I will attempt an answer, though we're nearing the basement bellow all of mathematics and logic, so it's not quite straight forward.
So I said "The number of things we can physically count", this was a gross over simplification. So I'm going to start at what ZFC is, from where it stems recursively.

ZFC is an axiomatization of Number Theory and Set Theory.
Number Theory is a direct abstraction of "Things we can count", so to speak.
Set Theory is the tricky part with reference to empiricism, so lets go deeper into what it's derived from.
Set Theory is the axiomatization of Naive Set Theory.
Naive Set Theory is just rudimentary definitions of the same concepts as Set Theory, described informally in natural language, rather than axiomatically. It is based on Discrete Mathematics.
Aaaannd Discrete Mathematics deals with Countable Sets, which expands into discrete integers, graphs, and logical statements.

The basis of ZFC is

1. boolean logic: yes/no, existence/non-existence.
2. discrete integers, i.e. "Things we can count"
3. graphs, i.e. "Things we can relate to one another in some way"

So I'll append my simplistic claim to something more inclusive of ALL of Mathematics, since what I originally said didn't quite make sense. Even mathematics starts with presuppositions and tautologies stemming from conditions in reality. From countable objects to abstractions of said numbers, relations of objects in reality to the abstracted graphs of hypothetical objects, and finally to what truth can be derived from said relations and the derivative algebra that's generate there of.

All deep systems of thought start with simple observations of reality, and as long as the derivative works there of are accurate and consistent I think that small tether to reality can lead to the sum total of the knowledge of mankind.

edit* added one "s". Also had to reformat because I lost my line-breaks.