A new kind of Neo-Fregeanism?
Frege argued that number concept are, in the first place, second-order predicates. When we talk about numbers as objects, we use a logical device of "nominalization" that introduces object-level representations of higher-level properties. In Grundgesetze, he assumed that every first-order predicate can be nominalized: for every first-order predicate F, there is an associated object – the "extension" of F – such that F and G are associated with the same object iff ∀x(Fx ↔︎ Gx). The number N is then identified with the extension of 'having an extension with N elements'. Unfortunately, the assumption that every predicate has an extension turned out to be inconsistent, so the whole approach collapsed.
But we can introduce more cautious principles of nominalization. Button and Trueman (2024) show how one can conservatively (and hence consistently) extend a theory in a Fregean typed language, without singular terms for properties, to a theory in an extended language in which all predicates expressible in the original language have a nominalization. I'll give a brief summary of the construction.
Let T be the original theory. For any expression F of a higher type, we introduce an object-type expression nom(F). We also introduce an operation app so that app(nom(F), a) = F(a). Finally, we restrict all quantifiers in T by a new predicate real. To the resulting theory, we add some axioms governing nom and app and real. The most important ones, for present purposes, are these (slightly simplified):
Nom-real. ∀F(real(F) ↔︎ E!nom(F)).
Nom-nonreal. ∀F ¬real(nom(F)).
Nom-inj. ∀F∀G(nom(F) = nom(G) ↔︎ F = G).
B&T prove that the new theory is a conservative extension of the original theory.
B&T assume that statements like app(nom(F), a) are literally false, like metaphors or fictional statements. I prefer a different conception of their machinery. We can regard app(nom(F), a) as a convoluted way of saying F(a), which is true as long as the object a is F. The old language is a perspicuous representation of reality. The new language is a stipulative extension with no extra metaphysical commitments.
Now. Can we recover Frege's logicism with this machinery?
Not quite. We'd like to define N as the nominalization of 'having a nominalization that applies to N things'. But we can't do that: nom isn't defined for 'having a nominalization that applies to N things'.
We could follow Frege's idea in Grundlagen and define zero as the nominalization of 'being non-self-identical': 0 =df nom(λx.¬(x=x)). We'd then like to define 1 as nom(λx.x=1), but again we can't do that because nom is not defined for λx.x=1.
So we need to change the construction.
We could proceed in stages. At stage 1, we introduce nominalizations for all predicates in the original language. At stage 2, we introduce nominalizations for all predicates in the language of stage 1, and so on.
Since λx.x=1 is a stage-1 predicate, it can be nominalized at stage 2. So the number 1 can be defined at stage 2. In general, each number n is definable at stage n+1.
To allow quantification over all numbers, we need to add a transfinite stage:
- stage 0: the old theory in the old language;
- stage n+1: add nominalization of all predicates in stage n;
- stage ω: take the union of all previous stages.
If we generalize the last clause to all limit ordinals, will we reach a fixed point at which every predicate expressible in the language has a nominalization? I think so, because I think the language remains countable.
In any case, we have to adjust some other parts of the B&T machinery. The Nom-real axiom has to go. And I think we have to use different app and nom predicates for each stage, to avoid paradox.
It might be worth spelling out this construction. It seems to have a few advantages over traditional Neo-Fregeanism (as in Hale and Wright (2001)). In particular, it doesn't rely on the magic of abstraction principles (with the bad company of Axiom V). The introduction of numbers as objects is not an instance of abstraction, but of nominalization, and nominalization is a pervasive feature of our language. (We also don't have a special "Julius Caesar problem": that Julius Caesar is not a number is entailed by Nom-nonreal.)
(Thanks to Rob Trueman for discussion.)