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2+2 = not what you think

felipec.substack.com

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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I'd concur that this is more of an annoying semantic trick than anything else. It is never denied that 2 + 2 = 4 within the group of integers under addition (or a group containing it as a subgroup), a statement that the vast majority of people would know perfectly well. Instead, you just change the commonly understood meaning of one or more of the symbols "2", "4", "+", or "=", without giving any indication of this. Most people consider the notation of integer arithmetic to be unambiguous in a general context, so for this to make any sense, you'd have to establish that the alternative meaning is so widespread as to require the notation to always be disambiguated.

(There's also the epistemic idea that we can't know that 2 + 2 = 4 within the integers with complete certainty, since we could all just be getting fooled every time we read a supposedly correct argument. But this isn't really helpful without any evidence, since the absence of a universal conspiracy about a statement so trivial should be taken as the null hypothesis. It also isn't relevant to the statement being untrue in your sense, since it's no less certain than any other knowledge about the external world.)

Most people consider the notation of integer arithmetic to be unambiguous in a general context

But that is the point: most people make assumptions. In this particular case it's easy to see what assumption is made for people who do understand modular arithmetic, but that excludes the vast majority of people who don't.

The whole point of the article is to raise doubt about more complicated subjects which are not so easy to mathematically prove.

But that is the point: most people make assumptions.

Assumptions about the meaning of symbols, namely that symbols carry their conventional meaning unless denoted otherwise.

This is a necessary prerequisite of communication, and messing with it is merely a failure to communicate.

And the failure to communicate can be entirely on the listening side by assuming a meaning that was never there.

The fact that today people don't understand each other is a huge problem, and worse: people don't want to understand what the other side is actually saying.

In general? Yes. In this example? Absolutely the speaker's fault. If you're using non-standard symbols, you need to denote that.

You are assuming I'm the one who brought up the 2+2=4 factoid.

If the speaker who brought up 2+2=4 is using standard symbols, he's unambiguously correct, so that can't be what we're talking about.

If the speaker claimed that 2+2=4 is unequivocally true, he/she is wrong.

Absolutely not. The speaker knows what the statement means, what the symbols mean, in what structure we're operating. The rest is just basic arithmetic over the natural numbers.