Then I ask you to prove tI've been doing quaddition, not addition. — Patterner

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.

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. — RogueAI

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.

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. — RogueAI

See this comment I made in this thread 25 days ago:

You lay out 68 marbles and then you lay out 57 marbles in a separate row, then you ask the other "what are the names of the numbers of marbles in the two rows". Then you push them together and ask the other to count all the marbles and say what the name for that number of marbles is. — 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?

I think I'd respond by saying you're doing counting, which is neither addition nor quaddition. Counting is an entirely different, third rule where you count, rather than add or quus. If you were quussing you'd stop the moment you counted up to 5', the 5 beyond all 57's.

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.

As far as I can see this solution dissolves the supposed problem. Much ado about nothing... — Janus

:up:

Exactly. I was never into Wittgenstein. I'm having a hard time seeing what the problem is. Something to do with private language?

I think I'd respond by saying you're doing counting, which is neither addition nor quaddition. — Moliere

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?

Isn't counting adding 1 to the previous number? — RogueAI

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.

Rather, rules are post-hoc classifications and inferences we impose on our own behavior. — Apustimelogist

I don't think rules are imposed, they describe behaviors which are entrenched and replete with their own logic.

I think I'd respond by saying you're doing counting, which is neither addition nor quaddition. — 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.

I don't think rules are imposed, they describe behaviors — Janus

Yes, that's what I said, you must have interpreted impose differently. But my point is we construct those descriptions.

Right. I'm not sure about "constructed" because it seems to carry an implication of arbitraryness. The formulation of rules seems to be motivated by and follow what we actually observe.

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.

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. — Apustimelogist

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.

so it is not we who construct, but we who are constructed from moment to moment — Janus

Semantics really, isn't it?

There is no self!

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? — 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.

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. — Janus

Yes, and what is in question is whether there is a fact of the matter about who is correct.

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. — Janus

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.

so it is not we who construct, but we who are constructed from moment to moment
— Janus

Semantics really, isn't it? — Apustimelogist

Not really, I think it is literally true that we are being created moment by moment—until we are not. We do not create ourselves. We don't even know what causes the thoughts we have to arise in awareness.

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. — 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.

Yes, and what is in question is whether there is a fact of the matter about who is correct. — 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.

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. — 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.

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. — Janus

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.

The article is paywalled on the links I found, so I guess we will have to take your word for it. — Banno

Here is an accessible version: "Kripke and Wittgenstein: Intention Without Paradox," by Paul Moser and Kevin Flannery.

Cheers. The literature on this topic is vast.

Basically I'd say that arithmetic is more complicated than counting. — Moliere

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.

Not really, I think it is literally true that we are being created moment by moment—until we are not. — Janus

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.

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. — 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.

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. — 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.

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. — 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.

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. — Apustimelogist

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.

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. — Apustimelogist

I don't see the phenomenological dimension of philosophy as "armchair speculation", but rather as reflection on what we actually do.

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. — 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.

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? — 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.

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. — Janus

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.

I don't see the phenomenological dimension of philosophy as "armchair speculation", but rather as reflection on what we actually do. — Janus

Well thats more or less what I mean.

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. — 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.

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. — 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.

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. — Janus

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)

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.

Probably, yes. But as the influential codger said:

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 — Kant

I keep harping on the square root of 2 — Moliere

I'm not sure how much the symbolic language matters. — 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.

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 — Kant

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.