[FOM] the stuff that [numbers] are made of

[Martin Davis]

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 
discussions.

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?

Martin