Against isotropy

We think of Euclidean space as isotropic: any two points in space are exactly alike both intrinsically and relationally, and if we rotated or translated space, the only changes would be to the bare numerical identities to the points—qualitatively everything would stay the same, both at the level of individual points and of larger structures.

But our standard mathematical models of Euclidean space are not like that. For instance, we model Euclidean space on the set of triples (x, y, z) of real numbers. But that model is far from isotropy. For instance, some points, like (2, 2, 2) have the property that all three of their coordinates are the same, while others like (2, 3, 2) have the property that they have exactly two coordinates that are the same, and yet others like (3, 1, 2) have the property that their coordinates are all different.

Even in one-dimension, say that of time, when we represent the dimension by real numbers we do not have isotropy. For instance, if we start with the standard set-theoretic construction of the natural numbers as

0 = ⌀, 1 = {0}, 2 = {0, 1}, 3 = {0, 1, 2}, ...
and ensure that the natural numbers are a subset of the reals, then 0 will be qualitatively very different from, say, 3. For instance, 0 has no members, while 3 has three members. (Perhaps, though, we do not embed the set-theoretic natural numbers into the reals, but make all reals—including those that are natural—into Dedekind cuts. But we will still have qualitative differences, just buried more deeply.)

The way we handle this in practice is that we ignore the mathematical structure that is incompatible with isotropy. We treat the Cartesian coordinate structure of Euclidean space as a mere aid to computation, while the set-theoretic construction of the natural numbers is ignored completely. Imagine the look of incomprehension we’d get from a scientist if one said something like: “At a time t₂, the system behaved thus-and-so, because at a time t₁ that is a proper subset of t₂, it was arranged thus-and-so.” Times, even when represented mathematically as real numbers, just don’t seem the sort of thing to stand in subset relations. But on the Dedekind-cut construction of real numbers, an earlier time is indeed a proper subset of a later time.

But perhaps there is something to learn from the fact that our best mathematical models of isotropic space and time themselves lack true isotropy. Perhaps true isotropy cannot be achieved. And if so, that might be relevant to solving some problems.

First, probabilities. If a particle is on a line, and I have no further information about it except that the line is truly isotropic, so should my probabilities for the particle’s position be. But that cannot be coherently modeled in classical (countably additive and normalized) probabilities. This is just one of many, many puzzles involving isotropy. Well, perhaps there is no isotropy. Perhaps points differ qualitatively. These differences may not be important to the laws of nature, but they may be important to the initial conditions. Perhaps, for instance, nature prefers the particles to start out at coordinates that are natural numbers.

Second, the Principle of Sufficient Reason. Leibniz argued against the substantiality of space on the grounds that there could be no explanation of why things are where they are rather than being shifted or rotated by some distance. But that assumed real isotropy. But if there is deep anisotropy, there could well be reasons for why things are where they are. Perhaps, for instance, there is a God who likes to put particles at coordinates whose binary digits encode his favorite poems. Of course, one can get out of Leibniz’s own problem by supposing with him that space is relational. But if the relation that constitutes space is metric, then the problem of shifts and rotations can be replaced by a problem of dilation—why aren’t objects all 2.7 times as far apart as they are? Again, that problem assumes that there isn’t a deep qualitative structure underneath numbers.

The many-worlds interpretation and probability

The probability of a proposition p equals the probability that p is true. I have argued that this principle refutes open future views. It’s interesting that it also refutes the many-worlds interpretation of quantum mechanics.

Suppose that I have prepared an electron mixed spin state 2^−1/2|↑⟩+2^−1/2|↓⟩ and we are about to measure whether the spin is up or down. The Born rule says that I should assign probability 1/2 to each of the two possible measurements. But by the many-worlds interpretation, the world splits into two (or more—but I will ignore that complication as nothing hangs on the number, or even the number being being well-defined) branches: in one an electron in a spin-up state is observed and in the other one in a spin down state is observed. Now consider these two propositions:

1. I will observe a spin-up state.

2. I will observe a spin-down state.

Given the many-worlds interpretation, metaphysically reality is symmetric with respect to these two propositions, as reality includes branches with both observes and with the observer standing in the same relationship to me. Hence, either both are true or neither is true on the correct reading of the many-worlds metaphysics: both are true if the observer in both branches counts as me, and otherwise both propositions are false. If the correct reading of the metaphysics is that both are true, then the probability of each being true is 1, and hence by the principle I started the post with, the probability that I will observe a spin-up state is 1 and so is the probability that I will observe a spin-down state. If the correct reading of the metaphysics is that neither is true, then the probability of truth for each will be 0, and hence the probability of my making either observation is 0.

So, the probabilities of (1) and (2) are 0 or 1. In neither case are they 1/2, which is what the Born rule stipulates.

This seems to me to be a stronger argument than the more common argument against the many-worlds interpretation that all branches should have equal probability, and hence would violate the Born rule in cases where the quantum state has unequal weights. For the usual argument depends on indifference, which is a dubious principle.

The Principles of Indifference and Sufficient Reason

The Principle of Indifference says that we should assign equal probabilities to outcomes that are on par. Why? The thing to say is surely: "Well, there is no reason for one outcome to be more likely than another." But the equal probability of outcomes only follows from this remark if the Principle of Sufficient Reason holds so that when there is no reason for something, it doesn't happen. So it seems that the Principle of Indifference presupposes some version of the Principle of Sufficient Reason.

Symmetry and Indifference

Suppose we have some situation where either event A or event B occurred, but not both, and the two events are on par: our epistemic situation is symmetric between them. Surely:

1. One should not assign a different probability to A than to B.

After all, such a difference in probability would be unsupported by the evidence. It is tempting to conclude that:

2. One should assign the same probability to A as to B.

From (2), the Principle of Indifference follows: if it's certain that exactly one of A₁,...,A_n happened, and the epistemic situation is symmetric between them all, then by applying (2) to the different pairs, we conclude that they all have equal probability, and since the probabilities must add up to one, it follows that P(A_i)=1/n for all i.

But while (1) is very plausible (notwithstanding subjective Bayesianism), (2) does not follow from (1), and likewise Indifference does not follow. For (1) is compatible with not assigning any probability to either A or B. And sometimes that is just the right thing to do. For instance, in this post, A and D are on par, but the argument of the post shows that no probability can be assigned to either.

In fact, we can generalize (1):

3. One should treat A probabilistically on par with B.

If one of the two has a probability, the other should have a probability, and the same one. If one of the two has an imprecise probability, the other should have one, and the same one. If one is taken as maximally nonmeasurable, so should the other one be. And even facts about conditional probabilities should be parallel.

Nonetheless, there is a puzzle. It is very intuitive that sometimes Indifference is correct. Sometimes, we correctly go from the fact that A and B are on par to the claim that they have the same probability. Given (1) (or (3)), to make that move, we need the auxiliary premise that at least one of A and B has a probability.

So the puzzle now is: Under what circumstances do we know of an event that it has a probability? (Cf. this post.)

Indifference is dead, again

I've noted that given the Axiom of Choice, the Hausdorff Paradox kills the principle of indifference. But we don't need the Axiom of Choice to kill off Indifference in this way! Hausdorff's proof of his paradox[note 1] also showed, without at this point in the proof using the Axiom of Choice, that:

• There are disjoint countable subsets A, B and C of the (surface of the) sphere and a subgroup G of rotations about the center such that: (a) U=A∪B∪C is invariant under G, and (b) A, B, C and B∪C are all equivalent under rotations from G.

Suppose the only thing we know about some point x is that it lies in U. Now A, B and C are rotationally equivalent under rotations that U is invariant under, so our information about whether x lies in A or in B or in C is exactly on par. By Indifference, hypotheses that are equivalently related to our information get equal probability. Thus P(x∈A)=P(x∈B)=P(x∈C)=1/3 (that they equal 1/3 is due to the fact that the three hypotheses are mutually exclusive and jointly exhaustive). But A and B∪C are also rotationally equivalent under rotations that U is invariant under, so again by Indifference P(x∈A)=P(x∈B∪C)=1/2. And so 1/2=1/3, which is absurd.

The principle of indifference and paradoxical decomposition

There are many paradoxes for the Principle of Indifference. Here's yet another. The Hausdorff Paradox tells us that (given the Axiom of Choice) we can break up (the surface of) a sphere into four disjoint subsets A, B, C and D, such that (a) D is countable, and (b) each of A, B, C and B∪C is rotationally equivalent. This of course leads to yet another paradox for the Principle of Indifference. Suppose our only information is that some point lies on the surface of a sphere. By classical probability, we should assign probability one to A∪B∪C (and even if that's disputed, because of worries about measure zero stuff, the argument only needs that we should assign a positive probability). By Indifference, we should assign equal probability to rotationally equivalent sets. Therefore, since P(A∪B∪C)=1, we must have P(A)=P(B)=P(C)=1/3. But by another application of Indifference, P(B∪C)=P(A). So, P(B)=P(C)=P(B∪C)=1/3, which is absurd given that B and C are disjoint.

Does this add anything to what we could learn from the other paradoxes for Indifference? I doubt it.