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

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. — T Clark

You haven't been doing addition. It was quaddition. — frank

Then I ask you for a fact about your previous behavior that shows that the rule you were following was addition rather than quaddition. — frank

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): — schopenhauer1

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. — schopenhauer1

Yes, definitely. The challenge ends up being about the meaning of any word. — frank

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." — T Clark

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. — schopenhauer1

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. — schopenhauer1

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? — frank

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? — T Clark

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. — frank

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. — frank

I give up. — T Clark

Your challenges still helped me flesh it out, so thank you. — frank

Anytime you need somebody to be confused, I'll be happy to help. — T Clark

You and Kripke may be excused. Return to dinner when you have stopped playing your little games. — jgill

Following our failure to deliver a fact that distinguishes our historic use of "plus" vs "quus," — frank

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? — flannel jesus

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? — Count Timothy von Icarus

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." — Count Timothy von Icarus

Still, I get the point. Defining systems only in terms of past use seems to miss something. — Count Timothy von Icarus

201 ... For what we thereby show is that there is a way of grasping a rule which is not an interpretation, but which, from case to case of application, is exhibited in what we call “following the
rule” and “going against it”.

That’s why there is an inclination to say: every action according to a rule is an interpretation. But one should speak of interpretation only when one expression of a rule is substituted for another.

202. That’s why ‘following a rule’ is a practice. And to think one is following a rule is not to follow a rule. And that’s why it’s not possible to follow a rule ‘privately’; otherwise, thinking one was following a rule would be the same thing as following it.

Who is to say that this [quus] is not the function previously meant by '+'? (9)