[FOM] the stuff that [numbers] are made of
martin at eipye.com
Tue May 13 17:50:40 EDT 2003
I am amazed at the extensive discussion on the constitution of the subject
matter of mathematics. Everyone agrees that mathematicians don't care in
the least about determining what the number 3 "really" is, or in what sense
the rational number 1/3 is the same as the real number 1/3. But at least
some philosophers see these questions as worthy of seemingly infinite
The objects of mathematical investigation are best viewed as being abstract
patterns that have objective properties some of which we are able to
ascertain. A story: J. walks into a tile shop to buy tiles for flooring a
bathroom. Dialogue follows:
Clerk: Here is our catalog with pictures. Will you want THE pattern of
square tiles or THE pattern of hexagonal tiles? Those are the most popular.
But more complex patterns are possible.
J: I'll go for THE hexagonal pattern.
Clerk: Half inch or full inch tiles?
J. Half inch. My bathroom's dimensions are ......
Note that there is no problem with the definite article here even though
the words "the hexagonal
pattern" encompasses various different physical embodiments, both in layout
and even size of individual tiles. (I wonder whether philosophers thinking
in a language like Russian that manages without a definite article would
have the same concerns.)
One could prove various things about this pattern. If one chooses to
formalize the matter in the language of set theory, then of course our
pattern will have to be represented by some rather complex set-theoretic
construct which will perforce have some (Benacerraf style) idiosyncrasies
having nothing to do with the hexagonal pattern. If one has the notion that
this construct IS the pattern, then this could give rise to misgivings. But
once one understands clearly that this construct is just one of infinitely
possible legitimate representations to be selected only on the basis of
convenience, what philosophical issues remain?
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