Nominalism vs. Syntax

It is difficult to maintain, consistently, the following two claims:

(i) There are no abstracta.
(ii) There are syntactical entities (and they behave as our standard accounts say they do).

Consider, for example, how one defines a language $L$. Beginning with two building blocks, $A$ and $B$, we say that $\{A,B\}$ is the alphabet. It's usually implicit, but sometimes needs to be stated, that $A \neq B$. (One has to state this in a formalized theory of syntax.)

For the many of the usual purposes of syntactical theory, it does not matter what these building blocks $A$ and $B$ are. They could be two eggs. They could be the numbers 7 and $\aleph_{57}$. They could be the letter types "a" and "$\aleph$". Or they could be two of my guitars. Or they could be two tokens of the letters "a" and "$\aleph$''. It does not matter. And the fact that it doesn't matter plays an important role in Gödel's incompleteness results, where the leading ideas involve the structural interplay between the properties of numbers, sequences, syntactical entities and finitary computations (plus, times, exponentation, and so on).

Let our alphabet $\Sigma = \{A,B\}$. Next we consider the set $\Sigma^{\ast}$ of finite sequences drawn from $\Sigma$. Finite sequences drawn from an alphabet are usually called,

• strings
• words
• expressions

These are the syntactical entities that one is discussing, quantifying over, referring to, etc. The crucial point is that these are sequences from the alphabet. In particular, $\Sigma^{\ast}$ is closed under sequence concatenation. So,

if $\alpha, \beta \in \Sigma^{\ast}$, then $\alpha ^{\frown} \beta \in \Sigma^{\ast}$.

And:

$|\Sigma^{\ast}| = \aleph_0$.

This means that there are $\aleph_0$-many syntactical entities. The terms $a_0, a_2, \dots, a_n$ occurring in a sequence $\alpha = (a_0, a_1, \dots, a_n)$ may well be concreta. But the sequence $\alpha$ itself is a (possibly mixed) abstractum. More exactly, a sequence is usually understood as a function:

$\alpha : \{0,1,\dots,n\} \to \Sigma$.

This is not mandated. What is mandated is that sequences are individuated in a certain way:

$\alpha = \beta$ if and only if $\alpha$ and $\beta$ have the same terms, in the same order.

So, e.g,, if $(a_0, a_1, \dots, a_n) = (b_0, b_1, \dots, b_k)$, then $n = k$, and $a_0 = b_0$, $a_1 = b_1$, and so on.

Normally, one goes on to define certain special subsets $X, Y, \dots$ of $\Sigma^{\ast}$. Perhaps these are the formulas, or terms, and whatnot. Usually, the definitions satisfy certain computational constraints: e.g., perhaps an inductive definition. So, $X$ might be, e.g., a recursive set or a recursively enumerable set. But for this discussion here, these subsets don't matter. They're subsets, and we discussing the enclosing set, of all strings from the alphabet.

Return to (i) and (ii). Suppose (i) is true. So, there are no abstracta. Hence, there are, a fortiori, no mixed abstracta; and therefore, there are no sequences; and, therefore, there are no strings; and therefore no syntactical entities, except a very, very small number of tokens, which are not closed under concatenation. Hence, (ii) is false.

One might suggest that these claims (i) and (ii) are "really" consistent under some reinterpretation $I$. But what exactly is this $I$? How is $I$ defined? Is it a secret?

I think that the optimal nominalistic responses to the inconsistency of (i) and (ii) are:

• either to accept the inconsistency and thus simply accept that (ii) (i.e., syntax) is false (see Quine & Goodman 1947, "Steps Toward a Constructive Nominalism"), 
• or to reinterpret (ii,) to make it "true under a reinterpretation", so that "syntactical entity" refers perhaps to possibilia (i.e., possible concrete tokens: see Burgess & Rosen 1997, A Subject with No Object, for some discussion of this) or perhaps to some kind of physical entity (such as perhaps spacetime regions), assuming there are sufficiently many.

I'm not optimistic about either kind of approach. The first approach is an error theory, and is too damaging to science. An error theory for morality is one thing; an error theory for science is another! The second, "hermeneutic", approach invokes possibilia and this raises similar sceptical and metaphysical worries as abstracta do. (See the final chapter of Shapiro 1997, Philosophy of Mathematics: Structure and Ontology.) It also raises the question of what grounds one might give for the reinterpretation. A classic discussion of some of these topics is Burgess 1983, "Why I am not a Nominalist".

So far as I can tell, the more recent "weaseling" approach to nominalism---which I think is extremely interesting---proposed by Melia 2000 ("Weaseling Away the Indispensability Argument", Mind) and endorsed and developed recently by Yablo 2012 ("Explanation, Extrapolation and Existence", Mind) doesn't seem to apply in the syntactic case. But I'm not sure.