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• CommentRowNumber1.
• CommentAuthorTom Leinster
• CommentTimeNov 28th 2011
• (edited Nov 29th 2011)

Joel Hamkins and Andy Putman made some comments about the nLab on MathOverflow, beneath an answer by Andrew.

It’s interesting to know what people’s perceptions are, even if they’re wrong. (And I would think that Andy P’s perception is wrong.) I don’t know what Andrew S has in mind when he says that Joel’s point is extremely easy to answer.

• CommentRowNumber2.
• CommentAuthorAndrew Stacey
• CommentTimeNov 28th 2011

(Technical note: I know that this really does belong in the “General Discussions” category but I’ve moved it to “Latest Changes” purely to allow guest posts in the hope that Joel joins in.)

• CommentRowNumber3.
• CommentAuthorAndrew Stacey
• CommentTimeNov 28th 2011
• (edited Nov 29th 2011)

Here’s the comment thread from MO (with comments that were too long merged into a single paragraph):

But does the nLab really welcome contributions that might not conform with the nLab point of view? In light of nlab.mathforge.org/nlab/show/nPOV, I would think not. – Joel David Hamkins

Yeah, I’ve always wondered why the nLab people are so aggressive in trying to get people whose work has nothing to do with category theory to post there. Shouldn’t their suggestion be “start your own subject-specific wiki”? - Andy Putman

If I actually thought that answering either Joel’s or Andy’s comments would make any difference, then I would as those are extremely easy points to answer. But instead I think I’ll get on with writing a page on the nLab that lies on the interface between differential topology and functional analysis and has absolutely no categorical content whatsoever. – Andrew Stacey

Andrew, I am sorry if my comment has irked you; this was not at all my intention. Rather, it was a serious question. Namely, the nLab seems perfect for the use you mention in your answer, provided that the topic of the post is in harmony with the nPOV. And to be sure, this includes a lot of excellent mathematics, which I have enjoyed at the nLab. But meanwhile, there is also a lot of work arising from a perspective at odds with the nLab view, such as most research in set theory, and my sense is that such posts are not really wanted at the nLab. Isn’t this correct? – Joel David Hamkins

Apology accepted, and insert standard comment about comments at MO being lousy places for discussions! What defines the nLab as far as I’m concerned is that it is for working mathematicians. If you wanted to make some notes there on a topic in set theory then you’d be most welcome to do so. There is a danger that someone might come along and add something a bit nPov-y to it (note the “add”, no-one would take away) but to avoid that all you’d have to do is register on the nForum and say, “I’m writing a page on X and I’d like to keep it as ’pure set theory’.” then the most that would happen is that someone would maybe rename it to make it clear that it was from a particular perspective, and probably someone else would write a different page with the nPOV version and link the two. If that’s not distinct enough, then we actually have several distinct “wikis” at the nlab which can be interlinked but have different flavours (most are “personal”). You could ask to start one - I think you’d be welcome, though I only speak for myself - and then you’d get the benefit of cross-linking when you wanted it without fear of your stuff being overwritten. – Andrew Stacey

@Andrew Stacey : I also was not trying to offend. Like Joel, my own research (on geometric topology and geometric group theory) seems at odds with the nPOV. This is not meant as a criticism of the nPOV, which I understand as a statement of shared interest rather than a statement that “all math that doesn’t fit into this box is bad”. Is the statement of the nPOV not reflective of the current intentions of the nLAB (which even has n-categories in its name)? I’ve just always been puzzled by the insistence by nLAB people that people whose research has no connection to nLAB type stuff would want to post there instead of some more appropriate venue. Maybe you should encourage the creation of other wiki type places? – Andy Putman

(the above comments were being composed when Andrew Stacey posted his reply to Joel) – Andy Putman

• CommentRowNumber4.
• CommentAuthorMike Shulman
• CommentTimeNov 29th 2011

I basically agree with Andrew S. I think some confusion may lie in what it means for something to be “at odds with” the nPOV. I would not describe any mathematics as being “at odds with” the nPOV, at least the way I understand the nPOV. The nPOV is that category theory and higher category theory are “true and useful”, as the page nPOV says, and that when a subject can be viewed through those lenses, it is often useful to do so. How can any other part of mathematics be at odds with this? The most that could be said about some other part of mathematics is that it, itself, does not (yet) have any relationship with, or use for, category theory – but this can say nothing about the usefulness of category theory elsewhere.

The only thing that can be “at odds with” the nPOV is another philosophy, such as “category theory is useless in regards to subject X”. It might be factually the case that category theory has not yet found any (or many) uses in subject X, but this is not at odds with the nPOV, unless one insists that it never will and such uses should not be looked for. That is the only situation in which I would advise someone not to contribute some piece of mathematics to the nLab: if you do not want someone else to ever come along and try to see it from a category-theoretic point of view.

For instance, take Joel’s example of pure set theory. The only reason not to write on the nLab about some argument involving set-theoretic forcing (say) is if you don’t want anyone else to come along and add something to it explaining the connection to sheaf toposes or making the argument constructive.

If others share a similar understanding of the nPOV to this, then perhaps the page nPOV should be clarified.

• CommentRowNumber5.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
Andrew invited me here, and I am happy to participate in this discussion. Thank you for the invitation. (I regret that my comment on MO might have been taken as outright criticism, when it was not intended that way. Also, please note that the unattributed comment after mine in the quoted text of Andrew's answer is due to Andy Putman.)

Let me try to explain my perspective.

One view of the nLab would be that it is a site dedicated to providing the category-theoretic take or n-category-theoretic take on diverse topics in mathematics. Indeed, it seems to be admirably living up this vision, with numerous posts expanding on any number of different topics and particularly how they relate to the category-theoretic or n-category-theoretic perspective. It is succeeding very well with this, and I find it to be a valuable mathematical resource. Indeed, I completely support this enterprise; please keep up the good work!

Another view of the nLab would be that it is a site welcoming of posts on diverse mathematical topics, from any part of mathematics, from any knowledgeable contributor. It is on this view that I take Andrew's answer to the MO question under discussion, and I have seen this perspective expressed also elsewhere. Surely it would be a laudable goal and a valuable mathematical service to provide such a forum for general mathematical use.

But the difficulty for me with this second view is that it seems to be fundamentally in conflict with the nPOV as described at http://ncatlab.org/nlab/show/nPOV. The nPOV appears instead to describe a focused philosophical perspective, rather than a generally welcoming neutral perspective. For example, I think it is fair to say that the vast majority of researchers in set theory would object to the view expressed by the nPOV. And in light of the strongly worded nPOV, one would not expect posts from an opposing viewpoint to be welcomed here. That is, the nPOV seems to express a strong philosophical position, useful in order to direct and motivate a group of like-minded individuals for concerted action, rather than an all-encompassing big-tent perspective. This would not matter much, except that the nPOV expresses a philosophical position that is in many ways in fundamental disagreement with the standard philosophical perspectives for researchers in set theory and much of the rest of mathematical logic. The difficulty of this fact is complicated by the unfortunately loaded language, such as in the nLab concept of Evil, labeling fundamental features of the opposing philosophical position. After all, how can one rationally defend evil? Furthermore, how can it be seen as welcoming to say that one can make posts, if the promise is also made to follow them up with parallel opposing and presumably critical posts from the corrected philosophical perspective?

Thus, my suggestion is that the nLab give up on trying to advertise itself as an all-encompassing forum, welcoming of contributions from all perspectives, and instead focus on doing what it already does, and does so well: providing with precision and expertise the category-theoretic perspective on diverse mathematical topics. If it does this well, then who knows? Perhaps in a few decades, we evil-doers may be overcome with remorse.
• CommentRowNumber6.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
I forgot to indicate in my previous post that it was I who made it as a guest. - Joel David Hamkins
• CommentRowNumber7.
• CommentAuthorTom Leinster
• CommentTimeNov 29th 2011

Joel’s comments prompted me to read the nPOV page properly, which I don’t think I’d done before. In fact, I’d never noticed the extraordinarily strong boxed statement: “nothing in mathematics makes sense except in the light of higher category theory”. There can’t be many mathematicians who agree with that.

• CommentRowNumber8.
• CommentAuthorYemon Choi
• CommentTimeNov 29th 2011
Tom, please don't take this the wrong way, but I am surprised to find that you hadn't seen the boxed nPOV tenet before. It's things like that which used to make me leery of contributing - and I hasten to add that, as JDH has said, the nLab might *want* to take that line, because it would seem to make sense for many of its core users.

(Of course, I don't contribute at the moment because I'm disorganized and snowed under with self-inflicted backlog, so I can't blame any nPOV for that)
• CommentRowNumber9.
• CommentAuthorTodd_Trimble
• CommentTimeNov 29th 2011

Well, then, you can be surprised at me too, Yemon.

In my opinion, the boxed statement is absolutely ridiculous, and should be stricken from the record. I would further say that anyone should feel free to ignore the nPOV at any time, and feel free to contribute however one wishes.

Further, despite the fact that many might see me as a category-theory partisan, there are plenty of things I’ve written on the nLab that are not particularly categorical. For me, the most useful way to think of this site is that it is a highly cross-linked repository of working notes on mathematics.

• CommentRowNumber10.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
Perhaps we can take this to mean that the statement on that page is just the position of the person who added it (Urs), and that other regular contributors haven't bothered to read the page since that statement was added.

S. Carnahan
• CommentRowNumber11.
• CommentAuthorMike Shulman
• CommentTimeNov 29th 2011

Yeah, shame on us for not paying closer attention to that. I’m with Tom and Todd. Somehow I missed that box, because I registered the rest of the page as expressing a fairly reasonable attitude.

• CommentRowNumber12.
• CommentAuthorUrs
• CommentTimeNov 29th 2011

@Mike in #4

If others share a similar understanding of the nPOV to this,

I do.

• CommentRowNumber13.
• CommentAuthorUrs
• CommentTimeNov 29th 2011
• (edited Nov 29th 2011)

I have removed that boxed statement.

(Previous discussion of this was here.)

• CommentRowNumber14.
• CommentAuthorSridharRamesh
• CommentTimeNov 29th 2011
• (edited Nov 29th 2011)

For whatever it’s worth, I also am in agreement with Mike’s post #4.

• CommentRowNumber15.
• CommentAuthorTim_Porter
• CommentTimeNov 29th 2011

As a quite regular contributor to the nLab, let me just say that I use it as a place to put material on things I am trying to get my head around. Often there is no specific question that I need answering but I find that the nLabbers in the Lab and also here in the nForum, are good at adjusting an entry so that other points of view do come through and this is very useful for MY learning process. I am usually forced to work by myself and so this place is invaluable for the interaction with others that is so necessary. Specific questions are suited to MO, so that serves another useful function. I wish there was more coverage of other areas of mathematics from a similar perspective (not meaning the NPOV but rather accessible sets of discussions , definitions and development), either here or in other similar projects.

My own reason for trying to get others to contribute to the entries is that they are likely to have knowledge that I do not have and which could be useful to the topics that are already here (and are of interest to me). They will help clarify points that are obscure in existing entries, precisely because they are not the experts in the existing n-fields.

Joel mentioned geometric topology and there, for me, I would find it great to have a list of current items of interest in that area. Joel and Andy did mention encouraging people to contribute to set up their own similar enterprises. Please do, but if you find the nLab useful perhaps you could put an entry telling us about the other projects and creating some links in entries here to entries in the others (and possibly vice verse if it helps).

• CommentRowNumber16.
• CommentAuthorAndrew Stacey
• CommentTimeNov 29th 2011

Joel (#5):

Also, please note that the unattributed comment after mine in the quoted text of Andrew’s answer is due to Andy Putman.

Whoops! I edited out the timestamps as I judged them to be irrelevant, but must have edited out Andy’s name as well. I’ve put it back in now (thus meaning that your comment doesn’t look right anymore, but hopefully you can live with that!).

The original comments did irk me, but on reflection I think that I was too easily irked, especially once I’d read the nPOV page again. I also, again on reflection, will agree that I can be a bit “aggressive” in my promotion of the nLab (though I’d prefer “enthusiastic”!). My defence for that is that on MathOverflow, I tend to feel that I have to shout to be heard. The moderators (and community) do a great job in keeping the noise down, but it is still quite a rowdy place (particularly in comparison to here).

I’m glad that others brought up the inconsistency between what the nPOV page says and what people actually do, and I’m glad that it was Urs who changed it. The reason that I haven’t taken so much notice of what it said there is simply that the impression that it appeared to give is not what happens in practice. So as it wasn’t getting in the way of what I wanted to do, I didn’t particularly think about it.

My own view of the nLab is that it is more about the people than the subject. As an active contributor, then I would put all my online notes here, whether they are categorical or not, because it seems simply daft to have to think each time “This is about manifolds, so it goes on the ’Topology Lab’, this is about algebra so it goes on the ’Algebra Lab.’”. We divide mathematics from time to time to make it easier to think about in small chunks, but the boundaries are so vague that the division doesn’t make sense at the wiki-level. Of course, there can be other reasons for having separate wikis, but subject matter doesn’t make sense to me.

I joined the nLab shortly after it began not because I thought that n-categorical thinking was the way to go (I didn’t), but because I thought that the idea of a collaborative research wiki was the way to go. We’ve seen many initiatives that are concerned with making better use of the internet in research mathematics, and of all of them then it’s my view that the nLab has the most potential. Exactly how I think of the nLab is probably best explained by the page explaining how to think of the nForum on nlabmeta: about the nForum (nlabmeta).

My enthusiasm for the nLab is based on two things. Firstly, a selfish reason. The more people that contribute to the nLab, the more chance there is for me to learn the things that I need to learn. Secondly, an unselfish reason. I’ve found being a member of the nLab community to be extremely valuable in my work (at many, many levels) and I’d like to encourage others to try it out.

The reason that I don’t suggest others create their own wikis is fairly practical. It takes a lot of time and energy to set one up, both in terms of keeping it going with contributions and in terms of the technical details. Why go through all that angst when you can join ours? There are plenty of small wikis that people maintain for their own notes, but none (that I know of) has the same momentum as the nLab. Joining ours wouldn’t be an irreversible step either, you could see how it went and if you felt you had enough momentum to start your own, then do so. If not, all that you have done is still of value and is still available to others. As a half-way house, consider applying for a “personal web”. These are linkable to and from the nLab, but are separate and can have their own flavour.

I don’t feel qualified to answer Joel’s more mathematical points. But I think I can say something on his last paragraph. I completely agree with the phrase:

[the nlab should] focus on doing what it already does, and does so well

but the preceding and following phrases are not right. As I hope I’ve explained, for me what the nLab does and does so well is to be there for the use of its users. That may be a bit circular! But it works in practice. The nLab does not have a “higher purpose”, it has a low-down, mundane, boring purpose: to be useful. That is what it does, and what it does so well.

• CommentRowNumber17.
• CommentAuthorDavid_Corfield
• CommentTimeNov 29th 2011

Along with the removal of

nothing in mathematics makes sense except in the light of higher category theory,

from some of the comments it sounds as though some of you want people to take this very lightly:

In particular, the nLab has a particular point of view, which we may call the nPOV or the n-categorical point of view.

Personally I wouldn’t want this to be taken too lightly. There should be at least a hope that some nPOV sense can be made of the material, and a willingness to see if that can happen. Ideal cases would have someone write on, say, Kantorovich duality, and then the nPOV emerge showing it to be describable in terms of enriched profunctors. (Hmm, I wonder how much further that example could have been taken.) Of course, if this doesn’t happen, we may learn something too, but I’d want there to be the willingness to try.

• CommentRowNumber18.
• CommentAuthorTim_Porter
• CommentTimeNov 29th 2011

@Andrew: !00% agree.

• CommentRowNumber19.
• CommentAuthorUrs
• CommentTimeNov 29th 2011
• (edited Nov 29th 2011)

I’m glad that others brought up the inconsistency between what the nPOV page says and what people actually do, and I’m glad that it was Urs who changed it.

It is clear now that most of you read the sentence

Nothing in mathematics makes sense except in the light of category theory.

entirely differently than I do. To me, this sentence describes precisely what people actually do on the $n$Lab.

Notice that this is a reference to the classical statement

Nothing in biology makes sense except in the light of evolution.

What does this sentence mean? Of course it does not mean that only evolution theorists are proper biologists, nor that it does not make sense to study, say entymology or molecular biology. What it says is that only in the light of evolution do all the many facts and theories unify to a coherent global story where all pieces find their proper place with respect to each other.

To my mind this is exactly what category theory is for mathematics and what the $n$POV is all about.

But I am glad that you (all) pointed out that you disagree with the statement.

• CommentRowNumber20.
• CommentAuthorAndrew Stacey
• CommentTimeNov 29th 2011

David, I think that - in light of the comments of “outsiders” - then what I would change it to is that it is not the nLab that has that point of view, but that there are those amongst the main contributors have that point of view. So when making a contribution to the nLab one should be aware of the fact that a “hardline nPOVer” might come along and add to the page considerable detail on how to reinterpret everything in light of the nPOV. In my experience, when this sort of thing happens then it is always an addition and not a rewriting and I’d hope that that would continue.

As an analogy, Toby is well-known (around here) for his interest in constructive mathematics. From time to time, I’ll find that something I’ve posted to the nLab has gained an addition commenting on whether or not it works constructively. For myself, I simply note it as an “interesting fact” but don’t really think much more about it. It is certainly never intrusive, and I can see that for others it could be a useful thing, so I’m pleased that he does it. Now, the nPOV stuff is a little more substantial than a “This does/doesn’t work constructively” statement, but the wiki-format makes it easy to organise material so that it is similarly an addition and one that can be safely ignored if one, for whatever reason, wishes to ignore the nPOV for the time being.

But then that’s the main point of putting stuff on a wiki: that others can comment, extend, and refine what you write. If I really want to claim something as, to quote Gollum, “Mine, all mine” then I’ll write it on a computer that is disconnected from the internet, with an encrypted drive, and publish it in an obscure journal behind a massive paywall. But that’s not what I want. I want my ideas out there, easily accessible, to see what others make of them.

So if someone doesn’t want their contributions examined from the nPOV then they shouldn’t contribute here. However, their contributions elsewhere will still end up being examined from the nPOV because there are people interested in the nPOV who do their work here! Being a direct contributor only means that the person is involved in the process - the mathematics will always be so!

• CommentRowNumber21.
• CommentAuthorAndrew Stacey
• CommentTimeNov 29th 2011

Urs, I think that what is missing from the old box is the equivalent of:

Of course it does not mean that only evolution theorists are proper biologists, nor that it does not make sense to study, say entymology or molecular biology.

We have to remember that people trying to work out what the nLab is from the outside don’t see the same picture as we do from the inside. In the absence of the mathematical equivalent of the above, then the old box reads:

Nothing in mathematics makes sense until it has been reinterpreted in the light of higher category theory.

and it carries with it the implication that any contribution to the nLab will be rabidly reworked until it starts with something like “The theory of Banach spaces is nothing more than the working out of the theory of $(\infty,3)$-snarks for the specific case of a boojum.”. Whether or not you secretly dream of that being true, it simply is not the case in practice and from the comments that started this discussion, it would seem that the fear of that happening is putting off some people from contributing.

As an example, take the recent discussion about Banach spaces. The original page was written from a categorical point of view and bore many hallmarks of that. Some functional analysts came along and said, “This isn’t how things are thought of in functional analysis” and we pretty much said, “That’s really interesting. We’d like to know more, please edit the page to make it correct.”. Now, it has both points of view sitting quite happily side by side.

So the nLab is a place where one will not be able to avoid the nPOV, and hopefully many will be brought round to the point of view that it does provide a coherent explanation of many strange things. But at the same time, no-one will be thrown out for not being a fully paid-up member of the nPOV philosophy (I hope! Otherwise, I’m in trouble.).

• CommentRowNumber22.
• CommentAuthorUrs
• CommentTimeNov 29th 2011
• (edited Nov 29th 2011)

Andrew writes:

I think that what is missing from the old box is the equivalent of:

To my mind (but not to your minds, I understand), it was all there in the analogy. It started out saying in that box

Similar to the statement that “Nothing in biology makes sense except in the light of evolution” […]

and for those who don’t know what this statement means, it was hyperlinked to the page which explains it:

The fact that evolution occurs explains the interrelatedness of the various facts of biology, and so makes biology make sense.

I thought this makes it all clear. The statement implied is hence that

The fact that category theory exists explains the interrelatedness of the various facts of math, and so makes math make sense.

Do you guys find this “ridiculous”? This is what the $n$POV is all about for me and this is what I thought was the message of that box. But I understand that the message miserably failed, so I am glad I removed the box now.

I wish somebody had removed it back then when I asked to do so if it sounds too strong.

• CommentRowNumber23.
• CommentAuthorAndrew Stacey
• CommentTimeNov 29th 2011

I wish somebody had removed it back then when I asked to do so if it sounds too strong.

It (probably) didn’t back then when we discussed it last. But (if so) then it would have been because those discussing it were on the inside. Now we’ve been asked to look at it as if from the outside and are seeing that it could be interpreted in another way.

The bit that does irk me, and I think a bit justifiably, is the expectation that things here are in a “finished state”. Of course, that’s not true of stuff on the wiki itself (we even say so!) but it’s also true of the whole set-up that we have. It’s continually changing as we figure out what does and doesn’t fit (look at the nPublications - that’s a new thing). Even when we have the main direction figured out (if we ever do) there will always be smaller bits to work out as new ideas, new ways of doing things, new people turn up.

• CommentRowNumber24.
• CommentAuthorTodd_Trimble
• CommentTimeNov 29th 2011

@Urs: I’m sorry I wasn’t more alert when you asked before if the boxed statement sounded alright.

And maybe I shouldn’t have said “ridiculous”. But if I could indulge in some parody: nothing in the boxed statement makes sense except in the light of further explanations. In other words, read on its own, “nothing in mathematics makes sense except in the light of higher category theory” is sheer dogma, and is practically asking to be misunderstood. Older generations accused category theorists of the 60’s and 70’s of that kind of excessive dogma – maybe due to overly strong-sounding pronouncements like that. And it certainly didn’t do category theorists any good.

I don’t think the link quite solves the problem. I can imagine a lot of people just reading the statement and turning away in disgust. I can also imagine people freely quoting the sentence (without the link), as an example of what sorts of fanatics these n-categorical types are.

I could get behind a more tempered expression of the idea being placed in another box, however.

@David: with regard to “from some of the comments it sounds as though some of you want people to take this very lightly:

In particular, the nLab has a particular point of view, which we may call the nPOV or the n-categorical point of view.

Personally I wouldn’t want this to be taken too lightly.” –

Actually, my position is simply for those “outside, looking in” not to agonize over whether what they write fits in with the nPOV – they should feel free to contribute material however they see fit. We nPOV types may then feel free to consider such material in a categorical or n-categorical light, whenever it makes sense to do so. I hasten to add that those people who don’t want any categorical glosses to eventually appear on a page they started probably shouldn’t post here. In other words, those who submit material should be aware that there is this thing called the nPOV (like a kind of “Holy Ghost”) which might assert itself anywhere in the nLab – but not to the point where it would erase or override sound contributions from those who don’t subscribe to it.

I hope this clarifies my position.

• CommentRowNumber25.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
Interesting discussion (btw this 'Guest' is quid some might know from MO):

IMO if one would real want to have an 'open' wiki for all kinds of mathematicians, I think one would simply have to add the o to the nPOV.

It's fine if some people are enthusiastic about CT, but it might also not be unknown to them that some others (actually I personally not so much) have a serious dislike of this entire approach to math and some things I consider as related.

I mean there are plenty of (classical) conflicts/discussion vaguely around this: say Siegel--Lang, also later parts of the Grothendieck--Serre correspondence are interesting related to this, or some analytic number theorists are highly sceptical much will come out of these abstract approaches to RH, mentioned in some linked discussion to point out the usefulness of CT to ANT, and consider them as let's say a bit misguided, and so on.

Due to my personal math development I live somehow between the worlds of say abstract and concrete, and (so and in general) do not have strong personal convictions related to this. But if there is a site where some core members have a strong POV that is controversial (perhaps they are 100% right but still it is controversial) and this is somewhat visible, then I think it is to be expected that others are hesitant to contribute. Even if in daily practise it is a complete non-issue, they might not want to, as in some sense their work would promote this POV.
• CommentRowNumber26.
• CommentAuthorMike Shulman
• CommentTimeNov 29th 2011

The fact that category theory exists explains the interrelatedness of the various facts of math, and so makes math make sense.

I don’t agree with that either. I think the fact that category theory exists helps to explain some of the interrelatedness of some of the various facts of math, and so helps to make math make more sense than it would otherwise.

• CommentRowNumber27.
• CommentAuthorTodd_Trimble
• CommentTimeNov 29th 2011
• (edited Nov 29th 2011)

Even if in daily practise it is a complete non-issue, they might not want to, as in some sense their work would promote this POV.

I think that would be an unfortunate misunderstanding, but Andrew Stacey would be better placed to respond to that. Actually, I think he has already responded to that point, many times in fact.

If one looks around a bit, one finds lots of stuff that isn’t exactly suffused with category theory. But being aware of this may take more than a cursory acquaintance with the nLab.

Edit: I’ll repeat the advice that those who are actively hostile to category theory (cf. the first part of quid’s comment) should definitely give pause before contributing to this site, for reasons I gave in my previous comment (#24).

• CommentRowNumber28.
• CommentAuthorMike Shulman
• CommentTimeNov 29th 2011

@Guest 25 and Todd 27: I think it would be unfortunate, but I’m not sure it would be a misunderstanding. If someone actively thinks that category theory is bad (as opposed to not having any use for it themselves personally), then I think it makes perfect sense (within the framework of that mistaken view) for them to refrain from contributing to the nLab in an attempt not to encourage the use of category theory. Of course we should work to convince people that category theory is not bad, but I don’t think we can (or, really, should) do anything about the fact that if they do think it’s bad, they’re not going to want to participate here.

I also feel strongly, along the lines of Todd 24:

Older generations accused category theorists of the 60’s and 70’s of that kind of excessive dogma – maybe due to overly strong-sounding pronouncements like that. And it certainly didn’t do category theorists any good.

that overly grandiose claims about category theory don’t contribute to the goal of making people not think category theory is bad. Even if one holds such grandiose beliefs in one’s heart, it’s better to downplay them in public and emphasize only the many real, verified, and undisputable uses of category theory in mathematics.

• CommentRowNumber29.
• CommentAuthorMike Shulman
• CommentTimeNov 29th 2011

carries with it the implication that any contribution to the nLab will be rabidly reworked until it starts with something like “The theory of Banach spaces is nothing more than the working out of the theory of (∞,3)-snarks for the specific case of a boojum.”

I know for a fact that the impression that this sort of thing happens at the nLab even puts some people off of reading it. It is true that there are some pages that do start in such a way, though not because they have been reworked but just because they were written by someone who naturally thinks in that way and the Banach-spaces thing hasn’t happened to them yet. But I think it would be to the advantage of the nLab to put some effort into reworking such pages to be more accessible at the beginning to non-initiates, and perhaps even in future strive not to write only in that way to begin with.

• CommentRowNumber30.
• CommentAuthorTodd_Trimble
• CommentTimeNov 29th 2011
• (edited Nov 29th 2011)

I agree with that amendment, Mike (comment 28). I’m not worried about those people – I’m addressing more those open-minded potential contributors who might still somehow be dissuaded from contributing, due to some worry that they will be associated with promoting an nPOV (with which they don’t agree to the extent that some regular contributors do). My instinct would be to regard such worries as largely spurious.

For example, I’d love to see contributions from JDH, who I view as someone certainly not hostile to CT, but who presents himself as one who doesn’t embrace an nPOV.

• CommentRowNumber31.
• CommentAuthorAndrew Stacey
• CommentTimeNov 29th 2011

I’d like to respond to one thing that quid says:

IMO if one would real want to have an ’open’ wiki for all kinds of mathematicians, I think one would simply have to add the o to the nPOV.

That’s almost precisely what we don’t have. We have an open wiki for us. I don’t think anyone ever set out to create something that would benefit everyone - though we hope that people do find it useful. This is what I meant above about the primary purpose of the nLab being to be useful to those who use it.

This also has implication for the comments along the lines of what articles should and shouldn’t look like. Yes, it’s good if they start in an easy way and with “classical” stuff, but more important is that they exist and saying what they ought to look like is a sure way to stop them getting written because people don’t have the time to make them “look right”.

That’s not to say that we shouldn’t polish articles and reorganise them; of course we should. It’s a balance, and that’s probably what doesn’t come across so well to “outsiders”.

• CommentRowNumber32.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
quid again:

It is not my intention to start some heated discussion here, but somehow I also think it is perhaps useful to say some things directly.
(I explictly do not want to suggest that JDH or Andy P might think this.)

While it is perhaps unfashionalble to say so, I would say it is simply a fact that there is a certain competition between subfields of mathematics. And, which subfields are visible and (thus) considered important, can have immediate consequence. Why should somebody who (while not being hostile to it in anyway or thinking it is bad) does not have promoting CT among their priorities contribute to a wiki whose (it seems) to some extent explict purpose this is. (And even if the content is not CT it increases the visibility of nLab if somebody contributes here and than say links to this article in a paper. As opposed to a situation where there would be an analog resource in that persons subfield.)

The only reason I can see is that this resource exists (already) and the utility of contributing should take precedence over above mentioned considerations. And, a fragmentation would seem pointless. However, for example it is somewhat less clear why there could not be a neutral front-end and below that live nLab, TopLab, SetLab, AlgLab, NumbLab,... (easily interlinkable and not disjoint).
If one wishes to have a resource for all mathematicians (and physics and philosophy). Now, I understand the original idea and delopment was something else, and this is fine, and it is nice it exists. But if this resource is also promoted as some general purpose resource I think such question will come up at least implictly.

To give an analogy: if MathOverflow were called Vari(ety|ous)Overflow because Anton is an algebraic geometer and here and there it was written that the philosophy of the site is to provide a place for asking question related to Algebraic Geometry but since AG is a vast field and linked in one form or another anyway to essential everything in mathematics one should not worry too much over this but ask just anything wich is research level mathematics because likely it is AG anyway after suitable reinterpretation, but just be aware that there are lot of AG people on the site and might answer from their POV. Then this might also be an issue.

As said it is not my intention to start some heated discussion and I also do not want to tell the nLab community how to run this site, to which I have nothing contributed at all and likely won't at least in the forseeable future (for the same reason as Yemon) and do not even fully understand. I merely wanted to make something explict which I believe implictly is a (potential) issue, whether I say it or not.

@Andrew: the above was written before your comment. But who is 'us'? I assume those that use the site. But who should or might want to? Every mathematician or just those with a certain thematic focus? Also the later is fine. But then I do not understand why it is an issue if somebody says: this site is not for me as my thematic focus is different.
• CommentRowNumber33.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
I should like to say that my view is that the nLab should not significantly modify its nPOV in an attempt to be more inclusive. The value of the nLab is precisely that it is developing an account of mathematics under the perspective expressed by the nPOV, including the now-removed statement on evolution (which I understood anyway to be completely consonant with the views expressed in the rest of the nPOV, and which could be reinserted without harm, perhaps with some minor additional explanation as suggested above). The nPOV expresses a coherent, robust and productive philosophical perspective, which deserves to be fully worked out in the way that the nLab largely is working it out. To make significant changes in the nPOV in the hope of gaining a broad appeal would seem mainly to dilute the perspective, without really satisfying those who understand their work from a fundamentally different perspective. Rather, what the world needs is for the nLab community to be true to its nPOV.

But I take this also to mean that the nLab should not try to promote itself as a general purpose neutral math posting web site. You have a philosophy, and the work that appears here should be expected to fall under that rubric. To be sure, the philosophy is sweeping, in that it aspires to apply nearly universally in mathematics, and so the nLab community should aspire to have posts under its rubric from nearly all parts of mathematics. But the point of my original comment which led to this discussion is that this is not the same thing as welcoming mathematical contributions from all mathematicians, since at least some of them may find the nPOV to be alien or unhelpful.

As for my part, although I count as an outsider here, I have deep sympathies for CT, and indeed I am currently writing a paper on a cross-boundary topic between CT and set theory (regarding a question Mike Shulman asked long ago on MO), a boundary that in my opinion would benefit from greater interaction. In truth, 90% of the difficulty is merely translating concepts between the two realms.

- Joel David Hamkins
• CommentRowNumber34.
• CommentAuthorTodd_Trimble
• CommentTimeNov 29th 2011

@quid: abstractly I see the point you’re making. It is probably correct to say that if competition between subfields is a big concern to a mathematician, that person will probably not fit in well here. Not just because of a perception that an aim of this site is to promote category theory, but more interestingly because the very spirit that has developed here is extremely supportive, cooperative, and non-competitive. Or at least, so it seems to me.

For example, the author hilbertthm90 writes articles on algebraic geometry, and doesn’t write as if he has to fit in with the category theorists somehow. I hope he feels welcomed here; I think he does. He’s just one example. We strongly encourage more!

Another part of me wonders how CT could seriously be viewed as “competition”. As far as I know, it’s not at all well-funded or well-supported in the community. Politically, it isn’t very powerful (I mean, I don’t know of any committed category theorists who could be considered “big players” on the scene – a lot of those who might count are either dead or well-retired). Sure, the language is widely prevalent, but the general attitudes toward it are as skeptical or hostile as they ever were. Category theory gets such a tiny sliver of the pie that it’s a wonder people “worry” about it!

• CommentRowNumber35.
• CommentAuthorDavid_Corfield
• CommentTimeNov 29th 2011

I’m with Joel #33 here. I just would hope that it’s perceived to be a friendly, open-minded (and capable) enough place that people with an interest in seeing whether there might be some useful translation between their work and nPOV ideas would consider trying to do this with the community here.

• CommentRowNumber36.
• CommentAuthorUrs
• CommentTimeNov 29th 2011
• (edited Nov 29th 2011)

@Andrew #31: thanks for saying this. I agree very much.

• CommentRowNumber37.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
I think that Joel expressed my feelings on the subject very well, so I have little to add. However, I'll say a couple of things.

I don't think it is exactly a matter of competition between subjects. I view math as a pretty big tent, and while I personally am not interested in category theory, that doesn't mean that I am opposed to other people doing it, and I don't think that its success "hurts" me or my subareas in any way. I do have a slightly negative reaction to the idea that everyone (no matter what their mathematical interests/tastes) should be heavily encouraged/pressured to participate in the nLab. Upon reflection, this is probably related to my previous sentence -- I don't like the insinuation that there is only "one way" to do/think about mathematics, and I have a negative reaction to the idea that everyone should share one grand vision of the subject. Math is a whole lot bigger than category theory (or geometric topology, or analysis, or whatever!). That's one of the things that attracts me to it.

But in keeping with my "let a thousand flowers bloom" philosophy, I would strong encourage the nLab not to water down its philosophical positions to try to attract people like me! I suspect that one of the reasons that it is so successful is that it has created a community around a set of shared tastes and ideas, and I don't think it would be well-served by trying to put that aside.

best,

Andy Putman
• CommentRowNumber38.
• CommentAuthorTodd_Trimble
• CommentTimeNov 29th 2011

Thanks for your thoughts, Andy. I do have a reaction to

I do have a slightly negative reaction to the idea that everyone (no matter what their mathematical interests/tastes) should be heavily encouraged/pressured to participate in the nLab. Upon reflection, this is probably related to my previous sentence – I don’t like the insinuation that there is only “one way” to do/think about mathematics, and I have a negative reaction to the idea that everyone should share one grand vision of the subject.

as follows: (1) “pressured” – do you really feel pressured? (2) I don’t like that insinuation either, and I don’t feel that way, and I certainly don’t think everyone should share one grand vision!! If this is what the nPOV is taken to mean, then I simply don’t take the nPOV. And I doubt I’m the only one of the regular contributors who feels that way.

To be perfectly honest, “the nPOV” is hardly in my mind, ever, when I write here. I’m not out to defend a philosophical position; I come here to record mathematical thoughts as they come on my radar screen. Of course, categorical thinking has very interesting philosophical import for mathematics, and it’s also true that my own thinking is strongly guided by category theoretic considerations, but as far as philosophy goes, I’ll “defend an nPOV” to the extent that category theory can be a great aid in clarifying and simplifying and learning a great deal of mathematics (as partially and wittily summarized in the quote attributed to Freyd), but not to the more dogmatic-sounding statement in the original box.

• CommentRowNumber39.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
@Todd: perhaps my focus on competition was rather nonoptimal, sorry about that.
But I would have one more question, to you but also more generally.

Is this nPOV the (unique) POV here by coincidence or by design?

That is, say, if over time some 50 combinatolists started heavily contributing, and a significant subset of them has some other POV.
Would this one then be accepted on equal grounds? Say, could they quote Erdos and Zeilberger in some equaly visible meta page and so on?
• CommentRowNumber40.
• CommentAuthorGuest
• CommentTimeNov 29th 2011
@Todd_Trimble : On MO, no matter what the subject, there are often calls from people active on nLab for people to write up their thoughts about things they post on nLab. The tone suggests that the suggesters think this is something that they think all mathematicians should want to do as a matter of course (whether or not they care about category theory or related matters). Of course, one wants to promote one's point of view, but I perceive this as being pressure.

best,

Andy Putman
• CommentRowNumber41.
• CommentAuthorTodd_Trimble
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

@Andy: I actually interpret those calls to contribute to the nLab as being similar to a call to someone at MO to write up their insightful comment as an answer. I mean it’s obviously not quite the same, since the person is already tuned into MO. I just mean that’s a similarly useful thing to do. There can be some advantages: the nLab is better adapted for tracking down information than MO, it’s better interconnected, it’s easier to have a mathematical discussion away from those frickin’ comment boxes, etc. (what else am I missing, Andrew?).

But IMO it doesn’t matter if you don’t want to do it – if an nLabber feels strongly about it, then he’ll just create a stub or a remark on a page that points to the MO thread in question. It’s not like anyone has to write a full-blown article or anything. The idea is that it should be easy to do and no big deal.

• CommentRowNumber42.
• CommentAuthorSridharRamesh
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

Well, I do think there is value in having the nLab maintain a particular (lower case) nPOV; while continuing to welcome contributions from all, if disputes were to arise as to the best way to organize some article, with the broad lines of the conflict being between an n-categorical point of view being given prominence or shied away from in deference to tradition, we should have no shame at biasedly opting towards the former, no matter how popular the tradition may be. Anyone should feel free to write on the nLab about anything in mathematics(/physics/etc.) they like, from whatever viewpoint they like, so long as they are comfortable with the very significant possibility that it will be re-examined or even re-written according as to others’ n-categorical aesthetics.

There are plenty of useful resources on the net which present mathematics(/physics/etc.); Wikipedia, for example. One of the reasons to have nLab in addition to these is as a space for presentation of what would on Wikipedia be considered “original research”. But that is not the only reason; the nature of the audience at nLab also allows for articles to be written with a “voice” which would often not fly elsewhere, for sake of not being sufficiently mainstream in certain respects.

My goals in mathematics are not to understand everything through category theory; my goals in mathematics are to understand everything through whatever lens seems simplest, clearest, and most illuminating to the topic. But it is a fact of life that people often argue over what those lens are, and that in a number of areas where I feel the categorical viewpoint is helpful and traditional approaches more obfuscatory, others, in far greater numbers, would argue the exact opposite. That the differences of attitude exist is fine; that is always the way of things, and I enjoy having, in the world at large, access to all possible viewpoints on topics. But I also enjoy having a location where I do not feel so strong a need to defend my choice to use categorical presentations of certain topics. (And certainly, it does often happen in any location where mathematicians congregate, that the choice to use a categorical presentation of some topic is considered needlessly obscure or altogether misguided). I would not want the n in nLab to be considered so inessential in the end as to be thrown out altogether.

• CommentRowNumber43.
• CommentAuthorAndrew Stacey
• CommentTimeNov 30th 2011

Todd (#41) is absolutely correct. When I say “Put it on the nLab” on MathOverflow then what I’m really saying is “Put it somewhere accessible”. Sometimes I mean by that “more accessible than MO” since I don’t regard MathOverflow as a very good place to build up a repository of mathematics. To a certain extent, this is similar to the “This question doesn’t belong here, put it on a blog” or “Discussions don’t work here, try a forum” comments. The standard comeback is “Which blog? Which forum?”. So rather than say “Put it on a wiki” and have them say “Which wiki?”, I say “Put it on the nLab”. If they decide to put it on some other wiki, then of course that’s fine. But there aren’t, to my knowledge, currently wikis for other subject areas. Maybe I should say, “Put it somewhere online, for example on the nLab” instead. Would that be less aggressive?

It would be great to have other wikis that do similar things to the nLab but for other areas or viewpoints. Speaking for myself, it would be a reasonable idea to have them literally alongside the nLab to benefit from cross-linking where appropriate. It would even be possible to start within the nLab and then branch out if so desired. Setting up and maintaining a wiki like the nLab takes a fair bit of effort, both in terms of commitment to putting content on it and in terms of the technical know-how. By using the nLab, or by asking for a parallel “web”, you get all that for free (well, if it really takes off we might ask for a donation to hosting). It takes me no more effort to run John Baez’s azimuth wiki and forum than it does to run the nLab and nForum. We can add fully mathematically-enabled labs and forums in seconds. A “web” (or “lab”) requires the assent of the steering committee, a forum just requires mine.

Todd’s also right in that if there’s anything that I think ought to be “nlabbified” then I’ll just go ahead and do it (so long as I have time). I did that recently with isomorphism classes of Banach spaces. I’ve also just done it from something on the algebraic topology mailing list on embedding of smooth manifolds.

Sridhar (#42): I agree with everything you wrote. The key, for me, is the sentence:

Anyone should feel free to write on the nLab about anything in mathematics(/physics/etc.) they like, from whatever viewpoint they like, so long as they are comfortable with the very significant possibility that it will be re-examined or even re-written according as to others’ n-categorical aesthetics.

What gave rise to this discussion (as I see it) is that the way we have expressed ourselves on certain pages doesn’t convey that very well, particularly the first part. I would be tempted to change “re-written” to “added to”, though. I’ve yet to see an occurrence where the original point of view has been removed.

• CommentRowNumber44.
• CommentAuthorzskoda
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

I agree with Andrew 31 (supported by Urs 36 as well) and Todd 34. On the other hand, I think we should respect the traditional definitions/conventions to large extent even when a more natural but conflicting definition is more natural from $n$POV. At least in such cases one should say, if adopting a nontraditional extent of the concept, that the definition is actually different or, in another case, that it is just believed, but not proved, that the two definitions are equivalent in the classical case. I also think that the manner in which the $n$Lab community should be more strict about $n$POV (or $c$POV) standard are those entries which are closer in subject to the $n$Lab community, which are backbone of the $n$Lab. For example, it would be a pity if somebody would convolute the exposition of sheaf theory into some old-fashioned mess, while introducing in initially less modern way some other more peripheral to us, (but mathematically valuable) topic should be OK. I think this is silently accepted by the most active members.

• CommentRowNumber45.
• CommentAuthorzskoda
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

Say, could they quote Erdos and Zeilberger in some equaly visible meta page and so on?

Doron Zeilberger is a great mathematician and it would be good to have a program of exposing some of his nice mathematical ideas and programs, say about the umbral calculus here. On the other hand, Zeilberger is an extreme finitist, the promoter also of negativism againts infinite sets, against category theory, and against many ideas in mathematics which are not close to him but are fruitful or interesting to others. He wrote blog pages against Grothendieck’s approach to mathematics (with a shortcut that AG is out of math now because his approach to math was not sustaining in itself, unlike Gelfand’s, neglecting personal and social reasons beyond and more important than such narrow specialist’s analysis), against the value of the work of present Fields medalists (he cries that they should stop doing BORING stuff, literally) and so on. I do not think that $n$Lab should give space to such negative campaigns. Thus yes, we want Zeilberger’s math and visions here, but also do not want his negative campaigns here. And similar for other directions. At least I think so.

• CommentRowNumber46.
• CommentAuthorTodd_Trimble
• CommentTimeNov 30th 2011

Yes, I think Sridhar 42 summarized well attitudes I share regarding the nPOV question, and particularly in the last sentence of the first paragraph, as emphasized by Andrew (although I concur we need to be, and in practice are, cautious and respectful about overwriting correct material).

In general, I would like to see more nLab articles start off gently and not from the highest empyrean reaches of $\infty$-categorical thinking. I’m thinking particularly here of many of the Ideas sections in current articles – in quite a few cases I find them a bit too staunch and uncompromising in terms of promoting the nPOV – it might help rather to build up to the nPOV’s in these cases. But this is to be addressed on a case-by-case basis. This may be related to Zoran 44.

Also related to Zoran 44, I think there is no trouble at all in having a page on, for example, ultrafinitism. I agree that there is no compelling need to give space to negative emotions, as frequently expressed on Zeilberger’s well-known Opinions page. On the other hand, if there is some body of mathematics which proceeds from ultrafinitist axioms, then I don’t see the harm in describing what such a world looks like, how it differs from a classical world, or a “classically (i.e., Bishop-) constructive” world, or whatever. Done right, it would obviously be description, not prescription. Just because something is tied to a POV somewhat in conflict with the nPOV doesn’t necessarily make the subject taboo.

• CommentRowNumber47.
• CommentAuthorMike Shulman
• CommentTimeNov 30th 2011

I basically agree with everything being said. (-:

• CommentRowNumber48.
• CommentAuthorGuest
• CommentTimeNov 30th 2011
To summarize my impression. Andrew (and perhaps few others) would in principle allow a paralleity of other POV. Yet, it seems there would be considerable oposition to this from others. So, that somebody who does not in one form or another embrace the nPOV would always stay a 'second class' participant here. This is fine, but actually, this was not clear to me before this discussion. My impression of the nLab in fact was (from the way it was mentioned on MO) that this was a in principle general math wiki that just happened to have been started around (higher) category theory.

Regarding Zeilberger. This is funny. In my opinion most of the things that are received so critical are by and large a parody.
He insterchanges the fields/POVs and writes what is pretty wide spread (almost main-stream) opinion if the fields were not interchanged [not CT in a narrow sense, but the theory/terminology heavy fields]...that it works so well to provoke people proves his point. And, look, some here think everything in maths should be/could be/has to be/would be better viewed from a categorical POV. But, now, whenever somebody writes a proof or a definition it is in the end manipulating a finite string of finite symbols according to some rules. So: Everything is Combinatorics! If one follows this other POV too litterally this is not reasonable (at least not everywhere), yet IMO the same is true for any purist POV of view. (The difference is just that with Zeilberger I am pretty sure he means some things rather a bit in playful way, while here it seems some, not all, are very serious about this nPOV.)

While I would be tempted to elaborate a bit on Grothendieck too, I won't.

That all being said, while I actually have not read much of nLab I feel it could be from time to time interesting to read around here a bit more. And perhaps once I will have something to write where I am interested in some nPOV input, then I think here will be a great place.

quid
• CommentRowNumber49.
• CommentAuthorTodd_Trimble
• CommentTimeNov 30th 2011

@quid: since what has been said about Zeilberger so far in this thread has been pretty non-specific, let’s please not pursue a discussion about him. Let’s please consider any mention of him highly incidental, and not get sidetracked into controversial territory. Similarly, I’d rather not get started on Grothendieck in this thread (and am glad you held back).

I also take exception to your formulations. “So, that somebody who does not in one form or another embrace the nPOV would always stay a ’second class’ participant here.” That’s a conclusion you draw from this discussion, not from careful observation of how things actually work here (you admit you haven’t spent much time here). The formulation moreover puts participants here in an unkind and I think unfair light – I don’t think it’s at all accurate.

In practice, what often happens is that someone may write up just some straight mathematics. No nPOV. Then someone else may come along, and add a little from his expertise in constructive mathematics. Then someone else will come along, and add some remarks which puts matters in an abstract light. Now, if the first person would like to come back add a theorem or two that he knows from a more traditional point of view, he can, and everyone here will think that’s great. (Some creativity may be required in getting all the components to fit together nicely, but experience shows it’s doable.) There is no second-class citizenry.

If you think you can detect concrete instances of people being treated like second-class citizens around here, then please point them out. (I will discount cases where some people were provably not in command of the mathematics they wrote about.) But it may take more familiarity with this site than you currently possess.

Your points might (might) better apply in cases where this so-called nPOV comes into direct conflict with another POV being introduced. It could be interesting to see what happens. Right now I’m having trouble coming up with a concrete instance of that having happened. Until that point, I think the discussion is largely in a vacuum.

• CommentRowNumber50.
• CommentAuthorzskoda
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

So: Everything is Combinatorics!

You allude to the formalization of a proof (which is by the way in that form almost never attained in practice), not the intuitive imagery and content behind the proof – in which not everything is combinatorical in nature, nor even amounts to a complete or rigorous line of deduction. Much content can be detailed in many formal ways which are not very relevant for the crux of the matter, so the reductionist picture you quote is not fully approaching the totality of modern mathematical art. The same could be said about Matiasevich’s theorem about reducing “all math” to the diophantine equations, he reduction curious and deep but not really living up to the literary interpretation in real praxis (it would be too difficult to solve most problems by reducing them by Matiasevich to the diophantine equations).

I full agree with Todd 46.

• CommentRowNumber51.
• CommentAuthorMark Meckes
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

As Todd suggests, it would be good to have some more concreteness in this discussion. So I offer a description of my own recent experience as an “outsider” adding to the nLab some material which could be seen as not fitting into the nPOV.

In a recent discussion on meta.MO (sorry, too lazy to search for links right now), Andrew said he had created a new nLab page on isomorphism classes of Banach spaces — which makes no mention of categories or anything else nPOV-y — and suggested it would be nice to have some more details added. Since I knew some of those details, I decided to add them myself, and following some links, I found that the nLab page on Banach spaces defined an isomorphism of Banach spaces differently from what was meant in the page Andrew had created. Specifically, that page referred to a “usual” notion of isomorphism, which is in fact the usual notion among category theorists but not the usual notion among functional analysts. So I added a couple paragraphs to that page, clarifying that there are many different notions of isomorphism, and who calls which one what. After that Toby wrote a new paragraph explaining how the analysts’ usual notion of isomorphism can be viewed naturally from a category-theoretic perspective. Some interesting discussion about all these changes took place in other threads on the nForum.

At no point in all this did I feel like my contributions and (non-category-theoretic) perspective were anything other than welcome; noone suggested that the functional-analytic terminology and perspective I introduced was wrong or misguided; and noone rewrote my contributions — they were simply added to. (Not that I would have minded if what I wrote had been rewritten, and in fact I suspect that I, not being familiar with prevailing practice, did more rewriting of others’ material than is usually done here.)

• CommentRowNumber52.
• CommentAuthorGuest
• CommentTimeNov 30th 2011
@Todd: I will stop discussing Zeilberger, but zskoda was not non-specific. It is true I did not observe anything carefully, but you at the end say you do not recall any precedences either. So, I asked: what would happen if this and that to see the reaction. It is true I was perhaps a bit provocative, but this is needed to find something out. If one is not a bit provocative or overdoes this a bit, chances are one gets and abstarct 'yes, yes, no problem, don't worry' but then in practise things could be a bit different.

Also, it was not my intention to put anybody here in an unkind or unfair light. Sorry if I caused this impression. In some sense in retrospect I do not undestand this entire discussion. Some insiders and outsiders say there is a certain POV around here and it should be like this. And everybody said this is fine.

The only thing happening in addition is that some ask not to advertise this site as a general/neutral place for mathematics, since it is not.
This 'since' was not clear to me, so I asked some things to find out. Now I think I know it.

What did I mean with the 'second class' pariticipant, well for example it was said:

> if disputes were to arise as to the best way to organize some article, with the broad lines of the conflict being between an n-categorical point of view being given prominence or shied away from in deference to tradition, we should have no shame at biasedly opting towards the former

So if your personal preferences are typically in favor of the latter then you are sure to be on the lossing end in case of conflicting opinions. This seems like a permanent disadvantage, for potential users of the site with such preferences.
It seems they could create their own sites and they would be hosted, which is nice and friendly, but as soon as any relevant interaction would arise and it becomes relevant they are expeected to follow or at least not oppose the nPOV.

And to be clear: I have no problem that this is like this and Sridhar Ramesh even made clear that this is not meant as an absolute judgement of the merits but as what should happen locally here.
• CommentRowNumber53.
• CommentAuthorGuest
• CommentTimeNov 30th 2011
@zskoda (50): basically this was meant as a joke. In view of Todd's request I do not follow up further on this line of the discussion.

52 was me too.

quid
• CommentRowNumber54.
• CommentAuthorzskoda
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

I think, 52, that there is a confusion you have about neutrality vs. $n$POV which you partly misunderstood but can be very easily explained. In wikipedia one can not be original, everything has to be neutral and justified with published or accepted references etc. This would be a constraint to people who not only present here knowledge but also do research with some attitude, dose of trust, aesthetics and so on. Also one often sees that there is a traditional accepted way of doing things which is not the best as it is created before the concept was understood in a true light. Thus in understanding a concept’s presentation one wants to incorporate the insight, knowledge, usefulness and importance which come from the present state of understanding. Many ideas in modern mathematics made a sweeping change in view of geometry and so on, and they came with several revolutions, some of them closely related to homotopy theory, universal properties in category theory, and so on. Thus one often replaces a concept with its derived version and so on. This makes the “correct” version of the concept. This correct attitude of simplicity, universality, truthfulness is not reduced to the category nor to the higher category theory, though in search of foundamentals and simplicity it is closer in spirit to them then to some other programs of work, which were dedicated more to the initial attack on a problem than to clearing the conceptual truisms at the bottom of the things. And it is more general. So $n$POV is not necessarily about categories, but about choosing the right and essential insight, the conceptual backbone, rather than convoluted mass of data and ad hoc arguments. Thus many other mathematics can be done in a clean way, and not only clean but in a right level of generality, not artifically general, but not too specific to obscure its general and simple essence. And it is good to speak in a way tolerant to the variations in other fields and other approaches of mathematics where the concept may reappear in some form. I think this kind of spirit is $n$POV for the purpose of filling $n$Lab, not the $n$POV in the most literal sense.

this was meant as a joke.

You are welcome to joke, but I think for the reference of many people who might read such long discussions superficially it is good to be clear and conclusive, so I anyway answered to it. Most of us, do not have time to read the whole discussion and will take argumentation by default as serious one. This is not to ask for loss of enthusiasm to bring good spirits, but simply an insider impression that our work in $n$Lab and $n$Forum is a chase with time and the list to do is orders of magnutude much longer than any fully appreciating and conclusive argumentation would require. In other words, it would be too consuming to try to understand what is joke and what is not, unless it is entirely obvious.

• CommentRowNumber55.
• CommentAuthorGuest
• CommentTimeNov 30th 2011
@zskoda: IMO there is simply a huge difference between differeent approaches to mathematics. It's fine if you search for the best, true, correct, right,... But I think it is better to just do whatever seems interesting, practical, feasible at a given time.

quid
• CommentRowNumber56.
• CommentAuthorTodd_Trimble
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

Zoran’s mention of DZ was not based on specific quotations; he was just giving an example of someone whose POV might conflict with ours. I thought the mention was more or less incidental.

It is true I did not observe anything carefully, but you at the end say you do not recall any precedences either.

Right! And that was in support of my point, because I can claim intimate knowledge of how things are here.

To continue: so, we’re all okay with this place having a POV. Okay, good.

In explaining this “second-class” business, you quoted

if disputes were to arise [snip]

and fair enough, but permit me to assure you that in practice, that’s a mighty big if. Can I tell you some more how things actually work in practice? In some cases where I’ve thought to myself, “I would’ve written that differently”, I simply write a new article, and link back (or intend to link back; things are always in an incomplete state here) to the other article. A specific instance I have in mind is some stuff Andrew Stacey wrote on algebraic theories; I can look it up if you want. I wound up writing stuff myself on algebraic theories, from more of an nPOV. These articles co-exist in total peace. No political battles whatsoever, no imposing an nPOV on an article written from a more traditional perspective, nothing like that.

Here’s another example: Andrew Stacey pointed out a little blooper on a page which gave an abstract description of flat modules, and decided he would go in there and add a concrete description of what a flat module is. You could definitely say, this was not along nPOV party lines! And so what happened? Well, he told us what he did over at the nForum. Then I made a remark which, in the totally biased way we have around here, we might call “middlebrow”, meant to bridge the “lowbrow” to the “highbrow”. Then Andrew worked that remark into the article. So we wound up having three descriptions of flat module, each useful in its own right, together with explanations of how they are interconnected . Everyone won. (And then Andrew conducted a tongue-in-cheek straw poll, “How highbrow are you?” or something, and a good time was had by all.)

In general, we try to be very respectful of each other’s contributions. You seem to be projecting a very political spin on how things are (or might be, theoretically), and I just don’t think that’s really warranted.

Finally,

In some sense in retrospect I do not undestand this entire discussion.

I’m not sure why you’re so interested to begin with! You don’t strike me as having read the nLab much, or being actively interested in doing so. Sorry if I’m wrong about that.

Imagining how political conflicts might come up and the mighty nPOV hammer dealing with them is just not based on anything real I am aware of. The history suggests a very different and somewhat more creative and productive picture, I believe (cf. Mark Meckes’ comment).

• CommentRowNumber57.
• CommentAuthorzskoda
• CommentTimeNov 30th 2011
• (edited Nov 30th 2011)

Yes, $n$Lab has a specific approach. I pointed in 54 that it is far not limited to category theory, but the “practical” brute-force goals may be lower on the list of our preferences than the search for conceptual insights and understanding. I can personally add that much of what is considered practical in mathematics is often not really practical from the point of view of needed applications: often one has a way which works to generate some code and publish a paper and produce formulas for exhibition, but when faced with real needs in practice it is often equally artificial. If one wants to show that something is really practical, one has to go into understanding motivation and real needs and then much of the industry moderates true progress to the very original and insightful (and often simple) contributions.

• CommentRowNumber58.
• CommentAuthorGuest
• CommentTimeNov 30th 2011
@Todd: I admit to some idle and perhaps naive curiousity here (in this discussion). When I noticed it some points were not quite clear and I was interested in some clarification, and then well I was in the middle of it. It is true in retrospect I should have likely stayed out of it.

But, it's not so unlikely I will read it more in the future. What really made me use MO was participating in a discussion on anonymous usage when hardly using it. And, I have some sort-of plan which gets delayed all the time (as I discuss too much on the internet among others) to finally fill some gaps of mathematical knowledge which are quite close to what is happening here, so who knows you might see me around here more often; I promise to be more peacefull then. But I would also like to add that I did not want to troll here, I was truly interested in what I asked for.

Thank you for the information I got and I appologize for the noise.

See you on MO,
quid
• CommentRowNumber59.
• CommentAuthorTodd_Trimble
• CommentTimeNov 30th 2011

But I would also like to add that I did not want to troll here

I know that!! Don’t worry about anything.

I’d enjoy seeing any contributions you’d like to make! And in that case, I hope you will be pleasantly… is surprised the word? :-) Anyway, I hope future interactions are pleasant and productive. See you on MO…