I'm not seeing the relevance to deciding whether addition, subtraction, multiplication and division are basically derivable from counting operations. — Janus

If you can't derive addition from counting then how are you proving you are doing addition?

Neither do I see a real significance in the distinction between "natural"and "artificial" concepts. — Apustimelogist

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.

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. — Apustimelogist

Well, it's not what I mean. Armchair speculation I would class as metaphysics, not phenomenology.

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. — Apustimelogist

I don't see the relevance at all, and no one seems to be able to explain clearly what it is, so...

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. — 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.

If you can't derive addition from counting then how are you proving you are doing addition? — Apustimelogist

You can derive addition from counting. Counting basically is addition.

inevitably evolve out of experience — Janus

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.

Well, it's not what I mean. Armchair speculation I would class as metaphysics, not phenomenology. — Janus

I don't really see how phenomenology is not another form of armchair speculation in a similar way.

I don't see the relevance at all, and no one seems to be able to explain clearly what it is, so... — 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.

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. — 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.

You can derive addition from counting. Counting basically is addition. — Janus

I dunno, it seems to me with what

— "Moliere

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.

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. — Apustimelogist

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.

I don't really see how phenomenology is not another form of armchair speculation in a similar way. — 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.

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. — Apustimelogist

Some descriptions of some behaviors are more determinate than others, obviously.

I can demonstrate quus with objects just as well as I can with addition. — Apustimelogist

I don't believe you can.

You are not presenting any arguments, just baseless objections, it seems, so the conversation is going nowhere.

You keep mentioning objectivity, which has nothing to do with what I've been arguing — Janus

Well what have you been arguing?

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. — 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?

I don't believe you can. — 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 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? — Apustimelogist

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.

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. — Apustimelogist

Maybe...I remain unconvinced.

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. — Janus

Well I would say there is still a difference between know-how and know-that when worded like that.

Right, I also recognize a distinction between knowing-how and knowing-that.

Isn't all this similar to Descartes' evil demon? Sure quaddition could be different than addition. Sure there could be an evil demon messing with us every time we think 2 and 2 are 4 (spoiler: it's really 5). But there's probably no evil demon and quaddition probably gives the same answer as addition.

But not the significance that know-how doesn't give a determinate know-that

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. — Janus

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.

But not the significance that know-how doesn't give a determinate know-that — Apustimelogist

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.

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. — 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.

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. — Janus

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.

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. — Moliere

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.

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. — Moliere

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.

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. — frank

I'm not trying to do philosophy asof 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.

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. — frank

Heh. I don't think I'm that deep. I see a question, but I don't see a resolution.

I don't think I'd reduce rationality to rule-following either. — Moliere

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 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?" — Moliere

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.

Counting makes sense as a genesis of arithmetic. But is doesn't escape from the sceptical question. There is no fact of the matter that determines whether I have counted correctly - except the fact that others will agree with me. This reinforces me in my practice of counting, as my agreement with others about their counts reinforces their practice of counting.

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.

I think some people would assume that means I end up a behaviorist — frank

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!

Quaddition seems to arbitrarily countermand the natural logic of counting and addition; the logic that says there is neither hiatus nor terminus. — Janus

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.

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. — Janus

How do you respond here to @Ludwig V's point?

Counting makes sense as a genesis of arithmetic. But is doesn't escape from the sceptical question. There is no fact of the matter that determines whether I have counted correctly - except the fact that others will agree with me. This reinforces me in my practice of counting, as my agreement with others about their counts reinforces their practice of counting. — 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.

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. — 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.

It is therefore possible to use alternative concepts without any difference in behaviour. — Apustimelogist

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.

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 — Apustimelogist

I don't think you can give me a satisfying definition of counting or quantity, — 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.

the natural logic of counting and addition; — 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".

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. — 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".

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. — Apustimelogist

Counting makes sense as a genesis of arithmetic. But is doesn't escape from the sceptical question. There is no fact of the matter that determines whether I have counted correctly - except the fact that others will agree with me. — Ludwig V

As stipulated the rules of quaddition do provide different outcomes:

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."" — 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.

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. — 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?

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". — Ludwig V

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.

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. — Ludwig V

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.

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. — 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.

As stipulated the rules of quaddition do provide different outcomes: — 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.

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. — Apustimelogist

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.

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. — 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.

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. — Apustimelogist

But do they yield answers that are pragmatically workable?

To you maybe. It might be totally unintuitive to a different kind of being. — 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.