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felipec  unbelief  2yr ago (felipec.substack.com)   3603 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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Z4 (i.e. ℤ₄, man I love Unicode) is just another name for ℤ/4ℤ. ([edit for clarity: ℤ₄ is] a disfavored notation now, I think, because of the ambiguity with p-adic integers for prime p, but that's how I learned it)

The "/" symbol itself in the notation you're using is specifically referencing its construction as a quotient group, a set-of-cosets with the natural operation, as described above.

OK. I'm not a mathematician, I'm a programmer, but from what I can see the set {0,1,2,3} is isomorphic to ℤ/4ℤ that means one can be mapped to the other and vice versa. The first element of ℤ/4ℤ is isomorphic to 0, but not 0, it's a coset. But the multiplicative group of integers modulo 4 (ℤ/4ℤ)* is this isomorphic set, so it is {0,1,2,3} with integers being the members of the set. Correct?

Either way 2+2=0 can be true.

Either way 2+2=0 can be true.

Only because 4=0. So 2+2=4 is true, and the central claim of your substack post is wrong.

Correct?

No. Some basic mistakes:

  • Isomorphy requires preservation of structure, in our case the structure of respective additions. This is not the case: Addition in {0,1,2,3} works different than in ℤ/4ℤ.

  • We don't say an element in a structure is isomorphic to one in another.

  • (ℤ/4ℤ)*is an entirely different structure. For starters, it contains only 3 elements. (The * signifies we're excluding the 0.)

Only because 4=0.

So 2+2=4=0="not what you think". Therefore the claim of my post is true.

But 0 is what we think, because 0 is 4. You're just changing the representation. It's like saying "You think 2+2 is '4', but it's actually 'four'".

Also, the claim in your post was

So there you have it: 2+2 is not necessarily 4.

which is wrong whether or not 2+2=0 can be true.

But 0 is what we think, because 0 is 4.

Nobody thinks that 0 is 4.

Nevertheless it is the case. We think 4, 4 is 0, therefore "0=not what you think" isn't true.

"You" are less than 0.0001% of the population, so virtually nobody.

Did you just claim less than 0.0001% of people think 2+2=4?

4 is what everyone thinks, 0 is merely a different representation of the same object. So people are giving the correct answer, you're just insisting on a different formulation.

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