That makes a lot of sense. I asked about explicit declaration versus implicit assumption because assumptions of this sort do exist in social models. They’re just treated as unmodeled characteristics either of agents or of reality. We can make these assumptions because they either don’t inform the phenomenon we’re investigating (e.g. infinite ammunition can be implicitly assumed in an agent-based model of battlefield medic behavior because we’re not interested in the draw-down or conclusion of the battle in the absence of a decisive victory) or the model’s purpose is to investigate relationships within a plausible range (which sounds like your use case). That said, I’m very curious about the existence of models for which explicitly setting a boundary of infinity can reduce computational complexity. It seems like such a thing is either provably possible or (more likely) provably impossible. Know of anything like that?
I see your distinction now. That is a good classification.
To go back to the low-Mach/incompressible flow model, I have seen series expansions in terms of the Mach number applied to (subsets of) the fluid flow equations, and the low-Mach approximation is found by setting the Mach number to zero. (Ma = v / c, so if c, the speed of sound, approaches infinity, then Ma goes to 0.) So it seems that you can go the other direction to derive equations starting with the goal of modeling a low-Mach flow, but that’s not typically what I see. There’s no “Mach number dial” in the original equations, so you basically have to modify the equations in some way to see what changes as the Mach number goes to zero.
For this entire class of problems, even if there were a “Mach number dial”, you wouldn’t recover the nice mathematical features you want for speed by setting the Mach number to zero in a code that can handle high Mach physics. So, for fluid flow simulations, I don’t think an explicit declaration of infinite sound speed reducing computational time is possible.
From the perspective of someone in a fluid-flow simulation (if such a thing is possible), however, I don’t think the explicit-implicit classification matters. For all someone inside the simulation knows, the model (their “reality”) explicitly uses an infinite acoustic wave speed. This person might falsely conclude that they don’t live in a simulation because their speed of sound appears to be infinite.
It seems like such a thing is either provably possible or (more likely) provably impossible. Know of anything like that?
Btrettel’s example of ray tracing in thermal radiation is such a model. Another example from social science: basic economic and game theory often assume the agents are omniscient or nearly omniscient.
False: Assuming something is infinite (unbounded) is not the same as coercing it to a representation of infinity. Neither of those examples when represented in code would require a declaration that thing=infinity. That aside, game theory often assumes players have unbounded computational resources and a perfect understanding of the game, but never omniscience.
That makes a lot of sense. I asked about explicit declaration versus implicit assumption because assumptions of this sort do exist in social models. They’re just treated as unmodeled characteristics either of agents or of reality. We can make these assumptions because they either don’t inform the phenomenon we’re investigating (e.g. infinite ammunition can be implicitly assumed in an agent-based model of battlefield medic behavior because we’re not interested in the draw-down or conclusion of the battle in the absence of a decisive victory) or the model’s purpose is to investigate relationships within a plausible range (which sounds like your use case). That said, I’m very curious about the existence of models for which explicitly setting a boundary of infinity can reduce computational complexity. It seems like such a thing is either provably possible or (more likely) provably impossible. Know of anything like that?
I see your distinction now. That is a good classification.
To go back to the low-Mach/incompressible flow model, I have seen series expansions in terms of the Mach number applied to (subsets of) the fluid flow equations, and the low-Mach approximation is found by setting the Mach number to zero. (Ma = v / c, so if c, the speed of sound, approaches infinity, then Ma goes to 0.) So it seems that you can go the other direction to derive equations starting with the goal of modeling a low-Mach flow, but that’s not typically what I see. There’s no “Mach number dial” in the original equations, so you basically have to modify the equations in some way to see what changes as the Mach number goes to zero.
For this entire class of problems, even if there were a “Mach number dial”, you wouldn’t recover the nice mathematical features you want for speed by setting the Mach number to zero in a code that can handle high Mach physics. So, for fluid flow simulations, I don’t think an explicit declaration of infinite sound speed reducing computational time is possible.
From the perspective of someone in a fluid-flow simulation (if such a thing is possible), however, I don’t think the explicit-implicit classification matters. For all someone inside the simulation knows, the model (their “reality”) explicitly uses an infinite acoustic wave speed. This person might falsely conclude that they don’t live in a simulation because their speed of sound appears to be infinite.
Btrettel’s example of ray tracing in thermal radiation is such a model. Another example from social science: basic economic and game theory often assume the agents are omniscient or nearly omniscient.
False: Assuming something is infinite (unbounded) is not the same as coercing it to a representation of infinity. Neither of those examples when represented in code would require a declaration that thing=infinity. That aside, game theory often assumes players have unbounded computational resources and a perfect understanding of the game, but never omniscience.
A better term is “logical omniscience”.