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felipec  unbelief  1yr ago (felipec.substack.com)   3200 thread views

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 would actually double down and assert that 2+2=4 is a fact deeper than arithmetic. If 2+2=4 are elements in a modular ring, it holds true. If they are vectors, it still holds true. If they are abstract discrete topological spaces, it holds true. I have not encountered a situation in (non-joke) mathematics where the symbols '2', '+' and '4' are overloaded so as to not make this equation true.

There is an underlying concept of "twoness", "addition" and "fourness" that holds this property even as you generalize it to systems beyond integer arithmetic, almost like a fundamental structure of mathematics. This is not even about notational trickery. Even if you decide to use different symbols, it does not change the underlying mathematical relationships. You would just be expressing the same undeniable fact differently.

If 2+2=4 are elements in a modular ring, it holds true.

Integers modulo 4 (𝐙/4𝐙) is a modular ring which does not contain the number 4.

It contains the congruence class 4Z (= {...-8,-4,0,4,8...}) of which the number, more so the symbol, 4 is a valid representant.

The statement remains true.

So does the statement 2+2=0.

Which no one has doubted.

You accepted here that (4 (sa)) is not the same statement as (4 (mod 4)).

Thus conceding my point.

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