In philosophy, sometimes things seem to be the obvious thing, and then they just vanish. The Principle of the Uniformity of Nature (PUN, no pun intended) is a good example. There was a time when it was practically everywhere. It was the foundational principle of induction, and likewise the fundamental warrant for reasoning beyond our immediate experience. It was the key element in widely accepted proofs of determinism, of the impossibility of miracles, of the eternity of the world. Scientists appealed to it in scientific investigations of all different kinds. It perhaps began soaring in importance somewhere around the 1850s, and then, with some up-and-down, seems to have begun a slow collapse in the 1880s, until in the 1950s Wesley Salmon (if I recall correctly) pronounced it dead, and certainly nobody takes it quite seriously on its own terms today. To rise suddenly into prominence as one of the most important of all rational principles and then just to end up as an artifact of purely historical interest, and to do so in about a century, is an interesting career.
Of course, it's not quite so simple. The above rough resume is that of a principle specifically being called 'the principle of the uniformity of nature' or 'uniformity of nature', discussed specifically as such or casually referred to in the course of proving other things. The big question is what the principle actually was. Is the above resume the history of a label which made an already existing principle prominent? Is it the history of a particular use of a principle? Was there actually any such principle at all?
Ironically, given its once-sovereign importance, one could make an argument that there was never such a principle. Indeed, part of the downfall of the entire notion of a PUN is seen in the fact that people increasingly began to be confused about what it means, and by preface or apposition would remark (for instance) about the difficulty of finding an exact formulation. And when we look at attempts to more specific, we find that people trying to formulate the PUN more exactly do not come up with the same formulation.
We have the Naive Formulation, which is perhaps most often used, because it is naturally connected with the name: The course of nature is uniform. But even in the 1850s it was clear that this was not adequate. W. G. Ward, in his debate with J. S. Mill about a priori propositions in the mid-century, noted, without putting a lot of emphasis on it, that there was a lot of evidence that the course of nature was not so very uniform. Later figures also note that it's unclear what 'nature' is supposed to mean here. If we try to take the Naive Formulation seriously, it's difficult to get more out of it than that there are some kinds of general principles describing the things into which we require, which is perhaps true, but not so obviously helpful.
Perhaps the most successful and widely accepted specification we can call the Causal Sameness Principle: The same causes have the same effects and the same effects have the same causes. This is an attractive candidate, because it has a very venerable history, long preceding the PUN label; it tends to be the formulation used by more philosophically minded scientists, like James Clerk Maxwell; and it would mean that PUN was not some fluke, because it's still often used today, even if it is no longer treated as having the importance or centrality that it once did. Unlike the Naive Formulation, whose status as a fundamental principle gave people considerable difficulty, it is fairly plausible to argue that it is a necessary principle, and therefore not in need of any further explanation. But even at the time it was occasionally noted that the Causal Sameness Principle can't do most of the things attributed to the PUN. It's useless for disproving miracles or free will, for instance, because miracles and free will deal with things that are neither 'same effects' nor 'same causes'; they are explicitly dealing with different kinds of causes. While you can perhaps see vaguely how a theory of induction might be based on it, the details of how were surprisingly tricky to work out -- in particular, it's not clear that the Causal Sameness Principle gets you closer to a full theory of induction than any other causal principle. Perhaps more serious is a point noted by Mill, that not all 'uniformities of nature' could be causal in the way the Causal Sameness Principle required. (Mill's example is fundamental properties, which as fundamental could not have any cause, unless you count God, but even if you do, that plays no role in how we usually reason inductively about them.)
Mill, in fact, provides in A System of Logic what I think is probably the best alternative to the Causal Sameness Principle, although he does it almost in passing: What is true in one case is true in all cases of a certain description. This has the advantage of being consistent with the Causal Sameness Principle while have flexibility more like that of the Naive Formulation. Call it the Description Formulation. What is, I think, most attractive about the Description Formulation is that it is the only version of PUN in which it is very obvious what it actually has to do with induction. In induction you are, more or less, going from 'Some A is B' to 'All A is B', and the puzzle is what you are doing that lets you do this. The Description Formulation gives an answer: you are not doing so simpliciter, but only under a description. And what we think of as induction is the work of finding and ruling out candidates for that description. (Mill thinks of this as happening, Bacon-like, with the various methods like the Methods of Agreement.) I start with this case, in which X is Y, and then I, by various kinds of trial and error, find a description D that lets me say 'All X is Y' in the context of D. At the extreme you can do this trivially -- if some A is B, then all A that is B is B -- which shows that the principle is not just an arbitrary claim, since we know it is a legitimate move for at least one description, but of course, in induction we are looking for non-trivial descriptions. That there are non-trivial descriptions to be found is not something we know beforehand, but Mill wouldn't be bothered by this, because he doesn't think the PUN is necessary, a priori, or indefeasible; he just thinks it is a principle that we've found to work in a lot of different kinds of situations, and so is suitable for practical purposes. But he also would have to concede, on the same grounds, that the Description Formulation can't do many things that people wanted it to do -- you can't use it to establish determinism, for instance. All that's really salvaged by the Description Formulation is a general format for inductive inquiry.
So it seems like the best summary here is that, while there are a few things that could be considered a principle of the uniformity of nature, they are quite limited and don't have the features that the PUN was generally taken to have; the PUN, as such, despite all the fanfare, never actually existed in any form at all. It was almost entirely carried by the rhetorical force of the phrase, 'uniformity of nature', in a context in which people would describe scientific inquiry as being concerned with 'uniformities of nature'. People could point to particular 'uniformities of nature' -- in the period astronomical phenomena and Newton's Laws would have been the obvious cases -- which is what made it plausible. But a general principle was never really formulated; the phrase was a placeholder, a stand-in serving as an IOU, whose place was never filled with anything that could deliver even a portion of what was promised.
