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Published online by Cambridge University Press: 14 March 2022
The history of Physical Science appears to exhibit, periodically, a race between the acmulation of data and the ability of its codification (usually, a mathematical theory) to find a natural place (in the codifying scheme) for much of the empirical findings. If the codifying scheme is a mathematical theory, capable of interpolation and extrapolation, according to the rules of the particular branch of mathematics employed, the ablest handlers of the theory are frequently confronted with a situation in which mathematical computation alone does not suffice. In such a situation the introduction of symbols, whose interpretation is not direct and simple (from the point of view of the methodology whereby the data were procured) is resorted to. This symbol is usually involved in a mathematical relation by a species of reasoning, which may be somewhat unfamiliar and remote from customary habits of thought. It is generally accepted as physically valid, if the mathematical formulation which involves it can produce results which account for the mass of data uncovered by experimenters. As an example of such a procedure we may mention the case of the Schrödinger Ψ-symbol, a symbol which was involved in Schrödinger's famous matter wave equation. The successes which such a theory enjoyed, particularly, in its initial stages, in formally relating the relative positions and intensities of spectral lines, with the atomistic concepts of gross matter which were inferred from the data of macroscopic measurement, such as the atomic mass, the electronic charge, Planck quantum of action, etc.,—these successes, gave a certain physical dignity to the Ψ-function itself. This dignity was further enhanced by the disclosure, by Schrödinger, of four functions of the Ψ-symbol (and its complex conjugate) which held the same place in the “continuity equation” as the classical charge density and the three components of current density. The successes accompanying this discovery for spectroscopic problems, in particular, are well known to Physicists.
1. These two points of view raise the questions: What is a “good” codification; what is a “bad” codification? When it is fully appreciated that many different schemes of codification may be advantageously employed for the same set of accumulated data (the differences in the schemes of codification arising, chiefly, from the differences in the analytic elements which seem acceptible) the criteria for “good” theories and “bad” theories will probably be more frequently raised. This appears, to the writer, a problem of great moment, on account of its possible affect on the future course of the exact sciences. We require an Ethics of Codification, so to speak.
2. The simplified sketch given in this paper will be elaborated on the mathematical side in a paper to be presented for publication shortly, in one of the standard journals.
3. The term “amorphous” (coordinate system) becomes clearer when compared with the “scale” coordinate system discussed below. Every point of the amorphous coordinate system is associated with some kind of idealized clock. All these clocks are synchronized by some more or less arbitrary prescription which in the field of visual coincidences involves the use of light beams.
4. “Relativity,” Henry Holt & Co., 1921, p. 5 et seq.
5. The italics are our own.
6. We should say “amorphous system of coordinates” on account of the obliteration of distinguishing marks.
7. No finite body is free of distinguishing marks. A finite body, by postulation has boundaries and these boundaries are its distinguishing marks. For example, within a gas such as hydrogen, under constant temperature and pressure, there are no macroscopic distinguishing marks. The distinguishing mark system of such a matter configuration is formed by its boundaries; that is, for example, by the contours of the glass tube which contains the hydrogen.
8. As we shall see, these “names” become for us ordinal numbers.
9. It must be emphasized that the measured properties of matter are revealed by comparing the distinguishing marks on matter to a set of ordered distinguishing marks on another piece of matter, called a scale (discussed in the next section).
10. “The Logic of Modern Physics,” P. W. Bridgman—The Macmillan Co., New York, 1928. This essay, of remarkable lucidity and insight, we shall refer to again.
11. For the orientation of the term “visual coincidences” see “The Spacetime Scale As A Quantum Coordinate System,” Jour. Frank. Inst. Vol. 216, No. 2, August, 1933.
12. In the paper of reference 11, these spectra are called spacetime scales. Now we shall postulate here that: All scales are essentially spacetime scales, for in order to compare two (or more) coincidences, we must postulate simultaneity of record. Thus we are obliged to think of a comparator scale as having at each mark, a physical clock, which clocks must be synchronized according to some prescription which involves the agency which connects them. In the field of visual coincidences this agency is the light beam. Now a uniform scale may be called a galilean11 scale. It is with such uniform scales that we may, on last analysis, measure the macroscopic properties of mercury, such as its elastic, electric, magnetic, and thermodynamic properties. From this point of view it would appear that the quantum problem, at least, insofar as spectroscopy is concerned, involves a set of transformation equations which relate the ordering of spectrum lines (or Laué spots) by a galilean scale, to the ordering of the distinguishing marks in closer contact with gross matter (from which we codify its macroscopic properties) also with galilean scales. The quantum problem from this point of view thus appears to consist in a more or less deductive establishment of transformation equations between virtual contact scale systems, and direct contact scale systems.
13. In the procedure given here we have glossed over the whole troublesome question between the relationships of a priori to a posteriori probability. For a clear cut account of the problem involved see, e.g., “Probability and Its Engineering Uses,” T. C. Fry, D. Van Nostrand Co., New York, 1928. Chapter IV.
14. Even though not all of these parts may be necessary to define an electrometer for purposes of an operational-configurational16 physical geometry.
15. Reference 10, p. 63.
16. We have added to the term operational the term configurational throughout this paper. We have not space to go into the reason for this additional term. While Bridgman does not specifically use the configurational aspects of apparatus in the determination of dimensionality, the spirit of the essay appears to demand it. It is probably true that on last analysis the operational point of view will obtain. Nevertheless for a physical geometry, what does it matter whether two geometrically congruent parallel plate condensers (serving as part of an electrometer) are made on a lathe or by hand. The point is that a whole host of operations may be grouped under a particular class of geometric configurations of matter. Once the particular configuration is fixed the number of physical operations which determine dimensionality are enormously reduced.2
17. Not so much from the subject matter of this paper as the spirit underlying the procedure.