[mekoka]

Can you conceive of a universe where the abstract quantity 1 added to another abstract quantity 1 does not equal the abstract quantity 2? How would that work?

Edit: added the word abstract to make it clear that the context of the question is the realm of logic, and not the boundaries of nature (physics).

[less_less]

There could be a universe where "thingness" doesn't work like in ours -- e.g. where things aren't portable, or where the natural world doesn't divide easily into things. Maybe quantity wouldn't mean anything in such a universe.

More concretely, there are lots of ways that the mathematics of a different universe could be so different from ours that π is at best a theoretical concept.

There could be a universe where spacetime is discrete, à la Conway's Game of Life. We might even live in such a universe, but the discretization in ours is too small to probe.

There could be a universe with a different distance metric, such as a L1 ("taxicab") or L∞ (max difference), so that "circles" look like squares.

Or spacetime, or even "thingness" could be modulo some number p -- if p=2, then 1+1 = 0. Space could still be infinite, though -- it could have more dimensions, or it could be based on (e.g.) a complete field of characteristic p.

Or the metric could be p-adic, where two numbers are close if their difference is divisible by many powers of p -- kind of like the US highway system, where highways 80 and 880 are nearby. These metrics have rules like the triangle equality: the longest two sides of a triangle are the same length.

[mekoka]

You've mostly described physics i.e. nature, not logic.

[coldtea]

Logic doesn't exist except in 2 forms:

(a) as a theory in minds

(b) as a description of how we see the world (nature) work

[lordnacho]

Those are just representations of logic that have developed via evolution in the physical world.

The logic itself doesn't have a physical presence, it's just the inescapable conclusions drawable from any number of starting assumptions. There doesn't need to exist a universe with thinking things for logic to be logic.

Part of the explanation problem is that we tend to say "if you were to assume..." naturally in our language. But if course this causes a problem because logic is not part of nature and that phrase trends to assume some kind of thinking being doing the logical thinking.

[less_less]

This is a very strong philosophical assumption.

Logic definitely exists as a method in human philosophy. You could argue that it's somehow "baked in" to the universe, whether by a creator or by some other cause, and that we only discovered it and didn't invent it. But the claim that it transcends our universe and would exist and be accessible from all other possible universes, even ones with different physics, different kinds of space and time (two temporal dimensions? who knows?) etc doesn't seem obvious in the slightest.

This is especially true when there are so many types of logic used in mathematics and philosophy to begin with: first-order or higher-order logic, constructive logic, Peano arithmetic, Zermelo-Fraenkel, ZF with Choice, etc.

Even once you have the axioms, they may support different models. Are just the axioms true universally, but models varying across universes? Is it possible that there are universes where all statements are true? Are there universes with Choice and others without it?

[mekoka]

Again, it might be best not to conflate logic and physics. Having different universes with different laws of physics is very conceivable to the mind. Logic is pure philosophy. It's an exercise in reasoning. No physical observation is required. The various kinds of logic you cite are different subsets that coexist without ambiguity under the same big umbrella. That is, it's still the same logic. If one law were to create ambiguity with another (which is the mechanism used in proofs), then at least one would be declared void, or some more work would be needed to explain the conditions that lead to the paradox. Logical truths have no order of precedence. Our minds simply use the ones it's more comfortable to reason about as a scaffold to uncover new ones. But they all exist together and are all equally true without ambiguity, including those we have not uncovered yet. If we posit that another universe has logical truths that differ from our own, they will not only be foreign to us, but they will also be totally inaccessible to our imagination in a way that makes any kind of sense (unlike alternate laws of physics), as being able to reason about them in our own universe then gives them validity and creates the aforementioned ambiguity. 1 + 1 must be equal to 2 when adding those two quantities. If you say it doesn't, you're either mistaken, or you're talking about a different subset of the same logic (bases, set theory, etc) where the same symbols are used to mean something different. That doesn't qualify as conflicting logic.

[milesvp]

You may want to read up on fuzzy set theory. Most certainly there is a set of 1 that when added to itself results in the set of 3, 4 or really any arbitrary set to varying degrees.

Bivalant sets is only a special case subset of fuzzy sets where 1+1 must equal 2.

[V-2]

I (for one) can't conceive it, but I also cannot really conceive what's happening in a black hole, or at the quantum level. And the weirdness of those phenomenons is nothing in comparison to a "brand new" different reality.

[mekoka]

Black hole, quantum. Physics i.e nature, not logic.

[coldtea]

>the quantity 1 added to another quantity 1 does not equal the quantity 2

Our universe already fits the description, as there are algebras where that is not the case (similar as we have geometrics where triangle angles add up to > 180 degrees -- in fact geometry calculations on a sphere surface like earth works like that).

There are also probably physical examples where that's not the case (e.g. unlike adding 1+1 apple, adding two bodies of water gives you one unified body of water).

Could it hold even more fundamentally? Sure why not. Our "not being able to conceive it" doesn't mean much regardless to whether it's possible.