numerals, and language with an infinite lexicon

Here is an idea that I've seen in several places: "it is plain that if a language is to be learnable, the number of basic significant elements (words) has to be finite" (Tim Crane, The Mechanical Mind, 2nd ed. p.140).  (Is the locus classicus for this Davidson's "Theories of Meaning and Learnable Languages?")

But how does this square with numerals (words for numbers)?  There are an infinite number of numerals.  And arithmetical language is learnable. 

So how could/should Crane, Davidson, et al. deal with this apparent counterexample?  I'm not sure... perhaps (despite appearences) numerals are  not themselves genuine words; rather only the digits are genuine words, and the higher numerals are complex supra-word symbols. 


Jonathan Livengood said...

Obviously, they should just deny that arithmetic is learnable.

Greg Frost-Arnold said...

Note to self:
Upon further thought, it seems to me that '9,471' is not really different from 'SSSSS0' or '|||||' (as representations of the number 5). All three are singular terms, but none are proper names: they are rather function symbols (applied to proper names).

Greg Frost-Arnold said...

Wow -- Thanks for all of those resources, Kai! This looks very helpful.