## Monday, July 15, 2019

### Probabilistic propensities and the Aristotelian view of time

Consider an item x with a half-life of one hour. Then over the period of an hour, it has a 50% chance of decaying, over the period of a second it only has a 0.02% chance of decaying. Imagine that x has no way of changing except by decaying, and that x is causally isolated from all outside influences. Don’t worry about Schroedinger's cat stuff: just take what I said at face value.

We are almost sure that x will one day decay (the probability of decaying approaches one as the length of time increases).

Now imagine that everything other than x is annihilated. Since x was already isolated from all outside influences, this should not in any way affect x’s decay. Hence, we should still be almost sure that x will one day decay. Moreover, since what is outside of x did not affect x’s behavior, the propensities for decay should be unchanged by that annihilation: x has a 50% chance of decay in an hour and a 0.02% chance of decay in a second.

But this seems to mean that time is not the measure of change as Aristotle thought. For if time were the measure of change, then there would be no way to make sense of the question: “How long did it take for x to decay?”

Here is another way to make the point. On an Aristotelian theory of time, the length of time is defined by change. Now imagine that temporal reality consists of x and a bunch of analog clocks all causally isolated from x. The chances of decay of x make reference to lengths of time. Lengths of time are defined by change, and hence by the movements of the hands of the clocks. But if x is causally isolated from the clocks, its decay times should have nothing to do with the movements of the clocks. If God, say, accelerated or slows down some of the clocks, that shouldn’t affect x’s behavior in any way, since x is isolated. But an Aristotelian theory of time, it seems, such an isolation is impossible.

I think an Aristotelian can make one of two moves here.

First, perhaps the kinds of propensities that are involved in having an indeterministic half-life cannot be had by an isolated object: such objects must be causally connected to other things. No atom can be a causal island. So, even though physics doesn’t say so, the decay of an atom has a causal connection with the behavior of things outside the atom.

Second, perhaps any item that can have a half-life or another probabilistic propensity in isolation from other things has to have an internal clock—it has to have some kind of internal change—and the Aristotelian dictum that time is the measure of change should be understood in relation to internal time, not global time.

IanS said...

For what it is worth, the spontaneous emission of a photon from an excited atom is treated in quantum theory as the result of interaction of the atom with the quantized EM field. Even in its ground state, with no photons present, the field exists and has quantum fluctuations. This picture matches your first move: “empty” space is not passive - it plays an essential role in the decay of the excited state.

I assume (N.B. not an expert) that the fields of the Standard Model play a similar role in the radioactive decay of atoms.

To be sure, it is not so clear how the QFT-based Standard Model fits with an Aristotelian view of the world. But that is a different issue.

Michael Gonzalez said...

This is very interesting. Could it be that, in a world with nothing else, there just is no "metric time" (to use Newton's term), and that Aristotle was referring to "metric time" without realizing it? Indeed, would there really be any "seconds" or "minutes", since there are no regularities of any sort? It might even cease to have meaning to talk about the "half-life" of the atom anymore. Still, what Newton called "mere duration" would have meaning, and the atom's having come into existence is clearly followed by a period without change, which is followed by its decaying, just without any definite metric intervals....

Alexander R Pruss said...

Michael:

Maybe.

But if there is no metric time in the absence of external objects, then there will be no intrinsic difference between two atoms, one of which has a half-life of an hour and the other has a half-life of a minute. Yet surely there is a difference there.

Here is another worry for the no-metric-time move. Take any two worlds, each of which has exactly one such atom at the beginning of the world's time. Well, in both worlds, the atom decays (at least with probability 1). If there is no metric time, we can't say that in one it decays faster than the other. So, the two worlds are exactly alike. But that means that we don't have any significant indeterminism here: for the significant indeterminism that the atoms are supposed to have is indeterminism as to when (not whether) they decay. (By "significant indeterminism", I mean indeterminism where neither possible outcome has zero probability.)

Here's a thought experiment: Consider a universe consisting of a large finite number of causally isolated atoms, some of which are one-hour half-life (Hs) and some of which one one-minute half-life (Ms). On average, we'd expect that there would be sixty M-decays per H-decay in any given hour. But if metric time is constituted by external reality, and external reality is constituted by the behavior of other atoms, how does one of the Ms "know" to decay faster than the Hs, since by hypothesis it is causally isolated from them?

Michael Gonzalez said...

On the possible worlds issue, I'm very inclined to just bite the bullet and say that they are indistinguishable worlds. Indeed, the indeterminacy, from a quantum mechanical perspective, has to do with distribution of probability in a configuration space, with distinct units of time as one of the variables (I may be mistaken... I'm no physicist!). If that's the case, then the configuration space for both worlds will be identical, in that they will have no units of time at all across which to distribute the probability of decay, and they would both just say that the particle will certainly (probability = 1) decay.

Now the case of causally isolated particles in the same world seems trickier to me.... If we imagine the decay as a gradual process, but the Ms have a steeper curve on their linear decay rate, then the Ms would decay more quickly than the Hs, despite causal isolation. They don't have to "know" or "care" what the Hs are doing, they just decay more quickly. But, in the case of stochastic, instant decay.... I don't know. It would seem to me that each passing second is supposed to make the decay more likely (with a greater increase for Ms than Hs each second), but are there really "seconds" in this situation? Maybe this is an issue with the very concept of causal isolation and/or pure indeterminacy?