Physical Idealization as Plausible Inference, Vol. 4 (Classic Reprint)

Excerpt from Physical Idealization as Plausible Inference, Vol. 4

Let us make the simplifying assumption, for the moment, that the appropriate fix, here is to associate a small resistance with the wire. Thus, in most cases, we wish to approximate a wire as a short circuit; here, we wish to approximate it as a resistor of very low resistance. One way to accomplish this would be to change the basic model of a wire to be a small resistor; to apply this uniformly to all circuits; and then to use mathematical techniques to justify ignoring this resistance in almost all cases. (one might, for example, treat the resistance of a wire as an infinitesimal quantity, and use techniques like those of [raiman, 86] to show that its effect is negligible.) But the computational costs of this approach are non-trivial, and become worse as more tiny but non-zero characteristics are found to be occasionally important. For example, a physical wire has non-zero capacitance and inductance; and it is possible to construct systems where these have non-negligible effects. If we include these as very small, but non-zero, quantities in our analysis of all models that contain wires, the effect is to add a second-order differential equation for each wire in the system, and then to remove it through a perturbation analysis. Pursuing this further, one would end up setting up Maxwell's equations, or Schrodinger's equations, or quantum electrodynamics for all space, and then throwing out all the lower-order terms. This is not feasible. Rather, reasoners will proceed by invoking the idealized assumption that a wire supports no voltage difference as a plausible inference at the time when a mathematical model is constructed from the given physical description, and replacing it by a more accurate assumption if necessary.

Thus, given a physical description that specifies that there is a wire between nodes A and B, we will prefer to model this by the equation VA VB if this is possible; if it is not, we will use a more detailed model.

The remainder of the paper studies theories that incorporate such inferences. Section 1 exhibits a simple example that is difficult to handle correctly if physical idealization is treated as a default rule in a non - monotonic logic. Section 2 show that probability theory can be made to give the right answer to the problem, but doing so involves some anomalies. Section 3 discusses the possibility of using deductive rather than defeasible inference. This can be made to work for simple microworlds, but it relies on an implicit notion of causal directionality that is, itself, hard to formalize. Section 4 discusses the relevance of these difficulties to physical prediction programs. Section 5 discusses a variety of examples where the failure of the idealization can be detected only over an extended interval, not at a single instant; such examples are even more difficult to analyze.

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This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.

Excerpt from Physical Idealization as Plausible Inference, Vol. 4

Let us make the simplifying assumption, for the moment, that the appropriate fix, here is to associate a small resistance with the wire. Thus, in most cases, we wish to approximate a wire as a short circuit; here, we wish to approximate it as a resistor of very low resistance. One way to accomplish this would be to change the basic model of a wire to be a small resistor; to apply this uniformly to all circuits; and then to use mathematical techniques to justify ignoring this resistance in almost all cases. (one might, for example, treat the resistance of a wire as an infinitesimal quantity, and use techniques like those of [raiman, 86] to show that its effect is negligible.) But the computational costs of this approach are non-trivial, and become worse as more tiny but non-zero characteristics are found to be occasionally important. For example, a physical wire has non-zero capacitance and inductance; and it is possible to construct systems where these have non-negligible effects. If we include these as very small, but non-zero, quantities in our analysis of all models that contain wires, the effect is to add a second-order differential equation for each wire in the system, and then to remove it through a perturbation analysis. Pursuing this further, one would end up setting up Maxwell's equations, or Schrodinger's equations, or quantum electrodynamics for all space, and then throwing out all the lower-order terms. This is not feasible. Rather, reasoners will proceed by invoking the idealized assumption that a wire supports no voltage difference as a plausible inference at the time when a mathematical model is constructed from the given physical description, and replacing it by a more accurate assumption if necessary.

Thus, given a physical description that specifies that there is a wire between nodes A and B, we will prefer to model this by the equation VA VB if this is possible; if it is not, we will use a more detailed model.

The remainder of the paper studies theories that incorporate such inferences. Section 1 exhibits a simple example that is difficult to handle correctly if physical idealization is treated as a default rule in a non - monotonic logic. Section 2 show that probability theory can be made to give the right answer to the problem, but doing so involves some anomalies. Section 3 discusses the possibility of using deductive rather than defeasible inference. This can be made to work for simple microworlds, but it relies on an implicit notion of causal directionality that is, itself, hard to formalize. Section 4 discusses the relevance of these difficulties to physical prediction programs. Section 5 discusses a variety of examples where the failure of the idealization can be detected only over an extended interval, not at a single instant; such examples are even more difficult to analyze.

About the Publisher

Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com

This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.