Suppose time is in fact continuous and modeled by the real numbers.
It seems odd indeed to me that the real numbers should be the only possible way for time to run. The real numbers are a very specific mathematical system. There are other systems, such as the hyperreals or the rationals or even the integers, that seem to be plausible alternatives. I know of no argument that the time sequence has to be numbered by the real numbers.
Thus, given our initial supposition, it should be possible to have time sequences corresponding to ordered sequences numbered by the integers or the hyperreals. Here, then, is a further intuition. It is possible to have a multiverse with radically different spacetime structures in each universe of it. If so, then we would expect the possibility of a multiverse where different universes in a multiverse have time sequences based on very different ordered sets.
Suppose presentism is necessarily true. Then even in such a multiverse, there would be an absolute present running across all of these timelines in the different universes. And that would be rather odd. Imagine that in one universe the time-line is corresponds to the integers and in the other it corresponds to the reals, and both are found in one multiverse. What happens in the universe whose time-line is based on the integers when the line of the present moves continuously across the uncountable infinity of times numbered by the real numbers? Does it stay for infinitely moments at the same integer? But then at infinitely many moments of time it would be at one moment, which is a contradiction. Or does the universe with the integer time-line pop out of existence when the present doesn’t meet up with these integers? Maybe that’s the best view, but it’s a weird view.
Perhaps the presentist’s best bet is to say that there is a privileged mathematical structure that models what a time-line could be like. If so, my intuition says that the only candidate for that privileged structure would be a discrete structure like the integers. For there are arguments in the history of philosophy for time having to be discrete (arguments from Zeno through myself), but none for time having to be modeled by specifically the real numbers, or the rational numbers, or some specific hyperreal field.
5 comments:
> I know of no argument that the time sequence has to be numbered by the real numbers.
Hmmm, this is very interesting. What if there are a cluster of mutually reinforcing arguments? What do you make of the following arguments for thinking there are good, non-ad-hoc reasons to single out the real line once you assume “continuous” time?
a. Consider order-theoretic uniqueness. If you grant four very natural qualitative facts about moments ((i) they’re linearly ordered earlier/later, (ii) there is no first or last instant, (iii) between any two instants there is another (density), and (iv) every bounded set of instants has a least upper bound (Dedekind completeness)), then Cantor’s theorem kicks in: only one order type satisfies all four, and it is isomorphic to the usual real numbers.
b. Classical and quantum dynamics are written as differential equations. Those require limits to exist wherever Cauchy sequences converge. Completeness (the very property that picked out ℝ a moment ago) is exactly what guarantees such limits. On the rationals there are gaps, on the integers no limits at all, and on a non-Archimedean field you get infinitesimal “holes” that ruin standard proofs of existence and uniqueness. The real line is simply the smallest field in which the toolkit of calculus works exactly as our successful physical theories need it to.
c. What about symmetry considerations? Time translations form a one-dimensional Lie group, Stone’s theorem and Noether’s theorem both lean on that fact. Break completeness or introduce infinitesimals and the group ceases to be a smooth manifold or even a Lie group, so the tight link between symmetries and conservation laws disappears.
In summary, ℚ is incomplete; ℤ kills differentiability; hyperreals and other non-Archimedean fields break separability, so you spend your time projecting back to the “standard part” (i.e. ℝ) whenever you want predictions.
Wait, you defend the discreteness of time? If so, how certain are you of it as opposed to continuousness? Would the continuousness of time impact any of your other beliefs, such as causal finitism?
I think these are very good reasons to conclude that time IS modeled by the reals, but I think they are fairly weak reasons to conclude that time MUST be modeled by the reals.
(I also wonder if (b) and (c) wouldn't work with definable or computable reals. And maybe (a), too, with a suitable definability or computability restriction on the cuts.)
(Note, also, that the arguments presuppose a privileged model of set theory. Maybe that's OK.)
Continuous time is prima facie problematic on causal finitism (it seems then that a typical physical event has a cause at all the infinitely many times in the preceding second), but I do discuss ways of making the two go together in my infinity book.
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