I continue my discussion of the quantum theory of uncertainty. Having outlined the axioms of classical theory of uncertainty (probability), I describe which of them don't apply in the real world, and what they should be replaced by.
In classical physics we use probability as a numerical representation of uncertainty. Indeed, not only in classical physics: this is our standard way of understanding the world. Whenever a bookmaker quotes an odd, or we say there is a 50-50 chance of Donald Trump creating a diplomatic row with one of his tweets in any given day, we are using probabilities to parametrise uncertainty. Even those who don't understand the details of the mathematical description of probability will still intuitively think in a way consistent with it.
Probability is convenient because probabilities can be used to directly predict frequency distributions. A probability distribution is proportional to the frequency distribution expected on average over a large number of samples. Frequency distributions are natural and intuitive, because all they are is a count of different types of object. And counting is something which even babies know how to do. The mathematics behind probability is thus the same as the mathematics behind counting. And so, probability comes naturally to us.
However, when we come to quantum physics, things are a little bit different, and probability doesn't work as a representation of uncertainty.
In this post, I briefly introduce the axioms behind probability theory, so that we can see in later posts which of them are violated by quantum physics.