of additional ingredients such as the relation of *infinite proximity* and the closely related *shadow principle* for passing from a finite inassignable quantity to an assignable one (or from a finite nonstandard number to a standard one), as in $2x+dx\operatorname{\,{}_{\ulcorner\!\urcorner}\,}2x$.[^2] As far as Skolem’s nonstandard extension ${{\mathbb N}}\subset{}^\ast{{\mathbb N}}$ is concerned, anything involving the actual construction of the number system and the entities called *numbers* would go under the heading of the *ontology* of mathematical entities. Note that *the first order theories* of $\mathbb{N}$ and ${}^\ast\mathbb{N}$ are identical, as shown by Skolem (for more details see Section \[FOL\]). In this sense, not only is one not adding *a lot*, but in fact *one is not adding anything at all* at the level of the *theory*.

What about the claim that *Leibniz did not have this perspective*? It is true that he did not have our perspective on the *ontological* issues involved in a modern construction of a suitable number system incorporating infinite numbers, but this needn’t affect the *procedural* match.

What about the claim that *one has to build up a considerable conceptual machinery to emulate Leibniz’s probably rather modest arsenal of procedural moves; that is to say, we may 