Literature that helps explain what the theory of numerosities contributes with

Question

Since 2003 a group of Italian mathematicians (Benci, Di Nasso and Forti) has developed a new measure for infinite sets that satisfies the Euclidean principle: The whole is greater than the part. The theory has some interesting consequences. To mention some of them:

• It shows that the Euclidean 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 demands 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.

Answer 1

There is a clear way to express numerosities of sequences via $\omega$, which is the germ of the identity function at infinity or half the numerosity of $\mathbb{Z}$. It also can be considered a surreal number. This measure is more accurate than asymptotic density, because it gives the exact value.

Suppose you have a strictly increasing sequence $a_k\ge0$, where $k\in\mathbb{Z}, k\ge0$.

To find the numerosity, you have to

1. Perform operator $D\Delta^{-1}$ on your sequence, where $\Delta^{-1}$ is anti-difference (indefinite sum). This is quite sandard operator in umbral calculus.
2. Find inverse function of the resulting expression.

In other words, apply to your sequence the operator $N(a_k)=\left(D\Delta^{-1}a_k\right)^{[-1]}(\omega)$, where $f^{[-1]}$ is the inverse function.

The following Wolfram Language code does the thing:

a[k] := k^2
SolveValues[D[Sum[a[k], k], k] == \[Omega], k] /. C[1] -> 0 //
   FullSimplify // Expand

So, some examples:

• $N(2k)=\frac{\omega }{2}+\frac{1}{2}$ (we allow $k$ to be zero, but if you want non-zero naturals, subtract $1$ from this)

• $N(k+5)=\omega -\frac{9}{2}$

• $N(k^2)=\frac{1}{2}+\frac{1}{6} \sqrt{36 \omega +3}$ (again, subtract $1$ if you do not count zero)

• $N(1/3+k+k^2)=\sqrt{\omega}$

• $N(k^4)=\frac{1}{30} \sqrt{30 \sqrt{900 \omega +30}+225}+\frac{1}{2}$

• $N(7^k)=\log_7 \left(\frac{6 \omega }{\ln (7)}\right)$

So, yes, Galileo was right, the numerosity of naturals is greater than the numerosity of perfect squares, and we even know exactly, by how much.

Answer 2

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?