Willem de Sitter and Albert Einstein discuss the equations governing the dynamics of the universe

In a pair of earlier posts I discussed mathematician Warren Weaver’s opening address at the 1947 RAND conference of social scientists, in which he suggested that all the attendees shared a devotion to the “rational life.” Weaver made it clear that what he meant by the “rational life” was not a strict rationalism, but a kind of searching, open-ended approach to analyzing questions that decision makers were compelled to answer whether they analyzed them or not.

Weaver’s interest in such problems appears to have been primarily prompted by his experience in World War II, dealing with conundrums in the design and selection of military equipment. Weaver confronted these problems, first as an overseer of research on “fire control” (gun-aiming) devices, and then as chief of an organization called the Applied Mathematics Panel. He was particularly impressed by a body of analytical techniques first developed in Britain by a statistician named L. B. C. Cunningham, and referred to as the “mathematical theory of combat” or “air warfare analysis.” In brief, Cunningham’s theory combined expressions describing the specifications of alternative weapons systems and equipment configurations, the tactics of attackers and defenders, and the vulnerability of targets, and used them to derive expectation values for victory in combat.

Various pursuit curves a fighter might follow in making an attack on a bomber. The image links to a post with further context.

It is important to note that, although these expectation values might be checked against data from actual combat, they were not imagined to provide accurate predictions. Rather, they provided a means of comparing different choices of design by making explicit and interrogating previously tacit assumptions that engineers made about the virtues of their various designs. When Weaver spoke, RAND was beginning to elaborate on these methods and to apply them to the design of more complex and prospective military technologies under the new label “systems analysis” (a label that would shift significantly in meaning in subsequent years).

To clarify the intellectual value of this analytical activity, Weaver compared its epistemology to the then-nascent field of relativistic cosmology.

Here is an extended quote from Weaver’s speech:

[RAND] has a considerable group staffed largely, but not exclusively, with mathematicians trying to work along the general lines of seeing what a quantitative logical analytical theory, i.e. a mathematical theory, can accomplish in analysing general theories of air warfare. This work has a great deal to do with the concept, if indeed there be such a concept, of military worth, to see to what extent it is possible to have useful quantitative indices for a gadget, a tactic, or a strategy, so that one can compare that piece of hardware, that tactic or that strategy over against the accessible alternatives, guided by analysis to the extent that analysis can be shown to be relevant and possible.

To those people who think that these military problems are too complicated, that you never can get enough of the data, that you can never know the data accurately enough, that you cannot possibly have a theory for such a wide ranging field of human activity, I say, first of all, of course, I agree with you. But will you just think for a moment of what I seriously think is a reasonable analogy? Think of our friends, the cosmologists—they want to try to cast up a theory of the universe. All of you will agree that that is a reasonably complicated problem to attack—that knowledge of all the details lies beyond the possibilities of any person or group of persons. The amount of information they have about the universe as a whole is, in fact, pathetically small. And yet, what can they do with it? Well, they can do with it just what I think can be done from any of these problems. They can think awfully hard; and at the end of the thinking, they can make statements of the following sort; they can say: “Gentlemen, as regards possible conceptual models of the universe, we can’t tell you what kind of universe this is, but we can tell you that it cannot possibly be such and such kinds—nor (looking in other directions) such and such kinds. The possible kinds of universe, in other words, lies in between certain limits of likelihood.”

Now that is a remark of extraordinary interest and value and power. How narrow are those limits? Well, as yet, not very narrow, but they are considerably more narrow than the distance between the two end points of an infinite line. Furthermore, the cosmologist can say, “Gentlemen, we can tell you of those items of information which we do not have, which ones are of the most critical importance in narrowing this gap and sharpening our knowledge of what the universe really is. We can tell you the sensitivity, so to speak, of the boundaries of knowledge with respect to different lines of information. We can say to you, ‘Please don’t strain yourself indefinitely to try to find out this particular fact because that is not going to move the boundary of ignorance very far. The boundary of ignorance is surprisingly insensitive to that particular kind of information—so insensitive that we are satisfied with our present state of knowledge of that. Whereas this other factor and this factor and this factor are critically important factors, in that the boundaries of ignorance are extremely sensitive to them. The minute you find out anything about that, you sharpen down rapidly.”

That again is exceedingly important information, and at the very worst, I am sure that information of that that sort can be gotten in the most complicated of these military problems. That is my answer to the man who says, “Why attempt analytical theories of air warfare? Why attempt to struggle with the concept of military worth?”

The brand of cosmology that Weaver was referring to derived from Albert Einstein’s (1879–1955) general theory of relativity, which had been formulated primarily in the mid-1910s. Among its other features, the theory related the force of gravity to space-time curvature. In doing so, it was able to make some highly precise predictions—one of its early triumphs was accounting for a known anomaly in the orbit of the planet Mercury. However, Einstein quickly realized that the theory also implied that the universe must be closed as well as either expanding or contracting. To preserve the implication that the universe is stable, he introduced the now-notorious “cosmological constant” to avoid predicting a dynamic universe.

Helge Kragh is the foremost historian of relativistic cosmology.

By the late 1920s, further theoretical inquiry and observations established that the universe was indeed expanding, and in the 1930s George Gamow (1904–1968) and other theoreticians began to develop more detailed theories describing the energetic development of the early universe (what became known as the “big bang” theory). Observational corroboration of these ideas was at that time weak. Nevertheless, as Weaver indicated in his speech, the theory permitted very general claims to be made with some confidence because they followed logically from the theoretical premises.

Of course, when Weaver spoke, he would not have known that a competing “steady state” theory of the universe was about to be championed by Fred Hoyle (1915–2001), Hermann Bondi (1919–2005), and Tommy Gold (1920–2004). Even once controversy arose, though, the theoretical framework of relativistic cosmology shaped the terms of the debate, and ultimately delineated the experimental evidence that might resolve it. The discovery of the cosmic microwave background radiation in 1964 would not only deal the steady-state cosmology a heavy blow, observations of the cosmic microwave background’s features would provide an important check on further cosmological theory going forward. Weaver’s assessment of the power of the epistemology underlying relativistic cosmology was correct.

Weaver’s connection of the epistemology of mathematical theories of combat to relativistic cosmology was informed by his own experience in applying L. B. C. Cunningham’s methods. First, Weaver gathered a group primarily comprising mathematical statisticians to interpret the theory, which they did in two reports. In his final report as chief of the Applied Mathematics Panel, “Comments on a General Theory of Air Warfare,”¹ Weaver recalled:

These reports were made the basis for an extended conference with AAF [Army Air Forces], [the Navy] Bureau of Aeronautics, and RAF [Royal Air Force[ personnel; and we were then asked, as the second step, to apply these ideas to the problem of comparing the relative effectiveness of four 20 mm. guns versus eight .50″ guns as armament for a fighter making a stern attack on a defended twin-engine bomber.

The methodology of the analysis of this problem was rather carefully set down, and a list was made of the kinds and pieces of information (or of estimates, or of guesses) which would have to be made before the answer would be forthcoming. These questions (nature of combat, bomber and fighter armament, value and variations of accuracies, ammunition, vulnerabilities, etc.), were then discussed at very considerable length with the experienced officers. As a result estimates were arrived at which every one agreed were almost certain to bracket the true values, although in many instances the true values were admittedly unknown. Thus the analysis was based on five alternative assumptions of vulnerability, three assumptions concerning bomber fire power index (which takes into account rate and accuracy of fire) and four assumptions concerning fighter fire power index. For four of the five assumptions concerning vulnerability the eight .50″ were found to have an advantage (ranging from 27 percent to 76 percent) over the four 20 mm. guns. For the fifth vulnerability assumption, the 20 mm. armament was preferable, having about 20 percent advantage.

The study was thus necessarily inconclusive. It did, however, make clear just what sort of information was necessary to obtain a conclusive answer, and it furnished the necessary analytical methods. Furthermore the study rather strongly suggested the superiority of the .50″ armament especially since it was discovered that the use of an optimum ammunition mixture increased the superiority of the eight .50″ guns for the first four vulnerability assumptions and reduced the advantage of the 20 mm. guns to 10 percent in the case of the last vulnerability assumption.

Weaver was highly impressed by theory’s ability not to resolve the problem of design choice definitively, but to pinpoint to which elements of the problem the choice was sensitive, and to which ones it was insensitive. In doing so, it also narrowed down the vast range of possible equipment tests one might want to perform, and which aspects of combat one might gather data on. This was important when time and experimental resources were scarce.

The virtues of crude theoretical analysis as a means of clarifying the structure and nature of problems, and of calling attention to the desirability of certain kinds of data, would come to characterize large bodies of work in fields such as economics and operations research. I have also found that a very similar epistemology has operated in glaciology (the physics of large bodies of ice). I am sure we could rattle off a large number of other fields as well. Because such crude theories cannot rely on direct observational confirmation, but rely on their ability to approximate, or be otherwise consistent, with observation, they are perhaps best thought of as heuristic tools. This heuristic attitude to building understanding and making difficult choices was the essence of what Weaver considered to be the rational life.