FOM: replies to Shipman: compartmentalization; math vs physics

[Stephen G Simpson]

JoeShipman at aol.com writes:
 > Simpson has suggested that some of the hostility to f.o.m. in the
 > mathematical community may be a result of increasing
 > compartmentalization and specialization of mathematics, so that
 > research which tends to treat mathematics as a unified whole (or
 > more generally any attempt to breach the boundaries between areas
 > of mathematics) is viewed as an encroachment.

This isn't what exactly I was referring to.  When I mentioned
compartmentalization, I was referring to isolation of mathematics from
the rest of human knowledge, not isolation of particular specialties
within mathematics.  I find that many leading contemporary
mathematicians such as Connes actually have a pretty good feeling for
connections among various branches of core mathematics: analysis,
algebra, topology.  What is missing is an appreciation of the
interface between mathematics and the rest of human knowledge: applied
math, computer science, statistics, foundations of math, ....  In this
sense, we have declined a long way from Poincare, Hilbert, von
Neumann, ....

 > Even if mathematics is too big for anyone to expect to know all of
 > it, it certainly ought to be possible to work in more than one area
 > (even two is rare).  My guess is the reasons for today's
 > specialization of mathematicians are sociological and psychological
 > rather than intrinsic to mathematics.  I have some arguments to
 > support this guess, but I'd like to hear what others on FOM have to
 > say about this.

I agree that mathematicians today tend to be more specialized.  I
haven't thought much about the reasons for this.  Off-hand, my guess
at an explanation would be that mathematical knowledge has grown so
rapidly that it will take some time for the mathematical community to
discard the junk and absorb the good stuff.  In the meantime, there is
no general agreement or understanding of what is important, so people
simply pick a niche and stay there.

 > In her column in Parade magazine today, Marilyn vos Savant answers
 > a reader's query about the ultimate principles on which the
 > universe operates as follows:
 > 
 > "I'd say physics alone covers the territory.  Chemistry falls under
 > the heading of physics, biology falls under the heading of
 > chemistry, thought processes fall under the heading of biology, and
 > mathematics falls under the heading of thought.  I think."
 > 
 > ...
 > 
 > does mathematics really belong "at the other end" of her hierarchy
 > from where she places it, or is the situation more complicated?

I lean toward the Aristotelean view: Mathematics is a theoretical
science, hence it is indeed part of the ultimate principles on which
the universe operates.  Specifically, mathematics is the science of
quantity, i.e. quantitative aspects of the universe.  I don't see
mathematics as being in any way opposed to, or at the opposite end of
some spectrum from, physics.

-- Steve