[mkl]

> I'd argue it is impossible to imagine a world where 1 + 1 is not 2.

Actually, you've probably done that yourself, in a programming setting: integers modulo 2, where 1+1 = 0. It's useful in places and the consequences aren't too ridiculous in this case.

Following through figuring out the consequences of rule changes is a key thing mathematicians do. E.g. do we need this rule? What if this was weaker? What if this was reversed? What if we had this extra restriction?

[username90]

That is a number system where 1 + 1 isn't 2, not a universe. At least I can't imagine a universe where the concept of 1 + 1 equals 2 doesn't exist.

[mkl]

Numbers don't exist in any real sense, so we're clearly not talking about the actual physical universe. The universes we're talking about are the spaces of possibilities that arise from sets of rules. Examples include number systems and physics models built on them. Newtonian physics, built on Euclidean space; Einsteinian physics, built on space distorted by mass; quantum circuits, where modular arithmetic can show up.

[username90]

> Numbers don't exist in any real sense

I'd argue they do, numbers arise when counting and counting is definitely a part of our reality. It is pretty hard to imagine a universe where you can't count things.

[mkl]

We can count things, because we use the concept of numbers. Numbers aren't a physical thing, they are all imagined. How I see it is that physics models can be described using number systems, but the numbers aren't part of what's being described. E.g. the numbers describing properties of particles are categorically different from the particles themselves, and only the particles can interact with other physical objects. An electron can never bang into a 7.

[concreteblock]

That's because (imo) numbers aren't intrinsic to the physical universe. They are (imo) an abstraction we humans invented to describe certain phenomena. You may not agree but hopefully you can at least see why some people would have this PoV, and especially the more general PoV that mathematics is not about (physical) nature, but about abstract ideas.

[wizzwizz4]

I agree. Counting might not have any intrinsic relationship to the physics of the universe, but it's a strange universe where thinking beings can exist, yet are incapable of constructing the mathematical rules that would allow them to count.

[kergonath]

For a while, people thought that a universe in which Euclid’s postulates would be wrong would be very strange indeed. In my opinion it is short-sighted to go from “it’s weird (to us, right now)” to “it’s impossible”.

On the contrary, it’s very interesting to explore the consequences of something we take for granted being actually wrong or unnecessary.

[wizzwizz4]

In a universe where Euclid's postulates are wrong, like ours, you can still construct them. You're positing a universe where it's impossible to construct counting – i.e., set theory is impossible to imagine, Peano arithmetic is impossible to imagine, Church numerals are impossible to imagine…

In such a universe, how is conscious thought even possible?

[gjm11]

That isn't a situation where 1+1 isn't 2. It's a situation where 1+1=2 but also 2=0.

[camjw]

To me this is a non-example though because in this context 2 = 0. So 1 + 1 = 2 = 0.