[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 
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". 
   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

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