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Update: This post was originaly about the loose usage of even and odd, where people without providing a definition of what they meant by even or odd asked inaccurate questions like "is 0 even?". Although I tried to draw the attention to the fact the historically neither 1 or 0 was considered even or odd ( 0 was not even around when the concept of even or oddness was coined ), it missed the target by making the readers conclude that it was about claiming that 0 is not even. After continued downvoting without leaving constructive comments it was time for an update. So here it goes : if one uses equivalance classes to define 2 mutualy exclusive sets by

$Evens = \{ 2k : k \in \mathbb Z \} , Odds = \{ 2k+1 : k \in \mathbb Z \} $ Using this everyone can agree which set 1 or 0 fall in, regarding 1 and 0 being always considered even or odd there are plenty of math history sites.

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So what is the question? –  Yuval Filmus Dec 26 '10 at 22:23
Mariano's comment to your answer already addresses this issue. You are using an unnecessarily restrictive definition of even and odd. –  Qiaochu Yuan Dec 26 '10 at 22:23
Your posts in the linked thread and now this are going dangerously close to 'crank' territory. –  kahen Dec 26 '10 at 22:41
I don't understand this post/question at all. –  user02138 Dec 26 '10 at 22:52
@Arjang: in that case your question is a philosophical question (although, as stated, I do not find it a very compelling one) and as such is off-topic for this site. I have voted to close. –  Pete L. Clark Dec 27 '10 at 1:30

2 Answers 2

up vote 8 down vote accepted

You seem to be proposing that it is somehow an error to use a word beyond the context in which it was originally defined. This is a ludicrous linguistic claim, let alone a mathematical one. As the world and our understanding of it changes, the words we use to describe it must necessarily adapt to those changes. To invent a new word every time we need to label a new concept would be endlessly tedious, and it would ignore the significant compressive power of metaphor.

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@Arjang: the rigorous definitions change to suit our purposes. For me an integer is even if it lies in the ideal 2Z and odd if it doesn't. This definition has the significant benefit of isolating the key abstract property of evenness and allowing us to generalize it to arbitrary ideals in commutative rings. –  Qiaochu Yuan Dec 26 '10 at 23:41
@Qiaochu: right, but also not just for you. Like any mathematical terms, even and odd need to be formally defined somewhere, and when using these terms in discourse with each other we need to agree to both refer to this common definition. The definition of even and odd for arbitrary integers has been made in thousands of texts going back at least a hundred years. When I teach a sophomore level "introduction to rigorous mathematics" course, I include these definitions and work from them: e.g., I prove that every integer is either even or odd and no integer is both... –  Pete L. Clark Dec 27 '10 at 1:22
...The fact that in the distant past the definition was used in a more restricted way is irrelevant: it pretends that the (sometimes tacit and sometimes explicit) modern meaning of even and odd does not exist. As you point out, this makes no sense -- without such agreements mathematical discourse could not exist, and other forms of human discourse would be shaky as well: in your next political conversation, try pointing out that the word "citizen" is being used in a different and broader sense than it was in ancient Greece. See what kind of reaction you get! –  Pete L. Clark Dec 27 '10 at 1:26
@Arjang: yes, but I really don't understand why this is an issue. Look at almost any word in almost any language and it will not have the same meaning that it had hundreds of years ago. If you want to stick to ancient definitions, the price you pay is that no one will understand you. What's the point of that? –  Qiaochu Yuan Dec 27 '10 at 4:30
@Arjang: this is a completely different question from the one you asked above (which is not tagged math-history). Of course the answer is "it depends on what definitions they were working with at the time." –  Qiaochu Yuan Dec 27 '10 at 5:18

$\ldots,-4,-2,0,2,4,\ldots$ are even integers, $ 2,4,6, \ldots$ are even positive integers. $\ldots,-5,-3,-1,1,3,5,\ldots$ are odd integers, $ 1,3,5, \ldots$ are odd positive integers. What can be simpler than this?

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