Organizational Standard

[Barry R. Smith]

I am wondering if an organizational standard has been discussed for a mathematical concept that has successive generalizations to more and more abstract concepts. 

Today, I was thinking of beginning the "Chinese remainder theorem" article.  When one first encounters this, it is a statement about integers (and modular arithmetic), and probably for most non-specialists, this is the only context in which they will want to learn it.  However, it can also be formulated in a hugely more abstract context, that of commutative rings.  Would we include this on the same page, at the risk of scaring and or providing headaches to the non-initiated who stumble across the page?  Wikipedia does, but it seems bad form to me.  I would think a short sentence at the end of the introduction of the page talking about integers could say that there is a more abstract version and give a link to a page called "Chinese remainder theorem (ring theory)".

Similarly, even to formulate the statement of the Chinese remainder theorem, one must have an understanding of "relatively prime" integers in the first context, and "relatively prime" ideals in the second.  Once again, Wikipedia includes both topics on the same page.  This seems even more dangerous, since a lot of interest in relatively prime integers probably comes from schoolchildren (although I know we are aiming our pages at university educated folks).

If we do include both the familiar specific version and the abstract abstruse version of each topic on the same page, we cannot use this as a standard for all mathematics articles.  For instance, "vector space" and the more abstract concept "module" must have their own individual pages.

[Chris Day]

Check out the Quantum mechanics article and its more advanced sister subpage. is that the type of thing you had in mind? See more alternatives at http://en.citizendium.org/wiki/CZ:Subpages

[Paul Wormer]

I would write articles as you yourself feel most satisfied with,  because there is not really a common standard. There are some CZ mathematics articles that, although probably correct, are only readable by specialists (not even readable by most of the math editors).  Since it is near-impossible to convince some authors to write a more gentle introduction (most of them have left anyway), the articles stay here as they are, waiting for the off-chance that a specialist finds them useful. It is, of course,  better that you don't follow this lead (that is, don't always try to use the shortest argument  in the most advanced notation stating the most general form of the result),  but if it would be the only way you'd feel happy, then so be it.  Advanced math articles are to be preferred over no math articles.

There are also a few articles in existence that follow the WP "funnel strategy", they become more and more difficult when you read along, so that most readers don't make it to the end of the funnel. Personally, I don't object to this strategy, although I can understand that some readers don't like it if  they miss the ending (was Hausdorff the perpetrator, or was it Brouwer after all? Suspense, suspense).   

[Howard C. Berkowitz]

I am wondering if an organizational standard has been discussed for a mathematical concept that has successive generalizations to more and more abstract concepts. 

Freely admitting that the title of "CZ talk: Usability" is not very usable, there has been some discussion, I believe, on just the idea of such abstraction.

Today, I was thinking of beginning the "Chinese remainder theorem" article.  When one first encounters this, it is a statement about integers (and modular arithmetic), and probably for most non-specialists, this is the only context in which they will want to learn it.  However, it can also be formulated in a hugely more abstract context, that of commutative rings.  Would we include this on the same page, at the risk of scaring and or providing headaches to the non-initiated who stumble across the page? 

While many ideas about such abstraction, or, if you will, knowledge navigation, are still emerging, I believe we have the basics of a means of giving the larger context.

May I suggest that some of us hope it might be less a matter of stumbling across a page, but across a cluster of pages and subpages?

For example, you could create a page Chinese Remainder Theorem, but also create a page, Chinese Remainder Theorem/Related Articles. While we are not locked to a format, in general, we have three main headings: Parent Topics, Subtopics, and Related Topics. Increasingly, I find use for something I call (and am open for a better name), "Co-topics": things that are not proper subtopics of the main page, but are essential peer concepts.

At the parent topic level, you might have R-templates for {{r|Integer}} and {{r|Commutative ring}}. The argument after the r| is an article name, which does not need to exist as an actual page -- Related Articles are, I believe, a more effective, hyperlinked way of doing strawman outlines.

When you save, you will see, approximately (red assumes the article page does not exist)
*Integer insert definition
*Commutative ring insert definition

At this point, you could click either on the Article name, or on insert definition. Each will open a different edit page: the first for a main article page, the second for a main article/definition page.

With your example, I'd make the integerhave a definition that you see as making sense to the nonspecialist. Write a definition, however, for Commutative ring that talks about the more general meaning.

Rather than depending solely on the main page for Chinese Remainder Theorem for clues to the two levels of meaning -- I should mention that it is possible to start by explicitly creating Chinese Remainder Theorem/Related Pages without yet defining Chinese Remainder Theorem -- you give the reader of the Related Pages the idea that the topic can have a modest or extensive parent.

For the general reader, you could have modular arithmetic as a subtopic of CRT. I'd be inclined to use my subhead of "Co-topic", since modular arithmetic often will be the target of a link in other clusters.

Wikipedia does, but it seems bad form to me.  I would think a short sentence at the end of the introduction of the page talking about integers could say that there is a more abstract version and give a link to a page called "Chinese remainder theorem (ring theory)".

In the CZ style of introduction, I might mention that CRT exists at different levels of abstraction, and the Related Articles page (which can be a link) is especially important.

Similarly, even to formulate the statement of the Chinese remainder theorem, one must have an understanding of "relatively prime" integers in the first context, and "relatively prime" ideals in the second. 

Again, parent topics; you might have the introduction mention that Related Articles/Parent Topics contains prerequisite concepts.

[Barry R. Smith]

Chris:  I like the idea of using the "advanced subpage", except that it presumes just two versions of a topic exist:  an advanced one, and a dumbed down one.  Mathematics can very well have several levels of increasing difficulty, rather than just two -- I guess this is mostly due to "overloading" certain terms, giving them several disparate meanings, rather than having one concept like quantum mechanics that can be given a precise demanding description or a fluffy but more approachable one.  With my examples, I guess it probably is a duality, although now that I understand more about "related subpages" (thanks for the link), I don't see the advantage of using the advanced tab over having two separate pages listed as "co-topics of each other". 

Paul:  I personally often don't like the funnel approach.  I think it is totally inappropriate that Wikipedia's page on "relatively prime" includes a detailed discussion about ideals in abstract rings, rather than just talking about integers and referring users looking for the more hard-core stuff to another page.  I guess I'll try to avoid this, either by using the advanced tab or with a "co-topic" page about the more advanced stuff.

Howard:  Thanks very much for your help.  I knew about the related topics subpage, but hadn't read the details about what was supposed to be in it.  I like your idea of adding a fourth type of topic:  "co-topic".  However, I got confused about your description.  It sounds like you are saying, create a "Chinese remainder theorem" (CRT) related topics page, and include links to pages called "integer" and "ring (algebra)".  Either of these could be viewed as a parent topic for the CRT, being one of the two required background theories in which to discuss the CRT.  My original question was more about what should go in the CRT main page -- both the easier version with integers and the harder version with ring theory?  Or just the one with integers, and a link to another page, maybe with the title "Chinese remainder theorem (commutative ring theory)" for people who are looking for that page instead.  I guess I could solve it as suggested by Chris with the "Advanced" tab, but this won't always work in mathematics.

As a final note, I think we should try to reach some sort of consensus about the funnel approach -- at least certain instances where it will be deemed appropriate or not.  For instance, maybe we could create some policy that if there is typically a lag time of more than a year before one learns two versions of a topic, then they should be on separate pages.  This would prevent, for instance, having a paragraph about ring theory in the "relatively prime (integers)" or just "relatively prime" page.

[Howard C. Berkowitz]

Chris:  I like the idea of using the "advanced subpage", except that it presumes just two versions of a topic exist:  an advanced one, and a dumbed down one.  Mathematics can very well have several levels of increasing difficulty, rather than just two -- I guess this is mostly due to "overloading" certain terms, giving them several disparate meanings, rather than having one concept like quantum mechanics that can be given a precise demanding description or a fluffy but more approachable one.  With my examples, I guess it probably is a duality, although now that I understand more about "related subpages" (thanks for the link), I don't see the advantage of using the advanced tab over having two separate pages listed as "co-topics of each other". 

Howard:  Thanks very much for your help.  I knew about the related topics subpage, but hadn't read the details about what was supposed to be in it.  I like your idea of adding a fourth type of topic:  "co-topic".  However, I got confused about your description.  It sounds like you are saying, create a "Chinese remainder theorem" (CRT) related topics page, and include links to pages called "integer" and "ring (algebra)".  Either of these could be viewed as a parent topic for the CRT, being one of the two required background theories in which to discuss the CRT.  My original question was more about what should go in the CRT main page -- both the easier version with integers and the harder version with ring theory? 

Perhaps the basic version, but with a very, very strong suggestion, in the introduction, to look at the Related Articles page to understand the broader context.

Or just the one with integers, and a link to another page, maybe with the title "Chinese remainder theorem (commutative ring theory)" for people who are looking for that page instead.  I guess I could solve it as suggested by Chris with the "Advanced" tab, but this won't always work in mathematics.

We are stilll getting a sense of what we can do with Larry's original idea of Related Pages. Some of the ideas, such as co-topics, are experimental, but it;s valuable to know that someone else finds it useful.

Let me address both your observation to Chris, and your question about my suggestion about having multiple parent topics. Assuming the mathematical use of overloading terms is reasonably similar to the computer science usage, multiple parent topics, with the different definitions appearing on the related articles page, is a means of resolving the overloaded term. When looking at Related Articles, the reader would not simply see multiple parent articles, but multiple parent articles with different definitions.

There's no reason that the definition text could not explicitly say that this is more or less advanced, and to have more than one advanced perspective. Yes, I've studied the CRT, but at the level of the basic graduate discrete mathematical structures in a computer science program, so I don't have the background to say how it derives from different viewpoints in mathematics. In computer science, I can think of concepts that are introduced at an introductory to intemediate level, but, for example, there might be multiple parents, one discussing how a technique was developed to work around limits of hardware capability, while another independent development came from a scalability need in a particular area of application. This is very true of some networking principles; some methods of router design came independently from real-time hardware limits, and a more abstract approach to solving large-scale topological optimization.

Indeed, there's no reason why the definitions themselves could not reference one another:

• Parent 1 (simplest) definition: description of the Parent 1 concept, but include, in the definition, [[Parent2]] is a more generalized definition and [[Parent 3]] is even more abstract.
• Parent 2(more advanced) definition: description of the Parent 1 concept, but include, in the definition, [[Parent2]] is a more generalized definition  in [[Discipline A]]. [[Parent 3]] is also more generalized, but with an origin in [Discipline B]].
• Parent 3(more advanced) definition: description of the Parent 1 concept, but include, in the definition, [[Parent3]] is a more generalized definition  in [[Discipline B]]. [[Parent 2]] is also more generalized, but with an origin in [Discipline A]].

[Larry Sanger]

Based on the outcome of this discussion, we might want to change what our policy on this says--it currently says to include just the immediate parent topic(s), not parents, grandparents, great-grandparents, and so forth.  This might not be the best policy, but let us be explicit about what is in place and how we want to change it.

I'd also like to invite you to think of this not as an engineering problem, nor as an abstract system-building problem, but as a usability problem.  What will actual users be most likely to find most useful when they are researching a topic?

Howard, the name for what you call "co-topics" is "related topics"--i.e., that's where those topics are placed.  If in your mind there is a difference between your "co-topics" and "related topics" as described in CZ:Related Articles, I'd be curious to learn the difference!

[Howard C. Berkowitz]

Based on the outcome of this discussion, we might want to change what our policy on this says--it currently says to include just the immediate parent topic(s), not parents, grandparents, great-grandparents, and so forth.  This might not be the best policy, but let us be explicit about what is in place and how we want to change it.

I'd also like to invite you to think of this not as an engineering problem, nor as an abstract system-building problem, but as a usability problem.  What will actual users be most likely to find most useful when they are researching a topic?

Howard, the name for what you call "co-topics" is "related topics"--i.e., that's where those topics are placed.  If in your mind there is a difference between your "co-topics" and "related topics" as described in CZ:Related Articles, I'd be curious to learn the difference!

Co-topics, a name that I hate, is a distinct subset of Related Articles that I see as quite different than "Related topics". I can give specific examples in any number of disciplines. The problem, however, may be with the use of "subtopic". The even broader problem is that strict hierarchy doesn't always work -- there may be multiple inheritance.

There is an article, submarine-launched ballistic missiles. In its related article page, the Trident missile is clearly a subtopic, because the Trident is a specific SLBM, as opposed, say, to a Soviet R-29RM/Sineva. Assume a Trident article.  Tridents can only be fired from  Ohio-class (US) or Vanguard-class (UK) ballistic missile submarines, but I hesitate to call them "Related topics", because those submarines can do other things besides launch Tridents. Nevertheless, those principal reason those classes were built was to fire Tridents.

The R-27RM is a related topic to Tridents, in that they are both submarine-launched ballistic missiles. There is an interdependency between Trident and Vanguard/Ohio that I've called a "co-topic". It's a stronger bond than "related article", but I can't call Ohio boats a subtopic of Trident.

I can give any number of examples, in multiple disciplines, where the relationships are stronger than just "related", but not single-hierarchy. Ravioli, wonton, and pierogi are all subtopics of dumpling. Each has a traditional shape and a variety of traditional fillings. I can make them all "freehand", but I often use a kitchen gadget called a ravioli maker. If I use a pierogi filling but form it with the ravioli maker, I have been known to call the product "Polish ravioli", but an equally good argument could be made for "Italian pierogi". Tamales, quenelles, and gai bow are all dumplings, but share no filling or shape with the things that can be made with the ravioli maker.

[Larry Sanger]

Hi Howard, I'm still not getting it.  Parent, child, and "related" necessarily exhaust the relationship types, because "related" is anything that is, well, related, but is neither a parent nor a child.  You say "stronger than 'related'" but then I just have to wonder what you mean by "related" in that case.  There are also (formally definable) relations of sibling, cousin, etc....do you mean one of those, or all of them?  If you can't give an example that clearly shows your meaning, why not try an actual definition?

Also, is it really important that we track whatever distinction you're trying to draw in our canonical Related Articles categories?  Why?  (Please...if you want to answer, please try to answer that straightforwardly, not with more pierogis and missiles. :-) )

It seems obvious to me that the sorts of subs that fire the Trident missile, while neither parent nor child topics of the "Trident" topic, are related topics.  We can say exactly what the relationship is (the sub fires the missile) and why the relationship is important (being able to fire the missile is the main purpose of the sub).  But that is the sort of thing that the "Related topics" section of Related Articles page is for.  There's no way to exhaustively enumerate all the ways that topics can be "related" because they are as many as the relationships that exist between things.  The point anyway is that there are important relationships that are not parent-child or other hierarchical relationships.  We agree on that, and listing such relationships is the purpose of related topics sections.

What did you think the purpose of related topics sections was?

[Barry R. Smith]

The reason I said I like having a fourth "co-topic" category is because in mathematics, there are two ways that one article could be considered to be a subtopic of another.

The usual way certainly occurs -- one topic is a specific subject within the other broader field, i.e., algebra is a subtopic of mathematics, polynomial equations is a subtopic of algebra, quadratic formula is a subtopic of polynomial equations.

The other way is that one topic can be an abstraction, or generalization, of the other, as in the examples that started this thread.  The Chinese remainder theorem can be stated as a theorem within the context of modular arithmetic, hence is a subtopic of "modular arithmetic".  It can also be stated within the context of abstract ring theory, of which modular arithmetic could be considered a subtopic.  Let's call the first version CRT (mod) and the second CRT (ring), for sake of brevity.

Now if you take CRT (ring), whose formulation assumes you have a given ring (a certain type of set), and a family of certain subsets of that ring, then such-and-such is true.  If you then choose the particular ring to be the integers (the archetype for rings), and the certain subsets are chosen appropriately, the statement of CRT (ring) devolves precisely into CRT (mod).  So CRT (ring) is precisely a generalization of CRT (mod). 

However, I don't like calling CRT (ring) and CRT (mod) "parent topic" and "subtopic", because CRT (ring) is not what I would consider to be a broad field of knowledge containing the smaller field of knowledge CRT (mod).  Perhaps this is my own personal bias, but it strikes me as qualitatively different than the difference between algebra and polynomial equations.  There, the second is not a specialization of the latter by making certain assumptions more specific -- it is a much looser type of relationship.  CRT (ring) and CRT (mod) are so strongly related that, for instance, Wikipedia decided to put them both on the same page (an example of that "funnel" approach that I find inappropriate). 

On the other hand, "related topics" doesn't ring true to me either.  The relationship is much stronger than "related" -- one is a generalization of the other, in a very precise way.  As a mathematician, I must spend a lot of time precisely arranging the arrangement of the various concepts into a network of both generalizations/specializations and looser types of relationships, like the type that lets me know to look in a number theory book to find some information about modular arithmetic. 

Perhaps this biases my view of how information in the Citizendium math workgroup should be structured, but I'm not sure why each workgroup cannot have it's own organizational rules at a specialized enough level, like which headings to include on the related topics page.  Perhaps,if other math people agree with me,  a choice could be made as to whether "generalization/specialization" is more a "parent topic/subtopic" relationship, or a "related topic" relationship, and then the mathematics workgroup could make this into a formal subheading in the related pages tab.

However, I do not think this is necessarily a mathematics specific issue.  I should go look up, for instance, how biology distinguishes "gene" from the points of view:  piece of information inherited from parents, piece of a chromosome, and particular sequence of base pairs all serving one function.  Perhaps biology has no problem putting these all on the same page, because an educated person could probably follow a good enough explanation.  That is not the case with CRT (ring) and CRT (mod) -- a computer science student who just spent a year learning discrete math would be crushed trying to understand the general ring theoretic version with no other training.  But presumably, most readers will be more interested in CRT (mod), which is why I think it is inappropriate to stick CRT (ring) on the same page at the bottom.  I'm gonna start the CRT page now, and use the "advanced" tab I guess since this particular topic has only the two versions to distinguish.

[Howard C. Berkowitz]

Hi Howard, I'm still not getting it.  Parent, child, and "related" necessarily exhaust the relationship types, because "related" is anything that is, well, related, but is neither a parent nor a child. 

Ummm...that may work for basic familiar relatioships, but if you go beyond socially defined relationshps in English, such as genetic encoding and expression, the real world isn't that simple. Indeed, there are languages that define familial relationships in a much more fine-grained manner than Englsh; there are human languages in which they have single words for things that we might have to say "second cousin by marriage to my mother-in-law's brother-in-law", or things like that.

you[/i] mean by "related" in that case.  There are also (formally definable) relations of sibling, cousin, etc....do you mean one of those, or all of them?  If you can't give an example that clearly shows your meaning, why not try an actual definition?

Chris and I, for example, found an example of the combination of multiple inheritance and expression: To be C, one must be both a child of a parent with attribute A and an attribute B, although one might carry the A or B traits: autosomal dominant and recessive genes, which are Mendelian rather than much more modern concepts. Even so, since one easily can have multiple Mendelian traits, the expression becomes a collection of traits, dominant and recessive, inherited from parents and their relatives.

The simple answer is that quite a few decades of information sciences work have shown that basic hierarchy is not a terribly scalable or flexible means of showing relationships. Databases were once flat fliles, then hierarchies, then relational, and then various aspects of network, object-oriented, and semantic.  You have made the point that some philosophical concepts do not allow for simple, example-based networks.

Take a decent library card catalog. Yes, there is a strict hierachy of basic classification, such as Dewey Decimal, Library of Congress, or National Library of Medicine. In particular, take NLM, because certainly going back to manual Index Medicus that I first used in the sixties, the serious manual searching was keyword-based. The keywords were often, themselves, in hierarchies of more-general and more-specific. MEDLINE  came along and allowed even more powerful keyword searching, with Medical Subject Headings.

That still hasn't kept up, and semantic networks of some kind seem to be a major direction in information retrieval. They are certainly a leading direction in web research.  When I've been using some of these concepts, I am using some simpler ideas from semantic networks, which I use in my own personal work, because I find them efficient.
[quote author=Larry Sanger link=topic=2410.msg18322#msg18322 date=1226812420
It seems obvious to me that the sorts of subs that fire the Trident missile, while neither parent nor child topics of the "Trident" topic, are related topics.  We can say exactly what the relationship is (the sub fires the missile) and why the relationship is important (being able to fire the missile is the main purpose of the sub).  But that is the sort of thing that the "Related topics" section of Related Articles page is for. 
[quote author=Larry Sanger link=topic=2410.msg18322#msg18322 date=1226812420
There's no way to exhaustively enumerate all the ways that topics can be "related" because they are as many as the relationships that exist between things.  The point anyway is that there are important relationships that are not parent-child or other hierarchical relationships. 

We agree on that, and listing such relationships is the purpose of related topics sections.
[/quote]
But not in sufficient granularity, or the ability to show symmetrical and hierarchical relationships. If the Related Articles section had a way to show the relationship, not the term--metadata if you will--then it would be adequate.

The reason for having parent-child versus cotopic is they both show types of common relationships; the reason Chris suggested showing which Related Pages point to other Related Pages do show other kinds of relationships, which don't always have formal natural language names, that are stronger than a generic "related".

you[/i] think the purpose of related topics sections was?

A first attempt to capture what might be keywords, but not really reflecting current concepts in information retrieval. Again, I believe some of the discussion that Chris and I have had suggest that not all searching is necessarily manual. As an intermediate, in the ULMS semantic language, the arcs between concepts have meaningful labels. Propanolol is a "child", from a neurotransmitter standpoint, of beta-adrenergic antagonist. Propanolol  has a back parent link to beta-adrenergic antagonist. It also is has "treats" link to hypertension, but also to migraine and benign triggers, but "triggers" links to bradycardia and asthma.

Topic-subtopic shows inheritance; cotopic shows the common relation of mutual dependence, and related topics has other, less well defined relationships that some of us hope, either thrugh manual coding or bot assistance to be usable. Expressing such the set of such relationships for a concept is a key idea in the semantic web.

Useful counterexample: an orphan article does not have at least three other articles meaningfully pointing it, articles in, turn, which are pointed to by other articles until one can get to a "core article". Think classical analysis; the core articles are the axioms, the derivative theorems are the children, and, arguably, cotopics and what we've been calling, for want of a better term, "permanent stubs" are along the lines of lemmas.

Unfortunately, there are many areas of knowledge where the result can be shown, but the mechanism of producing it has to have the wizard behind the screen, or the use of specialized, not always natural human languages.

If I showed you a diagram with the cotopic, subtopic, etc., links among the various sets of missiles and submarines and superclasses of submarines and missiles, a lot of this would be more clear. It's hard to do it in natural language, but happens to work very nicely with graphic metaphors, or densely with formal language. I'm sure there are areas in philosophy in which you can show me useful conclusions, but the way someone got there took long and specialized study.

Library index cards may seem simple, but, as I learned when I was the network architect for the Library of Congress, the process of creating a good one, with all the systematic subject keywords, tends to take advanced training in library science, and formal references like Anglo-American Cataloging Rules. The cataloging in medicine is so complex that no one really expects to be able to write a specific description but rather to use semantic relationships among a controlled vocabulary of index terms. Chris is the expert here, but I'm sure he can give examples of how real-world genetic inheritance and expression are described. I have a big piece of paper where I'm writing out the relationships about several different hierarchies of computer science topics, so I can plan the article titles, which I'll start encoding in Related Pages.

[Howard C. Berkowitz]

However, I do not think this is necessarily a mathematics specific issue.  I should go look up, for instance, how biology distinguishes "gene" from the points of view:  piece of information inherited from parents, piece of a chromosome, and particular sequence of base pairs all serving one function.  Perhaps biology has no problem putting these all on the same page, because an educated person could probably follow a good enough explanation. 

Chris can speak to that.  At this point, from a military discipline standpoint, I really wish I could link to one of my favorite cartoons. Yes, I do read Playboy for the cartoons.

Picture: two many-starred generals holding up model rocket, wearing worried look.

Caption: Let me get this straight. Is this the one we send up the one to get the one that they send up to get the one that we send up to get the one that they sent up...

You'll see some of that in the Electronic Warfare article, which is a subset of what is now called Information Operations, and indeed is a constant evolution of technique, counter-technique, counter-counter-technique, variation on the original technique, etc. I am working out a lot of related articles and links among the variety of different components that go into real-world electronic combat. It just won't work as a hierarchical diagram, but it starts working with networks or semantic maps.

[Hayford Peirce]

My favorite military cartoon is from a long-ago New Yorker.  A large, upper-class matron is snapping to her equally large, very bemedalled, very angry uniformed husband, "Don't shout at me like I was one of your armies!"

PS -- by the wonderful Whitney Darrow, maybe?