Kripke allows that mathematicians can adequately specify the rules of addition. That's not being called into question. — frank

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.

Kripke allows that mathematicians can adequately specify the rules of addition. That's not being called into question. — frank

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.

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. — Ludwig V

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.

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? — sime

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.

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. — 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?

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. — Apustimelogist

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.

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. — Ludwig V

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 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? — frank

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.

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. — Janus

That may be true.

That may be true. — frank

I have seen no reason to think it is not true. I also see that fact as dispelling Kripke's skeptical challenge.

:up:

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 also see that fact as dispelling Kripke's skeptical challenge. — Janus

I don't see how.

It only dispells it if you think dogmatism is a valid way to objective truth. — Apustimelogist

Dogmatism has nothing to do with it; there is simply no reason that addition should terminate anywhere.

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. — 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.

Dogmatism has nothing to do with it; there is simply no reason that addition should terminate anywhere. — Janus

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.

This is nonsense: I haven't claimed that one could not use quaddition or any other arbitrary rule. — 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.

But then it is simply addition up that point, and so what? — Janus

But the question is whether it is also quaddition?

I don't disagree with the private language argument, at least as I interpret it, which is to say that if you tried to construct a private language, you would always be relying on the public language you know in order to tell yourself what your novel language means. So. I'm not seeing the relevance in this context.

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. — 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.

But the question is whether it is also quaddition? — 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?

I don't disagree with the private language argument, at least as I interpret it, which is to say that if you tried to construct a private language, you would always be relying on the public language you know in order to tell yourself what your novel language means. So. I'm not seeing the relevance in this context. — Janus

You don't see the relevance of the private language argument to Kripke's skeptical challenge? :chin:

If quaddition is the same as addition then it's not a different procedure but just a different name. — Janus

So what? If it differs, then how could it do so without arbitrarily stipulating that iteration must cease at some point? — 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.

I can just keep adding forever in principle. — 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?

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. — Apustimelogist

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.

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? — 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?

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. — Janus

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?

The rules are obviously different; you just need to give me something that distinguishes whether you are using one rule or the other. — Apustimelogist

How are they different?

If you cannot give me an intelligible explanation then how are you going to differentiate whether you are using plus and quus? — 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.

So, if I add two numbers and the sum is more than 57, I am not doing quaddition, but ordinary addition. And as I said before if I am working out how many guests will be at my daughter's wedding and there are 35 from one side and 75 from the other, quaddition will be of no use, because it will tell me that I only need five places at the dinner table and five meals.

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.

So, if I add two numbers and the sum is more than 57, I am not doing quaddition, but ordinary addition. — Janus

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.

And as I said before if I am working out how many guests will be at my daughter's wedding and there are 35 from one side and 75 from the other, quaddition will be of no use, because it will tell me that I only need five places at the dinner table and five meals. — 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.

and wherever they are the same then there is no point using another name for what amounts to being just ordinary addition. — Janus

Well thats dogmatism like I said because wherever they are the same you can easily use quus.

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 — Janus

Demonstrate it, give a definition that tells me you will always give the correct answers for plus and not quus.

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. — 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.

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, — Janus

It's probably the most widely discussed angle on the private language argument. :wink:

Your comment might be useful if it was augmented by some explanation as to how this discussion relates to the private language argument.

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. — 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.

Well thats dogmatism like I said because wherever they are the same you can easily use quus. — Apustimelogist

Demonstrate it, give a definition that tells me you will always give the correct answers for plus and not quus. — 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.

I think my view about dogmatism is valid. — 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.

Your comment might be useful if it was augmented by some explanation as to how this discussion relates to the private language argument. — Janus

From the OP:

This challenge comes from Saul Kripke’s Wittgenstein on Rules and Private Language (1982) — frank

:grin:

OK, I'm afraid I don't get it, but I might if you were to explain the connection clearly in your own words. If you cannot, or are not willing to, do that, then I think we are at an impasse.

then I think we are at an impasse. — Janus

Sounds good. :up: