WHAT IS MAXWELL'S METHOD?

In very general terms, it is a method, or strategy, or procedure to be used when developing and defending a theory about “unobservables,” when whatever experimental evidence exists is not sufficient to establish the theory.

It is a method designed for, or at least particularly appropriate for, theories in which the “unobservables” comprise a microsystem of which some observable macrosystem is claimed to be composed, and in which the claim is that the behavior of the microsystem causes or determines that of the macrosystem. In what follows I will offer a general characterization of the method that goes well beyond what Maxwell himself provides. In doing so I will distinguish four components and illustrate each by reference to what Maxwell actually does in his “physical speculations” about molecules.

Independent Warrant

First, whatever reasons one can offer should be given in favor of the existence of the postulated unobservables that determine macrobehavior, in favor of the central and distinctive principles introduced, and in favor of supposing that such principles are applicable to these unobservables. Such reasons can be of different sorts and may include: (a) appeals to experimental results and observations, arrived at independently of the theory in question, usually from other domains; these may provide a causal-inductive or an analogical basis for supposing that the macrosystem is composed of some type of unobservables that produce some of the observed behavior of the macrosystem;^[204] (b) a methodological appeal to the “fundamental” character and simplicity of the principles being applied to those unobservables; and (c) an inductively based appeal to the success of these principles in other domains when applied to objects with the same or similar properties as those attributed to the unobservables. The reasons offered may vary in their strength, but they are not of the form “if we make these assumptions then we can explain and predict such and such phenomena,” and they are not sufficiently strong to prove that the theory is true.

In other writings I have said that such reasons supply “independent warrant.”^[205] They provide some epistemic reasons for believing the hypotheses in question that are independent of the explanatory and predictive power of the assumptions.

Maxwell seeks to develop and defend a general molecular theory of gases and liquids that governs, relates, and interprets properties and phenomena such as pressure, volume, temperature, density, specific heat, and diffusion. In accordance with the “method of physical speculation,” the first thing he wants to provide are some reasons for making the molecular assumptions he does, including, most important, the assumption that bodies are composed of molecules, and that these satisfy classical principles of dynamics. He offers three different sorts of reasons.

A reason he proposes for assuming that bodies are composed of molecules of the sort postulated is that “whatever may be our ultimate conclusions as to molecules and atoms, we have experimental proof that bodies may be divided into parts so small that we cannot perceive them” and that by “particle” he means a small, possibly unobservable, part of a body, not some ultimate or indivisible “atom.” In his 1875 paper Maxwell does not say what such “experimental proof” is, but it is likely that he is thinking of various claims, made in his book Theory of Heat (first published in 1871), starting with the idea that it has been experimentally established that heat is not a substance (caloric) but a form of energy.¹¹ The energy of a body, he continues in that book, is either kinetic energy due to motion, or potential energy due to the body's position with respect to other bodies. But, he claims, heat cannot be the latter, because the presence of another body is not necessary for heat radiation. So it is due to motion, but not that of the body as a whole, since a body radiates heat even when stationary. He concludes: “The motion which we call heat must therefore be a motion of parts too small to be observed separately....

We have now arrived at the conception of a body as consisting of a great many small parts, each of which is in motion. We shall call any one of these parts a molecule of the substance.”^[206] ^{^[207] Maxwell offers two sorts of reasons for applying dynamical principles to the postulated set of unobservables. The first, which is empirical, is that such principles have been successful in astronomy and electrical science. Maxwell does not explicitly draw the inductive inference from this that such methods will therefore be successful for the kinds of phenomena he is concerned with. But this does seem implicit in his thought. The second involves claims that are methodological or conceptual. One is that, on his view, and that of most nineteenth-century physicists, dynamical explanations of phenomena are complete so that no further explanations are “necessary, desirable, or possible.” Another is at least an implicit appeal to simplicity, when he says that “of all hypotheses as to the constitution of bodies, that is surely the most warrantable which assumes no more than that they are material systems, and proposes to deduce from the observed phenomena just as much information about the conditions and connections of the material system as these phenomena can legitimately furnish.”^[208] Here the idea is that the basic molecular assumptions he is and will be making will satisfy a standard of simplicity by explaining macrosystems composed of bodies in terms of microsystems composed of bodies, and so introduce no new ontological category.
Having presented some reasons in support of the assumption that gases and liquids are composed of molecules and that they are subject to dynamical principles, Maxwell proceeds by formulating a virial equation derived by Clausius from classical mechanics as applied to a system of particles constrained to move in a limited region of space, and whose velocities can fluctuate within certain limits. The equation relates the pressure and volume of a gas or fluid to the total kinetic energy of the system of particles of which it is composed, the forces of attraction or repulsion between the particles, and the distances between them.
Maxwell writes the equation as follows: pV = 2/3T-2/3XX(1/2Rr).^[209]
In using this equation, Maxwell and Clausius are assuming that gases and fluids are composed of unobservable particles; this is not something which is proved by proving the equation itself. (In his discussion Maxwell uses the terms “particle” and “molecule” interchangeably.) Since the equation is derived from classical mechanics, support for which comes from observations of the behavior of observable bodies, he is also supposing that such observations provide some independent warrant for the claim that if the postulated particles exist they satisfy the virial equation as well. (As we will see next in considering the second part of Maxwell's method, he uses this equation to explain and give molecular interpretations of known gaseous phenomena.)
None of the facts Maxwell cites as independent warrant, separately or together, establishes Maxwell's initial hypothesis that gases are systems of particles or molecules satisfying the Clausius virial equation. But they do constitute at least some reason in favor of such a hypothesis. In Maxwell's own terms, such a hypothesis cannot “be derided as mere guess-work.”
Derivations and Explanations of Known Phenomena
Second, known properties, laws, and experimentally established deviations from these laws of the macrosystem should be explained by invoking properties of, and principles governing, the unobservables that comprise the postulated microsystem. More specifically, known properties of the macrosystem should be characterized as determined by, or identical with, certain properties attributed to the microsystem. And laws governing the macrosystem, and deviations from them, should be derived from assumptions regarding the microsystem defended in the first component. If they are so derived, then whatever justification for the assumptions is claimed can be claimed not just on the basis of the independent warrant but also on the basis of known laws and phenomena that are derivable from those assumptions.^[210]
Maxwell considers a gas with observable properties of temperature, volume, pressure, specific heat, and the like, which is subject to known laws and known deviations from them, and explains these properties, laws, and deviations using known properties and laws governing dynamical systems in general involving bodies in motion.
Employing Clausius's virial equation, and assuming that the pressure and volume of a gas are simply the pressure and volume of the postulated molecular system and that the temperature of the gas is proportional to the mean kinetic energy of the molecules, Maxwell derives Boyle's law for gases; and using the virial equation, he explains why known deviations from the law occur at low temperatures and high densities (see the appendix to this chapter). He considers the derivation of Boyle's law, and of deviations from that, to count in favor of the theory, even to provide at least some reason (or part thereof) for thinking the theory is true. But this is so only if there is some independent warrant for basic assumptions in the theory. Theoretical Development
The postulation of the set of unobservables satisfying the properties and principles introduced will suggest a range of questions about what properties and principles in addition to those introduced in components 1 and 2 these unobservables satisfy. To the extent possible, the theorist should attempt to develop the theory further by formulating and answering these questions. Doing so will usually require the introduction of new theoretical assumptions about the unobservables for which there may or may not be independent warrant, and derivations of new results that may or may not be testable by known means. Judging from the amount of time Maxwell devotes to it, this “theoretical development” of the theory—which can go well beyond what is contained in the two components discussed here—is a crucial part of Maxwell's idea. Its focus is on providing more and more information about the postulated microsystem, whether or not this yields testable predictions and explanations of properties of the macrosystem.
Maxwell introduces a series of questions about the unobservable molecules he postulates, including these: What is the mean distance traveled by a molecule before striking another molecule (the mean free path)? What is the motion of molecules after collision? Are all directions of rebound equally likely? What is the distribution of molecular velocities? He introduces various new assumptions which enable him to answer these and many other ques- tions.^[211] In the case of the last question, Maxwell derives a distribution law, now bearing his name, that relates the number of molecules with velocities between given limits to the total number of molecules in the sample of gas and to the velocities themselves.
In doing so he makes various new assumptions about molecules, including that molecular components of velocity in different directions are independent, and that the fraction of molecules in a unit volume does not depend on their direction but only on their speeds. He had no way of experimentally verifying these assumptions, or experimentally determining any of the quantities in the law, and hence no way of experimentally verifying the law.^[212] It is a “purely theoretical” conclusion.^[213] Unsolved Problems
In addition to formulating, defending, and developing the theory in accordance with the three points noted previously, problems with the theory should be noted. These can include a reference to known laws and properties of the macrosystem that have not yet been explained, as well as to experimental results that are not in accord with certain consequences of the theory. This, of course, is not a way of defending the theory. But it is a way of suggesting aspects of the theory that need further development, and of defending the “theorist” by showing that he is aware of these aspects.
Maxwell derives some conclusions from his theoretical assumptions that are contradicted by experiments. The most important of these he considers to be a derivation (first done in 1860) of the ratio of the specific heat of a gas at constant pressure to its specific heat at constant volume. According to theoretical calculations, in the best case, assuming that molecules are mere material points incapable of rotation, the ratio is 1.66, whereas the observed value is 1.4. This difference Maxwell considers “too great for any real gas.” And if we suppose that molecules can vibrate, so that there are at least six degrees of freedom, the theoretical calculation of specific heat ratios will be a maximum of 1.33, which is too small for hydrogen, oxygen, nitrogen, and several other gases. Maxwell says that he considers this “to be the greatest difficulty yet encountered by the molecular theory.”^[214] In addition to this problem Maxwell mentions several properties of gases, including electrical ones, that neither he nor anyone else had explained in molecular terms.
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Source: Achinstein P.. Evidence, Explanation, and Realism: Essays in Philosophy of Science. Oxford: Oxford University Press,2010. — 344 p.. 2010

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