Not a physisist, but I found this link [0] to be understandable. The Heisenberg uncertainty principle says that the more accurately we know position, the less accurately we know momentum. When you formalize this, the result is that as the size size of a particle you are trying to measure decreases, the energy necessary to measure it increases. Since energy causes gravity, there is a length, l, where measuring a particle would require so much energy that the resulting gravity creates a black hole.
I think it is fair to say that l is just a mathematical construct, since nothing really happens at this point. But it is a useful mathematical construct that is, theoretically, relevant to our ability to observe the universe.
Having said all of that, all of this is at a much smaller scale than we have ever been able to observe. Seeing as we needed to invent new physics (relativity) to explain scales as small Mercury's orbit [1], and as large as atoms, and that these two theories are still incompatible, it is highly likely that describing physics near the plank scale would require another reinvention of physics.
Yes I too can contrive a thought experiment that combines a bunch of physical constants. It's hardly a construct. It's just a length. It might mean something. It might mean something when multiplied by pi or a tenth or whatever. It's the scale at which quantum gravity might maybe possibly could be important. Nothing more nothing less.
I would not consider the explanation I describe to be a thought experiment, but rather a mathematical derivation; although I might showing my roots as a mathematician instead of a physicist there.
Additionally, all the argument, as I presented it, shows is that there is some length beyond which we cannot observe. It turns out that we can calculate this length, and the result happens to be precisely the plank length. As a mathematician, this seems highly unlikely to be a coincidence; however I am not familiar enough with physics to know if this is a deep result, or a trivial consequence of it's definition.
To clarify the disagreement: do you agree that our current theory predicts that there is some length beyond which we cannot measure?
Yeah, that would make sense, but I don't imagine it's a hard cutoff. It may be that you get diminishing information per unit energy at a length scale related to the Planck length or something.
I think it is fair to say that l is just a mathematical construct, since nothing really happens at this point. But it is a useful mathematical construct that is, theoretically, relevant to our ability to observe the universe.
Having said all of that, all of this is at a much smaller scale than we have ever been able to observe. Seeing as we needed to invent new physics (relativity) to explain scales as small Mercury's orbit [1], and as large as atoms, and that these two theories are still incompatible, it is highly likely that describing physics near the plank scale would require another reinvention of physics.
[0] http://backreaction.blogspot.com/2012/01/planck-length-as-mi...
[1] Arguably smaller, since we have confirmed relativity without leaving Earth orbit.