[FOM] Mathematical explanation

[Neil Tennant]

On Fri, 28 Oct 2005, Arnon Avron wrote:

> Note: this will be my first and last posting on this subject.

I hope not ... :-)

> I find it hard to believe that Searle and LW contended something so
> stupid.

Agreed. You say it more bluntly than I did.

> If they did then it only means that they did not understand 
> anything about mathematics in general, and about the meaning of 
> 3+4=7 in particular. This last proposition has two "literal" meanings,
> both of which are necessarily true:
> 
> 1) The sum of the natural numbers "3" and "4" is "7". 
> [material omitted--NT]
> 2) If S and T are two *disjoint* sets, the cardinality of S is 3,
>    and the cardinality of T is 4, then the cardinality of their union is 7.

I would add a third reading (which could be regarded as a slightly more
general than your (2)):

3) If S and T are two sets (whether disjoint or not), and the cardinality
of S is 3 and the cardinality of T is 4, then 
card(T union S) + card(T intersect S) = 7.

One gets (2) from (3) because when T is disjoint from S one has
card(T intersect S) = 0.

Any (neo-)logicist account of arithmetic, which treats of natural numbers
both in their 'pure' role as finite successors of 0, and in their
'applied' role as cardinalities of finite sets (or, more generally, as
objects #xF(x) associated with concepts F of finite extension), and
which provides a definition of addition, must be able to furnish [the
obvious generalization of] (3) [with m, n in place of 3, 4] as a theorem.

FOMers might be interested to learn that Frege himself, in the (two
volumes of the) Grundgesetze, never actually gave a satisfactory
definition of addition on the natural numbers that would have met his own
stringent criteria for adequacy of definitions. His comments in Vol.II,
section 33, which Avron has echoed in his (2) above, do not amount to such
a definition.

Neil Tennant