Modern Science and Its Philosophy
mechanical "explanation" or mathematical description?
WHEN the physics of Galileo and Newton put an end to the animistic period, it received the honorary title "mathematical." At that time Newton wrote the Mathematical Principles of Natural Philosophy. A pronounced antimaterialistic tendency was ascribed to this work, because in it palpable impacts of masses upon one another were replaced by a pure mathematical formula, the law of attraction. The physics of Galileo and Newton did not come to be called "mechanistic" until it had become customary to use the treatment of mechanics as a model for every other field.
The Newtonian mechanics was looked upon more and more as the standard type of a theory. Helmholtz, for example, said:
To understand a phenomenon means nothing else than to reduce it to the Newtonian laws. Then the necessity of explanation has been satisfied in a palpable way.
It was completely forgotten that in Newton's own day his theory was looked upon as a set of abstract mathematical formulas, which needed a mechanical explanation to satisfy man's desire for causality. Newton himself recognized this need, but he declined to take part in satisfying it himself, when he made the now well‑known remark, "Hypotheses non fingo." But men like Huyghens and Leibniz never considered the Newtonian theory a physical explanation; they looked upon it as only a mathematical formula. Thus even at the very beginning of the development of mechanistic physics, it was not easy to define precisely the distinctions between a "mechanistic" and a "mathematical" theory.
Much later a contradiction began to be noticed between "mathematical" and "pictural,"  and it was asserted that only the reduction to mechanics guaranteed a theory that provides a pictural representation and that without this pictural character no real understanding was possible. It was claimed, for example, that Maxwell's field equations of electrodynamics are not pictural, if they are not illustrated with a mechanical model.
There were two motives involved in the resistance to the abandonment of mechanism. First, there was no inclination to renounce the “explanatory" value that was ascribed to mechanistic theories only; second, it was feared that the abandonment of mechanistic explanations would lead to a return to medieval animistic anthropomorphic science.
But in speaking of this desire for picturization we ought to be clear as to the actual meaning of the term. The mechanical laws describe for us the ordinary experiences of daily lifethe use of tools, automobiles, firearms, as well as the movements of the planets. We find it desirable to interpret all other experiences by analogy with those that are most familiar.
The physics of the nineteenth century showed that this wish cannot be fulfilled. Electromagnetic phenomena cannot be reduced to the same mechanical laws that govern guns or tools. Nevertheless, the laws concerning electromagnetic phenomena may be considered in a broader sense as also pictural. We can verify experimentally the validity of such physical laws as, for example, Maxwell's field equations if we can derive from them a result that is directly observable experimentally. The experiment consists in observing the position of the pointer of an ammeter or of some other measuring apparatus, or the displacement of some colored spot. But these are precisely the kind of observation that we use to verify the laws of ordinary mechanics. In the end only gross mechanical events, which certainly are pictural, are derived from the equations of the electromagnetic field, and this holds in all physics, including that of the twentieth century. In this sense physical laws must be pictural if they are to have any scientific meaning at all, for otherwise they are not experimentally verifiable. The fundamental equations by themselves need not be pictural, since they cannot be submitted to any direct experimental verification.
For a long time efforts were made to set up a mechanical explanation of Maxwell's theory. There was always the feeling that without it, something essential to the understanding of electromagnetic phenomena was missing. Heinrich Hertz finally cut the Gordian knot, so to speak, when he said: "Maxwell's theory is nothing else than Maxwell's equations. That is, the question is not whether these equations are pictural, that is, can be interpreted mechanistically, but only whether pictural conclusions can be derived from them which can be tested by means of gross mechanical experiments."
These words gave birth to what we call today the "positivistic conception” of physics. Positivistic physics thus replaced mechanistic physics. The mechanistic explanation could now be abandoned as a foundation, without at the same time renouncing the achievements of the epoch of Galileo and Newton. If a positivistic conception of physics was accepted, the rejection of medieval animism was as complete as in mechanistic physics. In place of the mechanical model there was the mathematical formula with its experimentally verifiable results. In this sense, it may be said that the positivistic conception replaced the mechanistic interpretation with a mathematical one. Before anything was known about the relativity or the quantum theories, before, therefore, even the "rebirth of idealistic physics in the twentieth century," Hertz, Mach, Duhem and others had already seen that the essential point in every explanation of nature is not the mechanical model but rather the construction of mathematical relations.
The historical error is often made of connecting the struggle of Mach and Duhem for the positivistic physics with their aversion for atomism, so that a victory for atomism was considered a defeat for positivism. In reality the champions of atomism, Maxwell and Boltzmann, were exactly of the same opinion concerning the general nature of a physical theory as Hertz and Mach. The difference in their views about the value of atomistic theories arose only because they differed in their estimates of the convenience with which the actually known physical phenomena could be derived from these theories.
A few quotations from the writings of Maxwell and Boltzmann will at once make clear what they thought concerning the structure of physical theories and their connection with experience.
In the introduction to his treatise, On Faraday's Lines of Force, Maxwell expressed himself quite clearly on these questions. Boltzmann says in the notes to his German translation of this paper:
Maxwell's introduction proves that he was as much of a pioneer in the theory of knowledge as he was in theoretical physics. All the new ground in the theory of knowledge that was broken in the next forty years, is already clearly marked out in these few pages; indeed the very ideas are illustrated. Later epistemologists treated all this more fully, but also with greater bias. They set up rules for the future development of a theory after it bad already developed and not before, as was the case with Maxwell.
Maxwell describes how he first found a convenient mathematical formulation of the laws of electricity and magnetism that were already known, and then used the mathematical concepts so created for the construction of the new laws:
In order therefore to appreciate the requirements of the science, the student must make himself familiar with a considerable body of most intricate mathematics, the mere retention of which in the memory materially interferes with further progress. The first process therefore in the effectual study of the science, must be one of simplification and reduction of the results of previous investigation to a form in which the mind can grasp them. The results of this simplification may take the form of a purely mathematical formula or of a physical hypothesis. 
But Maxwell by no means thinks that there is an essential antithesis between a "purely mathematical formula" and a "physical hypothesis." He judges them only according to their practical exchange value, and finds that each of them has its advantages and disadvantages. Hence he looks for a kind of theory that is more general than either and that combines the advantages of both without the disadvantages. This more general sort of theory Maxwell finds in the physical analogy. It comprehends both the physical hypothesis, in which an analogy is drawn between electromagnetic events and a mechanical model, and a mathematical formula, which points out an analogy between the phenomena of electricity and certain mathematical relationships that are given by the electromagnetic‑field equations.
In the first case [of a purely mathematical formula] we entirely lose sight of the phenomena to be explained; and though we may trace out the consequences of given laws, we can never obtain more extended views of the connections of the subject. If, on the other hand, we adopt a physical hypothesis, we see the phenomena only through a medium, and are liable to that blindness to facts and rashness in assumption which a partial explanation encourages . . . In order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other. Thus all the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. Passing from the most universal of all analogies to a very partial one, we find the same resemblance in the mathematical form between two different phenomena giving rise to a physical theory of light. 
In the introduction to his lectures on the theory of gases Boltzmann's view is brought out very clearly. In one place he says:
The question as to the fitness of the atomistic philosophy is naturally wholly untouched by the fact stressed by Kirchhoff that our theories about nature bear the same relation to it as symbols to the things symbolized, as letters to sounds or as musical notes to tones, and by the question whether it may not be expedient to consider our theories as pure descriptions so that we may always recall their relation to nature. Therefore the question is whether the pure differential equations or atomism will one day turn out the more complete descriptions of phenomena . 
1 We here translate the German word anschaulich by the English pictural, which we take to imply, as the German does, both the ordinary visual perception of objects and the mental visualization of them. See also Chapter 7. [> main text]
2 J. C. Maxwell, "On Faraday's Lines of Force," Transactions of the Cambridge Philosophical Society 10, part 1 (1855); Scientific Papers of James Clerk Maxwell (Cambridge University Press, 1890), vol. 1, p. 155. [> main text]
3 Ibid., pp. 155 f. [> main text]
4 L. Boltzmann, Vorlesungen über Gastheorie (Leipzig: Barth, ed. 3, 1923), Vol. 1, p. 6. [> main text]
SOURCE: Frank, Philipp. Modern Science and Its Philosophy. Cambridge, MA: Harvard University Press, 1949. Reprint: New York: George Braziller, 1955. Chapter 6, Mechanical "Explanation" or Mathematical Description?, pp. 138-143.
Modern Science and Its Philosophy: Contents
Vienna Circle, Karl Popper, Frankfurt School, Marxism, McCarthyism & American Philosophy: Selected Bibliography
Positivism vs Life Philosophy (Lebensphilosophie) Study Guide
Salvaging Soviet Philosophy (1)
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