[tutfbhuf]

"Mathematical objects arise from definitions, not measurements and observations."

Abstract logical frameworks are not pulled from thin air. Human brains develop such frameworks with a physical brain that receives sensory input from a physical world, and hence there is a deep connection between the physical world and the abstract logical frameworks in many ways.

[613style]

Definitions are created to be useful or interesting or both, and sometimes those specifiers overlap with physical reality. But that doesn't mean they're created to match reality or to tell you anything about it.

I remember struggling as an undergrad to come up with a metaphor for visualizing what a group homomorphism is, aiming to develop better intuition when working with them. It's hard because the real world doesn't contain any, and all the "it's kind of like X" types of examples you can think of aren't really useful when mapped back to the mathematical domain. If math described the real world, I'd expect this type of thing to be much easier.

[tutfbhuf]

An example of a group homomorphism in physics is the relationship between the rotational symmetry group of a physical system (often the group SO(3) for three-dimensional rotations or SU(2) for spin systems) and the representation of that group by the angular momentum operators in quantum mechanics. The angular momentum operators in quantum mechanics form a representation of the symmetry group because they satisfy the same commutation relations as the infinitesimal generators of the group. This means that the algebraic structure of the rotational symmetry group is mirrored by the algebraic structure of the angular momentum operators, and the way these operators transform states in the Hilbert space of the quantum system reflects the symmetry properties of the system.

[613style]

I'm not sure if you intend this as a helpful analogy for an undergrad or as an argument in favor of groups/homomorphisms being created to match reality, but I don't think it applies very well to either.

(Though it is very interesting!)

[samus]

An important part of mathematics is indeed formalizing such objects. Mathematicians go further, question these assumptions, and abstract from them. For example, removing constraints on fields gives us rings, groups, semigroups, and categories, and exploring what can still be said about them.

But I'm sure pure mathematicians would also be totally fine studying randomly generated rulesets. The only restriction is that they should not lead to immediately obvious contradictions, since those would probably be kind of boring!

[tutfbhuf]

"But I'm sure pure mathematicians would also be totally fine studying randomly generated rulesets."

I do agree, but that we don't do that in reality is exactly my point.