What I talk about when I talk about numbers

Here is a valid argument:

(1) The number of Front national MEPS is worrying. 

(2) The number of Front national MEPS is 24. 

(3) 24 is worrying.

At least it’s valid if you think, as almost all philosophers who think about mathematical language seem to, that (2) refers to a number.  Contrast (2) with

(2*) There are 24 Front national MEPS.

(2*) is a statement about Front national MEPS, but (2) and (2*) are treated as being equivalent; not in the sense that they have the same meaning (one refers only to a political party, the other refers to a number) but in the sense that given (2) we can always infer (2*) and given (2*) we can always infer (2).  We can do this because we accept the abstraction principle:

(*) There are n Fs if and only if the number of Fs is n.

I’m not sure what to make of this.  I used to think that claims like (2*) were true because they predicate a property of something real, whereas claims like (2) were literally false because they make reference to something that doesn’t really exist—a number.  Making inferences using (literally false) claims like (2) was, I thought, fine, because doing so wouldn’t lead us astray with respect to how things stood with what really existed.  Similarly we could accept (*), not as being literally true, but as being “nominalistically adequate”, i.e. unable to lead us astray with respect to how things stand with what really existed.  (Compare: we accept ‘There is a dent in the car’ not because dents really exist or because they are an extra bit of the furniture of reality over and above the car, but because saying this doesn’t lead us astray with respect to the topographical properties of the car.)  But here is a problem with this: (3) is absurd.  A convenient fiction that aids inference-making is one thing; an absurd convenient fiction that aids inference-making is something else.  This is disastrous for the platonist who thinks that numbers really exist.  For the platonist (3) is true.  But it’s also bad news for the fictionalist who accepts that (2) refers (or at least purports to refer) to a number, because although fictionalist take (3) to be false, they’re still left with a problem: (3) isn’t even nominalistically adequate.  (3) can be used to infer falsehoods about the concrete world:

(3) 24 is worrying. 

(4) The number of Tunnock’s Teacakes in a four-pack is 24. 

(5) The number of Tunnock’s Teacakes in a four-pack is worrying.

(5) is about the concrete world and is false.  Maybe the only option is to drop the claim that phrases of the form ‘The number of Fs is n refer to the number n.  In this case the ‘is’ can’t be the ‘is’ of identity; the phrase can’t mean ‘The number of Fs = n’.

Propositions cannot exist

What a proposition is, or is supposed to be, can be grasped through abstraction principles. An abstraction principle is something of the form:

\[\forall \alpha \forall \beta (\Sigma(\alpha) = \Sigma(\beta) \leftrightarrow \alpha \sim \beta \]

Where \(\Sigma \) is an appropriate term-forming operator and \(\sim \) an equivalence relation. In the case of propositions, the abstraction principle will be something like:

The proposition expressed by u₁ = the proposition expressed by u₂ if and only if the content of u₁ is the same as the content of u₂

where u_i are appropriate tokenings such as utterances or inscriptions. A proposition is what is expressed by a sentence, written or spoken, and, furthermore, propositions are taken to be truth-bearers: they are the sorts of things that can be true or false. All this tells us that propositions are intentional entities: they are about or of the world; they pertain to things, and so on. The problem for propositions arises when one starts to consider what intentionality consists in, or what it is for something to be about, to be of, or to pertain to the world. It’s often (rightly) said that whatever aboutness propositions or sentence tokens have must be derivative from the fundamental intentionality associated with intentional agents. But something stronger can be said. Being about something essentially requires being responsible to that thing—not, it is worth emphasising, be responsive to a thing: lumps of wax are responsive to heat but are not about heat, thoughts about things outside our light cone are not responsive to those things but are about them; something different is required. If I think about the Empty Quarter I make my thinking responsible to the Empty Quarter itself. If I think of the Empty Quarter that it is the largest expanse of sand in the world then my thinking goes wrong—is subject to negative normative assessment—if it is not the largest expanse of sand in the world, and my thinking goes right—is subject to positive normative assessment—if it is the largest expanse of sand in the world. This isn’t an accidental feature of intentionality, it’s an essential one. So the only entities that can be about things are those that can be responsible to those things. Only persons are responsible in this way, abstract objects like propositions can’t be. But propositions are defined as things which are about the world; the result being that propositions would have to possess an essential property they cannot possibly have. Propositions then, cannot exist.