Thomas Hofweber

		 Numbers and Number Determiners


To understand arithmetic one has to understand what the relation is
between ``two'' in the following:

   (1) Three plus two is five.
  
   (2) Two apples are on the table.
  
   (3) The number of apples on the table is two.
    
It seems clear that they have to have some close relation, but, on
reflection, it is rather unclear what this relation is. There are a
number of substantial differences between them. For example, ``two''
in (1) and in (3) seem to be of low type, the type of objects (e),
whereas ``two'' in (3) seems to be of high type, the type of
determiners ((e,t),((e,t),t)). On the other hand, (2) and (3)
certainly are very closely related. After all, they are truth
conditionally equivalent.
    
A number of proposals about their relation have been made (Field,
Hodes, Wright and others). However they either are in conflict with
insights from natural language semantics, or they overemphasize that
number determiners are first order definable, or they introduce an
implausibly big gap between them. (Or so it seems to me).
     
In this talk we will outline a new account of the relation between
numbers in arithmetic and number determiners in natural language. We
will take recourse to evidence from the process of how number concepts
are acquired and how basic arithmetic truths are learned. Of
particular importance will be the uses of bare determiners
(determiners without explicit first argument), both in the plural and
the singular.

Overall, I will defend a view of arithmetic according to which every
arithmetic statement is either objectively true, or objectively false,
even though it is not about any particular objects.