Reduction and Elimination in Philosophy and the Sciences

Counterfactuals, Ontological Commitment and Arithmetic
Paul McCallion, St Andrews, Scotland, UK
1.
On the face of it, utterances of arithmetic sentences in
which number words or numerals occur as singular terms
carry commitment to the existence of numbers. Yet there is
no easy path from the syntax of an uttered sentence to a
claim about ontological commitment. For it might be that
the sentence is uttered with non-assertoric force, for example
as in make-believe. Alternatively, it might be that the
sentence is to be understood non-literally.
An assignment of ontological commitment to an
utterance should mesh with the beliefs that may be
reasonably attributed to the speaker (for a brief discussion
of the case of applied arithmetic see (Rayo forthcoming)).
It would, for instance, be implausible to attribute a
commitment to the existence of the average Scot to
someone who asserts
216
(1) The average Scot has 2.3 children,
because, for one thing, it is implausible to suppose that the
speaker believes that any Scot has a non-natural number
of children.
In (Yablo 2001) it is noted that the willingness of
speakers to assert
(2) The number of moons of Jupiter is four
seems to turn on whether they believe that Jupiter has
exactly four moons, and is apparently independent of their
belief in the existence of arithmetical objects. There is
therefore a prima facie case for denying that an utterance
of (2) carries commitment to the existence of the number
four, just as an utterance of (1) does not carry commitment
to the average Scot.
Yablo suggests that sentences such as (2) are
uttered with non-assertoric force, as part of the pretence
that there are numbers. An alternative explanation is that
utterances of (2) are to be understood non-literally. On that
view, the surface syntax of (2) is misleading, and it is used
to express the proposition that
(3) Jupiter has exactly four moons
An attractive feature of both Yablo’s suggestion and of the
alternative just mentioned is that they promise to remove
the difficulties surrounding the epistemology of mathematical
objects. One might therefore ask whether these analyses
can be extended to pure arithmetic, for pure arithmetic
is replete with numerical singular terms. The aim of this
paper is to investigate whether the latter analysis can be
so extended (for an extension of the pretence view to pure
arithmetic and set theory, see (Yablo 2005)).
The surface syntax of simple arithmetic sums (such
as ‘5 + 7 = 12’) suggests that their assertion will carry
commitment to numbers. Nevertheless, the willingness of
speakers to assert simple arithmetic sums seems to turn
on their belief in some sort of generalisation (evidence for
which is obtained by counting), and is apparently
independent of their belief in the existence of arithmetical
objects. So it is prima facie plausible that simple arithmetic
sums may be paraphrased by sentences whose literal
assertion would not carry commitment to arithmetical
objects.
2.
This may sound familiar. There have been previous attempts
to paraphrase (or otherwise provide a reduction of)
the sentences of pure arithmetic in terms of generalisations
constructed using the adjectival (or quantificational
determiner) occurrences of number-words or numerals (for
an overview, see (Rayo forthcoming)). These have typically
been undertaken in the context of a defence of nominalism
or logicism (or both). The following works are
broadly in defense of an adjectival strategy: (Bostock
1974); (Gottlieb 1980); (Hodes 1984); (Rayo 2002).
A standard starting point for such reductions is the
paraphrasing of arithmetic sums as generalizations
constructed using the material conditional. For present
purposes, it is more plausible for candidate paraphrases to
involve natural language conditionals. An initial suggestion
might be that (schematically)
(4) m + n = p
may be paraphrased as the indicative conditional
(5) for any concept F, and any concept G, if there
are exactly m objects that fall under F, exactly n objects
that fall under G, and no objects fall under both
F and G, then there are exactly p objects which fall
either under F or under G
It is a good question whether natural language sentences
involving the indicative conditional have the same truth
conditions as parallel sentences constructed using the
material conditional. If these particular indicative conditionals
do, then that creates a potential problem for the
candidate paraphrases: the standard distribution of truth
values will be preserved only if there are infinitely many
objects. For if there are less than m objects, the paraphrase
of m + n = p will be true even in a case where the
paraphrased utterance is standardly taken to be false.
It is standardly complained (for example in the
context of nominalist treatments of arithmetic) that such a
reduction would be theoretically flawed because, in the
absence of further assumptions, it would leave open the
epistemic possibility that there is a non-standard
distribution of the truth values of arithmetic sentences.
Moreover, the truth of any particular arithmetical sentence
would be contingent on there being enough objects around
(the only necessarily true sum would be ‘0 + 0 = 0’).
A related problem for the proposed reduction is that
the willingness of speakers to assert or deny arithmetic
sums in line with the standard distribution of truth values
should not be – but plausibly is - independent of their belief
in the infinitude of the world. Might then (4) be better paraphrased
as a counterfactual (or subjunctive) conditional? A
good candidate would be
(6) for any concept F, and any concept G, if there
were exactly m objects that fell under F, exactly n
objects that fell under G, and no objects fell under
both F and G, then there would be exactly p objects
which fell either under F or under G
This is an appealing paraphrase. On limited assumptions
concerning modality (firstly that each world accesses some
larger world, and secondly that the modal logic is at least