Thoughts on Time

This is an addendum to the essay On Creation and Time.

An age-old problem in cosmology - cosmology in the philosophical sense - is whether the Time is finite or infinite. To talk about the finitude or infinitude of time, a measure of time, a metric of the length of a time interval needs to be given. In the present age people typically measure time by means of setting a motion which is periodic and whose period is thought to be constant in time as a standard unit of time, e.g.

The second, symbol \(s\), is the SI unit of time. It is defined by taking the fixed numerical value of the caesium frequency, \(\Delta \nu_{\ce{Cs}}\), the unperturbed ground-state hyperfine transition frequency of the caesium 133 (\(\ce{^133 Cs}\)) atom, to be \(9192631770\) when expressed in the unit \(\mathsf{Hz}\), which is equal to \(\mathsf{s}^{−1}\).

But this leads to another problem: by means of what is it guaranteed that the caesium frequency is fixed? By this it is not meant that in an absolutely fixed time interval of one second the frequency is subject to change, but that the very one second that is the ground for the frequency is undefined, not necessarily fixed - and to fix it one needs another ground for its fixation and ad infinitum. The measure of time is strictly speaking conventional in a strong sense: it is not the case that merely the measuring standard is a convention, the thing that is measured by the standard is also, at least logically, determined by the convention. A question can be asked: (provided that event order is definable before a 'metric' is given) what was the length of one second, then, before the first emergence of \(\ce{^133 Cs}\) atom, namely, before, say, the Big Bang Nucleosynthesis, which also happened only hypothetically? It seems that only after accepting some form of fatalism (if materialism is the substratum) or Platonism can the measure or the definition of one second be backwardly extrapolated to the past.

But what does it matter even if an absolute measure of time cannot be given, provided that it can be shown that Time has a beginning and an end? If a being is outside of time, even when Time is structured like a circle, for that being Time is strictly speaking finite: any point of time is its end and its beginning. Or maybe only when Time is compact, e.g. structured like a circle, can it be said to be finite, since if the topology of Time is non-compact, it is not guarenteed that a covering of it by time intervals admits a finite subcover, and by calibration it is always possible to choose an infinite covering, which means that there is a way to cover Time with infinitely many durations that can be individually calibrated to be the atomic constituents of Time. If Time is seen to be homeomorphic to the open interval \((0,1)\) for this being, what are the beginning and the end, as seen by it, of this seemingly finite, in length, Time? Are the beginning and the end inside of Time, or are they outside of it? What does it mean that the beginning and the end are outside of Time?

Hence, putting the issue of giving an absolute measure of time aside, an important issue need to be addressed first: what, then, does a time inverval consist of? We have mentioned earlier that durations, or time intervals, can be seen to constitute Time. Now, given a, say, one second that is absolute, one can still talk about \(1/10\) second, \(0.192857\)…second, viz., time is not an amalgamation of one seconds in the sense that a basket of apples is an amalgamation of individual apples. The problem is essentially the question of what does a space consist of - is it a structured set with a topology, or something inherently different from a mere set? If a time interval consists of infinitely many instants as a space seen as a mere set consists of points (the elements of the set), then it is legitimate to say that the beginning and the end of Time with a topology homeomorphic to \((0,1)\) are both in Time: they're the immediate next or preceding instant of the delimiters - the 'prison walls' - \(0\) and \(1\) that are outside of time, but then one needs to face Zeno's paradox. If, on the contrary, a time interval consists of its subintervals, then the finitude of Time homeomorphic to \((0,1)\), and in general Time's with non-compact topologies, is questionable. Of course a topology needs to be defined first so that one can talk about the compactness of a set with the topology equipped, but as it is already discussed, to say that Time is a set with a topology on it is already a choice with strong consequences.

To be rid of Zeno's Paradox, it might seem desireable to say that a time interval consists of its subintervals, but then any time interval is infinitely divisible. Suppose that a linear order can always be given, then given events that occured at the instants \(t_1\) and \(t_2\) that delimits the interval \((t_1,t_2)\) but not inside any other time intervals as instants that constitute them, more events can happen in, say, the subinterval \((t'_1,t'_2)\), and by a similar token to the case when Time is said to consist of instants, a Paradox essentially not different from Zeno's arises. However this can only be a problem when the subintervals are thought to be existent before they're thought to exist: \((0,1)\) is by itself a time interval, it doesn't need to be made up of any subintervals, but it can be thought to be made up of them, e.g., when there are events happening in that interval. And further, notice that an event in Time now always are inside some intervals, and for an interval to be given there needs to be delimiters, so that causal order need not be isomorphic to time order. Also, Time as a continuum now should be seen as intuitionistic, so that the order structure is no longer transparent, e.g. the trichotomy no longer holds. The finitude or infinitude of Time, is also no longer a problem that can be given a definite answer independent of human perception and behaviour, since Time is not given as a set but constructed, and might be under the process of construction, where the past is still being affected by the present, and future can be partially but not wholly determined.

2023-11-15