What is wrong with this premise?

[Angrboda]

Where I disagree is the claim that they are made up. They are not made up descriptions but rather absolute descriptions. An alien might use a different "word" for two, but what ever "word" it uses is going to mean exactly the same thing as we mean when we say "two". The notion of a reality without quantities is nonsensical.

Only if they are deploying a system that uses the descriptor "two" in the same way as our system of counting. I realize it's counter-intuitive, but it really is a product of adopting a specific system of assumptions and rules. Even in our world if we define a number as a word having a certain number of letters, then two + two = twelve, because two has 3 letters and twelve has six letters. It all depends on the rules and conventions you adopt. There is no "natural" set of rules and definitions. (We could also use numerical symbols and become more abstract. 2+2 = 11, 2+2 = 14, etc. If we include the plus sign as a number, then 2+2 = 311, and so forth. And we would talk about it differently: "A basket with an apple, a donut, and a plum in it has 567 objects inside it," or perhaps, "A plum and a plum and a plum are 133,811,345,870 things," depending on where we stop "interpreting" the symbolic descriptors.)

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By abstracting away various amounts of detail, mathematicians have created theories of various algebraic structures that apply to many objects. For instance, almost all systems studied are sets, to which the theorems of set theory apply. Those sets that have a certain binary operation defined on them form magmas, to which the concepts concerning magmas, as well those concerning sets, apply. We can add additional constraints on the algebraic structure, such as associativity (to form semigroups); identity, and inverses (to form groups); and other more complex structures. With additional structure, more theorems could be proved, but the generality is reduced. The "hierarchy" of algebraic objects (in terms of generality) creates a hierarchy of the corresponding theories: for instance, the theorems of group theory apply to rings (algebraic objects that have two binary operations with certain axioms) since a ring is a group over one of its operations. Mathematicians choose a balance between the amount of generality and the richness of the theory.

. . . . . . .

Because of its generality, abstract algebra is used in many fields of mathematics and science. For instance, algebraic topology uses algebraic objects to study topologies. The recently (As of 2006) proved Poincaré conjecture asserts that the fundamental group of a manifold, which encodes information about connectedness, can be used to determine whether a manifold is a sphere or not. Algebraic number theory studies various number rings that generalize the set of integers. Using tools of algebraic number theory, Andrew Wiles proved Fermat's Last Theorem.

In physics, groups are used to represent symmetry operations, and the usage of group theory could simplify differential equations. In gauge theory, the requirement of local symmetry can be used to deduce the equations describing a system. The groups that describe those symmetries are Lie groups, and the study of Lie groups and Lie algebras reveals much about the physical system; for instance, the number of force carriers in a theory is equal to dimension of the Lie algebra, and these bosons interact with the force they mediate if the Lie algebra is nonabelian.

Wikipedia | Abstract Algebra