[FOM] Mathematical explanation

[Dean Buckner]

MJM
> > 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.

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

Nor did they.  Mjm has not supplied a reference, but there is a passage
in LW's Philosophical Grammar (1974 edition, p.339) where he considers a
circle containing 5 dots that overlaps a circle containing 4 dots, and
where two of the dots are the same.  LW points out that by "4+5" we
might mean the number of objects that fall under the concept "phi x or
psi x", where phi means being contained in the one circle, psi means
being contained in the other circle circle.  In which case 4+5 equals 7.

> 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!


Correct.  But such a person does not mean by "4+3" what we ordinarily
mean by the same expression.

I'm not sure what LW was getting at here.  It is in the same place as
his remarks that "if you want to know what 2+2=4 means, you have to ask
how we work it out".  And then "the difference between my point of view
and that of contemporary [1920's] writers on the foundations of
arithmetic is that I am not obliged to despise particular calculi like
the decimal system.  For me one calculus is as good as another.  To look
down on a particular calculus is like wanting to play chess without real
pieces, because playing with pieces is is too particularised and not
abstract enough".

D.