| From the article: > In their first paper, the mathematicians focused on what happens during the mixing process to two points of black paint that begin the process right next to each other. They proved that the points follow chaotic paths and go off in their own directions. In other words, the nearby points can’t ever get stuck in a vortex that will keep them close forever. > “The particles move together initially,” Blumenthal said, “but eventually they split apart and go in completely different directions.” > In the second and third papers, they took a broader look at the mixing process. They proved that in a chaotic fluid, generally speaking, the black and white paint mixes as quickly as possible. This further established that the turbulent fluid doesn’t form the kinds of local imperfections (vortices) that would prevent the elegant global picture described by Batchelor’s law from being true. > In these first three papers, the authors did the hard mathematics required to prove that the paint mixes in a thorough, chaotic fashion. In the fourth, they showed that in a fluid with those mixing properties, Batchelor’s law follows as a consequence. So no, they are not "proving something by not being able to disprove it." A better way of phrasing their strategy is, "proving something by proving that disproving it is impossible." In Computer Science, there is a similar concept for proving asymptotic bounds of algorithms called an "adversarial proof." The idea is, given some query that your algorithm performs (e.g. in a graph algorithm, a query could be "are two vertices connected") come up with a worst-case adversary that answers queries in the absolute worst way possible, that would necessitate even more queries to complete the problem. In this way, you can prove a universal lower bound for the cost of solving some problem. See [1]. In this case, the adversary is trying to come up with the worst-case initial conditions for this particular brand of turbulence. Basically they are saying, no matter what, you couldn't come up with an initial condition that challenges Batchelor's law more. [1] https://www.cs.cmu.edu/afs/cs/academic/class/15451-s20/www/l... Section 3.2 |
No, I take issue with this phrasing as well. There are things that can neither be proven or disproven (by godel's theorem), proving that disproving it is impossible would not have been sufficient.
Without having read beyond what is in the article, I imagine what they must have shown is that
1. For all systems x, if x does not obey Batchelor's law than neither would the thing they are talking about in the 4th paper.
2. The system they are talking about in the 4th paper obey's Batchelor's law.
The immediate corollary is all systems obey Batchelor's law, otherwise you would have a contradiction (the 4th system both would and would not).