# Mathematics, Science, and Blogs

Michael Nielsen wrote a lovely essay entitled “Doing science online” about  mathematics, science,  and blogs. Michael’s primary example is a post over Terry Tao’s blog about the Navier-Stokes equation and he suggests blogs as a way of scaling up scientific conversation. Michael is writing a book called “The Future of Science.” He is a strong advocate of doing science in the open, and regard these changes as truly revolutionary.  (The term “Science 2.0″ is mentioned in the remarks.)

Michael’s post triggered Tim Gowers to present his thoughts about massive collaboration in mathematics, and this post is also very interesting with interesting follow-up remarks.  Tim Gowers mentioned the n-category cafe as a place where a whole research programme is advanced on a blog. Terry Tao mentioned comments on posts on his open-problems series as having some value. He  mentioned, in particular, the post on Mahler’s conjecture. (Also I think some discussions over Scott Aaronson’s blog had the nature of discussing specific technical math (coming from CS) problems.)

Tim actually proposes an experiment: trying to solve collectively a specific math problem. This would be interesting!!! I suppose we need to give such an effort over a blog a longer than ususal life-span – a few months perhaps. (And maybe not to start with a terribly difficult problem.) (What can be an appropriate control experiment though?)

Ben Webster in “Secret blogging seminar”  mentioned, in this context, earlier interesting related posts about “Working in secret“.

Christian Elsholtz mentioned on Gowers’s blog an intermediate problem (called “Moser’s cube problem”) where you look not for combinatorial lines (where the undetermined coordinates should be 1 in x 2 in y and 3 in z), and not for an affine line (where it should be 1,2,3 in x y and z in any order), but for a line: it can be 1 in x 2 in y and 3 in z or 3 in x 2 in y and 1 in z.

Update: Things are moving fast regarding Gowers’s massive collaboration experiment. He peoposes to study together a new approach to the $k=3$ “density Hales -Jewett theorem”. A background post apears here. Hillel Furstenberg and Izzy Katznelson’s proof of the density Hales-Jewett theorem was a crowning achievement of the ergodic theory method towards Szemeredi’s theorem. Like the case for Furstenberg’s proof of Szemeredi’s theorem itself the case $k=3$ was considerably simpler and had appeared in an earlier paper by Hillel and Izzy. The recent extensions of Szemeredi regularity lemma that led to simpler combinatorial proof of Szemeredi’s theorem did not led so far to simpler proofs for the density Hales-Jewett case. If you look at Tim’s background post let me ask you this: What is the case $k=2$ of the density Hales-Jewett’s theorem? It is something familiar that we talked about!

Here is a particularly silly problem that I suggested at some point along the discussions: How large can a family of subsets of an $n$-element set be without having two sets A and B such that the number of elements in A but not in B is twice the number of elements in B but not in A?

Update: This problem was completely reolved by Imre Leader and Eoin Long, their paper Tilted Sperner families contains also related results and conjectures.

Massive collaboration in art (click the picture for details)

Q: What is the case $k=2$of the density Hales-Jewett’s theorem? A: It is  Sperner’s theorem! (that we discussed in this post.)

I will keep updating about news from Tim’s project. [Last update is from October 21].  More updates: Tim’s project is getting quickly off the ground. A useful wiki was established. More update: It is probably successful