(What follows is an attempt at an intuitive explanation of Bell’s inequality, initially for my own benefit. The Bell’s Inequality pages on wikipedia are an excellent example of the distinction between a definition and an explanation. There are some good simplified accounts out there, but to my mind they don’t quite get at what’s *going on*.)

The starting point is that quantum mechanics, *as a theory*, is indeterministic. All it can tell you is the probability that there will be a particle at a certain location at a certain time; it offers no way to say definitely whether it will be there or not.^{1} Even complete knowledge of all the quantum-mechanical details—which is humanly impossible for a system of any real size—would still only yield a probability.

The natural response, for the deterministically inclined (like Einstein), is that quantum mechanics must simply be incomplete. Some additional features of the physical situation, ones that aren’t described by QM, determine where the particles actually end up going. Once physicists understand those “hidden” variables, order will be restored and physics will be normal again. God doesn’t play dice; he can be mysterious, but he’s not *capricious*.

The alternative is that QM *is* complete, and reality is indeterministic. The reason QM can’t tell you where particles will go is because *nothing* determines where they go: it’s genuinely random.

Obviously, one way to resolve this debate would be to find the hidden variables. On the other hand, not finding them doesn’t prove the determinist wrong—perhaps we just need to keep looking. So how can the indeterminist win the debate? Scientists want to find explanations of things; that’s kind of what they’re for. How can the indeterminist persuade them to give up the search, and accept that QM is the best we can hope for?

Bell’s inequality, surprisingly, offers an answer: a way for the indeterminist to argue that no such hidden variables do or can exist.

The argument goes something like this.

• If, as the determinist believes, what we see is determined by factors that are fixed in advance, then our observations have to obey some simple rules of arithmetic.

• So, if the results of experiments vary in a way that breaks those rules, then what we’re observing must be genuinely random.

• The results of experiments do indeed break those rules.

So what are these rules? If you search up “Bell’s inequality” you will be shown some fiendishly complicated formulae. Fortunately, the arithmetical principle involved is simple; the complexity comes from wrestling it into a form that can be tested by quantum mechanical experiments.

Here’s the principle: if you have any three numbers a, b and c, the difference between a and c is equal to the sum of the difference between a and b, and the difference between b and c. In a formula:

(a - c) = (a - b ) + (b - c)

That’s *very* simple algebra: the right hand side has a positive and a negative b in it, which cancel out, leaving the left hand side.

Now, if you look at the absolute values of the differences, you get an inequality: the right side has to be the same or larger than the left.^{2} Formally:

|a - c| ≤ |a - b| + |b - c|

Informally, this is also quite intuitive. As an illustration, let’s say the difference between my height and my wife’s is two inches, and that there are three inches between my wife and my son. How much difference is there between me and my son? Well, the most it can be is if you stack the two differences on top of each other:

That’s the stereotypical family, with the wife having the middle height. If I have the middle height—as in reality—the difference between us will be less, just one inch:

But there’s no way the difference can be *more* than 5 inches.

That right there is the essence of Bell’s inequality. If you accurately measure a set of differences, and your readings don’t satisfy the inequality, and the laws of arithmetic remain in place, then the values of what you’re measuring can’t have been fixed before you measured them.

That doesn’t quite mean that the values are random. The values might simply have changed, in an ordinary deterministic sort of way, while you were doing all the measurements. So experiments testing Bell’s Inequality have to go to great lengths to ensure that nothing gets changed during the process—isolating the experiment from outside influences, and making sure that the process of measuring one value doesn’t affect the others. If you can rule out Skander growing during the experiment, and yet he’s still 7 inches taller than me when he stands next to me, then you’re left with the possibility that we don’t have determinate heights after all. The only other ways to avoid indeterminacy—the so-called “loopholes“—seem even less plausible than randomness. (Much more on that later.)

So why are the formal statements of Bell’s inequality so complicated? It’s to do with finding an instance of the basic inequality that contradicts QM’s predictions. (QM doesn’t say **every** measurement violates the inequality.) It also involves some more quantum mechanical absurdity to do with measurement. More on that next.

[1] Here I’m treating **location** as the property of interest. The same is true for other properties, such as spin and momentum.

[2] The "absolute value" of the difference between two numbers is just how large it is—regardless of which number is bigger, and so whether the difference is positive or negative.