[whatitdobooboo]

Not too well on mathematical proofs, but to me this sounds like the "proved" something by not being able to disprove Batchelor's law?

Also, the part about randomness makes sense in theory but the jump to an actual proof seems a little wide to me.

Are many mathematical laws proved in this way?

[flafla2]

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

[gpm]

> proving something by proving that disproving it is impossible.

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).

[JadeNB]

> 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.

Yes, but this is only a trivial mis-speaking in what is obviously meant to be a description of proof by contradiction: proving something by showing that its opposite is impossible (not that disproving it is impossible).

[flafla2]

Interesting, I'll have to take a closer look at the papers themselves (although I am a mere mortal with insufficient background to get it fully).

[diffeomorphism]

> but to me this sounds like the "proved" something by not being able to disprove Batchelor's law?

Not quite. The statement is more like, if Batchelor's law were to fail, then it has to be in one of the following specific ways. Then you show that these specific ways can't happen and get the result.

This is a common approach and needs a few ingredients:

- How could things go wrong?

- Show that these are all possibilities and that otherwise things work (hard problem).

- Isolate each scenario from the first step and show that things don't go wrong (hard problem again).

A classical example would be something like global existence for the two-dimensional Euler equations. If the solution were to fail after a finite time, then necessarily some quantity has to go to infinity, because otherwise we could find a solution for a small additional time (Beale-Kato-Majda criterion). We then show that this quantity does not go to infinity and we are done.

[whatitdobooboo]

I see, very helpful, thanks. But doesn't this leave room for what you potentially have not accounted for? I am just trying to wrap my head around how one can write a proof by saying something doesn't occur

[diffeomorphism]

Good question. That is part of the second point. You have to show that there is no room left.

For example say you want to show that a real valued solution stays bounded. Then you have to show that a solution always exists, starts at some small value and "it never occurs that the absolute value of the solution is bigger than 1000". Because you ruled out other scenarios this then implies that the solution is always bounded by 1000.