# [FOM] Mathematical explanation

Arnon Avron aa at tau.ac.il
Fri Oct 28 16:16:36 EDT 2005

Note: this will be my first and last posting on this subject.
>
> > Imagine two overlapping circles, A and B.  A contains three dots, B
> > four.  However, two of the dots fall into the area where A and B
> overlap.
> > Here, Searle and LW contend, A + B = 5.
>

I find it hard to believe that Searle and LW contended something so
stupid. If they did then it only means that they did not understand
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".
Here the sentences expresses a property of the natural numbers,
and sum is the primitive recursive function on the natural
numbers which mathematicians call "sum" (of course, if one claims
that this is context-dependent because one might interprete + as
multiplication, and the symbol 3 as dozen, then it will be difficult
to argue that in this sense every proposition is "context dependent",
but this would not be a very interesting philosophical claim...)
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.
Here the sentence expresses properties of sets and operations on sets.
Now if someone understands 3+4=7 as claiming the same, but without
the "disjoint" condition (who taught such a person basic
mathematics?) then for this person (if s/he knows at least logic)
3+4=7 is *false* -  and necessarily so!

Arnon Avron