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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.

But also:

  • 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.

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The theory of numerosities is more consistent with pre-mathematical intuitions of how collections (not to use the technical term set) should behave relative to each other. The fact that such an alternative is sorely needed is evidenced by the fact that a popular if not to say populist attempt to implement this at a naive level has apparently gained broad popularity (though perhaps not among pure mathematicians), see What is... A Grossone?

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  • $\begingroup$ From cursory reading it looks as though they discovered the category of injections (possibly order-preserving) to a standard fixed ordered set (eg N or R). If they can say something about the numerosity of a completely arbitrary set given on its own then I guess I'd retract that statement. $\endgroup$ – David Roberts Jan 6 '16 at 4:45
  • $\begingroup$ @DavidRoberts, I would suggest you take more than a cursory look. There are subtle issues involved here related to the properties of internal sets in Robinson's framework. If you have aesthetic objections to using internal sets, you can work in Ed Nelson's framework where there are no others. $\endgroup$ – Mikhail Katz Jan 6 '16 at 7:54
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    $\begingroup$ I did just that after writing my comment, and indeed there are some subtle and interesting things in defining defining a 'numerosity function'. Perhaps the spin/hype put on this is just too misleading for people who understand things like non-standard analysis and ultrafilters. $\endgroup$ – David Roberts Jan 7 '16 at 6:08
  • $\begingroup$ @DavidRoberts, we're in full agreement on spin/hype. Some of this touches on deeply seated beliefs. See e.g., math.stackexchange.com/questions/1602977 which is close to closing. $\endgroup$ – Mikhail Katz Jan 7 '16 at 12:17

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