Lorentz to Minkowski - Constructing the Principles

Soon after the sensational reports of the eclipse observations of 1919, the London Times asked Einstein to explain his theory of relativity to its readers. He accommodated the request with a short essay that is most notable for its description of a general classification scheme that Einstein proposed for physical theories. He wrote:

Notice that there are actually three different sets of alternative characteristics mentioned in this passage, but they all seem to be treated as representing essentially the same dichotomy. We're told that constructive theories proceed synthetically on the basis of hypothetical premises, whereas principle theories proceed analytically on the basis of empirical premises. Einstein cites statistical thermodynamics as an example of a constructive theory, and classical thermodynamics as an example of a principle theory. His view of these two different approaches to thermodynamics was undoubtedly influenced by the debate concerning the reality of atoms, which Mach disdainfully called the "atomistic doctrine". The idea that matter is composed of finite irreducible entities was regarded as purely hypothetical, and the justification for this hypothesis was not entirely clear. In fact, Einstein himself spent a great deal of time and effort trying to establish the reality of atoms, e.g., this was his expressed motivation for his paper on Brownian motion. Within this context, it's not surprising that he classified the premises of statistical thermodynamics as purely hypothetical, and the development of the theory as synthetic.

However, in another sense, it could be argued that the idea of atoms actually arises empirically, and represents an extreme analytic approach to observed phenomena. Literally the analytic method is to "take apart" the subject into smaller and smaller sub-components, until arriving at the elementary constituents. We regard macroscopic objects not as an indivisible wholes, but as composed of sub-parts, each of which is composed of still smaller parts, and we continue this process of analysis at least until we can no longer directly resolve the sub-parts (empirically) into smaller entities. At this point we may resort to some indirect methods of inference in order to carry on the process of empirical analysis. Indeed, Einstein's work on Brownian motion did exactly this, in so far as he was attempting to analyze the smallest directly observable entities, and to infer, based on empirical observations, an even finer level of structure. It was apparently Einstein's view that, at this stage, a reversal of methodology is required, because direct observation no longer provides unique answers, and thus the inferences are necessarily indirect, i.e., they can only be based on a somewhat free hypothesis about the underlying structure, and then synthetically working out the observable implications of this hypothesis and comparing these with what we actually observe.

So Einstein's conception of a constructive (hypothetically based, synthetic) physical theory was of a theory arrived at by hypothesizing or postulating some underlying structure (consistent with all observations, of course), and then working out the logical consequences of those postulates to see how well they account for the whole range of observable phenomena. At this point we might expect Einstein to classify special relativity as a constructive theory, because it's well known that the whole theory of special relativity - with all its observable consequences - can be constructed synthetically based on the exceedingly elementary hypothesis that the underlying structure of space and time is Minkowskian. However, Einstein's whole point in drawing the distinction between constructive and principle theories was to argue that relativity is not a constructive theory, but is instead a theory of principle.

It's clear that Einstein's original conception of special relativity was based on the model of classical thermodynamics, even to the extent that he proposed exactly two principles on which to base the theory, consciously imitating the first and second laws of thermodynamics. Some indication of the ambiguity in the classification scheme can be seen in the various terms that Einstein applied to these two propositions. He variously referred to them as postulates, principles, stipulations, assumptions, hypotheses, definitions, etc. Now, recalling that a "constructive theory" is based on hypotheses, whereas a "principle theory" is based on principles, we can see that the distinction between principles and postulates (hypotheses) is significant for correctly classifying a theory, and yet Einstein was not very careful (at least originally) to clarify the actual role of his two foundational propositions.

Nevertheless, his consistently viewed special relativity as a theory of principle, with the invariance of light speed playing a role analogous to the conservation of energy in classical thermodynamics, both regarded as high-level empirical propositions rather than low-level elementary hypotheses. Indeed, it's possible to make this more than just an analogy, because in place of the invariance of light speed (with respect to all inertial coordinate systems) we could just as well posit conservation of total mass-energy (with the conversion E = mc²), and use this conservation, together with the original principle of relativity (essentially carried over from Newtonian physics), as the basis for special relativity. As late as 1949 in his autobiographical notes (which he jokingly called his "obituary"), Einstein wrote

Here Einstein refers to "constructive theories based on known facts", whereas in the 1919 article he indicated that constructive theories are based on "a relatively simple formal scheme" such as the hypothesis of molecular motion (i.e., the atomistic doctrine that Mach (for one) rejected as unempirical), and principle theories are based on empirical facts. In other words, the distinguishing characteristics that Einstein attributed to the two kinds of theories have been reversed. This illustrates one of the problematic aspects of Einstein's classification scheme: every theory is ultimately based on some unprovable premises, and at the same time every (nominally viable) theory is based on what might be called known facts, i.e., is it connected to empirical results. Einstein was certainly well aware of this, as shown by the following comment (1949) in a defense of his methodological approach:

In view of this, what ultimately is the distinction between what Einstein called constructive theories and principle theories? It seems that the distinction can only be based on the conceptual level of the hypotheses, so that constructive theories are based on "low level" hypotheses, and principle theories based on "high level" hypotheses. In this respect the original examples (classical thermodynamics and statistical thermodynamics) cited by Einstein are probably the clearest, because they represent two distinct approaches to essentially the same subject matter, namely, thermodynamics. In a sense, they could be regarded as just two different interpretations of a single theory (much as special relativity and Lorentz's ether theory can be seen as two different interpretations of the same theory). Now, statistical thermodynamics was founded on hypotheses - such as the existence of atoms - that may be considered "low level", whereas the hypothesis of energy conservation in classical thermodynamics can plausibly be described as "high level". On the other hand, the premises of statistical thermodynamics include the idea that the molecules obey certain postulated equations of motions (e.g., Newton's laws) which are essentially just expressions of conservation principles, so the "constructive" approach differs from the "theory of principle" only in so far as its principles are applied to very low-level entities. The conservation principles are explicitly assumed only for elementary molecules in statistical thermodynamics, and then they are inferred for high-level aggregates like a volume of gas. In contrast, the principle theory simply observes the conservation of energy at the level of gases, and adopts it as a postulate.

In the case of special relativity, it's clear that Einstein originally developed the theory from a "high-level" standpoint, based on the observation that light propagates at the same speed with respect to every system of inertial coordinates. However, just three years later, Minkowski showed how the theory follows naturally from just a simple fundamental hypothesis about the metric of space and time. There can hardly be a lower conceptual level than this, i.e., some assumption about the metric(s) of space and time is seemingly a pre-requisite for any description - scientific or otherwise - of the phenomena of our experience. Kant even went further, and suggested that one particular metrical structure (Euclidean) was a sina qua non of rational thought. We no longer subscribe to such a restrictive view, and it may even be possible to imagine physical ideas prior to any spatio-temporal conceptions, but nevertheless the fact remains that such conceptions are among the most primitive that we possess. For example, the posited structure of space and time is more primitive than the notion of atoms moving in a void, because we cannot even conceive of "moving in a void" without some idea of the structure of space and time. Hence, if a complete physical theory can be based entirely on nothing other than the hypothesis of one simple form for the metric of space and time, such a theory must surely qualify as "constructive".

Einstein's reaction to Minkowski's work was interesting. It's well known that Einstein was not immediately very appreciative of his former instructor's contribution, describing it as "superfluous learnedness", and joking that "since the mathematicians have attacked the relativity theory, I myself no longer understand it any more". He seems to have been at least partly serious when he later said "The people in Gottingen [where both Minkowski and Hilbert resided] sometimes strike me not as if they wanted to help one formulate something clearly, but as if they wanted only to show us physicists how much brighter they are than we". Of course, Einstein's appreciation subsequently increased, when he found it necessary to use Minkowski's conceptual framework in order to develop general relativity. Still, even in his autobiographical notes, Einstein seemed to downplay the profound transformation of special relativity that Minkowski's insight represents.

In other words, Minkowski's contribution was merely the introduction of a convenient mathematical formalism. Einstein then added, almost as an afterthought,

This is rather a slight comment when we consider that, from the standpoint of Einstein's own criteria, Minkowski's insight that Lorentz invariance is purely an expression of the (pseudo) metric of a combined four-dimensional space-time manifold at one stroke renders special relativity into a constructive theory, the thing for which Einstein had sought so "desperately" for so long. As he wrote in the London Time article above, "when we say that we have succeeded in understanding a group of natural processes, we invariably mean that a constructive theory has been found which covers the processes in question", but he himself had given up on the search for such a theory in 1905, and had concluded that, for the time being, the only possibility of progress was by means of a theory of principle, analogous to classical thermodynamics. Actual understanding of the phenomena would have to wait for a constructive theory. I would argue that this constructive theory was provided just three years later by his former mathematics instructor in Gottingen.

From this point of view, it seems fair to say that special relativity has had three distinct incarnations, first in the form of Lorentz's (and Poincare's) constructive ether theory (1892-1904), second as Einstein's theory of principle (1905), and third as Minkowski's constructive spacetime theory (1908). Each stage represented a significant advance in clarity, with Einstein's intermediate theory of principle serving as the crucial bridge between the two very different constructive foundations of Lorentz and Minkowski.

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