[FOM] Natural Language and Mathematics (reply to Harvey)

[Richard G Heck]

Dean wrote:

>My background is the philosophy of language.  This requires a training that encourages existential conservatism: to prefer for example to explain "there are no unicorns" as the negation of "there is at least one unicorn", i.e. as denying the existence of something, rather than asserting of any existing thing (the set of unicorns e.g.) that it is empty.  Frustra fit per plura quod potest fieri per pauciera (Ockham).
>
Being a philosopher of language, I would have to disagree with this 
characterization. The sorts of ontological scruples Dean expresses do 
not have very much to do with the philosophy of language. Some 
philosophers (Quine, Field) put a great deal of stock in Ockham's razor 
and so construe the question what there is as the question what, in some 
sense, there /has/ to be. But not all of us do, and I dare say that most 
contemporary philosophers of language and semanticists have no more 
scruples about ontology than your ordinary mathematician would. Indeed, 
the most general semantical theory for quantification (the theory of 
generalized quantifiers) /does/ construe "No pigs waddle" as asserting 
that the intersection of the set of pigs and the set of waddlers is 
empty. Now there are semanticists and philosophers of language who would 
be troubled, as Dean is, by the thought that such sentences somehow 
involve reference to sets. And though I do not myself share such 
concerns, I am not here arguing that one should not. The point is that 
Quine's affinity for desert landscapes arises from some substantive 
philosophical assumptions, not from his being a philosopher of language.

Similar questions, I might add, arise in other areas. For example, it is 
now almost common wisdom that sentences like "The Grinch carved the 
roast beast with a knife" quantify over events. One might well be 
surprised by that and ask, as I believe John Wallace did, where the 
quantifier is supposed to be hidden. But the reasons for the claim go 
very deep, and there is strong syntactic evidence that the quantifier is 
in fact there.

One final point:

>Nor can the Goedel phenomena appear in the sematnics of ordinary language.
>
Goedelian phenomena are not semantic but proof-theoretic.

Richard Heck