To the algebraist, $\mathbb{Z}$ is just the free group with one generator. To the algebraic topologist, $\mathbb{Z}$ is just the fundamental group of the circle. To be glib, what do $\mathbb{Z}$ mean to you?
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$\begingroup$ He he: everyone's answering the wrong question! When translated into English, the question reads "To be glib, what do the integers mean to you?". $\endgroup$– Andrew StaceyCommented Jan 22, 2010 at 10:44
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3$\begingroup$ Why I don't like this question: (1) $\mathbb Z$ is not "just" anything. An "algebraist" very well might study "rings and algebras", for example. (2) It's discussion-y. How are you going to pick a "right" answer? $\endgroup$– Theo Johnson-FreydCommented Jan 22, 2010 at 16:29
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$\begingroup$ I'm not sure how appropriate this is to MathOverflow, but I've certainly wondered the same thing and I'm interested in seeing people's answers. $\endgroup$– Zev ChonolesCommented Jan 22, 2010 at 16:40
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2$\begingroup$ I agree with Theo. I don't really see what you expect to get out of asking this question. I don't like any of the answers, and it's hard to imagine somebody giving a good one. $\endgroup$– Anton GeraschenkoCommented Jan 22, 2010 at 18:36
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5 Answers
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Initial object in the category of commutative rings.
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$\begingroup$ (Or the category of Z-algebras) $\endgroup$ Commented Jan 22, 2010 at 10:42
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To a number theorist, shouldn't $\mathbb Z$ be the world?
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An infinite discrete subset of $\mathbb R$.
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2$\begingroup$ From the coarse geometry perspective, it is R! $\endgroup$ Commented Jan 22, 2010 at 21:17