What do we mean by “impossible”?
(I’m reposting this here from an old Dreamwidth post of mine, since I’ve seen people reference it occasionally and figure it would be easier to find here.)
So people throw around the word “impossible” a lot, but oftentimes they actually mean different things by it. (I’m assuming here we’re talking about real-world discussions rather than mathematical discussions, where things are clearer.) I thought I’d create a list of different things that people mean by “impossible”, in the hopes that it might clarify things. Note—because we’re talking about real-world things, time is going to play a role. (Yes, there’s not really any universal clock. Whatever.)
I’m listing these as “levels of impossibility”, going roughly from “most impossible” to “least impossible”, even though they’re not necessarily actually linearly ordered. Also, some of the distinctions between some of these may be fuzzy at times.
Level 0. Instantaneously inconsistent. The given description contains or logically implies a contradiction. It rules out all possible states at some point in time, in any universe. People often claim this one when they really mean level 2 or level 3.
Level 1. Instantaneously impossible (contingently). In the actual universe we live in, the given description is instantaneously impossible; it rules out all possible states at some point in time. I think in most discussion that isn’t about physics this isn’t actually strongly distinguished from level 0.
Level 2. Non-equilibrium. The described system fails to propagate itself forward in time; or, if a system extended in time is described, it contains an inconsistency. This is one that people often actually mean when they say something is “impossible”.
Level 3. Unstable equilibrium or possible non-equilibrium. The described system is not resilient to noise; it will not propagate itself forward in time unless exceptional conditions hold continually. This is another one that people often really mean when they say something is “impossible”.
Level 4. Unachievable. The described system is unreachable from our present state—it may make sense on its own, it may not be inconsistent with the way the world evolves in time, but it’s inconsistent with the initial conditions that hold in the real world. Yet another one that people often mean when they say “impossible”.
Level 5. Not “stably achievable”. The only path from the current state to the described state is not resilient to noise and requires exceptional conditions to hold, possibly for an extended period of time. We might also want to require that in addition that the failure modes of such a path leave us worse off than we were before or somehow prevent the same path from being used again (so that you can’t just try over and over for free).
I’m not sure that this is really complete; but, overall, the point is that when you say something is “impossible”, you should think about whether you’re actually talking about instantaneous impossibility, non-equilibrium / instability, or unachievability (and then yeah I’ve introduced some finer distinctions here).
Can you give some examples of levels 2-5?
0 and 1 are pretty obvious to me. Level 0 is “1+1=3” or “I have a program here that solves the Halting Problem”. Level 1 is perpetual motion machines and FTL.
But I’m not sure how firm the distinction is between 2 and 3, or 4 and 5.
How about these?
A dense ball of electrons with no confinement or a blob of molten tungsten not exchanging energy in a bath of liquid helium.
A spacecraft indefinitely “parked” at an unstable equilibrium (L1/L2 Lagrange points) without any course corrections.
Revive Julius Caesar exactly as he was, including his memories.
Land a rocket upright on a 1-meter target on Mars with zero controls/sensors/guidance. Everything depends on perfect initial calculations.
Yes, thanks; I think a thing worth noting here, that is a reason I originally wrote this in the first place, is that people often don’t speak in a way that strongly distinguishes “this state, instantaneously” and “this system, extended in time”. Thus I think a lot of the reason that 2⁄3 cause such confusion is that the debate often looks like Alice saying “Oh, we’ll just put a a spacecraft at L1 to deal with that”, Bob replying “But that isn’t possible”, and Alice being confused and going “What? Of course you can put a spacecraft at L1”. Yes, you can put a spacecraft at L1, but it can’t stay there indefinitely, and if the two parties don’t realize they may be using the same words in different ways—maybe the spacecraft would need to stay at L1 indefinitely to handle the particular problem being discussed, and so Bob here is thinking in terms of “does the proposed solution handle the problem”, whereas Alice is thinking in terms of the literal meaning of the word “possible” that Bob used—it can be confusing. (And of course it’s worse if the two parties don’t even agree on the instability claim that Bob here is presupposing!)