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Аbsolute, true, and mathematical time IN the project of the PRINCIPIAWe have seen that Newton had some empirical basis for making the distinctions between absolute and relative, true and apparent, and mathematical and common time (as I have interpreted that terminology), and that he had good empirical reasons for believing that the time parameter appropriate for mathematical astronomy is absolute, true, and mathematical, rather than relative, apparent, and common. I will now argue that each of these three conceptual distinctions suggested by mathematical astronomy is necessary for setting up the project of the Principia. This enables me to argue that the project of the Principia is a means for empirically investigating the characteristics of time that are associated with its being absolute or relative, true or apparent, and mathematical or common. As noted above, the end goal of the Principia is to establish the “system of the world”—that is, to decide between the traditional geocentric, the Tychonic geocentric, and the heliocentric systems by establishing the true motions of the bodies in our planetary system.^{[1]} ^{[2]} Successful completion of the project of the Principia thus requires true motion: without true motion, there is no determinate answer to the question of the system of the world. The project therefore requires the distinction between true and apparent motion. Newton’s proposed strategy for solving the problem of the system of the world was to systematically correlate forces with true motions. In the scholium to the definitions, Newton argues that true motion cannot be relative motion, and must therefore be absolute motion instead, by showing that on a relative conception of motion, true motions are not appropriately correlated with the presence/absence of forces. This is one purpose of the famous bucket experiment.^{[3]} Thus, insofar as true motion is necessary for the project, we also need absolute motion. But we cannot have absolute motion with relative time, since the resulting motion would then be dependent on the material bodies whose relative motions constitute relative time. Therefore, successful completion of the project of the Principia requires the distinction between absolute and relative time. True motion also requires true time: there must be a unique time parameter proper to the system of the world, for if there is not, then a second “time” might give different conclusions concerning the motions and forces, and therefore concerning the system of the world. Thus, setting up the project of the Principia requires that we distinguish between true and apparent time. If mathematical astronomy is the appropriate route for solving the system of the world, as Newton believed, then the need for distinguishing between mathematical time (with its metrical properties, and so forth) and common time is demonstrated by the need for an equation of common time. As noted above, Newton’s proposed strategy for solving the problem of the system of the world was to systematically correlate forces with true motions, which in turn demanded a mathematical treatment of forces and motions as set out in Books I and II of the Principia. The specific mathematical properties required of the time parameter in these treatments stands in need for further investigation by Newton scholars,^{[4]} but it is immediately clear from the outset of the Principia that the time parameter must be metrical: Law 1 relies on equal intervals of time for the distinction between uniform and nonuniform motion, and this is at the heart of the distinction between the presence and absence of forces by which we are to arrive at the true motions. The distinctions between absolute and relative, true and apparent, and mathematical and common time are therefore intimately tied to the project of the Principia. Newton had good reason to assert all three distinctions in setting up his project. The three distinctions are mutually independent of one another, and all three are needed for the purposes of his empirical project. Newton also had good reason to believe from the outset that the time parameter appropriate for successful execution of his project would turn out to be absolute, true, and mathematical, rather than relative, apparent, or common. However, setting Newton’s own position to one side, one cannot know at the outset whether the demands of the empirical project, as it unfolds, will indeed restrict the characteristics of the time parameter such that it turns out to be absolute, true, and mathematical. This is something that can be settled only by pursuit of the project. I therefore disagree with Schliesser (2013) that only absolute (mathematical) time is connected to an empirically open question capable of being addressed by the methodology of the Principia. As we have seen, Schliesser claims that absolute time (in the sense of a time parameter for the solar system) is needed by Newton for his dynamics in order for Newton to “identify and assign accelerations to moving bodies in a consistent fashion" but that Newton’s inclusion of something called “true time" (understood as an extension of absolute time from the “local temporal frame" of the solar system to spatial infinity) turns out to be a metaphysical commitment going beyond the demands of the Principia, rooted in Newton’s rational theology. I think that Schliesser is right about the need for such a time parameter for the project of the Principia, and about its empirical status, but I think that each of the three distinctions (as I have interpreted the terminology) represent open empirical questions concerning the characteristics of the time parameter, and questions that are capable of being addressed by the methodology of the Principia. The three distinctions are subject to empirical investigation in the first place through their connection to the problem of true motion. True motion is necessary for successful execution of the project of the Principia, but it is a contingent matter whether any such true motions exist: it might turn out that there are no true motions and thus there is no answer to the problem of the system of the world. Moreover, the project of the Principia enables us to probe the connections between true motion and time asserted above: for example, one might interpret Galilean relativity as indicating that true time is necessary but not sufficient for true motion, since there we have true time, and absolute (i.e., not relative) motion, but seem not to have true (i.e., unique) motion. This is an example of how the questions we are asking are transformed and become more finegrained in the process of addressing them through the project of the Principia. There is much more to be said here concerning the execution of the project in the details of the Principia, in the uses of time that Newton makes in his mathematical arguments, his search for empirical clocks, and the interplay between these and his construction of absolute, true, and mathematical time. It is by understanding these details, and their relationship to the empirical successes and failures of the Principia and later developments in physics, that we will find out the extent to which time turns out to be absolute and/or true and/or mathematical. The point that I want to stress here is that these aspects of the nature and structure of time are now tied to the details of empirical enquiry. All three questions of whether time is absolute or relative, true or apparent, and mathematical or common, have become empirically tractable.

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