FOM: various

[Randall Holmes]

On  20 Jan  J SHIPMAN wrote:

	To assail Hersh, we need a definition of
	mathematics better than his. 


Not really.  Hersh's definition is absurd, because it cannot be
squared with the range of applicability of mathematics.  There is no
reason to believe that a system which was _intrinsically_ a human
social construction would be applicable to the range of problems in
the real world to which mathematics is applicable.  Mathematics as
practiced by human beings is of course a social construction; so is
astronomy, but we do not say that the stars are a human social
construction.  Mathematics is about structure that is really found
"out there" (_and_ in human thoughts), and that has no special
relevance to human beings (this is not an assertion about the
independent existence of mathematical objects).

To take a mundane example, consider the hexagonal shape of the cells
of honeycombs.  There is a nice mathematical reason for this
(supported by the fact that physical space approximates Euclidean
space essentially exactly in this context).  I don't see how the view
that mathematics is a social construction can even begin to cope with
this kind of applicability of mathematics (unless bees are members of
the community :-) ).  The real world problem has a certain kind of
structure (because physical space approximates Euclidean space).  Our
independent investigations of Euclidean space "for its own sake" turn
up the mathematical optimality of the same configuration that bees
come up with.  Mathematical structure is seen here to be objective in
a very strong sense (not the socially dependent sense that Hersh would
have us see); every application of mathematics in science and
engineering demonstrates the same point.

It is a quite separate (and hard) question what this "structure" is
(e.g., are structures universals or do they inhere in particulars?)
But its nature has nothing special to do with the mathematical
community; we no more create it than the astronomical community
creates the stars.

And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the       | Boise State U. (disavows all) 
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes