Reduction and Elimination in Philosophy and the Sciences

112
Benacerraf and Bad Company (An Attack on Neo-Fregeanism) — Michael Gabbay
… no one actually gets their arithmetical knowledge
by second-order reasoning from Hume’s Principle
Rather, the significant consideration is that simple
arithmetical knowledge has to have a content in
which the potential for application is absolutely on
the surface, since the knowledge is induced
precisely by reflection upon sample, or schematic,
applications. [Wright 2000]
The schematic applications Wright has in mind is that of
drawing one-one correspondences when counting. The
thought might then be that Hume’s principle has one-one
correspondence, the potential for application of number
language, ‘on the surface’ whereas Benacerraf’s principle
does not.
But counting is not the only application of numbers.
An obvious application that has little to do with counting is
when we assign numbers to things to help identify them,
perhaps in some ordering. For example, rooms in a hotel
may be numbered in such a way as to indicate their
location in the building. In such an application, room
numbers may serve as no indication of how many rooms
there actually are. For example room 1729 on the top floor
may be so numbered, in part, because it is on the floor
numbered 17, which itself is numbered to indicate it is one
up from 16 (and there may not even be 17 floors in the
hotel, if there is no 13th floor). What is more important to
the hotel-room application of numbers is that they can
represent individual units in some successive ordering. It is
this potential for application that is absolutely ‘on the
surface’ of Benacerraf’s principle.
We can quite easily explain the relation between
Benacerraf-numbers and one-one correspondence. Oneone
correspondence is a learned application of
Benacerraf-numbers. The adjectival numerical quantifier
can be treated in the obvious way, analysing ‘there are n
apples’ as:
there is a one-one correspondence between
the apples and the Benacerraf numbers
less than nB.
*
(where ‘less than’ is formalised in terms of PreB , the
transitive closure of the predecessor relation on
Benacerraf numbers).
Now, who is to say whether drawing one-one
correspondences is part of the ‘schema’ for applying
numbers, or whether it is a further application of a simpler
schema relating to units and succession? There are
reasons to think of numbers being fundamentally tied to
one-one correspondence and equally good reasons to
think that they are tied to units and succession. But to
which, one-one correspondence or units, are numbers
really tied? I doubt we could answer one way or the other
without making unjustifiable or question-begging
assumptions about the psychology of learning a number
language. The role of one-one correspondence as an
application of numbers will not help us to decide whether
Hume numbers or Benacerraf numbers really are the
numbers.
I conclude this section with the claim that
philosophical analysis of the concept of number will not
help us decide between Benacerraf’s principle and Hume’s
principle as the true abstraction principle for cardinal
numbers. At least not without appealing to some
disputable intuitions about our psychology of number.
(19)
4.3 Distinguishing the abstracts
Perhaps Neo-Fregeans should not try to rule out the
Benacerraf-numbers as legitimate references of our
number language, but embrace them. There is nothing to
stop a Neo-Fregean accepting that in the abstract realm
there are at least two number-like sequences of abstract
objects. A good line for a Neo-Fregean to take might be
that the Hume-numbers are the referents of our numbersas-cardinals
language, whereas the Benacerraf-numbers
are the referents of our numbers-as-ordinals language. A
Neo-Fregean could then argue that there are two main
uses of number language, perhaps even two concepts of
number (cardinal and ordinal) and so see no reason to be
worried if there are two collections of abstract entities
associated with them. Indeed, such a result could be
regarded as a success of the Neo-Fregean programme.
The problem is that the Benacerraf-numbers are not
ordinals: Benacerraf’s principle involves no
characterisation of ordering or any criterion of position
correspondence. To understand Benacerraf’s principle we
need nothing that is not needed to understand Hume’s
principle. There is nothing about Hume-numbers that rules
them out as being ordinals, and there is nothing about
Benacerraf-numbers that rules them out as being
cardinals. The concept of a unit is no less important to that
of cardinality than the concept of one-one correspondence.
Benacerraf’s principle and Hume’s principle each could be
taken as allowing reference to the natural numbers as
cardinals. But then Neo-Fregeanism must account for why
our arithmetic language refers to the Hume-numbers rather
than the Benacerraf-numbers (or vice versa). I have been
arguing that that we stand in no significant relation to
Hume’s principle that we do not also stand in to
Benacerraf’s principle. So if the Neo-Fregean accepts that
the two abstraction principles allow reference to different
abstract entities, then he has made no progress
overcoming the objection of Section 3.3.
5 Conclusion
I have argued that there is no particular abstraction
principle that we can associate with the natural numbers.
At least two similar, but formally distinct, abstraction
principles are capable of lying at the heart of the Neo-
Fregean programme. The principles are distinct enough
that there is no natural way of equating the abstract
objects they give reference to. The principles are however
sufficiently similar that there is no principled criterion that
identifies one over the other as ‘the correct’ abstraction
principle for elementary arithmetic. I conclude that
numbers are not the abstract objects referred to by either
abstraction principle, or of any other abstraction principle.
The point to emphasise here is that neither the Humenumbers
nor the Benacerraf-numbers are really the natural
numbers. The whole Neo-Fregean framework of
abstraction principles is just another way of generating
sequences that encode the natural numbers. This
conclusion is independent of questions regarding the
metaphysics of abstraction and whether abstraction
principles really refer to any abstract objects at all.