Let’s try to make a profound statement about reality by thinking hard about baked goods.

I promise this is going somewhere.

A certain bakery has a special deal on muffins. They sell mystery boxes for those who like to live dangerously: mix-and-match sets of three muffins apiece. Each day, Alice, Bob and Charlie buy a mystery box together, and each day, Alice, Bob and Charlie take one muffin apiece back to their respective laboratories for analysis. They each have two testing devices — say, a device that can test whether a muffin is positive for dairy, and another device that can test whether it is positive for tree nuts. We’ll call these $X$ tests and $Y$ tests for short. Each day, Alice chooses either to do an $X$ test or a $Y$ test. Bob likewise chooses, independently of Alice, and so does Charlie. Importantly, each muffin can only be tested *once.* Maybe the test destroys the muffin, or maybe it takes so long to do one that they eat their muffins immediately afterward. Whatever the rationale, one test per muffin — that’s a rule of the parable.

We can write what they choose to do in a compact way. For example, if all three of them choose to do the $X$ test on their respective muffins, we’ll write $A_X B_X C_X$. If Bob and Charlie choose to do the $Y$ test but Alice goes instead with the $X$ test, we’ll write $A_X B_Y C_Y$. And so on. We can also write the results compactly, using $+1$ to stand for a positive result and $-1$ to stand for a negative one. (We could also record the outcomes with zeros and ones, or with trues and falses, greens and blues, etc. Using $+1$ and $-1$ is just a notation that will turn out to be helpful in a moment.) So, for example, if Alice chooses $Y$, Bob chooses $X$ and Charlie goes with $Y$, the results might be $(+1, -1, -1)$. Or they might be $(+1, +1, +1)$, or perhaps $(-1, +1, -1)$.

Over many days of muffin investigation, comparing their notes, they find a dependable pattern. Whenever *two of them* choose to do the $Y$ test, then the *product of their results* is always $+1$. The specific outcome varies randomly from day to day, but there’s never only one $-1$, and they never get all three results being $-1$. From this pattern, they can draw a couple conclusions. First, once two of them obtain their results, the result of the third is predictable. Let’s say their choices are $A_Y B_X C_Y$, as in the previous example, and both Alice and Bob get the result $+1$. Then we can predict that Charlie will get $+1$, because that’s the only way the product of the three numbers can be $+1$. Or, suppose their choices are $A_Y B_Y C_X$, and both Bob and Charlie get a $-1$. The two of them report their results and wait for Alice. Knowing Bob and Charlie’s results, we can predict that Alice will report a $+1$ outcome, because that’s the only way the product of the three outcomes is $+1$. A minus times a minus makes a plus, and so a third minus would spoil the plus.

If we had used a different notation for the outcomes, like “green” and “blue” instead of $+1$ and $-1$, then we could express this pattern by saying that whenever two of them choose to do the $Y$ test, an even number of the results will be blue.

Now, we deduce something else from the pattern. We can make a prediction about what happens under very different conditions. What about the days when all three choose to measure $X$?

Continue reading The Parable of the Muffins