Some equations are more equal than others
Here’s a strange math problem that I encountered as an undergraduate:
What is the solution to the following equation?
[Note: The order of exponents here is such that the upper ones are taken first. For example, you should read as and not as .]
As it happens, there’s a handy trick for solving this equation, and that’s to use both sides as an exponent for . This gives
From the first equation, though, the left hand side is just . So now we’re left with simply , which means .
Not bad, right? Apparently the conclusion is that
Where things get weird is when you try to solve an almost identical variant of this problem. In particular, let’s try to solve:
We can do the same trick as before, using both sides of the equation as an exponent for , and this gives
so that we’re left with . The solution to this equation is, again, .
But now you should be worried, because apparently we have reached the conclusion that
So which is it? What is the correct value of ? Is it 2, or is it 4?
Yes, but which one is the real answer?
Maybe in the world of purely abstract mathematics, it’s not a problem to have two different answers to a single straightforward mathematical operation. But in the real world this is not a tolerable situation.
The reasoning above raised a straightforward question — what is ? — and provided two conflicting answers: and . Both of these equations are correct, but which one should you really believe?
Suppose that you don’t really believe either of those two equations (which, at this point, you probably shouldn’t), and you want to figure out for yourself what the value of is. How would you do it?
One simple protocol that you could do with a calculator or a spreadsheet is this:
- Make an initial guess for what you think is the correct value of .
- Take your guess and raise to that power.
- Take the answer you get and raise to that power.
- Repeat that last step a bunch of times.
Try this process out, and you will almost certainly get one of two answers:
If you initially guessed that was any number less than 4, then you will arrive at the conclusion that .
If your initial guess was something larger than 4, though, you will instead get to the conclusion that .
The situation can be illustrated something like this:
Only if your initial guess was exactly 4, and if your calculator gave you the exact correct answer at every step, will you ever see the solution . A single error in the 16th decimal place anywhere along the way will instead lead you to a final answer of either or .
In this sense is a much better answer than . The latter is true only in a hypothetical world of perfect exactness, while the former is true even if your starting conditions are a little uncertain, or if your calculator makes mistakes along the way, or (most importantly) there’s some small additional factor that you haven’t taken into consideration.
Scientists adjudicate between mathematical realities
For the most part, this has been a silly little exercise. But it actually does illustrate something that is part of the job of a physicist, or anyone else who uses math as a tool. Physicists spend a lot of time solving equations that describe (or are supposed to describe) the physical world. But finding a solution to some equations is not the end of process. We also have to check whether the solution we came up with is meaningful in the real world, which is full of inexactnesses. For example, the equation that describe the forces acting on a pencil on my desktop will tell me that the pencil can be non-moving either when lying on its side or when balanced on its point. But only one of those two situations really deserves to be called a “solution”.
So, as for me, if you ask me whether or , I’ll go with 2.
Because, as Napoleon the pig understood, some equations are more equal than others.