My Faults My Own

One's ponens is another's tollens.

Quantum of Cake

This is a one-off resource for a conversation I'm in the middle of. It is not intended for general use. If you got here accidentally, please go away now.

Alex and Sam discover a crate full of mooncakes. Each is circular, but has an arrow on the top pointing from one edge to the other. And they're wrapped in pairs.

They discover that if they take an individual cake and cut it in half with a knife, sometimes the filling is red, and sometimes it's blue. (They can't tell from looking at the outside, and once it's open it never changes.)

Being scientists, they decide to experiment. They unwrap many pairs and cut them at random angles (relative to the cakes' arrows), and record the number of reds and blues. They get about as many reds as blues overall, with no correlation observed between the colors observed within a pair. That is, 1/4 of the time, both cakes in a pair have red filling; 1/4 of the time, both have blue; and 1/2 of the time, one is red and the other blue.

random directions25%50%25%

And if you take one cake from a pack and break it open, there is no perceptible pattern to its color, even considering angle of the cut.

single cake (any angle)50%50%

But...they find that if they take two cakes from a set and cut them both along the direction of the arrow, they find that the cakes are always the same color. If they cut both perpendicular to the arrow...same thing: always the same color. Or exactly 17 degrees clockwise from the line of the arrow, or...

same direction50%0%50%

(Note that there's no such thing as cutting "along the arrow backwards"; you just pick a line and chop along it -- they experimentally verify that "direction of cut" makes no difference to the expected result.) (Also, imagine that they're forced to cut along diagonals, for whatever reason floats your boat.)

This isn't the only odd thing: When they take a pair and cut them at perpendicular angles (e.g. one along the direction of the arrow; the other across), they always find different colored fillings inside -- one red, the other blue. Or if they cut one at a 45 degree angle clockwise from the arrow, and the other of the pair 45 degrees counterclockwise...different colors, always.

perpendicular directions0%100%0%

And if they cut a pair at angles that are exactly 45 degrees apart, they find that the cake fillings seem to be distributed independently: 1/4 red-red; 1/4 blue-blue; 1/2 red-blue (or blue-red).

45 degrees apart25%50%25%

Says Alex: "Aha! I will now form a hypothesis! Each cake has a 'true red direction', and if you cut within 45 degrees of it (or 45 degrees of its exact opposite), you get red -- otherwise, you get blue. And each pair has the same 'true red direction', so this explains our results!" (Take a minute to explain to yourself why it does -- assume a prior that allows the "true red direction" to be equidistributed around the circle, and update on finding red by cutting one cake at a particular angle.)

But Sam offers to test this hypothesis by experiment: "So, you're saying that if we cut a pair at angles 22.5 degrees apart, they should agree...75% of the time?" Alex: "Yeah!"

22.5 degrees apart~42.7%~14.6%~42.7%

Oops. They try it for 5 degrees, 10 degrees, 15 degrees, and so on, and eventually their results suggest that the relationship is as follows:

\[P(\text{cakes agree}) = (\cos(\text{angle difference}))^2\]

(example graph)

They double-check their results, but it's definitely correct, and Alex's "true red direction" hypothesis is definitely wrong. (Note that if the "true red direction" hypothesis is correct, this curve should have straight lines from (0,1) to (π/2,0) to (π,1), like example graph)

"Okay," says Alex, "maybe we can explain this with a 'fuzzy red direction', where the probability that you cut the cake open and find red inside is some wonky function of the angle between the angle of your cut and the angle of the 'fuzzy red direction'..."

But there isn't. Because (and you have to trust me on this, dear reader) there isn't any function for probability of finding red given angular distance from the "red direction" such that:

  1. If your prior for the 'fuzzy red direction' is equidistributed around the circle,
  2. and you update on Alex having cut along the arrow and found red,
  3. your posterior is that Sam, cutting the other cake of the pair at an angle \(\theta\) to the arrow, will find red with probability \(= (\cos(\theta))^2\).

So there isn't any direction baked into both cakes simultaneously that tells you what you'll find when you cut them open.

Of course, once you're in the situation of [Alex has cut the first cake along the arrow and found red.], now you can define a function for probability of Sam finding red given angular distance from "Alex's red direction"...but that's not the same thing as having one function which worked for both Alex and Sam, based on some baked-in directionality.

The superdeterministic story is "Well, whoever baked these cakes must just have put normal red and blue filling in them in just the right way so that our experiments turned out like this...and all of our future experiments will, too!"

The Copenhagen story is "Well, when Alex cuts open their cake, it instantaneously makes Sam's cake acquire some property that drives its angle-to-probability-of-red function!"

The many-worlds story is...well, Eliezer tells it best. (sequence S in RAZ.)