Hyperreal worlds

In a number of papers, I argued against using hyperreal-valued probabilities to account for zero probability but nonetheless possible events, such as a randomly thrown dart hitting the exact center of the target, by assigning such phenomena non-zero but infinitesimal probability.

But it is possible to accept all my critiques, and nonetheless hold that there is room for hyperreal-valued probabilities.

Typically, physicists model our world’s physics with a calculus centered on real numbers. Masses are real numbers, wavefunctions are functions whose values are pairs of real numbers (or, equivalently, complex numbers), and so on. This naturally fits with real-valued probabilities, for instance via the Born rule in quantum mechanics.

However, even if our world is modeled by the real numbers, perhaps there could be a world with similar laws to ours, but where hyperreal numbers figure in place of our world’s real ones. If so, then in such a world, we would expect to have hyperreal-valued probabilities. We could, then, say that whether chances are rightly modeled with real-valued probabilities or hyperreal-valued probabilities depends on the laws of nature.

This doesn’t solve the problems with zero probability issues. In fact, in such a world we would expect to have the same issues coming up for the hyperreal probabilities. In that world, a dartboard would have a richer space of possible places for the dart to hit—a space with a coordinate system defined by pairs of hyperreal numbers instead of pairs of real numbers—and the probability of hitting a single point could still be zero. And in our world, the probabilities would still be real numbers. And my published critiques of hyperreal probabilities would not apply, because they are meant to be critiques of the application of such probabilities to our world.

There is, however, a potential critique available, on the basis of causal finitism. Plausibly, our world has an infinite number of future days, but a finite past, so on any day, our world’s past has only finitely many days. The set of future days in our world can be modeled with the natural numbers. An analogous hyperreal-based world would have a set of future days that would be modeled with the hypernatural numbers. But because the hypernatural numbers include infinite numbers, that world would have days that were preceded by infinitely (though hyperfinitely) many days. And that seems to violate causal finitism. More generally, any hyperreal world will either have a future that includes a finite number of days or one that includes days that have infinitely many days prior to them.

If causal finitism is correct, then “hyperreal worlds”, ones similar to ours but where hyperreals figure where in our our world we have reals, must have a finite future, unlike our world. This is an interesting result, that for worlds like ours, having real numbers as coordinates is required in order to have both causal finitism true and yet an infinite future.

Reflection and null probability

Suppose a number Z is chosen uniformly randomly in (0, 1] (i.e., 0 is not allowed but 1 is) and an independent fair coin is flipped. Then the number X is defined as follows. If the coin is heads, then X = Z; otherwise, X = 2Z.

• At t₀, you have no information about what Z, X and the coin toss result are, but you know the above setup.

• At t₂, you learn the exact value of X.

Here’s the puzzling thing. At t₂, when you are informed that X = x (for some specific value of x) your total evidence since t₀ is:

• E_x: Either the coin landed heads and Z = x, or the coin landed tails and Z = x/2.

Now, if x > 1, then when you learn E_x, you know for sure that the coin was tails.

On the other hand, if x ≤ 1, then E_x gives you no information about whether the coin landed heads or tails. For Z is chosen uniformly and independently of the coin toss, and so as long as both x and x/2 are within the range of possibilities for Z, learning E_x seems to tell you nothing about the coin toss. For instance, if you learn:

• E_1/4: Either the coin landed heads and Z = 1/4, or the coin landed tails and Z = 1/8,

that seems to give you no information about whether the coin landed heads or tails.

Now add one more stage:

• At t₁, you are informed whether x ≤ 1 or x > 1.

Suppose that at t₁ what you learn is that x ≤ 1. That is clearly evidence for the heads hypothesis (since x > 1 would conclusively prove the tails hypothesis). In fact, standard Bayesian reasoning implies you will assign probability 2/3 to heads and 1/3 to tails at this point.

But now we have a puzzle. For at t₁, you assign credence 2/3 to heads, but the above reasoning shows you that at t₂, you will assign credence 1/2 to heads. For at t₂ your total eidence since t₀ will be summed up by E_x for some specific x ≤ 1 (E_x already includes the information given to you at t₁). And we saw that if x ≤ 1, then E_x conveys no evidence about whether the coin was heads or tails, so your credence in heads at t₂ must be the same as at t₁.

So at t₁ you assign 2/3 to heads, but you know that when you receive further more specific evidence, you will move to assign 1/2 to heads. This is counterintuitive, violates van Fraassen’s reflection principle, and lays you open to a Dutch Book.

What went wrong? I don’t really know! This has been really puzzling me. I have four solutions, but none makes me very happy.

The first is to insist that E_x has zero probability and hence we simply cannot probabilistically update on it. (At most we can take P(H|E_x) to be an almost-everywhere defined function of x, but that does not provide a meaningful result for any particular value of x.)

The second is to say that true uniformity of distribution is impossible. One can have the kind of uniformity that measure theorists talk about (basically, translation invariance), but that’s not enough to yield non-trivial comparisons of the probabilities of individual values of Z (we assumed that x and x/2 were equally likely options for Z if X ≤ 1).

The third is some sort of finitist thesis that rules out probabilistic scenarios with infinitely many possible outcomes, like the choice of Z.

The fourth is to bite the bullet, deny the reflection principle, and accept the Dutch Book.

Zero chance events

A standard thing in the philosophy of science to say such stochastic explanation questions is that one can given an answer in terms of the objective chance of the event, even when these chances are less than 1/2.

But consider the question: Why did this atom decay exactly at t₁?

Here, the objective chance may well be zero. And surely that an event had zero chance of happening does nothing to explain the event. After all, that the decay at t₁ had zero chance does not distinguish the atom’s decaying at t₁ from the atom’s turning into a square circle at t₁. And to explain something we minimally need to say something that distinguishes it from an impossibility.

Here, I think, the causal powers theorist can say something (even though I may just want to reject the presuppositions; see the Response to Objection 2, below). Stochastic systems have a plurality of causal powers for incompatible outcomes. The electron in a mixed-spin state may have both a causal power to have its spin measured as up and to have its spin measured as done. Normally, some of the causal powers are apt to prevail more than others, and hence have a greater chance than others. But even the weaker causal powers are there, and we can explain the event by citing them. The electron’s spin was measured as, say, up because it had a causal power to that outcome; had it been measured as, say, down, that would have been because it had a causal power to that outcome. We can give further detail here: we can say that one of these causal powers is stronger than the other. And the stronger causal power has, because it is stronger, a higher chance of prevailing. But even the weaker causal power can prevail, and when it does, we can explain the outcome in terms of it.

This story works just fine even when the chances are zero. The weaker causal power could be so weak that the chance associated with it has to be quantified as zero. But we can still explain the activation of the weaker causal power.

So, going back to the decay, we can say that the atom had a causal power to decay at t₁, and that’s why it decayed at t₁. That causal power was of minimal strength, and so the chance of the decay has to be quantified as zero. But we still have an explanation.

The causal powers story about the atom encodes information that the chances do not. The chances do not distinguish the atom’s turning into a square circle from the atom’s decaying exactly at t₁. The causal powers do, since it has a power to decay but no power to turn into a square circle.

Objection 1: Let’s say that the atom has twice as high a chance of decaying over the interval of times [0, 2] as over the interval of times [0, 1]. How do we explain that in terms of causal powers, given that there are equally many (i.e., continuum many) causal powers to decay at precise times in [0, 2] as there are causal powers to decay at precise times in [0, 1]?

Response: It could be that just as the causal power story carries information the chance story does not, the chance story could carry information the causal power story does not, and both stories reflect aspects of reality.

Another story could be that there are causal powers associated with intervals as well as points of times, and the causal power to decay at a time [0, 2] is twice as strong as the causal power to decay at a time in [0, 1]. There are difficulties here, however, with thinking about the fundamentality relations between the powers associated with different intervals. I fear that there is no avoiding an infinite sequence of causal powers that violates causal finitism, and I am inclined to reject the possibility of exact decay times—and hence reject the explanatory question I started this post with. I don’t see much hope for a measurement of an exact time after all. But someone with other commitments about finitism could have a story.

Objection 2: This is just like a dormitive power explanation of opium making someone sleepy.

Response: Opium’s dormitive power is fundamental or not. If opium has a fundamental dormitive power, then the dormitive power explanation is perfectly fine. That’s just the kind of explanation we have to have at the fundamental level. If the dormitive power explanation is not fundamental, then the explanation is correct but not as informative as an explanation in terms of more fundamental things would be.

Likewise, the power to decay at t₁ either is or is not fundamental. If it is fundamental, then the explanation in terms of the power is perfectly fine. If it is not, then there is a more fundamental explanation. But probably the more fundamental explanation will also involve minimal strength powers with zero activation chances, too.