thermodynamics - Definition of "intensive" and "extensive" properties

Today I was asked what does it mean for a physical property of a system to be intensive.

My first answer, loosely speaking, was:

"It is a property that is local."

I was specifically thinking about density and, by "local", I meant "that is unaffected by the dimension of the system". Ofcourse this is a very ambiguous answer, so after that I said (shifting to extensivity's definition):

"A property is extensive if it depends on the volume of the system observed."

To be honest, I said if it's proportional to the volume, but I'm not sure the this is correct. Now, that I'm still thinking about it, I've come to the conclusion that a good definition could be:

"A property is extensive if it depends on the quantity of matter of the system observed."

Looking on wikipedia I realized that this is exactly the definition given. But I'm somewhat still uncomfortable with that: if a gas is kept in a recipient of volume $V$ at a temperature $T$, his pressure is function of the number of moles of the gas:$$p=n(RT/V).$$ And, as we know, pressure is an intensive property. So (to me) it is not really clear what does "does not depend on the quantity of matter" mean.

I also thought that one could use an operational definition (if this is the good term) of extensivity/intensivity: one example might be:

"Suppose to measure a quantity $q(S)$ relative to a system $S$. Now reproduce a copy of $S$ and measure the same quantity for the system $S+S$ given by the two identicaly systems joined. If $q(S+S)=q(S)$, then $q$ is an intensive quantity."

This seems to give a more precise sense to the "does not depend on the quantity of matter" in the above definition, but there are gaps to fill. Maybe I will try to develop better this in a second time. Now, ofcourse, the question is: what is the definition of extensivity/intensivity in rigorous and unambiguous terms?

Answer

Personally, your last example is exactly how I would define intensive quantities:

"Suppose to measure a quantity $q(S)$ relative to a system $S$. Now reproduce a copy of $S$ and measure the same quantity for the system $S+S$ composed of the two identical systems considered as a single system. If $q(S+S)=q(S)$, then $q$ is an intensive quantity."

I edited it only slightly, because it's important that the two identical copies of the system remain independent and non-interacting.

I would add to this that

If, for two different systems $S$ and $T$, $q(S+T)=q(S)+q(T)$, then $q$ is an extensive quantity.

Note that this does indeed mean that extensive quantities are proportional to the system's volume.

These two definitions leave room for quantities that are neither intensive nor extensive. That's OK - there are indeed many such possible quantities, although we don't use these terms to talk about them.