A simple example where 1 + 1 + 1 = 1 is ℤ₂, the group of integers modulo 2 under addition.

In fact, in any group with binary operation, say +, the identity element 0, and a non-identity element a, we have a + a + a = a if and only if a + a = 0 (i.e. a has order 2).

There are plenty of groups with elements a satisfying a + a = 0. ℤ₂ as mentioned above has its unique non-zero element of order 2. The Klein group V₄ has three non-identity elements, each of order 2. Dihedral groups D₂ₙ (the symmetry groups of regular n-gons) contain reflections, all of which have order 2. Symmetric groups Sₙ (n ≥ 2) contain transpositions, each of which has order 2.

For example, in the dihedral group D₈, if we let a be a reflection of the square, then a + a = 0 and a + a + a = a. But this is conventionally written in multiplicative notation as a² = the identity element, so a³ = a.

Similarly, in the symmetric group S₃ under the binary operation of composition, if a denotes the transposition (12), then (12)(12) is the identity element and (12)(12)(12) = (12). In other words, applying a transposition three times is the same as applying it once.

In the last two examples, it is conventional to use product notation instead of +, although whether we use + or · for the binary operation does not matter mathematically. It is conventional to use + in some subjects (coding theory, additive groups of integers modulo n, etc.) and · in others (permutation groups, dihedral groups, etc.). Often + is used for the binary operation in abelian groups and · in non-abelian ones. I'm sure none of this is particularly insightful to someone who has studied group theory, but still I wanted to share a few concrete examples here.

>there are other operations that aren’t quite involutions – “near-involutions”, one might call them2 – that nonetheless have the property that thrice is the same as once, four times is the same as twice, etc. ... Unlike mod-two arithmetic, which is about counting “zero, one, zero, one, zero, one, …,” the kind of counting that governs these operations goes “zero, one, two, one two, one two, …”

Interesting! Earlier I had a shower thought about "what would be an variant of idempotence?" That's where an operation has the same effect whether done one or many times.

One variant would be "has the same effect whether one two or many times". Another would be "can be in any one of two possible states after done one or many times" (as opposed to one possible state for idempotence). This looks like the latter!

The intuitionistic negation example is so mind blowing. I can only wrap my head around it when I think about it in terms of functions and types:

    not : Type -> Type
    not P = P -> ⊥
    
    modus-ponens : P -> ((f : P -> Q) -> Q)
    modus-ponens p = λf. f p
    
    -- p implies not-not-p
    not-not : P -> ((f : P -> ⊥) -> ⊥)
    not-not p = λf. f p
    
    -- not-p implies not-not-not-p
    not-not-not : (P -> ⊥) -> (((P -> ⊥) -> ⊥) -> ⊥)
    not-not-not np = λf. f np

    -- not-not-not-p implies not-p
    not-p : (((P -> ⊥) -> ⊥) -> ⊥) -> (P -> ⊥)
    not-p nnnp = λp. nnnp (not-not p)

Beautiful math tricks. For things like these I think math education should start with sets and groups instead of numbers.

https://d1gesto.blogspot.com/2025/11/math-education-what-if-...

In tropical geometry, tropical multiplication (⊙) is replaced by standard addition (+), and tropical addition (⊕) is replaced by the minimum (min) function, so 1⊕1⊕1=min{1,1,1}=1.

1 + 1 + 1 ≡ 1 (modulo 2)

In the modulo 2 congruence, 1 + 1 + 1 is the same element as 1.

1 as a boolean? true

I'm not a mathematician but I don't think that's right.

Terrence Howard has entered the chat

modulo 2

Interesting. I always associate "1 + 1 = 1" with idempotency. Here, "1 + 1 + 1 = 1", but "1 + 1 = 0".

I'm not a math whiz, so I'm just stuck with "1 + 1 = 2."