Imagine two objects, *M* and *H*, where *M* has the intrinsic causal power of emitting some sort of a pulse once per minute and *H* has the intrinsic causal power of pulsing once per hour, and suppose *M* and *H* are causally separated from the rest of the universe. How do *M* and *H* “know” how quickly to pulse?

The problem seems to me to be particularly pressing on Aristotelian theories of time on which time is defined by the changes of objects. Let’s imagine that in addition to *M* and *H* there are a thousand identical clocks running in the universe, with all objects causally isolated from each other, and that there is nothing else that is changing besides what I have described. Presumably, then, the changes that define time are the changes in the positions of the clock hands. Then on our Aristotelian theory, to say that *H* pulses every hour just means that *H* pulses once per revolution of the minute hand on a typical clock, and to say that *H* pulses every minute means that *H* pulses once per revolution of the seconds hand on a typical clock.

But, first, how do *M* and *H* know about the movement of the hands of the clocks, so as to keep in sync with them, if all the objects are causally isolated?

And, second, let’s imagine this. God speeds up the clocks one by one by a factor of two: today he speeds up clock *C*_{1}, tomorrow he speeds up clock *C*_{2}, and so on, until eventually all the clocks have been sped up. After a week, seven clocks, *C*_{1}, ..., *C*_{7}, have been sped up. This doesn’t matter for the definition of time: seven clocks out of a thousand are negligible, and the time standards are set up by *C*_{8}, ..., *C*_{1000}. According to *C*_{1}, ..., *C*_{7}, *M* pulses once per two rotations of the seconds hand, and *H* once per two rotations of the minutes hand. But according to *C*_{8}, ..., *C*_{1000}, *M* pulses once per rotation of the seconds hand and *H* once per rotation of the minutes hand, since these rotations correspond to *real* minutes and *real* hours.

However, after 993 days, clocks *C*_{1}, ..., *C*_{993} are running at the same rate and setting the time standards. The clocks *C*_{994}, ..., *C*_{1000} are the outliers. So, now, *M* pulses once per rotation of the seconds hands and *H* once per rotation of the minutes hands of *C*_{1}, ..., *C*_{993}. And *M* pulses twice per rotation of the seconds hands of *C*_{994}, ..., *C*_{1000}, and *H* twice per rotation of their minutes hands. So at some point in the process of speeding up clocks *C*_{8}, ..., *C*_{1000}, *M*’s pulses go from once per two rotations of the seconds hand of *C*_{1}, ..., *C*_{7} to once per rotation, and similarly for *H*. But how can the speeding up of clocks other than *C*_{1}, ..., *C*_{7} affect the correlations between *H* and *M* when everything is causally isolated?

(Note that talk of “speeding up clock *C*_{n}” makes sense even if the clocks jointly define time. If *n* = 1, we can understand it as speeding it up relative to clocks *C*_{2}, ..., *C*_{1000}. If *n* > 1, we can understand it as bringing clock *C*_{n} in sync with clock *C*_{n − 1}. What may be a bit ambiguous is what “day *n*” means, but nothing hangs on the details of how to resolve that.)

But even without an Aristotelian theory of time it is puzzling how *M* and *H* “know” how quickly to pulse.

I think there is a nice solution that lets one keep a part of the Aristotelian theory of time: (external) time is not just the measure of change, but the measure of change in causally interconnected things. There is no common timeline between *M*, *H* and the clocks when there are no causal connections between them. This, I think, requires the rejection of any theory on which there is a metaphysically deep “objective present”.

Another move is to deny that there can be intrinsic causal powers that make reference to metric external time.