Re: a bit more on maths

From: Jerry Balzano (gjbalzano@ucsd.edu)
Date: Fri Oct 26 2001 - 16:47:28 PDT


OK, this thread has been quiescent for some time, and it may even be dead
by now, but let me try my belated two cents worth. I gave an informal talk
at LCHC a couple of weeks ago, following Joe Goguen's talk on "math
objects", and I proposed two conceptions of mathematics, both borrowed,
that move away from the idea of "objects". One, borrowed from Michael
Resnik (and others) conceives of mathematics as "the science of patterns",
and the other, borrowed from Brian Greer, treats math as the study of
"models of situations". I don't know if one of these conceptions can be
assimilated to the other, but neither one invites thinking of, say, a
number as an "object".

Numbers, as entities that can be added and subtracted, are better thought
of, for example, as parts of a model that we apply to different situations.
Most of these situations are intuitively obvious, but it is interesting to
think about applying the numbers/addition model to situations where this is
not the case. So "1+1 = 2" is a universal truth? Maybe if we are counting
beans but what about a situation where we are looking at clouds in the sky
or piles of sand on the beach and we think of "combining" them? In such
cases we would not be so quick to say that 1+1=2. Now we could challenge
the appropriateness of the "+" operator, but I don't think this criticism
will go through, since if we were combining sand piles and weighing them
(instead of counting them) then addition is once again appropriate for
modeling the outcome.

Patterns and models are in any case not Platonic forms that exist in an
abstract universe, and we might try to say that these things are our math
"objects". The tendency to try to "object"ify everything is just something
we do, and is part of the reason why "object oriented programming" seems to
so many of its practitioners to be a more natural way to program a
computer. So too with math, even though, as Jay has said (and I agree),
mathematics is really relations "all the way down". It's just really hard
to sink your teeth into, or even localize, a relation. As Bateson, I
think, has pointed out, Here is a black patch and Here is a white patch,
but Where is the relation between them? As real as this relation is, and
as perceivable -- it's surely as real as the items that constitute it -- it
defies localization. Numbers are convenient "objects" that, in their
standard interpretation, work well as place holders in an asymmetric
transitive relation, but I can also stick twelve of them on a clock and do
a different kind of "addition" with them that no longer obeys the simple
ordering relation that we take to be primitive and defining for "the number
line" (indeed, it's a "number circle").

I do believe that conceptions of math like these have implications for
education, although I agree that if all we are debating is formalism vs
logicism vs. platonism, nothing at stake makes much sense or much
difference for any kid learning math (or even most grownups).

                                - Jerry Balzano

-------------------------
Dr. Gerald J. Balzano
Dept of Music
Teacher Education Program
Laboratory for Comparative Human Cognition
Cognitive Science Program
UC San Diego
La Jolla, CA 92093
(619) 822-0092
gjbalzano@ucsd.edu



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