I have almost no idea what to do with this observation, but I think there’s a point of disanalogy between geometric and temporal continua, even if we take geometric continua to be ‘directional’.
Take a geometrical line from A to B. Here, we have a pair of limits and the extension between them. Now take a temporal continuum. But let’s understand the temporal continuum as the time of some particular change. So a block of wood bleaches in the sun, going from dark to pale. The temporal continuum we’re concerned with is the time that this change takes (say, one week).
So suppose this temporal continuum also has two limits, C (at the beginning) and D (at the end of the change), and an extension between them. Geometrical continua needn’t actually have minima and maxima, the ‘line on a paper’ is a case where they do. If this is so, then geometrical and temporal continua are in this sense analogous. But there’s a problem with the idea that temporal continua have a minimum. Suppose C is an indivisible moment. Has any change been accomplished at C? If yes, then given that no change can be accomplished in a moment (since there is no temporal difference), there must be previous moment at which some change had been accomplished, and therefore C is not early enough to be the first moment of the change.
If no change has been accomplished at C, then C is too early to be the first moment: it is not properly called part of the time of the change. So no matter what, the moment C cannot be the first moment of change.
Nothing prevents us from having a last moment, but temporal continua (so long as they are considered the time of some particular change) cannot have a first moment. Temporal continua can have greatest lower bounds, but no actual minimum.
I have almost no idea what to do with this observation, but I think there’s a point of disanalogy between geometric and temporal continua, even if we take geometric continua to be ‘directional’.
Take a geometrical line from A to B. Here, we have a pair of limits and the extension between them. Now take a temporal continuum. But let’s understand the temporal continuum as the time of some particular change. So a block of wood bleaches in the sun, going from dark to pale. The temporal continuum we’re concerned with is the time that this change takes (say, one week).
So suppose this temporal continuum also has two limits, C (at the beginning) and D (at the end of the change), and an extension between them. Geometrical continua needn’t actually have minima and maxima, the ‘line on a paper’ is a case where they do. If this is so, then geometrical and temporal continua are in this sense analogous. But there’s a problem with the idea that temporal continua have a minimum. Suppose C is an indivisible moment. Has any change been accomplished at C? If yes, then given that no change can be accomplished in a moment (since there is no temporal difference), there must be previous moment at which some change had been accomplished, and therefore C is not early enough to be the first moment of the change.
If no change has been accomplished at C, then C is too early to be the first moment: it is not properly called part of the time of the change. So no matter what, the moment C cannot be the first moment of change.
Nothing prevents us from having a last moment, but temporal continua (so long as they are considered the time of some particular change) cannot have a first moment. Temporal continua can have greatest lower bounds, but no actual minimum.