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You could say that math is the part of laws of physics which humans can't imagine being different. You can pretty easily imagine a world where newtons laws are

You could say that math is the part of laws of physics which humans can't imagine being different. You can pretty easily imagine a world where newtons laws are different, but I'd argue it is impossible to imagine a world where 1 + 1 is not 2. However being impossible for us to imagine doesn't mean that such worlds can't exist, it would just have completely ridiculous consequences we can't imagine, so those rules are a part of our universe and not a fundamental logical truth.

mkl 6y ago
> 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 6y ago
  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 6y ago
    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 6y ago
      > 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 6y ago
        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.
      - wizzwizz4 6y ago
        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 6y ago
        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 6y ago
        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?
- camjw 6y ago
  To me this is a non-example though because in this context 2 = 0. So 1 + 1 = 2 = 0.
- gjm11 6y ago
  That isn't a situation where 1+1 isn't 2. It's a situation where 1+1=2 but also 2=0.
ddxxdd 6y ago
A universe where 1 + 1 != 2 is a universe filled with self-contradictions and so cannot exist. It's like imagining a universe where True == False. It's not a hypothetical, it's a logical impossibility.
- mkl 6y ago
  In modulo arithmetic, 1+1 != 2 can work just fine. We're not talking about the literal universe, but the space of possibilities that opens up when you change the rules. E.g. all the amazing power and complexity that comes from imagining the existence of a number i with the property that i^2 = -1. This was initially thought to be a logical impossibility. True == False does seem pretty broken though. Not sure that can go anywhere.
- username90 6y ago
  1 + 1 = 2 is based on conservation of particles. You put a marble in a bowl, then another marble in the bowl, you now have two marbles in the bowl. If you remove conservation then there is no reason why 1 + 1 should equal 2. 1 + 1 being equal to anything could just be nonsense in that universe, such a construct wouldn't exist and there would be no way to reason about quantities. That isn't a logical inconsistency, so such a universe could exist.

mkl 6y ago

> 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 6y ago
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 6y ago
  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 6y ago
    > 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 6y ago
      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.
    - wizzwizz4 6y ago
      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 6y ago
        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 6y ago
        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?
camjw 6y ago
To me this is a non-example though because in this context 2 = 0. So 1 + 1 = 2 = 0.
gjm11 6y ago
That isn't a situation where 1+1 isn't 2. It's a situation where 1+1=2 but also 2=0.

ddxxdd 6y ago

A universe where 1 + 1 != 2 is a universe filled with self-contradictions and so cannot exist. It's like imagining a universe where True == False. It's not a hypothetical, it's a logical impossibility.

mkl 6y ago
In modulo arithmetic, 1+1 != 2 can work just fine. We're not talking about the literal universe, but the space of possibilities that opens up when you change the rules. E.g. all the amazing power and complexity that comes from imagining the existence of a number i with the property that i^2 = -1. This was initially thought to be a logical impossibility. True == False does seem pretty broken though. Not sure that can go anywhere.
username90 6y ago
1 + 1 = 2 is based on conservation of particles. You put a marble in a bowl, then another marble in the bowl, you now have two marbles in the bowl. If you remove conservation then there is no reason why 1 + 1 should equal 2. 1 + 1 being equal to anything could just be nonsense in that universe, such a construct wouldn't exist and there would be no way to reason about quantities. That isn't a logical inconsistency, so such a universe could exist.