# Definition of x=y **URL:** \ **Category:** Factual Questions\ **Created:** [October 22, 2005, 1:07am UTC](https://boards.straightdope.com/t/definition-of-x-y/327367 "2005-10-22T01:07:28Z")\ **Posts on this page:** 8\ **Page:** 2
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [October 23, 2005, 4:08pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/21 "2005-10-23T16:08:22Z")
> [@Polerius](#): > > [I don’t think so](http://mathworld.wolfram.com/IrreducibleFraction.html) From your link: > [@](#): > > An irreducible fraction is a fraction p/q for which GCD(p, q) = 1. IOW, the fraction p/q is irreducible iff GCD(p, q) = 1. Like I said, a statement about a pair of integers. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 24, 2005, 4:43pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/22 "2005-10-24T16:43:37Z")
> [@ultrafilter](#): > > This is false. > > The fraction 1/2 is not identical to the fraction 2/4, yet they are considered to be equal. Because this _isn’t_ GD I’ll stop there and point out to the OP that even professionals don’t agree on an answer to your question. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 24, 2005, 4:50pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/23 "2005-10-24T16:50:01Z")
> [@DarrenS](#): > > While we’re on the subject, and I hope this isn’t irrelevant - what is the difference between “equivalence” and “identity”? > > From what I recall, an “identity” is a statement that is true for all values of the variable - informally, something that is “trivially true”. > > I’m referring to the concept denoted by a triple “=” sign, which I don’t know how to generate in a browser-neutral way. > > Thus, x^2 \ x \* x > > Isn’t it also valid to write x ^ 2 = x \* x using a regular ‘=’ though? > > Is “identity” really just a hint to other mathematicians that “this is an equation but I’m going to express it as an identity to show you that I consider it obvious” ? “Equivalence” is highly context-dependent and you have to read the text (or be familiar with the field and its conventions) to know exactly what it means. Usually, it has a connotation of “I’m not asserting that they’re _identical_ (the exact same thing) but that there’s a sequence of moves from this list that transform one into the other”. I warn everyone around there that we’re in danger of having to run a seminar in higher category theory to untangle equivalence/isomorphism/identity issues, at least as far as the professionals have. ---
**Author:** ![AWB](https://avatars.discourse-cdn.com/v4/letter/a/bb73d2/32.png) [@AWB](https://boards.straightdope.com/u/AWB)\ **Post date:** [October 24, 2005, 5:12pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/24 "2005-10-24T17:12:03Z")
> [@Polerius](#): > > i.e. what does “x=y” mean? To paraphrase William J. Clinton: “It depends on what your definition of ‘equals’ equals.” :D:D ---
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [October 24, 2005, 7:08pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/25 "2005-10-24T19:08:12Z")
> [@Mathochist](#): > > I warn everyone around there that we’re in danger of having to run a seminar in higher category theory to untangle equivalence/isomorphism/identity issues, at least as far as the professionals have. I can see how this could get deeply philosophical. Can you point me to some useful terms to Google, or even some websites? Or perhaps start another thread here, or in GD? This is fascinating. It reminds me a lot of the concept of “identity” in object-oriented programming. An object X may have the same “contents” as object Y, in that all its fields have the same values, yet it is a distinct object. However, you can very well have two object references that point to the _same_ object. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 24, 2005, 8:58pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/26 "2005-10-24T20:58:50Z")
> [@DarrenS](#): > > I can see how this could get deeply philosophical. Can you point me to some useful terms to Google, or even some websites? Or perhaps start another thread here, or in GD? This is fascinating. It reminds me a lot of the concept of “identity” in object-oriented programming. An object X may have the same “contents” as object Y, in that all its fields have the same values, yet it is a distinct object. However, you can very well have two object references that point to the _same_ object. Well, all the references I know of assume a good background in mathematics. The first section of Baez and Dolan’s paper in “Higher Category Theory: Workshop on Higher Category Theory, March 28-30, 1997, Northwestern University, Evanston, Il (Contemporary Mathematics)” (edited by Ezra Getzler and Mikhail Kapranov) is pretty readable, though. Basically the idea is that a lot of concepts are “decategorifications” of more fundamental concepts. Consider the collection of all finite sets. Each one has a cardinality – the number of elements in the set. Two sets with the same cardinality are isomorphic, but not necessarily identical. We say that two (cardinal) numbers are equal if they are the cardinalities of two isomorphic sets. This keeps track of _which_ sets are isomorphic, but leaves out any information about a specific isomorphism. It gets more technical from there. ---
**Author:** ![DSYoungEsq](https://avatars.discourse-cdn.com/v4/letter/d/c6cbf5/32.png) [@DSYoungEsq](https://boards.straightdope.com/u/DSYoungEsq)\ **Post date:** [October 24, 2005, 10:58pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/27 "2005-10-24T22:58:02Z")
> [@Mathochist](#): > > Well, all the references I know of assume a good background in mathematics. The first section of Baez and Dolan’s paper in “Higher Category Theory: Workshop on Higher Category Theory, March 28-30, 1997, Northwestern University, Evanston, Il (Contemporary Mathematics)” (edited by Ezra Getzler and Mikhail Kapranov) is pretty readable, though. > > Basically the idea is that a lot of concepts are “decategorifications” of more fundamental concepts. Consider the collection of all finite sets. Each one has a cardinality – the number of elements in the set. Two sets with the same cardinality are isomorphic, but not necessarily identical. We say that two (cardinal) numbers are equal if they are the cardinalities of two isomorphic sets. This keeps track of _which_ sets are isomorphic, but leaves out any information about a specific isomorphism. It gets more technical from there. I’ve said this before about your postings. It looks like English, it spells like English, but that ain’t quacking I’m getting from reading it! 😛 ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [October 24, 2005, 11:13pm UTC](https://boards.straightdope.com/t/definition-of-x-y/327367/28 "2005-10-24T23:13:36Z")
> [@DSYoungEsq](#): > > I’ve said this before about your postings. It looks like English, it spells like English, but that ain’t quacking I’m getting from reading it! 😛 Okay, consider the classic story of the shepherd who collects a stone for each sheep he lets out of the pen in the morning and drops one for each sheep going back into the pen at night. What he’s doing in the morning is considering two sets (his sheep and the bunch of stones) and constructing an explicit one-to-one map between them. The sets are “isomorphic”, though obviously not identical. In the evening he constructs another isomorphism from the set of stones to the set of sheep going into the pen. If he exactly runs out of stones he’s assured that the sets of sheep leaving in the morning and returning at night are _isomorphic_, but not that they’re identical. Passing from sets to their cardinal numbers is a process of “decategorification”, in which we remember an object’s isomorphism class (in this case its cardinality) but nothing else about it. So, when we say #{sheep leaving in the morning} = #{stones in collection} = #{sheep returning at night} we use the equals symbol to denote equality of cardinal numbers, which is _not_ the same as equality of sets. As far as the OP goes, “=” is always heavily context-dependent and the validity of its interpretation as identity hinges mostly on one’s notion of ontology (what “exists”). If I take the position that sets exist, but cardinal numbers are “just” abstractions, then I can’t say that two numbers are “identical” because they aren’t _things_ with identities in the first place. As far as most applications in day-to-day life (and, frankly, even in technical discourse) of arithmetic and algebra are concerned, this makes no difference. The increase in understanding gained by a thorough investigation of the nature of equality/equivalence/isomorphism/identity is simply too minor to justify the enormous difficulty in exposition. In fact, insisting on a deeper understanding of the nature of numbers was one of the huge sticking points of the New Math. It’s just not worth it for most people. [Previous page](https://boards.straightdope.com/t/definition-of-x-y/327367.md?page=1)