# Mathematical Time

Newton’s third distinction between mathematical and common time brings us back to the relationship between time and change. Newton’s assertion that time is mathematical includes the claim that it is metrical. What this means is that time is a measure of change: changes occur regularly or irregularly, and faster or slower, with respect to the metric of time, and the comparative length of time between pairs of events is determined by this metric structure. (This is independent of whether time is absolute or relative, and so of whether there are any material clocks, or not. It is also independent of whether time is true or apparent.) As we have seen, time without this much structure seems to be insufficient for the needs of the project of the Principia. Thus, the pursuit of the project of the Principia is in part an investigation of the claim that the time parameter of the Principia must be metrical, and in this way the question of whether time is mathematical has been transformed into an empirical question. Indeed, just how much structure the time parameter must have, in order to fulfill the needs of Descartes’s project, is an open question to be explored through the process of empirical enquiry, and it is one that remains an open question today.

There are some interesting conceptual intricacies that arise in the wake of the assertion that time is metrical, and I will finish by offering some comments on these. Following the assertion that time is metrical, there remains the possibility of a gap between time as a measure of change, in and of itself, and time in relationship to actual material changes taking place in the world: why think that the two tick in harmony? Relationists close the gap by identifying units of time with the rhythm of repeating material phenomena, but if time is absolute, as Newton believed, then it is not clear why the ticking of clocks should tell us anything about the metric of time. While time is a measure of change, we measure time by means of material change, and if the metric of time were capricious in relation to material change, then material clocks would not measure time. This is not as silly as it sounds: Newton was very alert to the question of whether a unit tick of a clock at one time measures a unit of duration at another time; it is a live possibility for Newton that the length of time between any two ticks of a clock may not be equal. In order to see the significance of this more clearly, it is instructive to consider the difference between Newton’s treatment of spatial interval and his treatment of duration.

Spatial intervals are measured by rulers; rulers are bodies, and according to Newton, place is the part of space that a body occupies. A body of unit length by definition occupies a region of space of unit length. There is therefore no distinction between the metrical characteristics of bodies, as occupiers of space, and the metrical characteristics of the parts of space that they occupy. All rulers are, in this sense, perfect rulers: no question arises as to whether a unit ruler measures a unit interval of space, and indeed whether that unit ruler at one location and at one time measures a unit interval of space when moved to another spatial location and/or at another time.

By contrast, not all clocks (indeed perhaps no clocks) are in an analogous sense perfect clocks. And this is the point about the possible capriciousness of time: a unit tick of a clock at one time may not measure a unit of duration at another time. For Newton, there is a gap between duration and our measure of duration that does not arise for length and our measure of length. This is why Newton need not specify explicitly that space is mathematical, whereas he must—and does—do so for time.[1]

The gap between duration and our measure of duration therefore presents us with an epistemic problem, in that we cannot measure duration directly, but only indirectly by means of motion. Closing this gap involves two steps. First, we have to specify that time is metrical, and this is something Newton does. Then, we have to establish a relationship between this metric of time and the “ticking” of material clocks, so that our means of measuring duration is not utterly unreliable as a guide to the metric of time. Newton does this by stipulating that time “flows equably”: the metric of time is not capricious in relation to physical processes.[2] To say that time

“flows equably” is to say not only that it is a measure of change, but that the processes and changes that we experience as regular are regular, at least approximately, with respect to time. Were it otherwise, then the metric of time would be epistemically utterly inaccessible to us.

Thus far, Newton has been making clear what demands are placed on our concept of time by the requirement that material clocks enable us to measure intervals of absolute time. The evidence that time flows equably is the practical success achieved by astronomers with the equation of time, and by Huygens with the pendulum clock. There may be no perfectly equable motions to be found in the material world, since there may be no material clock that ticks precisely in accordance with the metric of time, but we have good empirical reasons in support of the claim that absolute and true time flows equably.

I think that the issues here are more complex than those raised by absolute and true time, and that the subsequent history of work on space and time bears this out. The relationship between material processes and space and time continues to be rich philosophical territory in foundations of physics, in which a wide array of positions is available. There are disputes over the priority of space and time versus matter, over the minimal structure that we must attribute to space and time, over the nature of the relationship between space-time and matter, and of how it is that rods and clocks, such complex material systems, overcome that complexity to somehow tell us something about the structure of the spacetime in which they are situated and move. It is not at all obvious how to resolve these questions empirically, and what this makes vivid is the sophisticated reasoning that is involved in bringing the empirical to bear on such difficult philosophical questions. Nevertheless, what is also clear is that it is through the process of this very engagement with the details of empirical enquiry that progress on these questions is to be made. Again, the methodological point is the one that I wish to emphasize, not the answer to the question of whether or not time is mathematical. It is because of the moves that Newton makes that we uncover all the philosophical complexity associated with this claim in the first place.

• [1] Moreover, this marks a difference as compared to mass, as a measure of matter, and momentum, as a measureof motion, for example, where Newton is able to formulate quantities that do not suffer from the same kindof epistemological gap. In the case of time, it seems that no amount of conceptual work will remove the gap.
• [2] Arthur (1995) argues, though for different reasons, that we should regard Newton’s assertion that time is mathematical as asserting that it flows equably, as well as that it is metrical.