Talk:Cryptomorphism

Proposed merge

Searching for (Cryptomorphism OR Cryptomorphic) on Google Scholar only returns 69 hits using either term which raises my doubts as to whether this term is actually notable enough to warrant its own encyclopedia article. Is Homomorphism the same thing? --  Netsnipe  ^(Talk)  05:02, 29 July 2006 (UTC)

no it's not the same (as far as I understand). I'll change the tag to suggest a merge to isomorphism, but in any case I don't think this term is in wide use (even in the math community). Pascal.Tesson 05:24, 29 July 2006 (UTC)

Let's see. Two things in math are called "isomorphic" if they are only superficially different: for example, the set of natural numbers written in decimal notation is isomorphic to the set of natural numbers written in binary. It literally means "same shape", and I think it was around even in the late 19th century. The word "homomorphism" came much later. It's not a very deep generalization of "isomorphism" but I can't think of a ten-word explanation. Besides being a not-very-deep generalization of "isomorphism", the homomorphism concept is enormously, earth-movingly important. And it has a technical mathematical definition.

The word cryptomorphism is not so much a generalization of as a riff on "isomorphism": "cryptomorphism" is what you get when you mash "crypto" and "isomorphism". Two things are cryptomorphic if they're "secretly" the "same shape": that is, if it's "difficult to see" that they're the same. The concept is not very important, and it does not have any technical mathematical definition (possibly a reason it doesn't appear in the literature very often), but it is standard jargon among (some) people who work on matroids, and I think it's of some pedagogical value: there are a zillion different definitions of "matroid", and as a result lots of advances in the subject have a zillion different equivalent formulations. I think it's nice that there's a name for the phenomenon.

Anyway, I vote "no" on merging with either "homomorphism" or "isomorphism". Changbao 06:18, 29 July 2006 (UTC)

I don't think I made this clear: isomorphisms (and homomorphisms) have precise, technical definitions. "Cryptomorphism" is not a technical concept but a word people use when talking about math. Changbao 06:24, 29 July 2006 (UTC)

But then wouldn't a footnote in the isomorphism article be enough? The cryptomorphism article will never be more than a two-liner so a simple redirect to isomorphism should do. Pascal.Tesson 06:45, 29 July 2006 (UTC)

I think better that it be deleted entirely than merged with isomorphism; it seems to me that the two concepts are of different species. But I've just spent a few minutes trying to give the article some content, and though it's still not perfect, I think it just needs to be improved, not deleted. Let me know what you think. Changbao 07:50, 29 July 2006 (UTC)

I vote do not merge. I agree with Changbao on this. Cryptomorphism seems like a philisophical idea rather than a mathematical concept. - grubber 06:26, 31 July 2006 (UTC)

Actually, I am also tempted to change my position here. The article is cute now. By the way, shouldn't there be at least a wikilink to isomorphism? The cryptomorphism seems to be more general but there are certainly many isomorphisms which could be deemed cryptomorphisms. Pascal.Tesson 06:33, 31 July 2006 (UTC)

Yes, I think a wikilink back to isomorphism is certainly appropriate. It's a curious article, and I enjoyed reading it, so I hope it doesn't get deleted. But I don't think it belongs with isomorphism. - grubber 15:49, 31 July 2006 (UTC)

A cryptomorphism is not an isomorphism. It is a non-obviously equivalent definition of the same object. Thus, a matroid can be defined by bases, or rank, or duality, and these are not obviously equivalent. A simple matroid is not the same as a geometric lattice, so they should not be called cryptomorphic (in my opinion). Also, a group can be defined by division, instead of the usual operation of multiplication with id and inverses, and that could qualify as a cryptomorphism because it's not obvious to many mathematicians that such a definition exists. Therefore, cryptomorphism is not a kind of isomorphism. Zaslav (talk) 09:09, 3 August 2022 (UTC)

And yet, White's matroid book does call that exact example a cryptomorphism. So there is a mismatch between the definition of cryptomorphisms in your head and in the literature. We should follow the literature here. Even for the standard examples of cryptomorphism (matroid independent sets vs matroid rank, say) we have different but equivalent mathematical objects (a family of sets vs a function from sets to integers). —David Eppstein (talk) 16:22, 3 August 2022 (UTC)

Confusing

The problem with the term "cryptomorphism" is that it is used in a number of articles on matroids, and does nothing at all to aid in the understanding of what they are. When I read that something is cryptomorphic to something else, it does not help me understand what the (iso-)morphism is. It leaves me scratching my head, and makes me think that the author was somehow trying to show off in how clever they are, and how dumb I am. Articles in math need to be understandable, not crypto-understandable. 84.15.191.139 (talk) 09:41, 11 November 2015 (UTC)

I saw improper use of the term in some matroid articles, when "equivalence" would have been simpler and better. I agree the word should not be used without explanation. Zaslav (talk) 09:04, 3 August 2022 (UTC)

It seems to me, that "multiple different definitions giving the same objects" appears in other places as well, but it didn't receive a special name. For example Axiomatic foundations of topological spaces lists many ways to define topological spaces. Or boolean algebras can be defined over many different signatures (in the sense of universal algebra) as well.

So it might not be surprising that matroids enjoy a similar property. Columbus240 (talk) 10:51, 14 March 2023 (UTC)

Arts

Marcus Gheeraerts the Elder: Allegory of Iconoclasts, (c. 1567)

Hidden Face of the "Baker" in an illustration by Henry Holiday to Lewis Carroll's The Hunting of the Snark (1876)

The article should stay. Chryptomorphism may not be in wide use in the maths community. But it is in use in arts. Should the article be aded to that category?

http://artsciencefactory.fr/2011/02/04/par-gustave-dore-et-par-henry-holiday/#comments --DL5MDA (talk) 04:14, 29 March 2011 (UTC)

Cryptomorphism is a sign of importance

Somewhere I read that if a kind of mathematical objects has several cryptomorphic definitions, it is a sign that this object is important.

We need to state this after we find a citation for this. VictorPorton (talk) 22:13, 7 October 2022 (UTC)

On the etymology

The following is stated without any citation/sources:

The word was coined by Garrett Birkhoff before 1967, for use in the third edition of his book Lattice Theory. Birkhoff did not give it a formal definition, though others working in the field have made some attempts since.

In the third edition of his book, Birkhoff talks about a "cryptoisomorphism" being a way of defining the same abstract algebra in a "nonpolyisomorphic way". He explicitly defines polymorphism, polyisomorphism and then cryptoisomorphism. The statement above is correct in that "Birkhoff did not give it a formal definition" in his lattice theory book. But there are multiple questions which are raised by the above quoted two sentences.

1. How do we know it was coined before 1967? Did he write about it somewhere? Is there a source to this?
2. Are there sources of "others working the field have made some attempts since"? Is it only in the matroid community? Is this coming from an algebraic community? Who are these people?

Thecaligarmo (talk) 07:23, 19 October 2023 (UTC)