A geometric object can possess continuous or approximate symmetries that no discrete subgroup of spatial automorphisms captures exactly (Nozick's objection to structural definitions).

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Continuous symmetry(as used in geometry and physics): A property where an object looks the same after being rotated or shifted by any amount (not just specific angles), like a perfect circle that looks identical no matter how much you turn it.
Discrete subgroup(as used in abstract algebra and group theory): A specific collection of distinct, separate operations (like rotating by exactly 90 degrees or 180 degrees) that follow mathematical rules, unlike continuous operations that can be any amount.
Geometric object(as used in mathematics and philosophy of mathematics): A shape or figure in space, like a triangle, square, or sphere, that has specific mathematical properties.
Nozick: Robert Nozick was an American philosopher best known for developing a theory of individual rights and limited government based on the idea that people own themselves and the fruits of their labor. He argued that most government actions, including taxation for welfare programs, violate people's fundamental rights unless everyone freely consents to them. His work became highly influential in libertarian political philosophy, offering a compelling alternative to both socialism and traditional conservative views about the proper role of government.

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