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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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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.)
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.
Assumptions are a basic building block of cognition, necessary in order to make even the very simplest possible of conclusions. Literally everyone makes assumptions whenever they have literally any thought or take literally any action. To take a single step forward is generally to assume your eyes don't deceive you, the world isn't about to explode, you won't suddenly quantum phase to the sun when you take the step, you still remember how to take steps, etc.
There are certainly more and less justified assumptions, and perhaps you mean to call attention to less justified assumptions, but don't do so using an example of an extremely justified assumption.
Yes, but not everyone realizes they are making an assumption. Just like virtually nobody realizes they are making an assumption when answering the
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