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I improved magma. Entry quasigroup is reworked with some new ideas incorporated and part of the entry delegated to new entry, historical notes on quasigroups which also feature (terminological, historical and opinionated issues on) other nonassociative binary algebraic structures. This delegated also part of what Tom Leinster called in another occasion mathematical bitching (in his example used about categories, here about quasigroups), i.e. opinionated attack on some field of mathematics. Some parts of theory of quasigroups and loops are now very hot in connection to new classes of examples and applications. In particular, analytic loops (like Lie groups) appear to have rich tangent structures, Sabinin algebras (sorry, the entry still under construction) and (augmented) Lie racks (=left distributive left quasigroups) appeared as a solution to local Lie integration problem for Leibniz algebras (nonassociative algebras which satisfy the Leibniz identity, just like Lie algebras, but without skew-symmetry, with lots of applications and relation to the Leibniz homology of Lie algebras and to the conjectural noncommutative K-theory envisioned by Jean-Louis Loday).
I encountered that Borceux-Bourn call magma what wikipedia and Lab would call unital magma. I discussed origin of word groupoid at historical notes on quasigroups (which are now a proposed subject of discussion) and created a related name entry Øystein Ore.
Improvements at Moufang loop, new stub Bol loop and new entry identity of Bol-Moufang type.
Some minor edits at identity of Bol-Moufang type, including links to loop and quasigroup.
I did some editing at historical notes on quasigroups. Also, I added a note of disambiguation at the top of groupoid.
Why is the remark beginning “Some consider the concept of quasigroup to be an example of centipede mathematics” at historical notes on quasigroups rather than at quasigroup?
And wouldn’t the page historical notes on quasigroups be better named something like “historical notes on binary operations”, or maybe even merged with magma? If it stays separate, then we should at least link to it from magma and quasigroup and rack and so on.
Zoran created the page, so obviously he’d be the best person to reply. But besides #1 here, I think there are some clues to how Zoran feels about mixing “centipede mathematics” into the quasigroup article in comment #4 of this earlier long thread here. I think he wanted to relegate such opinions to a separate article.
The statement that “some people consider quasigroups to be centipede mathematics” is not itself an opinion but a fact. If one disagrees with such an opinion, then it seems to me the best response it is not to hide the fact that some people have that opinion, but to refute that opinion by examples, connections, and applications, which the “Applications” section at quasigroup already has a good start on. And I don’t see any reason not to include general remarks on historical and alternative terminologies on the main page of a concept; on the contrary, it may be very valuable to a newcomer to be told about variations in terminology.
Centipede mathematics is itself a non-mathematical term which just takes our concentration away from looking toward genuine mathematical content. It is about terminology, history and personality, that is, in my opinion, for the mirror pages on terminology, history and alike. It requires centipede activity to parse the statement “something is a centipede mathematics” as the expression is unknown to the most of the English-speaking mathematical community,
Maybe what Zoran means is that overly vivid mathematics terminology (centiped mathematics, evil mathematics, walking structure, etc.), while potentially fun at the beginnig, may tend to become, much like old jokes, an endless distraction from what it refers to, beginning to feel like a burden in daily work.
Elsewhere one refers more soberly to “reverse mathematics” for what is much the same idea as “centipede mathematics”. Incidentally, it might be beneficial to adopt the terminology “reverse mathematics” in category theory in order to emphasize that the idea of abstracting any context by removing unused axioms as far as possible is by no means an idea genuine to material set theory, but on the contrary at the core of what category theory is all about.
I do not feel that there should be “mirror” pages on terminology/history at all. In my opinion, all such discussion belongs on the main page of a concept (in appropriately labeled sections).
I can understand a dislike for the phrase “centipede mathematics”. If we want to invent a new phrase, I wouldn’t object, as long as we can come up with a good one. (I do think we shouldn’t try to take over “reverse mathematics”; that has a fairly standard meaning referring specifically to weak subsystems of set theory, and moreover even in that situation I don’t feel that it is a very correct term.) But that’s a separate question from what should be discussed where.
I do not feel that there should be “mirror” pages on terminology/history at all.
I may agree that the issue at stake is not a sufficient reason for it, but for many web pages on nLab, eventually it would be good to have such specialized pages as the historical versions of many concepts often have zillion of variants and twists which are definitely burdening the main concept which should be present in clean form whenever possible with only a minor pointer to historical variants. Do you disagree with this as well ?
I should also add that the readability of nLab is great problem for many people. For example, I have almost no success in pointing my own graduate students to nLab as they consider most entries written in too advanced way. So just saying let us include everything of relevance is not going to improve this situation. Especially for pages which will eventually become very long anyway, even without the literature discussion lists. Moreover, the history pages can have a tag “history” (I forgot what was my convention when I used such pointers) which is useful for those who do surveys or study similar issues.
Mike 5 wrote:
Why is the remark beginning “Some consider the concept of quasigroup to be an example of centipede mathematics” at historical notes on quasigroups rather than at quasigroup?
It is simply not true that the quasigroup entry does not have the statement. The current version #27 from December 2014 of quasigroup has this very statement at the end of the idea section. Just the elaboration, including the conference anecdote, is in the specialized entry on the historical and terminological issues on quasigroup-like structures.
It is simply not true that the quasigroup entry does not have the statement.
You’re right; I’m sorry, I shouldn’t have assumed from the fact that it was on one page that it wasn’t also mentioned on the other page. But I actually think that’s even worse; why say the same thing in two places?
The problem of readability is not that there is too much material; it’s that the introduction is not written in a readable and simple way. Whether historical/advanced material is placed on a separate page or in an appropriately labeled subsection lower down on the main page has nothing to do with it.
There is also a bit of tension even in an introduction between writing a textbook for a student versus writing a reference for a researcher. I don’t know the right answer to that; possibly it is to have separate pages for “X” and “introduction to X” based on the intended audience. But I don’t think the history needs its own page, because it’s not written for a different audience. Think of Wikipedia articles; they have (when appropriate) sections about History or Controversy or whatever, not separate pages for them.
Good section structure and a meaningful table of contents helps a lot when the information is all on one page. Urs’ insistence on doing this all along even when the page starts out small has been IMHO the correct viewpoint.
A side comment:
But I actually think that’s even worse; why say the same thing in two places?
Generally speaking, I think it’s okay to repeat some things across the nLab. (Of course, this occurs frequently anyway.)
Okay, yes, it’s okay to repeat if there’s a good reason for it. But here I don’t think there is.
I think it is a good reason to say an intro sentence in one entry with pointer to details in a specialized entry. This is the way wikipedia does. I see no reason to write anecdotes and alike in a central entry on an elaborate branch of mathematics. But I think it is good to have it in some side track.
I have hard time going through long pages with rarefied entries, lots of subtitles etc. while I have more photographic memory (research shows that majority of people memorizes visually!) when studying material in compact denser entries. Beside I often experience internet problems, as much of my work in nLab I do from very weak connections and on devices like Kindle where mere scrolling of a complicated page often kills the browser and I need about 2-3 minutes to get back to the browser and its required page. I find discouraging to print such pages and so on. I know some of you live in front of huge monitors and internet highways and do not share such concerns.
The problem of readability is not that there is too much material; it’s that the introduction is not written in a readable and simple way.
Different people need different introductions and what is simple depends on a reader. Urs obviously likes to have condensed mantras of hi-level definitions to remind him of things which he already learned. I sometimes share this point but depends on subject. Some people understand traditionally written material and so on. I can not understand people who like to learn basic algebraic topology from Hatcher as I feel his book is baroque with lots of inessential side tracks and notation. But some people like that way, going through forest of routine detail guided by a book which does all that detail explicitly. Some people have trouble finding in a long page which of the many approaches involved will eventually be digestable to them (for a newcomer deciding which chapter will be readable is a very nontrivial task, and the point is usually made only after the fact of multiple attempts!).
I should also add that, at least in my case, the long pages are a turn off for adding new material. I sometimes have a thing which I would like to add and when I open a huge nLab page on the subject I usually give up, while proceeding when the page is moderate. This has many reasons, including time needed to find out/decide where to integrate the new material, longer response of the system when editing long pages till the aesthetic feeling that the addition will be cumbersome or inessential and even fear that the material will be burried lost in material from the point of my own future use. All of these are weak reasons, but somehow the hugeness makes all these add up to turn off.
Probably the problem when my students do not want to read nLab is also partly in that, they have hard time finding the part which is readable to them: I usually tell them that such and such material is in nLab about such and such pages, they get there spend 10 minutes reading statistically incomprehensible part and then giving up and saying me not to recommend them nLab in future and ask instead for some other reference. I got used to such a response from most of the students.
But I don’t think the history needs its own page, because it’s not written for a different audience. Think of Wikipedia articles; they have (when appropriate) sections about History or Controversy or whatever, not separate pages for them.
In my understanding it is written for a different audience. I side here with Urs (and I explained that partly in the thread above) that the modern point of view should be cleaned from the various peculairities of historical forms of definitions. So it is a rare reader who is more concerned with relation to various historical versions of the concept (and other side tracks like opinions about the field from the past, schools of thought etc.). Wikipedia is here quite different: it is very bad in presenting material with the research passion and accuracy of motivation and purpose. On the other hand it is very accurate and often up to date in side track issues like bigraphies, numbers of trivia. Like there is a meeting of G8 and they will quote how many fireworkers, police etc was involved, how much money spent for such and such security and all such issues, but what is a deeper political point of such a concrete political meeting wil be missed. So wikipedia puts as central those things it is good at: documented trivia. nLab is good in modern and research vivid point of view rather than in accuracy and detail of historical account of all possible recorded stuff on the subject and we are here to read mainly such material. If the stuff gets cluttered with issues on personal biographies of Lobachevsky or disection of Gauss’ letters when discussing say non-euclidean geometry it will cease to be nLab, IMHO.
This is an interesting (and also sobering) discussion.
It seems our different styles point to different needs. Zoran brings up Hatcher and the forests of routine detail in his book. I think I also tend to include a lot of details of proofs, except that in my case it’s because I am old enough to realize that I might not be able to read myself twenty years later if I just jot down a bare skeleton. In other words, I write as if for a future senescent self.
But in view of the problems encountered by Zoran and his students, which are partly technological in nature, it might be well for each of us to think more about how to combat “page bloat”. (I also have some psychological problems dealing with mammoth pages.)
One idea that has been floated in the past, for example here, is the idea of “zoomable proofs”, being structured in a hierarchy where if more detail is wanted, you click on a link to a page where detail is provided. Somewhat in the spirit of Lamport’s Structured Proofs, I guess. Of course, the zoomability idea applies to more than just proofs; it could be whole research programs. Etc.
Probably the problem when my students do not want to read nLab is also partly in that, they have hard time finding the part which is readable to them: I usually tell them that such and such material is in nLab about such and such pages, they get there spend 10 minutes reading statistically incomprehensible part and then giving up and saying me not to recommend them nLab in future and ask instead for some other reference. I got used to such a response from most of the students.
Would it be too unseemly to give examples?
I have sympathy with bad Internet connections, and with computers that have small screens (my laptop is 12”) — indeed, the latter is part of the reason I get annoyed by wasted vertical space, like adding in lots of extra blank lines in the source code — but less sympathy with Kindles, iPads, and smartphones. I’m generally quite unhappy with the dumbing down of the Internet to accomodate unreasonably tiny devices with buggy browsers.
Personally, it annoys me to have to do a lot of clicking on links to get to what I’m looking for. I want a big picture and to know where I am in that big picture, and the table of contents of a page can more or less do that, whereas we don’t have as effective a way to do that when splitting into sub-pages. I very much like the overall organizational philosophy of the nLab that each mathematical notion has a page named after itself which contains all the information about that notion. Among other things, it means I usually know exactly what page to go to to find what I want to know, and I can often get there by directly typing in a URL because I know the page name is just the name of the mathematical thing.
I should also add that, at least in my case, the long pages are a turn off for adding new material. I sometimes have a thing which I would like to add and when I open a huge nLab page on the subject I usually give up, while proceeding when the page is moderate.
I have that problem too. But I have trouble imagining that it would be any better if the huge page were split up into many sub-pages, because I would then have to spend the extra time to figure out which sub-page my material should go on!
I side here with Urs … that the modern point of view should be cleaned from the various peculiarities of historical forms of definitions.
I agree. The modern point of view should be presented at the top of the page. The historical peculiarities — if we think they are important, which is not always the case — go in a separate dedicated section at the bottom of the page. When I said “a different audience”, I meant “an audience with a different background”, not just “a different set of people”. Of course not everyone will be interested in everything that appears on a page; some readers of localization will care that it is a coinverter, while others will not. But that doesn’t justify writing a separate page called “localization is a coinverter”.
Nobody’s talking about documenting trivia or writing biographies. The particular (brief) discussion in question is about the history and sociology of the mathematics, which can often be useful for mathematicians to know in the course of research.
(Mike 21) Nobody’s talking about documenting trivia or writing biographies. The particular (brief) discussion in question is about the history and sociology of the mathematics, which can often be useful for mathematicians to know in the course of research.
Why not ? The page on history of quasigroup was, when I created it just an abbreaviation of its true content roughly envisioned at that moment as notes on history of quasigroups, loops, racks and related structures and subject to change with content. This history is very interesting and quite different from “centipede” theory on that field. Such an integrated history is not belonging to the quasigroup or loop or rack or “centipede mathematics” entry. Unfortunately, I had written just about two screens not coming to the real start, just the stub version of what was planned (but will do MUCH more in future, as I am concerned with the role of those in classical geometry, as sprniging from the school of Blaschke (at the moment not yet mentioned on the page). Having the page “quasigroup” or any of the mentioned as the page I would feel discouraged (as your opinion classifying me as NOBODY clearly confirms) to continue and endlessly expand having in mind balance on the page.
(Mike 21) I’m generally quite unhappy with the dumbing down of the Internet to accomodate unreasonably tiny devices with buggy browsers.
In a way i agree: I hate that the multiwindow OSes of previous generation have been replaced, even on huge desktops with OSes resembling functions of the smartphone so that only one window is manifestly active on the screen at the time, so that one needs steps and steps going back and forth to read from one application and paste into another and similar cooperative tasks.
(Todd 20) Would it be too unseemly to give examples?
For example groupoid and algebroid entries talk and refer a lot about what he nLab calls (as I just recently realized) the algebroid of a groupoid, commonly called tangent algebroid of a groupoid. The entry tangent algebroid refers to the algebroid contructed from a tangent bundle of any manifold. There is lots of information on generalized Lie theory in the nLab, but it is very hard to find what is the tangent construction to a groupoid (it is in nLab though not in much detail).
Similarly, open the big page covariant derivative. It even has a subsection “definition” but the classical, simple definition from the textbooks of differential geometry what are the axioms and where this operator acts is impossible to find on that page and takes some ingenuity to find it on nLab, though I am sure it is there multiple times. I should point out that connections are one of the most well covered topics of nLab and that I think we should be proud (and Urs who created much of it especially) for reliable and comprehensive and often innovative material on that topic.
I know these two examples are not of the kind I described above (the hidden info is on the SAME page) but these are the last two cases I remember. Next time I have a case I will post.
Of course, it is easy to get fooled here. I directed one student to find the universal property of the ind-completion in the entry on it as I remember discussing it with nLab people and “seeing it” there. The ind-completion entry says something indirectly. First of all, it redirects to ind-object which has two definitions in 1-categorical case which are both existence theorems from the point of view of the universal definition. Now the first has the “proof” which derives the rule for the diagram realization of ind-objects using the properties of the functor C to Ind(C). Not quite the ones which are in the universal property, but partly there. The entry also ppoints that it is just a version fo the free completion where its universal property is stated and for an experienced nLaber it is enough, as it is easy to modify to the Ind-case. But too much to give to newcomers and students. Explicily saying what is in 5.3.6 of Lurie’s HTT is much better (I will do it next time). But you see, when we know that it is essentially there we have an expresssion that this will be seen by an outsider and we can fool ourself that we stated it in an explicit form, what we did not. But this is outside of the scope of this thread. (But I hope a useful remark for internal usage).
@Zoran: I’m sorry, I did not know that you intended to write more about history! I have no objection to having more extensive history on the nLab, and I agree that a more extensive history would deserve its own page. My comments have been based only on what is actually written right now on the history page, which I don’t think is on its own sufficient to merit a separate page; and my comment “nobody is talking about” was a sloppy way of saying “I am not, and I don’t know of anyone else who is”. Now that I know that you are talking about that (and you are, of course, somebody!), I retract my opposition to a separate history page. (But if we adopt some convention for sub-pages such as proposed in the other thread, then it ought to be renamed something like “quasigroup/History” and linked from the main page in the standard way.)
No problem (except the usual time problem with the realization of started projects).
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