> A binary distinction requires a binary definition No it doesn't. See absolute value of x on real numbers - it is usually defined as: abs(x) = x if x >= 0

> A binary distinction requires a binary definition No it doesn't. See absolute value of x on real numbers - it is usually defined as: abs(x) = x if x >= 0, else -x but it can also be defined as sqrt(x * x). You can use this trick to define arbitrarily complex "binary" f(x) as a combination of these. And there are many other tricks of course. But math is one thing, we see spectrums everywhere in reality and yet quantum mechanics is discrete (quantized :) ).