I'm at the Joint Mathematics Meetings in Seattle. One of the first talks (8:00 AM!) of the first day was given by Prof. Tim Gowers on How might Mathematics be better disseminated (slight change in the title from what was originally published). Prof. Gowers highlighted several themes on better ways of doing mathematical research as well as better ways of disseminating the same, which he has been working on, and popularizing on the web, in recent years. Here are some of my own interpretations of what he talked about.
In the current setting of mathematical research, most emphasis is on being the "first to prove the theorem", and the basic unit of discourse is the peer-reviewed journal article. There are more than a few things wrong with this set up, the main one being that the wheel is reinvented repeatedly! Here is a type of result which could be very useful. If Lemma A is true, then BIG RESULT B is true, which would be fantastic news!. But, I have a counterexample for Lemma A :-(. Unfortunately, such a negative result cannot be "published" in a peer-reviewed journal article. Hence, others fumble around and reinvent the same result!
Another major drawback of the current system is that mathematical conversation happens at an inordinately slow pace. Someone proves a theorem, which appears in the journal in two years time. Then someone else reads it, comes up with a modification or a simpler proof, which appears in another journal two more years later! But in the current internet age, it's only natural to expect conversations occurring at a much faster pace (live tweets, any one?).
While electronic publication has made all research easily searchable, the search capacity is strong only in one direction, so to speak. If you know what you're looking for in terms of keywords, then it's easy to find it by search, e.g., you want to know what Szemerédi's theorem states, a quick search pulls up multiple relevant web pages. What would be very useful is a way to search "in reverse" using some limited keywords or partial statements (and not the name itself!) to see if such a result is already known. In other words, the community needs a mathematics research database that allows semantic search. Today, forums such as the mathoverflow often gets us accurate answers to questions of the form "has this been done before?".
In current times, mathematics research should use the internet - both for conducting it as well as to disseminate it. But someone having a high rating on the Math StackExchange for posting numerous answers, or who writes popular blog articles on otherwise difficult to understand math papers, is not rewarded for these activities in the current system. To get tenure, you better have the required number of papers in the top(-enough!) journals! One could have a potentially huge impact by writing easy-to-understand expository blog posts on otherwise hard-to-read set of mathematical papers written by other authors. These blog posts could in turn spur new contributions from others, who would not have had the inclination otherwise to digest the original papers. As such, this effort could potentially be worth much more than publishing a paper with a new theorem! But then again, the current system has no means of rewarding such expository efforts.
In follow-up conversation, Prof. Gowers agreed that senior/reputed mathematicians such as himself could afford to spend more time and effort on such endeavors on the internet without worrying about the rewards or evaluations. For tenure-track faculty or other junior researchers, the best course might well be to do both - publish via the conventional means, but also spend some effort on internet-based activities. Further down the line, we as a community would want to be able to judge how "impactful" a series of blog posts have been, so as to reward the same. But we must be careful not to go down the same path as overusing journal impact factors (so, don't judge the impact of a blog post by the number of times it has been re-tweeted, or +1-ed!).
In the latter part of the talk, Prof. Gowers highlighted four of his personal efforts in this regard - a reform of the journal system (Discrete Analysis), informal mathematical writing (via his blog), polymath projects, and automating proof discovery. He also presented his ideas of what mathematics research would be like in 20-30 years from now, and then in 50-60 years from now. Not surprisingly, computers would be expected to do most of the heavy (and light:-) lifting in the future.
A question from the audience inquired about the place of real, i.e., face-to-face, conversations on mathematics that happen at conferences. Would there be less of a place for them in mathematics research in the future? While agreeing that such conversations have their place, Prof. Gowers observed that may be the back-and-forth postings on a polymath project (with time-stamps!) allows the posters the time to understand the subject better, and think through before posting what they want to post. An instant face-to-face conversation would not give that luxury!
In summary, much need to be changed for mathematics research to be done right and be impactful in the internet age. Individual researchers need to be a bit bold, and not worry too much about rewards and ratings when spending time and effort on the internet. The more people who do so, the sooner the inevitable transition would happen!