Professor Jaynes writes:
Why must degrees of plausibility be represented using real numbers when there are alternative number systems, which don’t have 1-to-1 correspondence to the reals, that satisfy transitivity and universal comparability?
Kevin S. Van Horn writes in his paper, “Constructing a Logic of Plausible Inference: A Guide to Cox’s Theorem”:
What’s so special about ? If we want our theory to have no holes then why don’t we use hyperreals or complex numbers? They also satisfy transitivity and universal comparability and I would argue that they have less holes than
I think the point is that
a) The reals are most convenient for representing plausibility.
b) Any results that hold under the transitive and universal comparability laws will still apply if we represent plausibility with reals, so you might as well use them.
So axiom 1 of Cox’s theorem is just “plausibilities can be represented using reals”, not “plausibilities must be represented using reals”?
Complex numbers don’t have an ordering, which seems counterintuitive as usually you can talk about things being more or less plausible. (Though they do get used in quantum mechanics for something that could be said to resemble plausibility, but that introduces a whole bunch of complicating factors that are not relevant here.)
When it comes to number systems with infinitesimals, you could often in principle use them, but in practice they aren’t relevant because infinitesimal values will almost always be outweighted by non-infinitesimal values.
(See amplitude if you want to look at the quantum generalisation of probability from R to C)
If it is not necessary to use reals, can you still say that probability theory as extended logic follows from common sense?
I think so. Complex numbers and infinitesimals are IIRC the only possible alternatives to the reals, but complex numbers only apply to certain limited quantum contexts (roughly speaking, complex numbers apply when information is perfectly preserved, while real numbers apply in contexts where there’s information leakage into the environment), while infinitesimals can be approximated perfectly by real numbers. So in everyday contexts (which is presumably where “common sense” applies), plausibility is captured by real numbers.
The thing that is special about R is that it is the unique (up to relabelling) complete ordered Archimedean field.
You’ve already said that it has to be ordered. It also needs to be a field so we can add, multiply, and divide things corresponding to the various properties that arise from Cox’s theorem. Completeness is exactly the “no holes” property.
The final one is the Archimedean property. This one says that for any element x>0 there is some natural number n such that nx>1. In terms of plausibility, it says that anything “infinitesimally small” in plausibility should be represented by zero, and not one of some infinite set of things that are for all finite purposes equivalent to zero.
If you relax the Archimedean property, you get systems such as hyperreals, surreals, and such. They all necessarily extend R, and so you do still end up needing to deal with R anyway. They also have awkward properties when used as numbers by which to measure things (such as countable additivity failing to be sufficient, but uncountable additivity being too restrictive to work).
So sorry! I assumed hyperreals and complex numbers are bigger than reals like a complete amateur!