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Similar Systems and Dimensionally Invariant Laws

Published online by Cambridge University Press:  14 March 2022

R. Duncan Luce
R. Duncan Luce*
Affiliation:
The Institute for Advanced Study, Princeton
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Abstract

Using H. Whitney's algebra of physical quantities and his definition of a similarity transformation, a family of similar systems (R. L. Causey [3] and [4]) is any maximal collection of subsets of a Cartesian product of dimensions for which every pair of subsets is related by a similarity transformation. We show that such families are characterized by dimensionally invariant laws (in Whitney's sense, [10], not Causey's). Dimensional constants play a crucial role in the formulation of such laws. They are represented as a function g, known as a system measure, from the family into a certain Cartesian product of dimensions and having the property gφ = φg for every similarity φ. The dimensions involved in g are related to the family by means of certain stability groups of similarities. A one-to-one system measure is a proportional representing function, which plays an analogous role in Causey's theory, but not conversely. The present results simplify and clarify those of Causey.

Type
Research Article
Information
Philosophy of Science , Volume 38 , Issue 2 , June 1971 , pp. 157 - 169
Copyright
Copyright © 1971 by The Philosophy of Science Association

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Footnotes

1

In working out these ideas, I have benefited greatly from several conversations and an extended correspondence with Robert Causey. He pointed out difficulties and ambiguities with my earlier formulations, and he has aided greatly in my understanding of the issues. I believe that he agrees that the present theorems are correct, but we have never been able to reach complete agreement as to their relation to his results. I also wish to thank a referee for a number of useful comments which helped to clarify the exact relation of this work to Causey's and eliminated an error.

References

REFERENCES

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