Axiomatic Density Heuristic of a Domain

What defines expertise ?

Let's consider a building. Is it the ability to build quickly? Is it the ability to envision a building to begin with? Or is it the ability to plan out the details to build the building?

I don't know yet, but let me log my thoughts anyway, under the hope that my forthcoming thoughts will help address this in some way.

Let's imagine I am building a first principles reasoning system. One that derives the axioms of a domain from sample data points.

Eg - Consider Planar Geometry. Let's say I have a bunch of proven statements(Human verified data points in Euclidean Geometry) such as :

The sum of exterior angles of a convex polygon is 360° or this beautiful result in planar geometry used extensively in competitive math problems

Can I build an RLHF system with the end goal being "Derive the 5 axioms of planar geometry" ?

RLHF system would need to reason about reasoning (meta reasoning) with the end state being the list of 5 axioms of Planar Geometry. Note that I used planar geometry as an example as the axioms of planar geometry are well established.

In any other domain, we might not know the axioms, so there is intuition involved in which data points to keep as part of the axiom set.

Now, if you ask me, we have answered our initial question.

What is expertise? If not the ability to figure out the data points to keep? Figure out the axioms via intuition deep within?

But here is an interesting follow up question.

Can we make this intuition quantifiable ? This hunch ? This guesswork ?

Let's call it Axiomatic Density Heuristic of a Domain.

We aren't defining it yet, just assigning a name to it.

Now, if I know the axioms, we can determine it rigorously.

But in non-axiomatic domains, can we make a rough estimate of the same without knowing the axioms?

Because if we can, that can be used as the heuristic(h value)in the A* algorithm while searching for the axiom space from the data points. Noting for reference:

f(n) = g(n) + h(n)

where g(n) = Cost from Start To Current Node and h(n) = estimated cost from current node to end node

Can we figure out a common axiomatic density metric for every domain ?

Is Axiomatic Density Heuristic of a Domain correlated to Information Density? Is it one of the following?

• Shannon Entropy

• Renyi Entropy

• Coleman-Liau Index

Or is it something else?