The best text for understanding Duhem's view on relativity is German Science. There have been many complaints about this book; for instance, it has been called an unfortunate piece of (World War I) war propaganda. What offends about the work are what are usually called its caricature of the 'German mind'. There's no doubt that the work presents us with something of a caricature; but a caricature is not a wholly inaccurate portrait. A caricature involves some distortion, but only for the purposes of bringing out particularly recognizable or distinctive points. And Duhem himself is quite clear that in talking about the 'German mind' he merely wants to indicate a tendency that arises from the way the Germans teach and learn science, not to make a universal statements about Germans. As he notes, there is no trace of the exclusively 'English mind' in Newton, and no trace of the exclusively 'German mind' in Gauss. His interest in the subject, actually is that it provides a useful context for investigating the mentality that is ideal for scientific work. The Germans just happen to be the concrete case that (he thinks) comes closest to a pure case of one element of this mentality.
We often tend to talk as if scientific progress were unilinear, as if all scientists had one type of mentality in their scientific work, one methodical approach. Duhem does not. Duhem has a Pascalian view of the human mind, which means he thinks there are two (major) kinds of mentality. The first is the esprit de finesse, the intuitive mind; the second is the esprit de géométrie, the geometrical mind. All human beings have both to some extent. A rare few have close to the perfect balance of both. Most of us, however, tip strongly to one side. Some of us are primarily intuitive, some primarily geometrical. There are several different sorts of both. For instance, one sort of intuitive mind (Duhem calls it the 'English mind') is heavily imaginative -- it relies on models, picture-thinking, metaphors. Another (the 'French mind' -- but Duhem is very clear that it is the French mind as it used to be, in the days of Pasteur or Ampère) is very formal; it eschews the messiness of models and pictures in favor of formal structures that lay things out neatly and clearly. No doubt there are other possible variants. The 'German mind', Duhem thinks, is geometrical.
Duhem insists that the healthy progress of science requires the active participation of both intuitive and geometrical minds. In other words, there are two lines of progress in science, each correcting the excesses of the other; both are essential if science is not to lose its way. The intuitive mind (and we are talking here chiefly of the formal-intuitive mind) is the mentality that allows for definite discovery; it is what keeps us grounded in reality. It calls us back to common sense, and provides the general background principles for rational discussion. Its two great characteristics are clarity and good sense. When the intuitive mind adds something to science, the addition illuminates. It articulates an explanation that makes sense because it is clearly linked to the common principles rational human beings have in common. The geometrical mind, on the other hand, is the mentality that is deductive, rigorous, and precise. It follows reasoning wherever it goes. It is disciplined and patient in a way the intuitive mind is not. Whereas the intuitive mind is often the source of a new scientific discipline, it is the geometrical mind that takes the principles provided by the intuitive mind and sets them into a rigorous logical or mathematical order so that their consequences can be followed to the very end.
Duhem's constant worry throughout German Science is the imperialism of the geometrical mind. The geometrical mind is very rigorous and logical; but in another sense it is very unruly. The geometrical mind is impressed by reasoning as such; it is careless about the starting points of the deduction. Indeed, these are treated as almost insignificant; the geometrical mind just posits whatever starting points are convenient for whatever it is doing. There is no absolute problem with this; but there is the danger that the geometrical mind, carried away with following out a line of reasoning to its bitter end, will stifle or completely ignore the intuitive mind. It is the intuitive mind, remember, that keeps reasoning grounded in reality; it is also the intuitive mind that has the real skill to recognize when our reasoning has brought us to a genuine absurdity. Duhem's worry is that science is in danger of being highjacked by the geometrical mind's tendency to be seduced by sophisticated reasoning, thus losing sight of the reality it is really supposed to be explaining.
Nonetheless, even when the geometrical mind gets carried away, it is making genuine contributions to the progress of science. It is only if the intuitive mind is pushed out that we have serious problems. So for Duhem, a step forward in the progress of science can be a step forward either by the intuitive mind or by the geometrical mind. Duhem considers the theory of relativity to be a useful step forward along the geometrical line of progress. It tells us how you can go about preserving Maxwell's equations in the face of a number of perplexities; it allows us to make precise and accurate predictions we could not otherwise make. There is no question that Duhem considers this to be a valuable step forward.
However, what Duhem wants, and what he's not getting, is for the geometrical mind to allow the formal-intuitive mind to look at the theory of relativity and say, "OK, use it insofar as it is useful. But notice that we come up with several conclusions down the road that seem counterintuitive. Let's see if we can take what we've learned from the theory of relativity and go back to re-analyze the foundations from which it set out, in order to see if we can develop a theory that does not have these counterintuitive conclusions but preserves much of what is valuable about the theory of relativity. If we can find such a theory, that would be even better than the theory of relativity." Clearly, we can be wrong about the general principles of good sense or common sense, and sometimes have been; but Duhem finds it worrisome that so many people are willing to say, "By positing this starting point (the principles that will maintain the form of Maxwell's equations) and rigorously following our deductions through to useful effect, we have proven that such-and-such common-sense principle is false."
He recognizes that there is a practical value in the particular posited starting-point of the theory of relativity, and that the theory of relativity has numerous other practical values that show that it is, indeed, a major contribution to scientific progress: beauty, simplicity, predictive power. But it is the geometrical mind that is interested in these pragmatic values in the first place. The geometrical mind is interested in what you can do with scientific theories; it is interested in how they can facilitate the deductive processes so central to its approach. The formal-intuitive mind, however, is much less interested in pragmatic values like the beauty, simplicity, and predictive power of the theory. The formal-intuitive mind is not so much interested in what you can do with the theory, but in what it makes obvious. The epistemological goal of the formal-intuitive mind is not a pragmatically valuable theory; it is the theory that makes things clear and obvious. The geometrical mind likes that you can use the theory of relativity to calculate satellite orbits; thinking in terms of clocks and rubber sheets and elevators might perhaps enchant the imaginative-intuitive mind for a while; but the formal-intuitive mind is left in the dark if it is not allowed to use the theory of relativity to progress along its own line of interest. The formal-intuitive mind can accept the theory of relativity as a valuable contribution of the geometrical mind, but only on its own terms, which require using what we learn from it in order to find a more common-sensical theory. Duhem is worried about the tendency of the geometrical mind to try to shut this down entirely. This heedlessness, this refusal even to take into account the fact that not all minds can be satisfied with what satisfies the geometrical mind, is Duhem's real irritation when it comes to the theory of relativity -- it is not the theory itself, but the refusal to recognize even the existence of the formal-intuitive mind and its needs. It is only in the cooperation of the geometrical and the intuitive minds that ideal science exists (German Science, p. 110):
French science, German science, both deviate from ideal and perfect science, but they deviate in two opposite ways. The one possesses excessively that with which the other is meagerly provided. In the one, the mathematical mind reduces the intuitive mind to the point of suffocation. In the other, the intuitive mind dispenses too readily with the mathematical mind.
Science needs the geometrical mind for rigor; but it needs the intuitive mind for truth. Such is Duhem's view, anyway. As he insists, "For science to be true, it is not sufficient that it be rigorous; it must start from good sense, only in order to return to good sense" (p. 111).