Introduction

Bell’s theorem is based on a small amount of math. The main things to know are that there are two ways to compute a certain number in physics, and that it isn’t always clear which way to use.

With some numbers as inputs, those two ways are, more or less:

Add all of the numbers, then square the result.
Square all of the numbers individually, then add those results.

If the inputs are 2 and 3, A) gives (2 + 3)² = 5² = 25, and B) gives 2² + 3² = 4 + 9 = 13. There may be more than two inputs, but the important point is that A) and B) sometimes produce different results.

The relevant experiments use numbers other than 2 and 3, and an operation that is slightly different than squaring, but I will stick with those, to keep things simple.

The Theorem

A proof of Bell’s theorem then mostly boils down to the following steps:

In a certain experiment with inputs of 2 and 3, we know that the output of applying either A) or B) should be 25.
When we apply B), we get 13, which is less than 25.
Physics therefore cannot be explained in a straightforward way.

Experiments have repeatedly shown that 1) does produce an output of 25. People usually interpret those results to mean that 3) is also true, due to the above proof.

Standard responses to 3) often involve one or more of the following ideas:

Things sometimes affect each other, immediately, across large distances.
Some things don't have a defined state before we interact with them.
The universe you're in frequently splits into multiple universes.
Every experiment that we do coincidentally produces misleading results.

A Problem

I think it’s reasonable to ask why we wouldn’t apply A) in 2), in order to end up with 25 = 25.

It seems like people mostly ignore that possibility, for reasons that are unclear to me.

The only argument I can think of is that there are guidelines for when to use A) or B), and maybe people think the guidelines say to apply B) in 2). In that case, I would say there is probably a mistake in the guidelines, or in someone’s use of those rules, given that the prediction in 2) is wrong, it can easily be fixed by using A) instead of B), and the guidelines are known to be tricky to apply.

If 2) is wrong, as it would be if we apply A) instead of B) there, then 3) doesn’t follow, and the proof fails. It seems like 3) could then still possibly be true, even if 1) and 2) didn’t justify it. However, it turns out that using A) in 2) provides a fairly direct path to a theory in which 3) is wrong.

Conclusion

I don’t know how Bell’s theorem holds up against what I’m proposing. It seems like the proof doesn’t work, for reasons that I think are pretty simple. People appear to take Bell’s theorem very seriously, and I guess the point of this post is that I don’t understand why that is.