Greek letter paradox

The Greek letter paradox is seeing the same symbol in two contexts and assuming it means the same thing. Maybe it’s used in many contexts, but I first heard it in the context of comparing statistical models.

I used the phrase in my previous post, looking at

α exp(5t) + β t exp(5t)

and

α exp(4.999 t) + β exp(5.001 t).

In both cases, α is the coefficient of a term equal to or approximately equal to exp(5t), so in some sense α means the same thing in both equations. But as that post shows, the value of α depends on its full context. In that sense, it’s a coincidence that the same letter is used in both equations.

When the two functions above are solutions to a differential equation and a tiny perturbation in the same equation, the values of α and β are very different even though the resulting functions are nearly equal (for moderately small t).

2 thoughts on “Greek letter paradox

  1. It doesn’t have to be Greek letters.
    I saw a problem on math.stackexchange where they were solving linked 2nd-order DEs, and they used C_1 and C_2 for the writing x(t), and then re-used C_1 and C_2 for writing y(t) – because, of course, the general solution, before fitting initial conditions, uses C_1 and C_2.

    The paradox is so strong that I couldn’t get people to recognize this error, and the accepted answer was “must have dropped a sign somewhere”.

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