What is a number?

Dit nummer levert Bahram Assadian een bijdrage aan onze rubriek 
de Pen. Hij doceert onder andere Text, Context en Debat: Kripke.
Voor Cimedart schreef hij een stuk waarin hij vertelt over
‘de taal van de wiskunde’.

Here is a problem I am interested in. It seems that mathematics
is a discourse we have to take seriously: we have to
respect its language, and take it at face-value – as opposed
to the language of, for example, fake sciences, superstitions,
and even some areas of philosophy. Mathematics tells us
that there are infinitely many prime numbers; or that there is
a natural number between 3 and 5. So, it seems that numbers
exist. But if so, what sort of entities are they? Where
are they? How do we know anything about them? These
questions are pressing, as numbers and other mathematical
objects, such as sets and functions, are not located in space
and time. We cannot see them, smell them, listen to them,
and touch them. In general, we cannot causally interact with
them, contrary to things such as tigers and tables. Mathematical
objects are abstract. So, there seems to be a tension
between a ‘good’ discourse (mathematics) and a domain of
‘bad’ objects (mathematical objects).

The problem has received an elegant treatment by Paul
Benacerraf in his very influential paper, ‘Mathematical Truth’
(1973). According to him, what makes mathematical truths
possible is the existence of abstract mathematical objects.
For example, ‘2 is prime’ is true if there exists an object (the
number 2) and a property (being prime) such that the former
belongs to the latter. On the other hand, what makes mathematical
knowledge possible is our ability to causally interact
with mathematical objects. But how is it possible to get in
touch with causally inert, abstract objects? It seems then
that what makes mathematical truth possible makes mathematical
knowledge impossible. Either mathematical statements
are not true, or else, we do not know such statements.
This is a hard dilemma!

This problem about the compatibility of a suitable account of
truth and knowledge has been formulated with respect to the
language of mathematics. But we can find its applications in
other discourses, which are committed, at least prima facie,
to the existence of abstract entities such as possible worlds,
fictional characters, universals, and propositions. I should
also say that this problem goes back to Plato’s Meno, where
he discusses our knowledge of abstract Platonic forms.

All the same, there is a more fundamental problem: the
problem of the explanation of our reference to mathematical
objects. How is it possible, if at all, that when we use a name
such as ‘2’, we refer to the number 2, and not, for example,
to a certain set, or a table? This question is an instance of
a general problem about reference to objects of any sort.
How is it possible, if at all, that when we use a name such as
‘Federer’, we manage to refer to the Swiss champion, and not
to something else? There must be something that could fix
the intended meaning and reference of mathematical terms.
But what’s that? This is not the traditional epistemic question
we discussed above: how do we know that our mathematical
beliefs are true? It is, rather, a question that arises prior to
this issue: how can we even have beliefs about mathematical
objects? How can we refer to them?

In the case of tables, chairs, tigers, and persons, it seems that
causal relations can play some role in determining reference.
When I use the term ‘Federer’, I refer to Federer, and not to
Nadal, because there is some causal relation between me
and him. But mathematical objects are abstract, and so, it is
not possible to causally interact with them. This is a serious
problem. Suppose we have different sequences of objects of
any sort, each one of which has the form or the structure of
the natural numbers, in the sense that they are arranged in
the way the natural numbers are (the first entity, the second
entity, the third entity, and so on...). For example, there may
be a sequence of sets, or electrons (assuming that there are
infinitely many of them), or even there could be a sequence
of the Roman Emperors: Augustus, Tiberius, Caligula, etc.
(assuming that the Roman Empire had never ended, and that
it will never end). There are infinitely many further alternative
sequences, while there is no principled method for privileging
one sequence over any other to be the unique referent of
our arithmetical vocabulary. So, what do we refer to when we
use, e.g., ‘1’? Do we refer to the first set, to the first electron,
or to Augustus? If each of these sequences has an equally
good claim to be the referents of our numerical expressions,
then how, if at all, can we privilege one over all others to determine
our numerical reference? If there is no such possibility,
then mathematical reference remains indeterminate and
inscrutable. Numerical terms such as ‘the number of apples’
or ‘1’ don’t stand for particular objects. There is no ‘fact of the
matter’ as to what particular object they refer to.

I do not think that referential indeterminacy is a serious
threat for mathematical language. It is a fact of our mathematical
life. Perhaps, referential indeterminacy is a fact of
life, in general. Perhaps, there is no fact of the matter at all
as to which objects proper names refer to, and which sets of
objects are in the extension of predicates. There is no fact of
the matter at all as to whether ‘Federer’ refers to the Swiss
tennis champion or to Amsterdam. I do not deny that there
is something utterly weird about this claim. But on the other
hand, this may tell us something quite fundamental about the
nature of the relation that we call ‘reference’. All the same, it
seems to me that at least in mathematical language, reference
is radically indeterminate.

In sum, the problems I am dealing with lie at the intersection
of mathematical truth, mathematical knowledge and reference,
and mathematical existence. I think that the best and
the most fruitful treatment of these philosophical questions
about mathematics is achieved when we assess them, more
generally, within the context of the philosophy of language,
epistemology, and metaphysics.