Since 2003 a group of Italian mathematicians (Benci, Di Nasso and Forti) has developed a new measure for infinite sets that satisfies the Euclidian principle: The whole is greater than the part. The theory has some interesting consequences. To mention some of them:
- It shows that the Euclidian principle can be captured in a theory of size in a consistent way.
- The set of numerosities has the same arithmetic as the natural numbers.
- We have that numerosity(even numbers)+numerosity(odd numbers)=numerosity($\mathbb N$), numerosity($\mathbb Z$)=2*numerosity($\mathbb N$)-1 and numerosity($\mathbb N\times \mathbb N$)=numerosity$(\mathbb N)^2$.
- It gives rise to nonstandard analysis.
- It can be extended to whole mathematical universes.
- It has applications in nonstandard probability theory, the foundation of nonstandard analysis and in number theory.
- Sizes depend on a choice of ultrafilter in the construction of the numerosities, e.g. if odd$\in U$ then numerosity($\mathbb N$) is an even number and if even$\in U$ then numerosity($\mathbb N$) is odd.
- It violates translation invariance, e.g. numerosity($\mathbb N+1$) < numerosity($\mathbb N$) (more generally it violates transformation invarianse for every transformation with an infinite orbit).
- Makes it hard to classify "similar" sets.
- The existence of numerosities in some cases demand the existence of selective ultra filters (they exists if we assume the continuum hypothesis).
- The construction is not simple, it demands a lot of knowledge about special ultra filters and a lot of technical machinery.
I have already considered Kitcher's (1984) idea of rational generalizations as a mean to explain how the contributions of the theory of numerosities differ from Cantor's theory of cardinalities (this is already done to some extend by Mancosu(2009)). But do you know any other (philosophical) theories or ideas that can help explain exactly what the new theory (more philosophical) contributes with and what the limitations of the theory may mean for these contributions? Specifically it would be nice, if there were some literature on what structure and classification means for the fruitfulness of a mathematical theory or a mathematical concept.