
Quoting Moliere
Maybe we should distinguish between what brings the rule into effect (I chose that word carefully because after it becomes effective it is correct to say that there is a rule that ...) Can we see conditions of assertability as comparable to the licence conditions for someone to perform a wedding? If so, laying down a rule is or at least is comparable to, a speech act. We then have to explain that in some cases, the rule is not formally laid down, but informally put into effect (as when language changes, and "wicked" comes to mean the opposite of what it meant before). Once the rule is in effect, there is a fact of the matter, as when your king is in check or 68+57=125.
Quoting Ludwig V
Yes, I would agree with this and agree with it myself. I don't know of there is anything particularly special about logic and am drawn in the direction of logical nihilism or pluralism.
A philosophical commonality of those approaches, is that mathematical objects are treated as being finite. So for example, plus is permitted to exist intensionally in the sense of an algorithmic specification but not in the extensional sense of a completed table (unlike with the quus function). Likewise, all sequences are treated as being necessarily finite and generally unfinshed. This apparent restriction is compensated by allowing objects to grow over time (technically, growable objects are describable by using what are now referred to as coalgebras and coinduction).
Intuitionism would call a sequence of n numbers x1,x2,...xn that were generated by iteratively applying the "plus" function as constituting a lawful sequence of n terms. But another sequence consisting of exactly the same numbers x1,x2,..xn, that wasn't assumed to be generated by some underlying function, would be considered a lawless sequence. In the case of quus, it can be considered to be a lawless sequence , since it is describable as a table of exactly 57 rows and 57 columns. So if that table is defined to mean "quus" as a matter of tautology, then skepticism as to what the function quus refers to can only concern operational assumptions regarding how the table should be evaluated - however skepticism of this sort is accommodated by intuitionism, since intuitionism doesn't consider mathematical truth to be a priori and the properties of unfinished sequences are allowed to change over time . The situation with quus is at least a partial improvement upon plus, whose table cannot even be explicitly stated. Quus is more or less a truncation of plus, and roughly speaking, intuitionism considers such "truncations" as constituting the basis of mathematical analysis.
Additionally, in Linear Logic terms and constants can only be used once. So two uses of a function demands two separate and distinct copies of that function. Linear logic includes a copy operation, (the so-called exponential rules), meaning that the logic can be used without loss of generality, but this forces the mathematician to justify and account for his use of resources in the same way as engineers who must always consider the possibility of numerical overflow and hardware failure.
Kripke allows that mathematicians can adequately specify the rules of addition. That's not being called into question.
I prefer pluralism coupled with pragmatism. Horses for courses. Logical analysis can give a kind of clarity.
I'm inclined to say there is a fact, but that it's not the fact which justifies, say, the license conditions for someone to perform a wedding. The rule is in effect, and in some sense then it produces facts -- but the production of facts is not justified by the facts so produced. The rule itself has no factual justification, though we could only judge if a person knows how to add if we know the fact we'd obtain by performing the rule. (at least, in my way of speaking where facts are true sentences. this could very well be me bringing in an inconsistency, though, whereas Kripke wouldn't bother with this notion of facts being true sentences. I'm not sure there)
Or, at least, I can't help but think that there has to be some distance between rules and facts for Kripke in order for the position to be philosophically interesting. The skeptic has to be pointing out that we're inclined to believe there's a fact where there is none in order for the skeptic to have a point at all, or else we're more or less just stating that the skeptic does not succeed in pointing out a skeptical problem.
There are rules for which the process that brings them into effect is quite clear. They are what we call laws, but there are other varieties. They are imperatives, not really different from the order given by the general. Other rules, like mathematical rules about how to calculate are different. There are proofs of such rules. What makes them effective? Which is to say, what justifies them? That's where the sceptical pressure (which W also applied) and his appeal to practices comes in. But that involves saying that the rule doesn't really of itself produce facts; human beings have to carry out the calculations (or psersuad machines to do it for them. Those results are facts, i.e. have the authority of facts? Only the calculation, which can't produce a wrong result. That means that if a result does not fit in to our wider lives in the way it is expected to, we look for the fault in the calculation and the calculator, not the rule.
At the bottom of this is the fact that "+1" can be applied to infinity. How can we know that? Certainly not by applying the rule to infinity. By definition, we can never exhaustively check that the infinite application of the rule will work out.
Quoting Moliere
If you mean a fact that justifies the rule and/or justifies how the rule is applied. I sometimes think that the quickest way to state the problem is to point out that the rule cannot be a fact, because the rule has imperative force and no fact can do that - a version of the fact/value distinction. For the same reason, no fact can, of itself, justify the rule.
If you mean the mathematical justifications of the rule, that's true - within the rules (practices, language games) of mathematics. But what justifies those? "This is how we do it. You need to learn that. Then we can discuss justification." It's not quite foundationalism and not quite some form of coherentism. As usual, he manages to not quite fit in.
That depends on the sense of adequacy you are referring to. The question is, how can an intensional definition of addition such as an inductive definition, that is finitely specified and only provides an inductive rule for performing a single step of computation, imply an unambiguous and extensionally infinite table of values? As Kripke himself pointed out (IIRC), as the numbers to be added get very large, there is increasing uncertainty as to what the meaning of a "correct" calculation is, for finite computers and human beings can only cognize truncations of the plus function. And a "gold standard" for the extensional meaning of addition up to a large enough arguments for most human purposes hasn't been physically defined by a convention as it has for the meaning of a second of time.
I am probably not understanding this at all correctly because its too technical for me but it sounds like its bolstering the Kripke's skepticism rather than really solving anything.
I think this misses out the point that this problem is supposed to only be an illustrative example of a generic problem that applies to all uses of language which would naturally include facts. And that's not even taking into account that I disagree that this cannot be looked at as a fact issue.
If you believe this is due to the fact-value distinction, then I think this kind of thought experiment would imply it applies to facts to: learning and inferring facts from evidence kind of implies an ought or imperative in the act of ascenting to some belief based on some evidence. No fact can then justify the belief. And I think yes, thats exactly what is being implied by the thought experiment; yes, I think the fact-value distinction is illusory or at the very least blurred since belief has what seems like a normative component (I am not a normative realist though). However, we can still distinguish facts and beliefs from normative concepts generally; the thought experiment I think must be about both, not just one or the other.
That's an interesting question, but it's not Kripke's skeptical challenge. His challenge is simpler: what fact is there regarding how you were using the word plus.
Quoting Ludwig V
The question of how mathematical rules are justified is also interesting, but Kripke's challenge is about the use of the English word plus. What fact is there about how you were using it?
Alternative foundations for general mathematics and computing can't solve Kripikean skepticism in the sense of providing stronger foundations that rule out unorthodox rival interpretations of mathematical concepts - but they can partially dissolve the skepticism by
1) Refactoring the principles of logic, so as to accommodate finer grained distinctions in mathematical logic, particularly with regards to a) Distinguishing intensional vs extensional concepts, b) Distinguishing between the process of constructing data and communicating it, versus the process of receiving data and deconstructing it, c) Distinguishing between various different meanings of finitism that are equivocated with classical logic.
2) Weakening foundations so as to assume less to begin with. This replaces skepticism with semantic under-determination. E.g, if "plus" is considered to be a finitary concept that does not possess a priori definite meaning to begin with, then Kripkean doubt about it's meaning doesn't make as much sense.
In summary, a good logic from the perspective of computer science describes the process of mathematical reasoning intuitively and practically in terms of a finite series of interactions between agents playing a partially understood multi-player game, in which no agent is the sole authority regarding the meaning and rules of the game, nor does any agent have omniscient knowledge regarding the eventual outcome of following a given strategy.
That seems to be the easiest way to parse things, I agree. Imperatives do not fit the form, so they cannot be either true or false.
I guess here we have to ask: is the reduction of addition to an imperative enough to satisfy the skeptic?
Can we state the imperative?
Is "68+57=?" a command? For the student it is, but when we are using the arithmetic it seems like we're actually asking something even if it's about numbers rather than some units of something (and perhaps this is what gives rise to the credulity Kripke's skeptic is pointing at). Perhaps we could rephrase all such instances as "If I were to perform addition on the constants a and b then what is constant c which all adders would agree to?"
Perhaps in general we could reformulate all arithmetical commands as "given this set of constants, and this set of operations, and this set of ordering the operations, find the correct constant"; which kind of highlights its game-like nature in that we have to have several stated "givens" before we're able to derive necessary conclusions.
The natural logic of addition includes infinitely many iterations simply because in principle there is no reason why you cannot just keep adding. Anything counter to that is a completely arbitrary stipulation.
That may be true.
I have seen no reason to think it is not true. I also see that fact as dispelling Kripke's skeptical challenge.
It only dispells it if you think dogmatism is a valid way to objective truth.
In addition, your point of view comes to the bizarre conclusion that under the conditions of underdetermination of the thought experiment where there is no fact of the matter that distinguishes someone's past usage of quus vs. plus, someone has to be using plus and not quus. Its impossible for someone to be using the rule quus because it would be too arbitrary.
I don't see how.
Dogmatism has nothing to do with it; there is simply no reason that addition should terminate anywhere.
Quoting Apustimelogist
This is nonsense: I haven't claimed that one could not use quaddition or any other arbitrary rule. If some rule of quaddition stipulates that addition must terminate somewhere then it is indistinguishable from addition up to the point of termination. But then it is simply addition up that point, and so what?
If you came up with some kind of rationalist attack on the private language argument that would be cool.
It has everything to do with it because you're adamant that even when the situation is underdetermined, you dogmatically lean on plus even though you have no further means that can disambiguate the actual rule was plus.
Quoting Janus
You are because from all our conversations so far, the final bastion you've decided to support yourself on is that quus is arbitrary and thats how you can somehow distinguish that you are using plus and not quus. Now if that is your only means of distinction then it follows that under the situation of the thought experiment, no one ever could be using quus because it is the arbitrary rule.
Quoting Janus
But the question is whether it is also quaddition?
Quoting Apustimelogist
Again nonsense. The logic of addition is not dogmatic, but simple: I can just keep adding forever in principle. Anything indistinguishable from that is just that and nothing more, so neither underdetermination nor dogma have anything to do with it.
Quoting Apustimelogist
If quaddition is the same as addition then it's not a different procedure but just a different name. So what? If it differs, then how could it do so without arbitrarily stipulating that iteration must cease at some point?
You don't see the relevance of the private language argument to Kripke's skeptical challenge? :chin:
Quoting Janus
Its about the fact that everything you have done so far is consistent with multiple different rules. The rules can then be different but your behavior so far has been indistinguishable.
Quoting Janus
You can keep adding forever but you then need to give me a definition of that which then naturally entails the results of addition and not quaddition, otherwise how would I know that you go on using your rule and then you just end up quadditing or any other rule?
I don't see human behavior as being relevant to the logic of counting or addition except insofar as it follows it. It's true that for finite addition (which all addition actually is) the logical possibility of endless iteration does not have to be kept in mind.
Quoting Apustimelogist
Can you tell me how quaddition differs from addition? If not, then there would seem to be no meaningful difference between them. If that is so, then why bother using the neologism?
The point here is that in order to show you are following a specific rule, you need to give me a reason to believe that it is one rule or the other. The rules are obviously different; you just need to give me something that distinguishes whether you are using one rule or the other. Your past behavior is possibly part of the evidence in terms of what answers you gave to previous addition problems. The issue is that they are identical to answers for quaddition so they fail to be useful evidence.
Maybe instead you can give me some kind of definition which tells me what you are going to answer next or in the future. But then again, if I can pose alternative rules that fit the data so far for your usage of plus, can I also not pose the same kinds of alternatives for the components of your definitions? To be honest, I am not sure you can give a definition of addition which can actually explicate what you are going to do next because it is one of those concepts that are so primitive, if you ask someone what it means, they tend to just give you another synonym.. but then what does that synonym mean? It goes on forever. Similar might be said for a concept like infinite or something like that. If you cannot give me an intelligible explanation then how are you going to differentiate whether you are using plus and quus?
How are they different?
Quoting Apustimelogist
If you cannot give me an intelligible explanation of how they differ then the question has no sense.
Why are you asking me how they differ when I know you know how they differ. quus operations are the same as plus except for numbers over 57 where it equals 5.
But again, this has nothing to do with the s
differentiating rules, its demonstrating one is using one rule and not the other.
The point is that wherever quaddition or any other arbitrary set of rules differs from addition then it is obvious which one I'm doing, and wherever they are the same then there is no point using another name for what amounts to being just ordinary addition.
This whole subject is a non-subject as far as I can tell, and no one has been able to come up with anything to convince me otherwise, so I think the time has come to drop it unless you have something new and substantive to say about it.
Not necessarily because there are other rules other than plus which are consistent with that sum also. There are no specific instances which where alternative rules cannot be applied.
Quoting Janus
Its not about picking which rule to use. I am going to assume you have been using addition putatively as plus only for your entire life; its about whether you can demonstrate that this rule you have been using and are still using is in fact plus and not quus.
Quoting Janus
Well thats dogmatism like I said because wherever they are the same you can easily use quus.
Quoting Janus
Demonstrate it, give a definition that tells me you will always give the correct answers for plus and not quus.
Quoting Janus
Well fair enough, I just don't think you have demonstrated a distinguishing fact yet and in view of that, I think my view about dogmatism is valid.
It's probably the most widely discussed angle on the private language argument. :wink:
Quoting Apustimelogist
Sure, we can make up any ad hoc set of rules to give the same answer as any result of addition, but insofar as it does give the same answer, then it is not saliently different than addition, and insofar as it doesn't yield the same answers (and there must be cases where it wouldn't, otherwise it would be no different than addition) it would be of no use.
Quoting Apustimelogist
Quoting Apustimelogist
It's not dogmatism: I'll change my mind if you can demonstrate that some rule could always yield the same result as addition and yet differs from it in the very part of it that does so. So, for example quaddition is exactly the same as addition up to any sum that does not exceed 57.
Quoting Apustimelogist
I think it is you, not I, being dogmatic because I have been providing arguments whereas you have
not addressed them and have provided no counter-arguments, but simply keep asserting the same thing over and over.
Anyway, I have gone well beyond exhausting my store of interest in this.
Quoting Janus
From the OP:
Quoting frank
:grin:
Sounds good. :up:
It has been of use though because all of the examples of addition you have used so far in your life have been consistent with some quus-like rule. If you could have used that rule so far, then clearly it could have been of use.
It isn't really about intentional use anyway. The premise is that you have been using the addition rule for the whole of your life and you know it intimately. Then someone comes a long and questions: "How do you know you are not actually using quadditon? give me a justification of this."
Use is not so much relevant in that you would have to demonstrate that you are in fact using the "useful" rule, and that somewhere a long the line the future you are not going to give an answer that other people might find totally inconsistent with addition (but consistent with the "useless" rule). How can you demonstrate that you are not going to do that and you are in fact using the "useful" rule?
Quoting Janus
I don't mean dogmatism in the sense of you not changing your mind, I mean dogmatism more in how it is used here:
https://en.m.wikipedia.org/wiki/M%C3%BCnchhausen_trilemma
You are defending the use of addition over other rules without demonstrating it. Your main justification so far seems to be that anything other than addition is arbitrary, but that in itself seems dogmatic. What do you mean by arbitrary other than that is just what you are used to, what seems natural... just what feels right? That seems to be dogmatism in the sense of the above wikipedia article.
Quoting Janus
Again, its not about the difference between the two rules - we know they are different. It is about whether you can justify that a single rule you have been using is addition and not quaddition.
It may be easier to think about it analogous to how theories compete in science. For instance, Special relativity and Newtonian mechanics are very obviously different. But from our perspective on earth right now it may not be apparent which one is correct because they yield more or less the same results in our everyday context. We need an experiment to demonstrate one is the case and not the other.
I am therefore asking for your experiment about this. Asking me to demonstrate that quus always yields the same result as plus yet is also somehow different is an impossible contradition. They are just different. Yet, in our difficulty in figuring out whether the laws of nature are obeying Special relativity or Newtonian mechanics, would you also ask me to demonstrate that Newtonian mechanics gives different results yet is also the same as Special relativity? No, because that isn't relevant. We know they are different models; the question is which one is being instantiated right now on earth, which has to be demonstrated by experiment.
For the quus example, where is the experimental demonstration that you have been using addition and not quus (and then not any other type of quus-like overlapping rule)?
Edit: I hope this last part has addressed your arguments in the sense of saying that your arguments are erroneous and not relevant to the problem just like how trying to demonstrate that Special relativity is both somehow the same and different to Newtonian mechanics is not relevant to the question of whether Newtonian or Special relativity is actually the case on earth. Only an experiment can differentiate the two, which is also what you have to analogously/metaphorically provide to differentiate quus and plus.
The connection is that: it is only in the context of public language as we use everyday, in a way that must be consistent with other people's language usage, that we find notions of rules and definitions determined - because we are checked by public consensus.
The point of the private language argument is that: without these checks, language seems redundant and there does not seem to be an inherent need for people to characterize the things they see in the world, or rules they apply, in one specific way or another (as illustrated by Kripke's quus rule-following paradox). Wittgenstein seems to suggest that giving things determinate labels via a private language seems to have no contribution on people's behavior and cognizance of the world.
I'm not defending the use of anything, all I've been saying is that addition seems to me to be a natural development of cognition-based counting, and there is no reason to say that counting is any different in principle regardless of how many things are counted, or addition any different no matter how many things are added.
I understand the logic of counting and addition, and I also understand how the logic of addition is consistent with the logic of multiplication, subtraction and multiplication. Do you think the logic of quaddition is consistent with those or some equivalents?
The other thing is that the logic of quaddition is the exactly same as the logic of addition up until its arbitrarily stipulated divergence regarding numbers over 57. There is no cognition-based logic to justify such an arbitrary stipulation, so I deem the whole thing a lame non-issue; I see no significance in it. And since no one seems to be able to tell me what the significance is, I will waste no more time on this unless someone does.
Thanks for trying, but none of that means anything to me I'm afraid.
I have addressed the thing you asked me to demonstrate, now I think you should try and address what I asked you to demonstrate.
You say you unddrstand the logic of addition; lay out for me that logic then and give me the facts that rule out that you will give a quus-type answer in future uses of addition.
Quoting Janus
And why can't I just question whether you have been quonting all along instead of counting?
Quoting Janus
There is nothing that logically forbids someone from just using quaddition either.
There is no cognition-based logic to justify such an arbitrary stipulation
You don't have a cognition based logic to justify it other than you are used to addition. Not really a justification imo... "it just is because it is and anything I am not familiar with is wrong"... That's how it sounds.
At the end of the day, in the thought experiment, the data so far is just as consistent with the use of quus as plus.
Quoting Janus
Well the significance is that you can't seem to refute it. Its very simple to refute Newtonian mechanics - for instance: under such and such conditions, time dilation occur; time dilation is impossible in Newtonian mechanics; Newtonian mechanics refuted. You don't seem to be able to use logic to justify addition at all.
This whole thing deep down is about the relationship between words and the world. The question is something like: do words have a fixed one-to-one relationship with the things that exist in the world in a way that they are intrinsically related? Does our behavior and thoughts prescribed in a rigidly defined, top-down manner by words and definitions, as if meaning has some kind of essence to it?
The alternative is: no, there is not a one-to-one fixed relationship between words and the world. Instead, we make labels and place them where we please and there are no fixed boundaries that force us to label things one way or another. We can, in principle, place the boundaries any way we like. Meaning is not essential in definitions but inferred from our behavior and how we use words in a bottom up manner. Our intractably, complicated behavior comes first.
If you think about it in this sense, what is in question is not whether we use quus or plus... we have a certain kind of mathematical behavior that we use very well for our own ends, but there is no single way to characterize it or label it or put boundaries around it. This underdeterminism has no consequence for our behavior because as I said just now, the behavior comes first, directly caused by the intractably complicated mechanistic behavior of our brains. And as our brains are just neurons communicating, there is nothing inherently semantically characterizable in what the brain is doing because its just mechanistic physics and there is probably not even a single way for brains to do any given task it is capable of.
I have already said that the logic of addition is unlimited iteration; in principle we can keep adding forever. The logic of quaddition like rules diverges from this when it stipulates some hiatus or terminus at whatever arbitrary point.
As long as such a quaddition-like rule does not diverge from the normal logic of addition, then there is no discernible difference and hence no need to use a different name to signify that procedure.
Quoting Apustimelogist
Of course, words don't have a fixed one to one relationship with the world. However, numbers do correspond to actual number as instantiated in the diverse and multitudinous world. Two is always two regardless of what word you use to signify the concept. In contrast the concepts /tree/ or /animal/ are not so determinate. So, introducing questions about ordinary language into a discussion of counting and addition is only going to confuse the issue.
Quoting Apustimelogist
I don't need to do that; I don't need to define some essence in order to know that I am counting or adding. I don't even need to define the rule because the logic of counting and adding accords with the logic inherent in the cognition of mutlitudinous things.
Nothing new regarding this is emerging from you, so I think we are done.
But what do you mean when you say "adding" or "forever". How am I sure you don't actually mean "qu-orever" instead of "forever"?
Quoting Janus
So what, this doesn't stop anyone using quaddition. It is both logically and literally possible to use the rule quaddition.
Quoting Janus
Yes, but equally someone could use that logic to say that quus should be preferred and there is no reason to use a different name of "addition" to signify it.
Quoting Janus
How is this any different from saying that the image of the world I see is the same regardless of the boundaries I wish to draw on it and the way I wish to partition my concepts that describe it?
Quoting Janus
But the image of a tree or an animal you see is determinate. Is the way you group different things as "trees" or "animals" much different from say describing things as prime numbers or odd and even or any other kind of mathematical concept? Can't addition, multiplication and subtraction all be grouped as operators?
Quoting Janus
No, because this whole issue is meant to be a generic property of all language. Quus was only given as a single example.
Quoting Janus
Okay, you know you're adding. But how do you know that what you are adding is not infact quadding, and how can you demonstrate that?
Quoting Janus
How can you say it accords with anything if you can't define it, meaning how do you know that other rules don't also accord with the logic inherent with cognition.
Quoting Janus
I wouldn't be still saying anything if you would just give me what I want, but you can't. If you could, you would have done it literally days ago. You cannot actually resolve the underdetermination inherent in the problem. There is no way you can rule out using various different rules instead of addition without being dogmatic i.e. declaring that it is addition for no evidence or reason other than "you feel it", and because you can't even demonstrate you're actually adding, you cannot even demonstrate that what you feel is actually truly addition and not quaddition. And with your choice of dogmatism, equally someone else could be equally dogmatic and just declaring that they are using quus just because thats what "feels right". You can say they're wrong. But they could say you're just wrong, and there's no way to resolve it... which is I guess where we are at!
Forever means there is no limit in prinicple. What does "qu-orever" mean? Tell me that and I'll tell you whether I meant that.
Quoting Apustimelogist
I don't even know what you want me to give, since you apparently are unable to articulate it clearly.
Quoting Apustimelogist
C'mon man, this is total bullshit. I know what adding consists in, and if you could tell me precisely what quadding consists in then I could point to how it is different than adding. Inosfar as it is not different it is a moot point.
This is my last response unless you can explain exactly what you mean and want.
I think it means, "Until you drop dead while adding 320 to 180 and only manage to say '5' before you keel over."
We will all stand around saying, "See, he was using quaddition!"
You've just repeated a synonym for "forever" so how does that help? What does "no limit" mean and does any of this really help without specifying what exactly has "no limit"? That would be "adding" presumably so you're back to where you started and probably should characterize that to me first.
"Quo-rever" is just an analogous concept for forever, exactly like quus where all your uses of forever so far are consistent with it but it differs in some way. But tbh, forever or infinite seems so abstract it seems difficult to point to what you mean anyway: how exactly can someone show they are referring to the infinite? hence why when I asked what forever meant, you just replied with a synonym effectively. But again, the target here isn't really forever but addition. This explanation youve given is essentially "I am adding forever" but "adding" was what was in question in the first place so how is saying " I am adding forever" resolving the issue?
You could have meant " I am quusing forever"
Quoting Janus
Well I already told you that is irrelevant. This is just like the Newtonian vs Special relativity example I already gave. Its not about what you think you mean, its whether you can prove a fact of the matter about what you think you mean.
Quoting Janus
Whats so hard? Prove that when you use "addition" at any given time you don't mean quus or some other quus-like word. A fact that unambiguously shows that every time you say "addition" you cannot be meaning any of these other alternative phrases.
Quoting Janus
I think I will jist have to leave a quotation from Kripke:
"Let us return to the example of 'plus' and 'quus'. We have just summarized the problem in terms of the basis of my present particular response: what tells me that I should say '125' and not '5'? Of course the problem can be put equivalently in terms of the sceptical query regardIng my
present intent: nothing in my mental hIstory establishes whether I meant plus or quus. So formulated, the problem may appear to be epistemological - how can anyone know which of these I meant? Given, however, that everythIng In
my mental history is compatible both with the conclusion that I meant plus and with the conclusion that I meant quus, It is clear that the sceptical challenge is not really an epIstemological one. It purports to show that nothing in the mental history of past behavior - not even what an omniscient God would know - could establish whether I meant plus or quus.
But then it appears to follow that there was no fact about me that constituted my having meant plus rather than quus. How could there be, if nothing in my internal mental history or external behavior will answer the sceptic who supposes that in fact I meant quus? If there was no such thing as my meaning plus rather than quus in the past, neither can there be any such a thing in the present. When we initially presented the paradox,
we perforce used language, taking present meanings for granted. Now we see, as we expected, that this provisional concession was indeed fictive. There can be no fact as to what I
mean by 'plus', or any other word at any time. The ladder must finally be kicked away.
This, then, is the sceptical paradox. When I respond in one way rather than another to such a problem as '68+57', I can have no justification for one response rather than another. Since the sceptic who supposes that I meant quus cannot be answered, there is no fact about me that distinguishes between my meaning plus and my meaning quus. Indeed, there is no
fact about me that distinguishes between my meaning a definite function by 'plus' (which determines my responses in new cases) and my meaning nothing at all.
Sometimes when I have contemplated the situation, I have had something of an eerie feeling. Even now as I write, I feel confident that there is something in my mind - the meaning I attach to the 'plus' sign - that instructs me what I ought to do in all future cases. I do not predict what I will do - see the discussion immediately below - but instruct myself what I ought to do to conform to the meaning. (Were I now to make a prediction of my future behavior, it would have substantive content only because it already makes sense, in terms of the instructions I give myself, to ask whether my intentions will be conformed to or not.) But when I concentrate on what is now in my mind, what instructions can be found there? How can I be said to be acting on the basis of these instructions when I act in the future? The infinitely many cases of the table are not in my mind for my future self to consult. To say that there is a general rule in my mind that tells me how to add in the future is only to throw the problem back on to other rules that also seem to be given only in terms of finitely many cases. What can there be in my mind that I make use of when I act in the future? It seems that the entire idea of meaning vanishes into thin air."
No, the point is the rule can never be determined, If he says 5, someone will just ask him to show he was using phlog-ddition!
:lol: Yeah, that's about how seriously I take this nonsense.
Quoting Apustimelogist
Any term can only be defined in other terms, so how does any term help? :roll: You know as well as I do what 'forever' in the context of 'addition can go on in principle forever'. You also know what 'no limit' means in the context of 'there is no reason to think there is, in principle, any limit to addition'.
Well this is the point, nothing helps. You may say that "you know as well as I do" but if I interpret "forever" in a non-standard way that is consistent with your past usage of the word forever then whats not to say that you mean something else other than "forever".
Kripke says -
"Here of course I am expounding Wittgenstein's wellknown remarks about "a rule for interpreting a rule". It is tempting to answer the sceptic by appealing from one rule to another more 'basic' rule. But the sceptical move can be repeated at the more 'basic' level also. Eventually the process
must stop - "justifications come to an end somewhere" - and I am left with a rule which is completely unreduced to any other. How can I justify my present application ofsuch a rule,
when a sceptic could easily interpret it so as to yield any of an indefinite number of other results? It seems that my application of it is an unjustified stab in the dark. I apply the rule blindly."
I think that this challenge is even less straightforward than we think it is and that the quoted portion must be evaluated first.
When one says that they've never dealt with a number over 57, does that mean that we do not know if addition will work when trying to add things to sums greater than 57? Or does it just mean that we haven't bothered to add that high but have the knowledge that addition will definitely continue to work?
This dilemma could allow for one or the other. I think that this challenge is interesting but the meaning of "you've never dealt with anything over that" seems to not indicate any clear constraints.
If it means the former, then the simple answer that mentions at the bottom of the first page does not apply. It does not matter if the rules of addition when handling sums over 57 must be consistent to preserve our knowledge of arithmetic if what appeared to be addition just stops working the way we think it would because really, we have been quadding - and this doesn't even mean the rules have changed as we have potentially been quadding this whole time. Or maybe addition sticks if we have knowledge that addition extends to (potential) sums that are greater than the greatest number we've ever encountered.
But I don't see that anywhere. So long as this uncertainty exists it seems to me we must side with the skeptic: you cannot prove that we can add 57 and 68 to 125, or that we haven't been quadding, because quaddition is one of an infinite number of equally valid rules that might dictate what happens when handling sums over 57 that could be consistent with the behavior observed when adding with sums less than 57.
Sorry Frank if you are over this thread already and have moved on.
The thought experiment is pretty contrived to get the point across. I think he picks 57 for the sake of explaining his criteria for a rule-following-fact. But there probably is a number above which you've never added. Your knowledge of addition is not under threat. The aim of the thought experiment is to examine your intentions regarding past rule following.
Kripke admits that you would tell the skeptic she's crazy.
:up: