Talk:Isomorphism

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[edit] Pseudo-philosophical gobbledygook

Therefore knowing what counts as an isomorphism is as good as knowing what we mean by structure, of a given kind.

I'm removing this until someone can rephase it. It sounds like pseudo-philosophical gobbledygook. In what sense is the one knowledge "as good as" the other? Who is this mysterious "we"? Is "structure" meant in a technical sense? If so, what's the definition? --Ryguasu 21:42 26 Jun 2003 (UTC)

There isn't a universal definition of structure that covers all mathematical structures that are discussed.

One way round this is to say 'we may not know what structure means in the abstract, but we can be given information about when two structures are the same'. It's like saying, faced with some unfamiliar type of money, this coin has the same value as this note, this pile can be exchanged for that one. Without trying to say what 'money' (or the value it represents ) is.

A simple example from arithmetic: we can write numbers in binary or base 10 notation. For the most part we don't care, since the results of calculations will be the same, after conversion. A represention of 'eigh't as 100 is 'as good' as 8.

A purely mathematical example would be a metric space X, which gives rise to a topological space in a standard way. If we are prepared to consider another topological space Y that is related to X by a homeomorphism as suitable for our purposes, that tells us that the open set structure is all we care about. If on the other hand we insist that Y be another metric space and the homeomorphism actually an isometry, that says we actually care about the metric.

That probably isn't the usual case, in fact: a constant multiplier in the metric is like changing your basic unit of measurement, say from metre to kilometre, and so yet another idea of 'isomorphism' can be brought it as a 'similarity'. That tells you that the structure that matters is the ratio of distances, e.g. similar rather than just congruent triangles in plane geometry.

Charles Matthews 07:58 29 Jun 2003 (UTC)


In the definition of the isomorphism, it is said that the functions f and f-1 should be bijective homomorphisms. But homomorphism is, as I remember, a group mapping, so I believe it ought to be morphism in that section, and that one should wait with the homomorphism until the definition of group isomorphism. Anyone here who could say whether I'm wrong about this? Mikez 10:38, 29 Jan 2004 (UTC)

Anyone who follows the homomorphism link gets an idea of the morphism concept immediately. I've added an informal note at that point, too.

Charles Matthews 10:48, 29 Jan 2004 (UTC)

Formally, an isomorphism is a bijective map f such that both f and its inverse f −1 are homomorphisms, i.e. structure-preserving mappings.

It seems like this sort of a statement should have a bit more context. Of course, this is a true statement in all of the familiar categories, but it's also a bit misleading in the sense that the important aspect of an isomorphism is that it is a morphism with an inverse. That this implies bijectivity in the usual algebraic categories is really a theorem, not the definition.

Personally I'd leave it like it is, but also expand the mention of the categorical definition (near the end) to be explicit. Most mathematicians would give a definition like this one unless specifically asked for the category theory definition and the same is true for most books that are not category theory books. --Zero 10:18, 25 Feb 2005 (UTC)

You may want to add a link to the First Isomorphism Theorem.

[edit] Merge with Cryptomorphism?

See Talk:Cryptomorphism. Qwertyus 21:04, 1 August 2006 (UTC)

No Way!!! Isomorphism is an extremely broad and widely used term familiar to every mathematician whereas cryptomorphism is highly specific to a particular context and is NOT in widespread use. In my opinion it would be a mistake to merge the two. Someone seeking information on isomorphisms doiesn't need to be told about cryptomorphisms. Hawthorn.

--Army1987 21:56, 24 November 2006 (UTC)

[edit] Lead cube versus wood cube

User:Army1987 removed the following physical analogy example –

A solid cube made of wood and a solid cube made of lead are both solid cubes; although their matter differs, their geometric structures are isomorphic.

– supplying this comment: "→Physical analogies - No, they don't. The lead cube contains less atoms, and two sets with different cardinalities can't be isomorphic."

Although the removed example was arguably not the best illustration of isomorphic structures (and ignoring the fact that there's no reason the cubes couldn't contain the same number of atoms), I don't think that the cardinality argument holds water. The isomorphism being illustrated was clearly the cube-ness of the objects, rather than their physical properties; and two cubes can certainly be viewed as isomorphic structures. (It is after all a basis of topology.) At any rate, non-mathematical use of the term isomorphism is imprecise; thus in biology, isomorphs are organisms with analogous structures – obviously failing a "count of atoms" test.

I believe that the term isomorphism finds a good deal of general non-mathematical (and thus non-rigorous) use. I will put a comment to that effect in the "physical analogies" section. Trevor Hanson 19:41, 12 November 2006 (UTC)

This article is about mathematical isomorphisms, see also Isomorphism (disambiguation).
Indeed. (Though the section on physical analogies stretches the discussion to fringe areas, by providing intuitive examples for the lay reader. This seems reasonable, since the term isomorphism links directly to this article, rather than the disambiguation page. But if you think my addition to the analogies section is off-topic, do feel free to remove it.) The distinction you make between isomorphisms between infinitely divisible Platonic solids and isomorphisms between the structures and atoms of physical objects is of course valid, and not at all 'unworthy'. I don't think that the original 'lead cube versus wood cube' example was intended to imply anything more than isomorphic shapes; but you're correct that it was ambiguous. Trevor Hanson 23:27, 12 November 2006 (UTC)
Two cubes can be put in biunivocal correspondence with each other, there is a function which nds each point of one to one and only one point of the other. If the function chosen is an isometry, it preserves distances between two points of one cube and the corresponding two points of the other. This is an isomorphism. (There are bijections from a cube and a line, from a cube and the powerset of N, etc., but these can't be obviously isomorphisms.)
I agree that the theoretical shape of a cube made of wood is isomorphic to that of a cube made of lead, but their theoretical shapes are just approximations. At a microscopic level they have very different structures. Yes, this does sound like a subtle distinction unworthy being done, and I myself would believe it is, if I didn't know that some people were fooled into believing that a gold ball could be duplicated via the Banach-Tarski paradox... --Army1987 20:30, 12 November 2006 (UTC)

Another problem in that section is this one: "A list of customer names and a list of telephone numbers, where each entry on either list corresponds to a single entry on the other list. Note that the lists exhibit the isomorphism, rather than the actual customers and telephones (which may or may not be isomorphic)." As well as it being written badly, I can't see that it is anything more than two sets and a bijection between them. If there is no structure in one set that is carried onto to structure in the other set, why bother calling it an isomorphism and why is it a useful example? McKay 03:57, 14 November 2006 (UTC)

They are isomorphic, for example, with respect with the measure d(x,x) = 0 and d(x,y) = 1 ∀ x ≠ y. Too bad that any two set with the same cardinality are. So that is not a very instructive example. However, it should be noted that, althought is is not a good example of isomorphism, there is a bijection (that sending every name to the phone number of the person with that name) being chosen among the n! possible ones. In contrast, there are as many quark flavours as many Rammstein members, but there is no bijection between them which is more 'important' or 'useful' or 'logical' than the other 719 possible ones. --Army1987 21:56, 24 November 2006 (UTC)

[edit] Consider changing main 'isomorphism' target to disambig page?

The latest revert suggests that perhaps the main entry for isomorphism should be the disambiguation page rather than the term in mathematics, and that this article should be renamed isomorphism (mathematics). Comments? Objections? That would seem the more normal Wikipedia model. Trevor Hanson 19:01, 24 November 2006 (UTC)

Almost all articles in "What links here" seem to be about maths, so I'm not sure this is the right thing to do... --Army1987 21:44, 24 November 2006 (UTC)
Hmmm, yes of course you are right. I wish I'd looked at "What links here" first. :) Many wrong-headed edits to this article (such as yesterday's involving Gestalt psychology, as well as recent edits of mine) seem to be good-intentioned attempts to deal with the facts that a) isomorphism, used without context, links here, and b) the term is widely used outside maths, with different meanings; and this seems potentially confusing, especially to non-English speakers. But as you point out, the term's richest technical usage, and no doubt its origin, is mathematical. We do always seem torn between two views of Wikipedia: as a union of specialized repositories, for use by experts; versus a trans-specialty gateway, for use by outsiders. And of course, it must be both things. I've been on the other side of this argument elsewhere. I guess I'd better leave this particular article to mathematicians. Trevor Hanson 05:24, 25 November 2006 (UTC)

[edit] Gestalt psychology

I was the person who added the Gestalt Isomorphism entry to the applications section of this page and was disappointed to see it removed by the author (whose right to do so I so not dispute). The Gestalt Psychology article includes a link to this Isomorphism article, but anyone following it would not really be well-enough informed as to the meaning Kohler and Wertheimer meant to convey by their choice (perhaps more accurately, the translator's choice, as the original works were in German)of this term for their idea. So the question is, where to put the Gestalt meaning of isomorphism? Perhaps in the Gestalt Psychology article, in place of the link, but then the author of that article might do what the author of the isomorphism article did. Mark Cole.

The 'normal' Wikipedia way is to have the source article (Gestalt Psychology) point to the disambiguated isomorphism (Gestalt psychology) page. I believe this is now how things stand.
Just to clarify: We are all "the author" of this page; I have no special claim to it (less than most in fact; this particular page is really the domain of mathematicians). I did step in, and moved the Gestalt Isomorphism text from this page to a new article: isomorphism (Gestalt psychology). I should point out that:
  • The material added was good, but it was not an application of mathematical isomorphism, any more than are the biological or computer science uses of the term. (This point was driven home to me when one of my earlier edits drew the same reaction.)
  • I moved this text to a new article because I saw that this text had already been added, removed, and added again, and I wanted to avoid a 'revert war' -- three reverts triggers bad things. Since creating a new article was the preferred Wikipedia solution, I decided to step in and do this – rather than just saying "this belongs on a disambiguation page." I'm sorry if this bruised any toes.
  • I have just changed the link in Gestalt psychology to point to that new disambiguated page, so I hope everything is satisfactory now?
  • I recently questioned (above) the decision of having isomorphism link to this article, rather than to the disambiguation page; but the valid point was made that there are innumerable mathematical articles referring to this page in a technical way, and very few articles from other fields. (I still think it could be argued either way...but not by me.)
Again, there is no such thing as "the author of an article" in Wikipedia. There is an "original author" who sets the style in such matters as citation format and whether to say "color" or "colour" (and those get changed, too). But all work is collaborative. Trevor Hanson 20:21, 26 November 2006 (UTC)

As the instigator of all this, I am guilty more of being a Wikipedia novice than of anything else. The only reason I added the Gestalt Psychology note to the Isomorphism article a second time was that I thought I had done something wrong (technically speaking) the first time when it disappeared. I certainly had no intention of starting any sort of "reversion war". The Isomorphism link in the Gestalt psychology article now points to my entry on Gestalt Isomorphism on the related disambiguation page. I am happy and no feelings were bruised along the way. Also, I am grateful for the heads up about global authorship. Mark Cole.

No worries, and I'm glad if things look OK to you now. BTW, in case you were thinking that a 'revert war' was like a childish 'flame war', that isn't the issue. Three reverts triggers automatic consequences; I decided to step in because I thought this might occur essentially by accident. Also: In case you don't know, you can type four tildes (~~~~) after your post to have the system automatically include your name and the time. Trevor Hanson 03:41, 29 November 2006 (UTC)


Thanks for helping me out in this expedition into uncharted waters, Trevor. 129.100.98.203 16:20, 29 November 2006 (UTC)

[edit] Clock analogy

"The Clock Tower in London (that contains Big Ben) and a wristwatch; although the clocks vary greatly in size, their mechanisms of reckoning time are isomorphic."

Would it be better to make a comparison between:

"The Clock Tower in London (that contains Big Ben) and a digital wristwatch; although the clocks vary in size and type of display, their mechanisms of reckoning time are isomorphic."

Or is this unneccessary, given the following dice example? —Matthew0028 05:29, 30 December 2006 (UTC)

[edit] Tic-Tac-Toe Analogy

When playing Tic-Tac-Toe X usually wins on the fourth move. Finding three numbers that adds up to fifteen is not the same game. There is an analogy for playing the most efficient game, but doesn't cover every possible way in which the game may be played, and won, with more than three moves as is often seen when inexperienced children play it. —The preceding unsigned comment was added by 130.36.62.141 (talk) 21:28, 19 February 2007 (UTC).

I think you misunderstand the 15-game. You aren't limited to just saying 3 numbers. But when you say more, only 3 of them have to add up to 15. The analogy stands. —The preceding unsigned comment was added by 82.146.104.183 (talk) 16:15, 9 May 2007 (UTC).

[edit] Matrix examples

I think this article would benefit from an example that an mxn matrix is isomorphic to an nxm matrix. Isomorphism has special application to linear algebra. Psyadam 20:42, 29 March 2007 (UTC)Adam Henderson, March 29, 2007

[edit] Isomorphic "sets"?

Hi.

Can you explain this?

"In a certain sense, isomorphic sets are structurally identical, if you choose to ignore finer-grained differences that may arise from how they are defined."

Huh? I didn't think that "sets" could be isomorphic. What, are we supposed to say that the sets A = { 1, 2, 3, 4 } and B = { Roswell UFO, Bill Gates's house, Microsoft, IBM } have some sort of "isomorphism"? That doesn't make any sense. Sets alone do not really have much "structure" to them, you need to set up relationships between elements of the set that create structure, and structure is what isomorphism deals with. But then it is misleading to call the resulting object just a "set". What gives? Is somehow, the Roswell UFO isomorphic to the number 3? Not unless you add some sort of relationship that puts a structure on those sets, but then just saying "sets" gets misleading! mike4ty4 06:16, 7 May 2007 (UTC)

I've changed "sets" to "structures". Sets can be isomorphic (which just means that they have the same cardinality, since the isomorphisms in the category of sets are simply the bijections), but that's not what this sentence is talking about. --Zundark 07:45, 7 May 2007 (UTC)

i think this article is really helpful. isomorphism confused me before, now i get it. thank you

[edit] pronunciation of "isomorphism"

I think it would be quite a good idea to add the IPA pronunciation of "isomorphism" (which is not really easy to find out, believe me). At least someone could write it here. Thanks. 77.202.221.15 23:20, 3 September 2007 (UTC)

[edit] What's this Hofstadter Business?

I liked G.E.B. as much as the next guy, but what's with the Hofstadter quote at the beginning of the article? Not only does it add nothing to the article, but it's also confusing. "Complex structures"? Stick with the well thought out description already agreed on. Quote removed. Rljacobson (talk) 03:11, 13 February 2008 (UTC)

[edit] Physical analogies?

Hello, I'm concerned that the "physical analogies" make it sound like "isomorphism" is a vague concept. This is misleading: one of the most important facets of abstract maths is that "isomorphism" is a precise thing and definitely not vague.

How can two decks of cards be isomorphic? A deck of cards is not a mathematical set, so what is "a function between decks of cards"? Or is there some "category of decks of cards"?

Similarly, how can two clocks be isomorphic? in what sense is a clock a set? and what is a function between two clocks? when is it structure-preserving? is there a category of clocks?

If "isomorphism" is a vague notion that exists in general language, and if we really need to mention that here, then it should be clear that this has little to do with "abstract algebra". Shall I cut the section? Or move it lower and make it clear that it is not mathematics? Or should we suggest what a "morphism of clocks" could be? Any thoughts? Sam Staton (talk) 15:49, 18 May 2008 (UTC)