# the concept of odd and even **URL:** \ **Category:** Factual Questions\ **Created:** [April 29, 2004, 5:54pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707 "2004-04-29T17:54:12Z")\ **Posts on this page:** 20\ **Page:** 1
**Author:** ![Thrillhouse15](https://avatars.discourse-cdn.com/v4/letter/t/ad7895/32.png) [@Thrillhouse15](https://boards.straightdope.com/u/Thrillhouse15)\ **Post date:** [April 29, 2004, 5:54pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/1 "2004-04-29T17:54:12Z")
I’ve heard this question repeatedly on a radio commercial from the Girl Scouts of America, in a commercial about encouraging young girls’ interests in math and science, but have never heard the answer. So I’m hoping some smart Doper can help me out: Why is the concept of odd and even a philosophical illusion? ---
**Author:** ![Kalt](https://avatars.discourse-cdn.com/v4/letter/k/b9e5f3/32.png) [@Kalt](https://boards.straightdope.com/u/Kalt)\ **Post date:** [April 29, 2004, 6:03pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/2 "2004-04-29T18:03:11Z")
Sounds like a loaded question to me. Who says odd and even numbers are a “philosophical illusion”? Seems that there is a pretty clear definition of odd and even, nobody is debating whether a given number is odd or even, and the concept is extremely straightforward and objective (as are most things in mathematics). ---
**Author:** ![Thrillhouse15](https://avatars.discourse-cdn.com/v4/letter/t/ad7895/32.png) [@Thrillhouse15](https://boards.straightdope.com/u/Thrillhouse15)\ **Post date:** [April 29, 2004, 6:08pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/3 "2004-04-29T18:08:03Z")
You’re probably right that there’s no real answer. But I thought there might be something I was missing, so I thought I’d ask. ---
**Author:** ![daffyduck](https://avatars.discourse-cdn.com/v4/letter/d/ec9cab/32.png) [@daffyduck](https://boards.straightdope.com/u/daffyduck)\ **Post date:** [April 29, 2004, 6:14pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/4 "2004-04-29T18:14:28Z")
You can define unicorns too, but that doesn’t make them any less imaginary. Many people hold that all of math is an invention, as opposed to a discovery, and in that sense, odd and even would be no less of an illusion than any other mental construct. The point I believe they are trying to make is that oddness and eveness are not inherent properties of anything but numbers and numbers exist only in human minds. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 29, 2004, 6:20pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/5 "2004-04-29T18:20:42Z")
No idea what they might mean. ---
**Author:** ![pravnik](https://avatars.discourse-cdn.com/v4/letter/p/8edcca/32.png) [@pravnik](https://boards.straightdope.com/u/pravnik)\ **Post date:** [April 29, 2004, 6:21pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/6 "2004-04-29T18:21:09Z")
Maybe because theoretically everything can be halved? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [April 29, 2004, 8:39pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/7 "2004-04-29T20:39:31Z")
> [@Thrillhouse15](#): > > Why is the concept of odd and even a philosophical illusion? Philosophers think up all sorts of crazy stuff (“categorical imperative”, my ass…), but rest assured that for integers the “concept of odd and even” is on _very_ firm ground. ---
**Author:** ![GreyWanderer](https://avatars.discourse-cdn.com/v4/letter/g/7cd45c/32.png) [@GreyWanderer](https://boards.straightdope.com/u/GreyWanderer)\ **Post date:** [April 29, 2004, 9:43pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/8 "2004-04-29T21:43:28Z")
It’s a definition, a concept. Nothing more. Just like a word can be used to describe something. ---
**Author:** ![Bippy\_the\_Beardless](https://avatars.discourse-cdn.com/v4/letter/b/ac8455/32.png) [@Bippy\_the\_Beardless](https://boards.straightdope.com/u/Bippy_the_Beardless)\ **Post date:** [April 29, 2004, 9:54pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/9 "2004-04-29T21:54:30Z")
Maybe they are getting at 'Why does the word Odd have negative conotations whilst the word Even has neutral or positive connotations?" Odd socks, odd looking, odd. Even handed, even-stevens, even so. What dis 1,3,5,7,… do to deserve such a fate as to be called odd? ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [April 29, 2004, 10:17pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/10 "2004-04-29T22:17:43Z")
> [@Bippy the Beardless](#): > > What did 1,3,5,7,… do to deserve such a fate as to be called odd? Their image under the canonical quotient map from **Z** to **Z** /2 **Z** is 1 rather than 0. As to the terms themselves, the O.E.D. saves the day yet again: odd: > [@](#): > > early Scandinavian: cf. Norwegian odde, Old Swedish odda, udda (Swedish udda), early modern Danish odde (Danish regional odde), all in sense ‘(of a number) odd’, and also Old Icelandic odda-, combining form of oddi (triangle, point or tongue of land, odd number); the adjectives in other Scandinavian languages are prob. of similar origin (cf. Norwegian oddetal, early modern Danish oddetal, both in sense ‘odd number’). > The senses ‘odd’, ‘odd number’ in early Scandinavian app. developed by metaphor from ‘triangle’ (as being three-cornered), and thence by extension from the third or unpaired member of a group of three, to any single or unpaired member of a group, and from three as the primary ‘odd number’, to all numbers containing an unpaired unit. even: > [@](#): > > Common Teutonic: OE. efen, efn, by assimilation emn = OFris. even, evin, OS. eban (Du. even, effen), OHG. eban, epan (Ger. eben), ON. iafn, iamn (Da. jevn, Sw. jemn), Goth. ibns:OTeut. \*eno-. > The word has not yet been satisfactorily connected with any other Teut. or Aryan word; hence it is uncertain whether the primary sense was ‘level’ or ‘equal, like’. And from there to the idea of a number divisible into two equal (“even”) parts. ---
**Author:** ![Bippy\_the\_Beardless](https://avatars.discourse-cdn.com/v4/letter/b/ac8455/32.png) [@Bippy\_the\_Beardless](https://boards.straightdope.com/u/Bippy_the_Beardless)\ **Post date:** [April 29, 2004, 11:47pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/11 "2004-04-29T23:47:20Z")
> [@Mathochist](#): > > Their image under the canonical quotient map from **Z** to **Z** /2 **Z** is 1 rather than 0. But 1,3,5… can hardly be blamed for that, and yet we say they are odd. I think there is a need for a more politically correct term to avoid this unpleasentness. I propose such numbers should be called ‘differently even’ to protect their feelings. But then we must think of all those poor fractions rationals and reals that are neither odd nor even, these we can call ‘specially even’. As for imaginary numbers and trans-finites, I say fuck the bastards, they arent even real numbers, free booting scum that just pretend to be numbers if you let them into your equations pretty soon the whole theorem is just packed with them and they just multiply like bleeding rabbits… (OK I’m losing it I admit it, wibble) ---
**Author:** ![moriah](https://avatars.discourse-cdn.com/v4/letter/m/b2d939/32.png) [@moriah](https://boards.straightdope.com/u/moriah)\ **Post date:** [April 30, 2004, 7:04am UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/12 "2004-04-30T07:04:58Z")
Define philosophical illusion. Because if you mean that oddness comes from a non-objective point of view, then they’re right. All of mathematics start with subjective axioms that hold no special place in objectivity… …unless one wants to define objectivity in math as being analogous to ‘reality’ with an unerring verisimilitude. In that case, integers and counting can be considered very objectively ‘true’ and without any illusion because they reflect the reality of counting individually real objects as perfectly as we can ascertain. So, depending on your definitions and initial stance toward mathematics and reality, the proposition of the OP can be true or false. On a side note, even-oddness is a property of modulo 2 mathematics. That’s a way of counting cyclically with two members in the cycle. It’s like a clock with only the 12 and 6 (or in this case only ‘odd’ and ‘even’). Starting with the first member of the counting numbers, one, you go round the cycle. Two is the second member and place in the cycle. Three is the third member, but now you’re back to the first place. Four is the fourth member, but is in the second place. All the numbers in the first place have a common characteristic we call ‘odd’ and the numbers in the second place are ‘even.’ Even numbers are divisble by two. Every member of the even set can be paired to a second member of the set with no member left over. Of course, with the odd set, you’d have a single member left over. And so, in this case, even-odd is not a philosophical illusion if we believe that modulo 2 mathematics is a good representation of reality. But, of course, someone can argue that our axioms are objectively unprovable, and thus, an illusion. Peace. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [April 30, 2004, 7:36am UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/13 "2004-04-30T07:36:29Z")
> [@moriah](#): > > Because if you mean that oddness comes from a non-objective point of view, then they’re right. All of mathematics start with subjective axioms that hold no special place in objectivity… Actually, structures and structure-types are very well objectively defined. The structure-type “ring” can be completely self-contained (in Lawvere’s Th( **Rng** ), for instance), after which there is a unique initial object in the natural category structure on the class of rings. This is **Z** : the integers. This unique structure has a smallest nontrivial quotient: **Z** /2 **Z** , from which oddness and evenness follow. Your impression of the philosophy of mathematics seems caught up in the ancient Platonist/Formalist debate, which lost all real relevance by the 1950s. > [@](#): > > …unless one wants to define objectivity in math as being analogous to ‘reality’ with an unerring verisimilitude. In that case, integers and counting can be considered very objectively ‘true’ and without any illusion because they reflect the reality of counting individually real objects as perfectly as we can ascertain. Or, as the philosophers call it, the “unreasonable effectiveness” problem. The problem is that while the Peano axioms specify a unique structure **N** (the natural numbers), most mathematical theories are not uniquely defined. Even Zermelo-Fraenkel set theory – which most mathematicians still cling to as their bedrock – is incompletely specified, as one may accept or reject the axiom of choice, the continuum hypothesis, and any number of other independant postulates to get different theories. Stepping away from set theory, one could work out a parallel mathematics in a topos very different from **Set** ; a non-Boolean topos might bring very interesting results in quantum logics. Which of these divergent strands is “the real world”? As far as we can tell we have no idea. Yes, the structure of the natural numbers seems to accurately represent the problem of counting macroscopic objects which are easily distinguishable, but I’d be hesitant about extending this notion to the full ring structure of **Z** (this issue is explored in [this thread](http://boards.straightdope.com/sdmb/showthread.php?t=251087)). Can any mathematics be “proven” to accurately model “reality”? I think people like Popper will always be able to worm their way out by denying physical induction. Still, one must be _very_ careful to draw the distinction between proving that a given collection of axioms obtain in “the real world” and proving them as mathematical objects. The language later in your post blurs this, though it’s more a problem for the reader than (I think) a reflection of any confusion on your part. > [@](#): > > So, depending on your definitions and initial stance toward mathematics and reality, the proposition of the OP can be true or false. Agreed, but I think you’re turned around as to which stance leads to which answer. ---
**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\ **Post date:** [April 30, 2004, 2:19pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/14 "2004-04-30T14:19:08Z")
> [@Mathochist](#): > > Your impression of the philosophy of mathematics seems caught up in the ancient Platonist/Formalist debate, which lost all real relevance by the 1950s. By this do you mean, “Everybody finally gave up and became a Formalist”? (As I suspect.) Or do you mean “Platonists and Formalists finally realized the debate was futile”? ---
**Author:** ![AHunter3](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ahunter3/32/368_2.png) [@AHunter3](https://boards.straightdope.com/u/AHunter3)\ **Post date:** [April 30, 2004, 4:55pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/15 "2004-04-30T16:55:49Z")
I assumed the commercials (which I heard on Air America Radio) were referring to the accident of 2. We don’t have special terms (the equivs of “odd” and “even”) to refer to “numbers evenly divisible by 3” and “numbers not evenly divisible by 3”. What’s special about 2 is that any string of numbers cleaves nicely in half when you divide it according to “numbers evenly divisible by 2” and “numbers not evenly divisible by 2”. Well, except for when the string of numbers contains a quantity of numbers not divisible by 2, in which case the string does not quite cleave nicely in half. But you know what I mean — the two sets of numbers are at least very nearly identical in quantity. When you split a string of numbers according to whether or not they are divisible by 3, you end up with one pile (the nots) being twice as big as the other pile (or very close to that, depending on the exact quantity of numbers in your initial string). Which is obviously less elegant and meaningful than if they were the same size, or almost the same size, because…uh… Actually, there’s nothing special about 2, or about being or not being divisible by 2, that should set it apart from being or not being divisible by 3. The sets, and the math, have certain characteristics that make 2 different, but only in the same sense that any other number is different from the rest when used as the differentiator. So odd and even are a philosophical illusion. (I don’t necessarily subscribe to this viewpoint, but I bet it’s what they’re hinting at) ---
**Author:** ![ouryL](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ouryl/32/6067_2.png) [@ouryL](https://boards.straightdope.com/u/ouryL)\ **Post date:** [April 30, 2004, 5:19pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/16 "2004-04-30T17:19:31Z")
Oddness or evenness has no real meaning when we talk of ordinal numbers. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [April 30, 2004, 5:25pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/17 "2004-04-30T17:25:09Z")
Actually, if you’re going to split the numbers according to divisibility by 3, you’d split them into numbers that are evenly divisible by 3, numbers that have a remainder of 1 when divided by 3, and numbers that have a remainder of 2 when divided by 3. You do end up with a similar structure to the split according to divisibility by 2 if you do it this way. In general, you partition the integers according to divisibility by k into sets such that the difference of any two members of a set is divisible by k. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [April 30, 2004, 5:31pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/18 "2004-04-30T17:31:25Z")
> [@Bytegeist](#): > > By this do you mean, “Everybody finally gave up and became a Formalist”? (As I suspect.) Or do you mean “Platonists and Formalists finally realized the debate was futile”? I mean that by the 1950s it was recognized that neither one works out very well in the long run. Just because all most people know about is Platonism and Formalism doesn’t mean that’s the end of the story. The current front runner (among people who pay attention to philosophy of mathematics _and_ how mathematics is done these days) is one or another variety of Structuralism. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [April 30, 2004, 5:36pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/19 "2004-04-30T17:36:14Z")
> [@AHunter3](#): > > Actually, there’s nothing special about 2, or about being or not being divisible by 2, that should set it apart from being or not being divisible by 3. The sets, and the math, have certain characteristics that make 2 different, but only in the same sense that any other number is different from the rest when used as the differentiator. One thing that’s definitely special about 2 is that 2 is a prime. **Z** /n **Z** is generally a ring, but for n prime, it is a field. Further, **Z** /p **Z** is the universal object in the category of fields of characteristic p. As anyone who pays attention to this sort of thing knows, characteristic 2 is _very_ special compared to any other prime characteristic. ---
**Author:** ![scotandrsn](https://avatars.discourse-cdn.com/v4/letter/s/a4c791/32.png) [@scotandrsn](https://boards.straightdope.com/u/scotandrsn)\ **Post date:** [April 30, 2004, 5:54pm UTC](https://boards.straightdope.com/t/the-concept-of-odd-and-even/242707/20 "2004-04-30T17:54:57Z")
I’ve heard that ad! Given that the whole thing is somewhat tongue-in-cheek anyway, I would say that they are referring to the idea that “divisibility by two” is no more important than divisibility by any number. The “modulo x” function (whose result is the remainder when some integer is divided by x) separates integers into what are called “equivalence classes”. If I apply modulo 3 to the non-negative integers, then 0, 3, 6, 9… are all in one equivalnce class, because they all have a remainder of 0 when divided by three. 1, 4, 7, 10… are in another class becuase their remainder is 1, and so forth. IMO, the “philosophical illusion” of odd and even arises when we apply a greater significance to the equivalence classes that result from applying modulo 2 to the integers than to any other modulo equivalence classes. Because our sense of math grew gradually throughout history, there are artifacts like this of our evolving understanding. It is hypothesized that it took some time to understand that doubling something and dividing it into two equal pieces were essientially two different aspects of the same thing, after which we had little trouble applying the same idea to larger numbers. Hence the fact that English (and several other languages) have words for “two” and “half” that have different linguistic roots, but similar roots for words signifiying other numbers (three and third, four and fourth, etc.)