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felipec  unbelief  2yr ago (felipec.substack.com)   3599 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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"2+2 = 4" is still actually true in Z4.

But not in 𝐙/4𝐙 (integers modulo 4).

Um, that's what I meant by "Z4" (I couldn't remember and didn't bother with the exact definitional name). The element of Z/4Z that is usually denoted "0" is that set I noted above and can also be correctly denoted "4".

Do you have any source for that? All the sources I've found say the elements of the underlying set of integers modulo 4 are integers.

https://math.stackexchange.com/questions/1556009/quotient-ring-mathbbz-4-mathbbz

Somebody was confused when defining Z/4Z and not getting integers; every response notes that Z/4Z is strictly not a set of integers, but a set of sets of integers.

I could go look for a (presumably pirate) online textbook if you really want (I learned this from lectures in uni, not from a textbook), but it'd be a pain.

(The elements of the underlying rings of the quotient ring - Z and 4Z - are of course integers, but the elements of Z/4Z aren't.)

OK. But in the answers it's claimed that this defines a new way to say what elements equals to what else, so 3=7. Therefore 4=0, and 2+2=0.

Yes. It is true that 2 + 2 = 0 in Z4; I've not disputed that. It's just also true that 2 + 2 = 4.

Yes. But the whole point of my post is to get people to reconsider what basic notions like 2+2 are.

And if I understand correctly in ℤ/4ℤ there is no 2 in the main set, it's {..,-6,-2,2,6,...}, so it's actually {...,-6,-2,2,6,...}+{...,-6,-2,2,6,...}={...,-8,-4,0,4,8,...}, or something like that. 2 is just a simplification of the coset.

Second part is right, yes.

OK. But then I do get it: 2+2 = 0 (mod 4).

Yes, without any other context 2+2 is assumed to be 4, but 2+2 (mod 4) is a different thing, because 2 and 2 (mod 4) are different (the latter is actually {..,-6,-2,2,6,...}). Correct?

I have updated the article to be more correct.

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.

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