# What is a number?

## Bahram Assadian

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.