(i) There areConsider, for example, how one defines a language $L$. Beginning with two building blocks, $A$ and $B$, we say that $\{A,B\}$ is thenoabstracta.

(ii) Therearesyntactical entities (and they behave as our standard accounts say they do).

*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*

*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.

*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.

I'm skeptical of your reasoning.

ReplyDeleteWhen we study a formal language, or try to define it, then we are doing mathematics and what we are defining is a mathematical idealization of language. So, of course, the entities in this idealization are mathematical entities and count as abstracta.

The question of the existence of abstracta, including those you discuss, still seems entirely distinct from the question of the existence of syntactic entities - the actual syntactic entities that are used, rather than their mathematical idealizations.