For the math geeks in my audience (and you know who you are), there's some followup to this post about Elin Oxenheilm and her claimed solution to the sixteenth Hilbert problem. Gustav Holmberg has several posts (here, here, here, and here) which discuss various aspects of Oxenhielm's claim. One of the things he notes is that press releases announcing big claims like this often go through a university's official channels, which conveys an official endorsement and thus acts somewhat like a "peer review" in and of itself. In this case, Oxenhielm sent out a press release on her own. Judge for yourself if this lowers her credibility.
An interesting side note is that a former teacher of Oxenhielm's who was thanked in the credits of the paper has issued an open letter in which she says "the paper is incomplete and includes serious mistakes, which I think any educated mathematician can easily see". Oxenhielm has a response to this, though you'll need to be able to read Swedish to get the exact meaning.
Another dissenter is Grigori Rozenblum, a professor of math at Chalmers University, who sent this letter to the editors of Nonlinear Analysis as well as leaving a comment on my original post.
In the end, Holmberg speculates about the role of the Internet in this discussion.
But let's leave the media aside for a moment and instead look at two comment threads on the blog unstruct.org; here and here. There are a number of people discussing the affair there and several seem to me to be knowledgeable in the field of maths, some are really advanced: Grigori Rozenblium, professor of mathematics at Chalmers technical university is one of the writers there, another one is a PhD &c. It is, thus, not the letters column of your average daily newspaper. Topics include womens' role in the academic system, proofs in mathematics, the peer review system &c.These winding threads contain, among other things, comments that used to be buried in other media: letters to colleagues, e-mail, gossip over beer at conferences, discussions in workshops &c. But where do the threads at unstruct.org fit in?
Now, I suspect that you could have at least two quite distinct paths taken in the development. You could have the peer review the peer reviewers scenario. Here, scientists decide to use the force of the Net, social software such as blogs, preprint archives and what have you, to make a more or less open discussion about the quality of papers, published or un-published. You would still have peer review, but since that is obviously not always to be trusted, you would have a semi-public discussion (yes, semi-public instead of public, not because it would be locked away behind passwords but rather because ordinary citizens would not be able to understand a word) about papers and other results. The Net result would be better science through public scrutiny and an opening up of scientific practice, just like when the guys at Royal Society decided to do things in the open some 360 years ago. It would not be a world without peer review. It would be a world with a better peer review.
Or you could have the don't rock the boat scenario. Anything that would take place outside of the classical arenas - peer review, publication, conferences, &c - would be deemed bad science. Some of the old school players among the publishing corporations could perhaps be interested in such a path.