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For example, why we study Group Theory to prove general results for all instead of specifically studying $\mathbb Z$ (or any other set) closed under some operation?

What makes algebra and those generalizations such an useful tool for practical and theoretical purposes? How abstract algebra emerged?

İbrahim İpek
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    See e.g. Leo Corry's book Modern Algebra and the Rise of Mathematical Structures. – Bill Dubuque Jul 14 '19 at 20:41
  • Just downloaded the book. It looks pretty interesting. Thanks for advice – İbrahim İpek Jul 14 '19 at 20:54
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    This is what mathematicians do. They inspect specific constructs, see that they have similarities to other objects. They look for the minimal amount of similarities that are enough so that the objects are still "alike". They define a term for objects with properties that make them similar. Then they work on this abstract level, guided by rules about how those objects behave. – B.Swan Jul 14 '19 at 21:05
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    This is a very good question, but much too broad for this site - entire books have been written on the topic. For example, I strongly recommend Gray's Plato's Ghost. I'll merely point out that you have to distinguish between a few fundamentally different aspects: why mathematicians (tend to) find the study of abstractions of concrete objects interesting for its own sake; why such abstractions (often) have applications to the original concrete objects; and how historically that interest became dominant and those applications became apparent. – Noah Schweber Jul 14 '19 at 21:23
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    @B.Swan This is what research mathematicians do nowadays. How this came to be is an interesting question. But I must agree that it is far too broad. – Servaes Jul 14 '19 at 21:51
  • I would like you people to evaluate my understanding so far: https://math.stackexchange.com/a/3310815/554493 – İbrahim İpek Aug 01 '19 at 20:46

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