From the preliminary results of a recent SurveyMonkey survey: out of 29 respondents (most of them people pursuing or qualified in mathematics), 24 claimed to have used Wikipedia often, and the remaining 5 to have used it occasionally for mathematical reference information. For Planetmath, 22 people claimed to have used it occasionally, and 7 to have heard of it but not used it. For Mathworld, 5 people claimed to have used it often, 16 to have used it occasionally, 7 to have heard of it but not used it, and 1 to have never heard of it.

In response to a question on how Planetmath and Mathworld compare with Wikipedia, the nine free responses were:

• Planetmath now loads too slowly, so I seldom use it. Even when I did use it often, I found that the articles tended either to be too incomplete, or too tailored to people who are already experts. Many Mathworld pages are still much better than many Wikipedia pages, but Wikipedia is more comprehensive.

• planet math is slow!!! mathworld has more graphics and less maths.

• Material is generally better presented on Mathworld, but the topics are more limited.

• Not as much information as Wikipedia and not as well arranged

• I usually find Wikipedia to be my first source and only go to Mathworld or Planetmath if wikipedia fails me. I guess that means Wikipedia is better.

• Planetmath is dying, Mathworld is static.

• I find wikipedia much more useful than Mathworld. Mathworld’s pages are very technical, which is not what I am looking for on the internet usually. Usually I am looking for someone’s nice conceptual understanding of a topic or definition (through nice examples), and Wikipedia usually has lots of these.

• Planetmath is quite useful to find proofs. Mathworld is very specialised, but it has a few nice bits of information sometimes. They seem to both be quite stagnant compared to Wikipedia.

• They can’t keep up. There was probably a time when PlanetMath was a better reference than Wikipedia, but it’s fading fast. I think their ownership model isn’t conducive to long term quality.

(See the responses to these and other questions here and take the survey here).

The general consensus does seem to confirm my suspicions. Why is Wikipedia gaining? Here are the broad classes of explanations:

1. It’s just a self-reinforcing process. The more people hear about and link to Wikipedia, the more people are likely to read it, the more people are likely to edit it and improve it. But if that’s the case, why did Wikipedia ever get ahead of Mathworld and Planetmath? Two reasons: (i) its more radically open editing rules (ii) Wikipedia covers many areas other than mathematics, so people come to the site more in general. Also, since it covers many areas other than mathematics, it can better cover content straddling mathematics and other areas, such as biographies of mathematicians, and historical information that is relevant to mathematics. This creates a larger, strongly internally linked, repository of information.

2. Planetmath’s owner-centric model (as mentioned in one of the responses) where each entry is owned by one person, does not create a conducive environment for the gradual growth and improvement of entries.

3. The appearance of content is better on Wikipedia. Prettier symbols, faster loading, better internal links, better search. This is definitely an advantage over Planetmath, which has slow load times in the experience of many users (as indicated by the comments above) though perhaps not so much over Mathworld.

4. Google weights Wikipedia higher (because of the larger size of the website and the fact that a lot of people link to Wikipedia). This is related to (1).

5. The people in charge of Mathworld and Planetmath simply lost interest. Mathworld is largely run by Eric Weisstein, an employee at Wolfram, who seems to have recently been trying to integrate metadata about mathematical theorems and conjectures into Wolfram Alpha. Developing Mathworld continually to a point of excellence does not seem to have been a top priorty for Weisstein or his employer Wolfram Research (that hosts Mathworld) over the last few years. The people running Planetmath also may have become less interested in continually innovating.

Given all this, is Wikipedia the best in terms of: (i) the current product or (ii) the process of arriving at the product? While I’m far from a Wikipedia evangelist, I think that the answer to (ii) is roughly yes if you’re thinking of broad appeal. Anything which beats Wikipedia will probably do so by being more narrowly focused, but it may then not be of much interest to people outside that domain. A host of many such different niche references may together beat out Wikipedia for people who care enough to learn about a multiplicity of references. For those who just want one reference website, Wikipedia will continue to be the place of choice in the near future (i.e., the next 3-4 years at least, in my opinion).

Currently, Wikipedia is an uneasy mix of precise technical information and motivational paragraphs. It makes little use of metadata to organize its information; on the other hand, it is easy to edit and join in. The mathematics entries cannot be radically changed in a way that would make them radically different in appearance from the articles on the rest of the site. This opens up many niche possibilities, some of which are being explored:

1. Lab notebooks, where people store a bunch of thoughts about a topic, without attempts to organize them into something very coherent. Here, good metadata and tagging conventions could allow these random lab notebook-type jottings to cohere into an easily accessible reference. This would be the mathematical version of open notebook science, a practice that is slowly spreading in some of the experimental sciences. nLab (the n-category lab) is one example of a “lab notebook” in the mathematical context. This is great for motivation, and also for understanding the minds of mathematicians and the process of mathematical reasoning.

2. Something that focuses on a particular aspect of mathematical activity. For instance, Tricki, called the Tricks Wiki, focuses on tricks. Other references may focus on formulas, others may focus on counterexamples, yet others (such as the AIMath wiki on localization techniques and equivariant cohomology) may focus simply on providing extensive bibliographies. Somewhat more developed examples include the Dispersive Wiki and complexity zoo (actually, a computer science topic, but similar in nature to a lot of mathematics). Some may focus on exotic tricks of relevance to a particular mathematical discipline. There is some cross-over with lab notebooks, as the tricks become more and more exotic and the writing becomes more and more spontaneous and less subject to organization into an article.

3. Highly structured content rich in metadata that is intended to provide definitions, proofs and clarify analogies/relations. Examples include the Group Properties Wiki [DISCLOSURE: I started it and am the primary contributor] which concentrates on group theory. The flip side is that the high degree of organization uses subject-specific structures and hence must be concentrated on a particular narrow subject.

There are probably many other niches waiting to be filled. And there may also be close susbstitutes for reference sites that weren’t created as references. For instance, Math Overflow, though not a reference site, may play the role of a reference site once it accumulates a huge number of questions and answers and adopts better search and specific tagging capabilities. Similarly, thirty years from now, the contents of Terry Tao’s weblog may contain a bit on virtually every mathematical topic, in the same way as Marginal Revolution have a bit on almost all basic economic topics (I say “thirty years” because economics is in many ways a smaller subject than mathematics).

http://www.surveymonkey.com/s/T2CBTZJ.

You can view the survey results here (but don’t do this before taking the survey!):

http://www.surveymonkey.com/sr.aspx?sm=aaYWJUl4DF9hjvM4AqrQxhgmev4PNRiYa8MJbwhLs9w_3d.

Surprising similarity of questions with questions I’ve asked in the past

Looking over the group theory questions, I found that a number of questions I had asked — and taken a long time to get answers to — had been asked on Math Overflow, and had been answered within a few days. The answers given weren’t comprehensive; I added some more information based on my past investigations into the topics, but it was still remarkable that these questions, most of which aren’t well known to most people in the subject, were answered so quickly.

• The question Are the inner automorphisms the only ones that extend to every overgroup? is a question that I first asked more than five years ago, when still an undergraduate. I struggled with the question and asked a number of group theorists, none of whom were aware of any pasy work on these problems. I later managed to solve the problem for finite groups, and then my adviser discovered, from Avinoam Mann, that the problem was tackled in papers by Paul Schupp (1987) and Martin Pettet (1990), along with the many generalizations that I had come up with (some variants of the problem seem to remain unsolved, and I am working on them). You can see my notes on the problem here and you can also see my blog post about the discovery here.

• The question When is Aut G abelian? was a question that I had been idly curious about at one point in time. I couldn’t find the answer at the time I raised the question, but stumbled across the papers a few months later by chance (all well before Math Overflow). It’s interesting that the question was so quickly disposed of on Math Overflow. See also my notes on such groups here.

• The question How can we formalize the naturality of certain characteristic subgroups is a more philosophical question with no real concrete answers, which I’ve considered for a long time too.

• The question Balancing problem in combinatorics that I had, based on a generalization of an Olympiad question I had seen, turns out to be part of something called rainbow Ramsey theory, as the answer suggests.

Two things stand out: (i) all these questions are questions whose answers are not well-known (the people I asked didn’t know the answers offhand) but are questions that many people do ask (ii) On Math Overflow, they were dealt with quickly.

I think the situation is similar in many other established areas of mathematics — the answers are out there, but they are not well-known, probably because these problems are not “important” in the sense of being parts of bigger results. But they are questions that may naturally occur as minor curiosities to people studying the subject. These curiosities may either go unanswered or may get answered — but the answers do not spread to the level of becoming folk knowledge or standard textbook knowledge, because they aren’t foundational for any (as yet discovered) bigger results.

Math Overflow now provides a place to store such questions and their answers — thus, next time you have one of these questions, a bit of searching on Math Overflow would yield the question and the answers provided and stored for posterity. Apart from the questions I had thought of, consider this one that somebody thought up and turned out to have been considered in multiple papers: When is A isomorphic to A^3?.

The situation is probably different for areas where new, cutting-edge questions are being asked — i.e., areas where the questions are charting “new” territory and are helping build the understanding of participants. Some people told me that this is likely to be the situation with areas such as topological quantum field theory or higher category theory.

Skills needed and developed

So what skills are needed to participate in Math Overflow, and what skills get developed? In order to answer questions quickly, a combination of good background knowledge of mathematics and the ability to search ArXiV, Mathscinet, JSTOR, Google Scholar, and other resources seems necessary. In order to ask questions, it seems that a combination of a lot of background knowledge and a natural curiosity — of the bent needed for research, is what is needed.

Pros and cons of posting on Math Overflow

The obvious pro seems to be that a lot of people read your question. The sheer number of readers compensates for the fact that most people, even experts in the area, may not immediately know the answer. Because of the larger number of people, it is likely that at least a few will either have come across something similar or will be able to locate something similar or will be able to solve the question because he/she gets curious about it. It was pretty exhilerating when a minor question that I didn’t feel equipped or curious enough to struggle with was answered by somebody who came up with a construction in a few hours (see collection of subsets closed under union and intersection), or when a question I had about elementary equivalence of symmetric groups was answered within a few days (see elementary equivalence of infinitary symmetric groups).

The potential con could be that people may be tempted to ask a question on Math Overflow without thinking too much about it. This probably does happen but I don’t think it is a major problem. First, the general quality of participants is quite high, so even if people ask questions without thinking a lot about them, chances are there is something interesting and nontrivial about the question — because if there weren’t, people of the profile contributing would have been able to solve it even without a lot of thought. Further, even if a question is a good exercise for a person specializing in that subject — so he/she should struggle with it rather than ask others, it may be good for a specialist in another subject to simply ask.

The voting (up/down) system and the system of closing questions that are either duplicates or not suitable for Math Overflow for other reasons (such as undergraduate homework problem), combined with a reputation system for users linked to the votes they receive, seems to be a good way of maintaining the quality of the questions.

A revision of some of my earlier thoughts

In a blog post almost a year ago titled On new modes of mathematical collaboration, I had expressed some concern regarding the potential conflict between the community and activity that is needed to have frequently updated, regularly visited content, and the idea of a steady, time-independent knowledge base that could be used as a reference. mathematics blogs, with regular postings and comments, and polymath projects, which involve collaborative mathematical problem-solving, are examples of the former. Mathematics references such as Mathworld, Planetmath, the mathematics parts of Wikipedia, The Springer Encyclopedia of mathematics, ncatlab, Tricki, and the Subject Wikis (my idea) are some examples of the latter, to varying degrees and in different ways. The former generate more activity, the latter seem to have greater potential for timeless content.

Math Overflow has the features of both, which is what makes it interesting. It has a lot of activity — new questions posted daily, new answers posted to questions, and so on. The activity, combined with the points system for reputation, can be addictive. But at the same time, a good system of classification and search, along with a wide participatory net, makes it a useful reference. I’m inclined to think of its reference value as greater than what I thought of at first, largely because of the significant overlap in questions that different people have, as I anecdotally describe above.

Math Overflow stores both mathematical data — questions and their answers, as well as metadata — what kind of questions do people like to ask, what kind of answers are considered good enough for open-ended questions, how quickly people arrive at answers, what kind of questions are popular, how do mathematicians think about problems, etc. This metadata has its own value, and the reason Math Overflow is able to successfully collate such metadata is because it has managed to attract high-quality participants as well as get a number of participants who are very regular contributors. Both the data and the metadata could be useful to future researchers, teachers, and people designing resources whether of the community participation type or the reference type.

On the other hand…

On the other hand, Math Overflow is not the answer to all problems, particularly the ones for which it was not built. Currently, answers to questions on Math Overflow are broadly of the following three types (or mixtures thereof): (i) an actual solution or outline of a solution, when it is either a short solution arrived at by the person posting or an otherwise well-known short answer (ii) a link to one or more pages in a short online reference (such as an entry on Wikipedia or any of the other reference resources mentioned above) (iii) a link or reference to papers that address that or similar questions.

For some questions, the links go to blog posts or other Math Overflow discussions, which can be thought of as somewhere in between (ii) and (iii).

With (i), the answer is clearly and directly presented, and the potential downside is that that short answer may not provide a general framework or context of related results and terminology. With (ii), a little hunting and reconstruction may need to be done to answer the question as originally posed, but the reference resource (if a nice one) gives a lot of related useful information. (iii) alone (i.e., without being supplemented by (i) or (ii)) is, in some sense, the “worst”, in that reading a paper (particularly if it is an original research paper in a journal) may take a lot of investment for a simple question.

In my ideal world, the answer would be either (i) + (ii), or (ii) (with one of the links in (ii) directly answering the question), plus (iii) for additional reference and in-depth reading. But there is a general paucity of the kind of in-depth material in online reference resources that would make the answer to typical Math Overflow questions by adequately dealt with by pointing to online references. So, I do think that an improvement in online reference resources can complement Math Overflow by providing more linkable and quickly readable material in answer to the kinds of questions asked.

Tricki

I blogged twice about Tricki, the Tricks Wiki, which went live in April 2009 (see the annoucement by Tim Gowers). Tricki held a lot of promise. Of late, the enthusiasm seems to have slowed down, though this might be a temporary phenomenon. The most recently created article and the most recent comments appear to be two weeks old as of today (October 27, 2009). According to Alexa data, the site has a rank of 1,200,000+ worldwide and about 550,000-600,000 in the United States. For comparison, subwiki.org, which I run, has Alexa data showing a site rank of 500,000-550,000 in the world and 150,000-200,000 in the United States, while Math Overflow has Alexa data showing a rank of 350,000 worldwide and about 60,000 in the United States (the numbers you see clicking on the links may be different if you don’t view this post within a few hours of my writing it).

Tricki also hasn’t been mentioned on Gowers’ blog since June 25, 2009 and on Terence Tao’s blog since August 2009.

Is the Tricki falling into disuse? Clearly, the initial spate of interst seems to have subsided, but it might well regain a slower and steadier momentum in some time.

Planetmath

I remember a time when Wikipedia had much less mathematical content than planetmath, which was one of the first places to check mathematics on the Internet. Planetmath appears to be going strong, though not as strong as before. While their message forum seems reasonably active, their latest addition was about a week ago, and they seem to be getting somewhere between 0 and 2 new articles in a day, and around the same number of revisions a day. Not exactly dead, but not bubbling with life. Their Alexa data indicates fairly steady performance with a traffic rank of around 130,000 over the last six months, but a decline over a longer timeframe — setting the drop-down parameter to “max” below the chart shows that their traffic rank and daily pageviews have been following over the longer run. Why? Decline in quality? Probably not — it’s more likely that people are increasingly using Wikipedia.

Are textbook prices soaring?

So I decided to look at the range of calculus books. The general impression from the things I read seemed to be that it would be hard to get a decent textbook for under $100. So, I went and typed calculus on Amazon, and looked at the first page of search results. Among these search results was Calculus for Dummies ($12.99), Forgotten Calculus ($11.53), Calculus Made Easy ($26.95), Schaum’s Outline of Calculus ($12.89), and The Complete Idiot’s Guide to Calculus ($12.89). Most of these books would seem reasonable for a low-level introductory semester or two quarters in calculus — admittedly, they may not be suitable for all calculus courses, but if price really is a primary consideration, it isn’t as if there are no options. There is also a wikibook on Calculus and an old public domain book on calculus. If you want somewhat more advanced stuff for free, you can try MIT’s OpenCourseWare course on single variable course, which includes video lectures, their course on calculus with applications, and their course on multivariable calculus.

Okay, so perhaps calculus is a bad example? Well, I decided to pick point set topology. The standard book for this is the second edition of Munkres’ book, which I think is one of the best, and it costs $107.73 on Amazon. But searching for topology on Amazon gives a number of other considerably cheaper books, such as Mendelson ($7.88), Gamelin and Greene ($10.17), Springer Undergraduate Math Series book by Crossley ($23.40), Schaum’s Outline of General Topology ($12.89), among many others. None of them seem as good as Munkres, but they all cover the basic material — and reasonably well, it seems.

Of course, I have picked on calculus and topology, both topics that are more than fifty years old, and where most of the material that should be included in an elementary textbook is widely known. In other words, the field for writing books is wide open. No publisher or author has significant scarcity power. When we are looking at exotic topics such as the theory of locally finite groups, then yes, you probably wouldn’t find cheap textbooks. But most undergraduate-level textbooks would likely be of the level of calculus or topology texts, and not exotic texts on locally finite groups.

Why do instructors choose expensive textbooks when cheaper alternatives exist?

Why do instructors choose $100+ calculus textbooks or Munkres’ topology textbook when there are so many cheaper books available on the market? One explanation, pointed out in a comment to Baez’s post, is the “moral hazard” explanation. This states that instructors do not need to bear the costs of buying the textbooks, so they just prescribe the “best” textbook based on their personal criteria rather than taking the price into consideration.

This is a reasonable-sounding argument, and is possibly true in many situations, but is it possible that even taking the cost into consideration, a slightly better book worth $100 might be better than a slightly worse book worth $10? For instance, at the University of Chicago, where I am pursuing my graduate studies (and now teaching calculus to first-year undergraduate students) the University charges students around $3000 per course per quarter. This means that students are paying something in the range of $100 per lecture. Even if you believe that only 25% of the money charged by the university as tuition is charged for the actual education, that still comes to $25 per lecture. For a calculus textbook that will be used for three quarters (and hence, about 75 lectures), is a price difference of $100 versus $20 worth worrying about? And even at universities where students pay smaller out-of-pocket fees due to more direct taxpayer subsidies, the cost of textbooks doesn’t come to a very large fraction of the total costs borne by students plus taxpayers, and it becomes even smaller considering all the other ways that students “pay” by working.

So, when a point set topology instructor chooses Munkres over a cheaper textbook, the instructor may well be making a smart choice! This isn’t to say that there aren’t situations where the better textbooks are the cheaper ones. But the point here is that the cost difference may simply not be worth worrying about in the grander scheme of things.

Is it evil to bring out new editions?

Another point raised in the material that Baez refers to is that textbook publishers are being evil by regularly bringing out new editions of their work. Certainly, one rationale for bringing out new editions of calculus textbooks is that people keep buying textbooks and don’t simply use the used books in circulation, but there are other reasons to bring out new editions, such as correcting things that were wrong, improving exposition, and so on. Even in a subject like calculus, terminology and best teaching practices change with time, student and teacher needs change with time, and new editions help keep pace with these changes. If new editions were completely useless, why couldn’t everybody just make use of calculus books old enough to be in the public domain, such as this one on Wikisource?

And if there is so much concern about bringing out new editions, all a college needs to do is buy enough copies of the current edition to keep in circulation for several years, offer students special deals whereby they can “rent” books for a year instead of buying them. If you buy enough books for three years’ worth of students and half the students in each year use the “rent” option, then a given stockpile of books can last five years. If such strategies seem ludicrous to contemplate, it suggests that the evil of new editions may not feel that threatening.

To be fair, there do exist book rental services such as Chegg (profiled in the New York Times PING column) and Campus Book Rentals. Also, publishers are offering to rent books to students.

Should legislators put price controls? And should textbook innovation be subsidized?

The Capital Campus News piece quoted by Baez says that a law passed in California state to control textbook prices lacks “teeth”. The piece sounds almost rueful about the fact that publishers cannot be stopped from bringing out new editions of their textbooks, since that would violate the First Amendment. While they don’t explicitly say so, they’re almost hinting as though this is a bad thing — an inconvenience. CALPIRG, which claims to “speak truth to power”, says that “Congress should require publishers to curb practices that drive up the cost of a college education.”

Let’s pass up here the issue that, in an issue of bargaining between two parties (students versus publishers) each with its own vested interests, one party is asking legislators to step in and tilt the tables in its favor. That kind of thing happens all the time in democracies. Let’s go instead to the question of whether price controls may be a good idea.

As Tyler Cowen says at the “Marginal Revolution” blog: “We know that textbook innovation saves lives and has a very high benefit to cost ratio. Thus, price controls or other restrictions that reduce prices are almost certainly a bad idea.”

What does this mean? The general idea is that price controls reduce the incentives for new entrants to enter the market. Those who are already writing and selling textbooks can handle the price controls — their profits may reduce, but they can still stay, but those who are thinking of whether to enter may choose not to stay put. Similarly, innovation into new kinds of product mixes might be drastically curtailed by price controls. (see also a similar argument against price controls on pharmaceuticals by Alex Tabarrok of Marginal Revolution, and the video titled “Price Gouging:Myth or Fact” on John Stossel’s videos on ABC).

Some people are skeptical that price-based incentives can result in better, or cheaper, calculus instruction. For instance, this comment on Baez’s post seems to suggest that if calculus textbooks became free, people would be motivated to write calculus for the love of the subject rather than in order to make money. The comment doesn’t say what this will do to the quality of textbooks.
Also, some mathematicians seem enamored with the idea of free instruction, in the sense of copylefted material that students can freely use and recycle. As Baez says:

One way to tackle this problem is to develop free online textbooks. I think a wiki-based approach could be good. People are trying it. Will it ever catch on?

It might also make sense for the NSF, or other funding agencies, to pay for scholars to write free online textbooks — or improve existing ones.

He then optimistically points out to a Chronicle of Higher Education blog post on a program at the Universoty of Wisconsin at Oshkosh using a $300,000 grant from the Fund for the Improvement of Postsecondary Education.

There’s definitely a possibility that such pilot projects, based on grants from the government or from private foundations, that aim to explicitly create freely available copylefted or remixable projects, will succeed. MIT OpenCourseWare has been pretty successful, though it doesn’t seem to have obviated the need for textbooks. Yale has also published open courses, and so have many other universities (see OCW consortium for more details). Independently, there have been various projects that aim to write “modules” — small units that can be fitted together into larger books and have Creative Commons licenses — the most prominent example of this being Connexions.

But while these are promising directions of investment, they’re probably not the only possible ones, and at this stage, it would be hard to predict what will catch on and to what extent new resources of this kind could replace textbooks. Besides, there are plenty of alternatives that may well be motivated by the lure of high prices. Some of these may start out even more expensive than textbooks, but may offer other features that textbooks don’t, and hence offer more value-for-money. Price caps could significantly deter investment into such tools.

For instance, Thinkwell, which was started over ten years ago, offers video textbooks. Their college calculus book includes 50+ hours of high-quality video instruction, with lecture transcripts, lecture notes, and practice exercises, and costs $140 + tax. This isn’t just video instruction in the sense of a person at the board — it is designed in a way that includes a lecturer writing stuff, side notes on what the important points covered so far are, and a useful presentation. It seems unlikely to me that this kind of investment into a “video textbook” would have occurred without the incentive to make money through selling online subscription — even though I’m sure there are non-financial motivators for them too.

The upshot? And broadsides

The upshot is that I don’t see any immediate and pressing reasons that the college textbook cost is a major issue or that politicians need to take action about it. Yes, there is a legitimate “moral hazard” argument, but the cost of college textbooks should probably be kept in perspective, and that perspective is the cost of college, both in terms of direct financial cost and the opportunity costs of attending college. Perhaps the appropriate standard against which to compare the rise in textbook costs is the rise in college costs, and not general inflation. It is rather ironical for colleges, that have been shamelessly raising fees well above inflation, and for faculty, who often make more money for every lecture they teach than the cost of a calculus textbook that lasts an entire year, to wax eloquent about the problems that students face because of rising college textbook costs. (For more on college costs, see this College Board report and this New York Times article on the report).

It’s always better for consumers to have things cheaper than to have them more expensive, so I’m all for cheaper books of the same quality or better books at the same price. But getting these is less about one group of vested interests trying to get support for their barganing position and more about encouraging innovation. And one way of at least not discouraging innovation is to not impose price controls and not try to determine the final shape or direction of things. And some of the innovations could drive average prices up, because they may be high-price-high-quality points on the price-quality spectrum.

In the meantime, Terence Tao started a polymath blog here, where he initiated four discussion threads (1, 2, 3 and 4) on deterministic ways to find primes (I’m not quite sure how that’s proceeding — the last post was on August 28, 2009). (UPDATE: A new post (thread 5) was put up shortly after I published my blog post).

Around the same time, Gil Kalai started a polymath on the polynomial Hirsch conjecture (1, 2, 3, 4 and 5).

Also, some general discussion posts on polymath projects: by Tim Gowers and by Terence Tao.

It remains to be seen whether any of these projects are able to reach successful conclusions or make substantial inroads into the problem. If there is another success for a polymath project, then that would be a major booster to the idea of polymath projects. Otherwise, it might raise the question of whether the unexpected degree of success of the first polymath project led by Gowers (which aimed for, and got, an elementary proof of the density Hales-Jewett theorem) was an anomaly.

Math Overflow is a website where people can ask math-related questions (the questions should be of interest to people at the level of Ph.D. student or higher), answer the questions, and rate the answers. It uses the Stack Exchange software, which is used for many other websites, such as Stack Overflow. Funding for the website is being provided by Ravi Vakil of Stanford University, and it has a bunch of moderators — but anybody who earns enough points through participation can rise to the status of moderator. For more information, see the Math Overflow FAQ.

Participation on the website has been increasing rapidly since the first post (September 28). Here’s the Alexa data, which seem to indicate that usage has been growing (Alexa is not very reliable for low-volume sites, since it uses a small sample of users and most Math Overflow users may not be using Alexa’s toolbar).

The software and site layout seem well-designed to encourage participation. The long-term performance seems unclear, since a lot depends on how effectively the site is able to allow users to fruitfully explore past questions and answers and discover things similar to what interests them. But, as of now, it has a bunch of interesting questions, and seems to have reached the ears of a lot of people who’re interested in asking good questions and giving good answers.

It’s been some time since I last wrote about the polymath project (see this, this for past coverage), and an even longer time since I wrote an extremely lengthy blog post about Michael Nielsen’s ideas about collaborative science.

The first polymath project, polymath1, was about the density Hales-Jewett theorem. This was declared a success, since the original problem was solved within about a month, though the writing up of the paper is still proceeding. The problem for the project was proposed by Timothy Gowers.

Terry Tao (WordPress blog) has now started a polymath blog discussing possible open problems for the polymath project, strategies for how to organize the problem-selection and problem-solving process, and other issues related to writing up final solutions and sharing of credit.

In this blog post, Jon Udell reflects on how the introduction of LaTeX typesetting into wordpress was a positive factor in getting talented mathematicians like Terence Tao and Timothy Gowers into the blogosphere, and leading to innovative user projects such as the polymath project. Udell notes that introducing existing typesetting solutions into new contexts such as Internet blogging software can have profound positive effects.

Christina Pikas offers her take. It seems to me that her post is a summary plus some cautious optimism and historical context.

Flip side?

This post on the backreaction blog considers the negative side of open science. Influenced by James Surowiecki’s book The Wisdom of Crowds, this talks about the pitfalls of too much openness leading to science developing only in certain directions that attract more immediate attention, compromising on independence during the decision-making process of what direction to pursue research in. The author says that it may in fact be the case that there is already too much collaboration in science leading to the underdevelopment of lines of research that have long-run promise but will not attract collaborators in the short term because others may not think that highly of them.

The backreaction blog post is a response, not specifically to the polymath project (in fact, it does not even mention polymath and is not specific to mathematics). Rather, it is a reaction to this post on the Open Science blog.

I think there are two somewhat differing and confused aspects here. The first is whether scientific findings should be open and widely available. The second is whether science should be done more collaboratively, with early sharing of findings and solicitation of inputs from others, rapid publishing, more “polymath”-style research. These are somewhat different issues.

Massive collaboration (of the kind that Nielsen talks about) is admittedly possible only when scientific findings are openly available. But the converse is not necessarily true. Greater open availability, accessibility, and ease of interaction with scientific content do not form a sufficient prerequisite for massive collaboration. Thus, it is possible to believe that openness and accessibility of findings is a good thing with few pitfalls while massive collaboration is a mixed blessing.

The key point is that open availability of research that is in progress can be achieved without making such research prominent to other researchers, except those who specifically express their interest in it. At the same time, well-established information should be made available in a manner that makes it very easy to find and use for all, as I’ve argued before. My paradoxical view is that the maximum openness can be achieved with the minimum downside if we concentrate on making sure that the material that is already well-known and clearly areed upon is made very widely available and “in-your-face”.

What about new and counterintuitive ideas?

For newer research and speculative findings, we need modes of availability that minimize the dangers of groupthink. One way of doing this would be to have incomplete information, betting, challenges, and competition.

For instance, in sharp contrast to polymath-style projects, one thing that could be done is that one person announces that he/she is going to conduct a certain experiment with a certain methodology and also explains how different outcomes for the experiment will result in different conclusions. This is put up openly and inconspicuously. The person then withdraws into his or her inner shell (or research lab) to do the experiment, and then puts up the results. In the meantime, the formulation of the experiment attracts some people (though not a lot) and they post their varying ideas on whether there could be problems with the methodology.

In this case, the people evaluating the experiment (who would be the select few with interest in that area) are not biased by the actual outcomes of the experiment when questioning the potential merits or demerits of the methodology. This could in principle reduce the possibility of herding.

This “betting” kind of approach could provide greater incentives to work on less apparently useful things. Suppose person A has an alternative explanation of some hypothesis, and designs an experiment to test it. But person A is uncertain about getting collaborators because nobody believes her hypothesis.

So she designs an experiment and puts up the methodology online (without actually carrying out the experiment). She challenges detractors to come up with possible flaws in the methodology. Detractors come up with such potential flaws. The methodology is accordingly fixed. Once the methodology is fixed to the comfort of the detractors, both she and the detractor are convinced that the experiment will yield opposing outcomes. This is an ideal time to place a bet. (This brings us to ideas of futures markets for ideas, which is an intriguing topic).

Here, openness contributes not so much by encouraging cooperative collaboration but by encouraging competitive collaboration.

Such a system could be adapted in many cases to mathematics, even though mathematics has fewer of these “experiments”.

Again, however, this runs the risk of getting people too much into wanting to find “counterintuitive” things rather than doing the good old solid work of progressing on the existing foundations by proving that epsilon more (too much cooperative collaboration probably leads to the opposite problem). Further, even with this, there may be some ideas where the problem isn’t that they’re counterintuitive but rather that they seem totally unimportant because people aren’t used to thinking in terms of them at all.

Here, a “futures market” type of idea may be useful. A person who is convinced that an apparently unimportant idea could be very important when developed further, can make such a claim. The exchange (or community, or market) then asks for something (the stake of the person’s reputation) in exchange for a large reward if that person turns out to be successful in developing the idea to a great level.

free, permanent, full-text, online access to peer-reviewed scientific and scholarly material.

In an earlier blog post, I discussed some issues related to open access. Here, I attempt to look at the matter more comprehensively.

Rationales for open access

There are many rationales for open access. The simplest rationale is that open access means reduced cost of access to information, which allows more people to use the same research. Since the marginal cost of making research available via the Internet to more people is near zero, it makes sense from the point of view of efficiency to price access by yet another person to the research at zero.

Another rationale is a more romantic one: making scientific and scholarly publishing available openly allowsfor a free flow of ideas and a grander “conversation”. Support for this rationale also indicates that open access should be more than just free (in the sense of zero cost) access to materials, but also a license that permits liberal reuse of research materials in new contexts. Academia already has strong traditions of quoting from, linking to, and building upon, past work, but this form of open access seeks to provide a legal framework that explicitly specifies reuse rights that go beyond the traditional copyright framework of countries such as the United States. An example of such permissive licensing is the Creative Commons licenses.

As shorthand for these two rationales, I shall use cost rationale and conversation rationale.

Open access policies/mandates

One of the major problems the open access movement has faced so far is getting people to publish papers in open access journals. As long as the best papers continue to be published in closed-access journals, academics who want to read these journals will pressure their university libraries to subscribe to these journals, even when the journals overcharge. Thus, librarians are unable to push open-access terms on publishers.

To overcome this problem requires collective action by major academic institutions. The most typical action is an “open access policy” undertaken at a department-wide or institution-wide level, that says that all papers published by people in that department or university have to be available in an open access repository. Harvard University has three open access policies. As Stuart Shieber explains in this blog post, the Harvard policy should not be viewed as a mandate (though it has been described in those terms) because Harvard has no effective enforcement mechanism. Rather, it should be seen as a powerful default rule that encourages faculty who may otherwise be indifferent to move towards open access (read here Richard Poynter’s blog post parsing and picking issue with this).

Open access has been accepted by many other universities, including an institution-wide policy by MIT.

At another level, the National Institutes of Health instituted a public access policy a little while ago, that requires all papers written with funding using NIH grants to be submitted to an open access repository a year after publication. The compliance rate for the NIH policy, when I last checked, was around 50%. This means that about half of all NIH-funded papers published a year or more ago are now freely available online. If other funding agencies such as the national Science Foundation and the National Security Agency introduce similar public access policies, the effect on open access is potentially huge.

Is this lowering the cost and increasing the availability of information?

Are the cost rationale and conversation rationale for open access justified? Do open access policies reduce the overall cost of accessing information?

Some of you may be saying at this moment that it has, by definition, reduced the cost of access. I don’t think it is completely obvious. The question here is whether there were people in the past who knew they wanted to access some content and were unable to do so because of the cost, but are now able to do so. People who have a demand but whose reservation price is less than the cost of the journal (and who are not in institutes that subscribe to the journal) would be the people for whom most value was created. The real question then is: to what extent are the open access resources being accessed? Is there a sudden, explosive increase in demand when something becomes freely accessible? Chris Anderson, in his book Free, argues that there is an explosion in demand when the price drops to the magical number of zero.

I am not so convinced that there will be an explosion of demand. If there isn’t, then I don’t think the conversation rationale is supported, and the best that can be said about the cost rationale is that it distributes income away from journal publishers towards academia and libraries, while possibly expanding access to others (even if very few actually use it). While an open access policy could still be supportable on these diminished grounds, the question then becomes one of whether publishers or libraries are most likely to “do good” with the greater money that they have.

Unfortunately, I do not have any idea of whether demand has indeed exploded thanks to the open access policies, or whether new demand segments have been created. What I do know is that in much of the discussion of open access, success is measured not in terms of the effect on increased demand but rather in terms of the compliance rate with the open access policy. This contrasts with initiatives to make things free outside academia, where we regularly hear about authors and artists putting up their work for free and getting millions of downloads and a boost in sales of their books and albums.

Better strategies

An extraordinary focus on “open access” from a principled perspective might obscure what I consider the larger goals: the cost rationale and the conversation rationale. Reducing costs and increasing availability are unlikely to be achieved by a few mandates without a change in the fundamental nature of incentives. My admittedly limited reading of academic papers published in journals makes it clear that these papers are often written for small audiences. For these audiences, they are often fairly user-friendly, but for others outside the cliche, they could be quite user-unfriendly. The nature and structure of the academic paper itself is a much greater issue when it comes to ensuring access to more people than the problem of cost.

Further, the fact that open access policies need to be made across-the-board in order to ensure compliance suggests that for many academics, their incentives make them largely indifferent to open access in practice, even while they may support it in principle. For instance, in this blog post, open access advocate and University of Michigan librarian Molly Kleinman writes:

Most of the time, I only hear from faculty seeking copyright advice after they have a problem. Until they have a problem, author rights and open access are simply not on their radar. I can send postcards and emails and speak at department meetings until I’m blue in the face, but it’s going to take an outside force to convince busy academics that this is something they should be paying attention to.

What then would be a better strategy? I don’t claim to have any concrete answers, but I will here outline some possibilities. First, something has to be done to make accessibility of materials more than just a matter of anybody freely being able to download one’s work. Rather, it should be a matter of understanding what kinds of other people might benefit from looking at one’s work, including segments within one’s profession and outside it. Then, it involves being able to reach out to these diverse audiences and increasing their awareness of, and hence their ability to benefit from, such work. Of course, this is already done in academia, but my limited experience suggests that it could be done a lot more.

Is the problem one of incentives? Possibly. More importantly, it is a problem of “thinking small”. Lecturers try to teach good courses, and some of them put up course notes online, that other lecturers can benefit from. But these course notes are heavily context-based, and their usefulness as a general resource for others is severely limited. It takes a whole new degree of effort to create something like MIT OpenCourseware. OpenCourseWare was possible because of thinking big, and in particular thinking deeply about the needs of people outside MIT to access course materials in a comprehensive and useful ways.

The localized incentive to teach a course well (often at a huge cost of preparation time) without a comparable incentive to work towards the design of a high-quality, widely available course is similar to the localized incentive to write a good paper to solve a problem in the field versus thinking about the larger, broader implication of making one’s research work and the many directions one has explored widely available and accessible to a larger audience including people working in totally different areas.

Some might argue that thinking “big” takes too much time and effort and it is best if such big thinking is restricted to a few individuals who specialize in such work while most others concentrate on tackling smaller problems. This is plausible. Even if this is the case, we need to have explicit roles for people who specialize in thinking big and who also have enough of the respect and admiration of their colleagues to be able to spread the best practices emerging from their work.

To some extent, the move towards open access may simply be a reflection of the bigger thinking that has emerged in a world where people have more ready access to a lot more information and ideas. In so far as it reflects part of a broader trend of this sort, the future looks promising for both the cost rationale and the conversation rationale. In so far as open access is treated as an end in itself, I think it has little promise to revolutionize research, teaching or academia, though it might definitely ease the budget burden on libraries.