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Changing someone's mind is very difficult, that's why I like puzzles most people get wrong: to try to open their mind. Challenging the claim that 2+2 is unequivocally 4 is one of my favorites to get people to reconsider what they think is true with 100% certainty.
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It's an assumption about the meaning of the question, not an assumption about the actual laws of arithmetic, which are not in question. The only lesson to be learned is that your interlocutor's terminology has to be aligned with yours in order to meaningfully discuss the subject. This has nothing to do with how complicated the subject is, only by how ambiguous its terminology is in common usage; terminology is an arbitrary social construct. And my point is that this isn't even a very good example, since roughly no one uses standard integer notation to mean something else, without first clarifying the context. Far better examples can be found, e.g., in the paper where Schackel coined the "Motte and Bailey Doctrine", which focuses on a field well-known for ascribing esoteric or technical meanings to commonplace terms.
There is no "arithmetic", there's multiple arithmetics. You are assuming that the "laws" of one particular arithmetic apply to all arithmetics, which is not true.
Here, I'm using "the laws of arithmetic" as a general term to refer to the rules of all systems of arithmetic in common usage, where a "system of arithmetic" refers to the symbolic statements derived from any given set of consistent axioms and well-defined notations. I am not assuming that the rules of integer arithmetic will apply to systems of arithmetic that are incompatible with integer arithmetic but use the exact same notation. I am assuming that no one reasonable will use the notation associated with integer arithmetic to denote something incompatible with integer arithmetic, without first clarifying that an alternative system of arithmetic is in use.
Furthermore, I assert that it is unreasonable to suppose that the notation associated with integer arithmetic might refer to something other than the rules of integer arithmetic in the absence of such a clarification. This is because I have no evidence that any reasonable person would use the notation associated with integer arithmetic in such a way, and without such evidence, there is no choice but to make assumptions of terms ordinarily having their plain meanings, to avoid an infinite regress of definitions used to clarify definitions.
There are no axioms that apply to all arithmetics. There are no such "laws".
Go ahead and try come up with one "law". I'm fairly certain I can point out an arithmetic where it doesn't apply.
There's a reason these fall under the umbrella of abstract algebra.
Also, you seem to be conflating "integer arithmetic" with normal arithmetic.
2.5 + 2.1is not integer arithmetic, and yet follows the traditional arithmetic everyone knows. I'm not even sure if normal arithmetic has a standard name, I just call it "normal arithmetic" to distinguish it from all the other arithmetics. Integer arithmetic is just a subset.Are you getting hung up on my use of the term "laws of arithmetic"? I'm not trying to say that there's a single set of rules that applies to all systems of arithmetic. I'm using "laws of arithmetic" as a general term for the class containing each individual system of arithmetic's set of rules. You'd probably call it the "laws of each arithmetic". The "laws of one arithmetic" (by your definition) can share common features with the "laws of another arithmetic" (by your definition), so it makes sense to talk about "laws of all the different arithmetics" as a class. I've just personally shortened this to the "laws of arithmetic" because I don't recognize your usage of "arithmetic" as a countable noun.
I was focusing on integer arithmetic since that was sufficient to cover your original statement. The natural generalization is group or field arithmetic to define the operations, and real-number arithmetic (a specialization of field arithmetic) to define the field elements. The notation associated with integer arithmetic is the same as the notation associated with real-number arithmetic, since the integers under addition or multiplication are a subgroup of the real numbers.
To repeat my actual argument, I assert that, without prior clarification, almost no one uses the notation associated with real-number arithmetic in a way contrary to real-number arithmetic, which implies that almost no one uses it in a way contrary to integer arithmetic. Therefore, I refuse to entertain the notion that someone is actually referring to some system of arithmetic incompatible with real-number arithmetic when they use the notation associated with real-number arithmetic, unless they first clarify this.
You made this claim:
The "laws of arithmetic" that are relevant depend 100% on what arithmetic we are talking about, therefore it's imperative to know which arithmetic we are talking about. People assume it's the normal arithmetic and cannot possibly be any other one. There is zero doubt in their minds, and that's the problem I'm pointing out.
Then please stop assuming that my uncountable usage of "the concept of arithmetic in general" in that sentence is secretly referring to your countable idea of "a single arithmetic". I've clarified my meaning twice now, I'd appreciate it if you actually responded to my argument instead of repeatedly hammering on that initial miscommunication.
Why should there be any doubt in their minds, if other systems of arithmetic are never denoted with that notation without prior clarification?
Where did I "assume" that in my last comment?
I don't know what argument you are talking about. If you are referring to this:
almost no one uses the notation associated with real-number arithmetic in a way contrary to real-number arithmetic
∴ I refuse to entertain the notion that someone is actually referring to some system of arithmetic incompatible with real-number arithmetic when they use the notation associated with real-number arithmetic, unless they first clarify this
That's not an argument, you are just stating your personal position. You are free to do whatever you want, if you don't want to doubt a particular "unequivocal" claim, then don't. Your personal position doesn't contradict my claim in any way.
Because that's what skepticism demands. I assert that 100% certainty on anything is problematic, which is the reason why skepticism exists in the first place.
You said, "The 'laws of arithmetic' that are relevant depend 100% on what arithmetic we are talking about," which is only meaningful under your usage of "laws of arithmetic" and does not apply to the term as I meant it in my original comment.
To quote myself:
To rephrase that, communication relies on at least some terms being commonly understood, since otherwise you'd reach an infinite regress. As a consequence, there must exist terms that have an unambiguous "default meaning" in the absence of clarification. But how do we decide which terms are unambiguous? Empirically, I can decide that a widespread term has an unambiguous default meaning if I have never heard anyone use the term contrary to that meaning in a general context, and if I have no particular evidence that other people are actively using an alternative meaning in a general context. I believe it reasonable to set the bar here, since any weaker criterion would result in the infinite-regress issue.
Sure, if someone writes "2 + 2 = 4", it isn't 100% certain that they're actually making a statement about the integers: perhaps they're completely innumerate and just copied the symbols out of a book because they look cool. I mean to say that it's so unlikely that they're referring to something other than integer arithmetic that it wouldn't be worth my time to entertain the thought, without any special evidence that they are (such as it being advertised as a "puzzle").
If you were to provide real evidence that people are using this notation to refer to something other than integer arithmetic in a general context, then I would be far more receptive to your point here.
Indeed, how do you know that your interlocutors are "100% certain" that they know what you mean by "2 + 2"? Perhaps they're "100% certain" that "2 + 2 = 4" by the rules of integer arithmetic, but they're independently 75% certain that you're messing with them, or setting up a joke.
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