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The single most defining feature of modern algebra (roughly from the 1890s–1930s, with figures like Dedekind, Hilbert, Noether, and Artin) is the deliberate abstraction from the nature of mathematical objects to the relations between them.
Instead of asking: “What is this thing (number, polynomial, matrix, etc.)?” Modern algebra asks: “How does this thing behave under given operations?” modern algebra and the rise of mathematical structures