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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?". $\endgroup$ Jan 22, 2010 at 10:44
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    $\begingroup$ 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? $\endgroup$ Jan 22, 2010 at 16:29
  • $\begingroup$ 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. $\endgroup$ Jan 22, 2010 at 16:40
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    $\begingroup$ 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. $\endgroup$ Jan 22, 2010 at 18:36

5 Answers 5

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Initial object in the category of commutative rings.

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  • $\begingroup$ (Or the category of rings.) $\endgroup$ Jan 22, 2010 at 7:22
  • $\begingroup$ (Or the category of Z-algebras) $\endgroup$ 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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Final object in the category of schemes.

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Just $\mathbb{Z}$ ? ;-)

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An infinite discrete subset of $\mathbb R$.

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    $\begingroup$ From the coarse geometry perspective, it is R! $\endgroup$ Jan 22, 2010 at 21:17

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