I think there's ambiguity here in what is a Mathematician. There's the
research mathematician that's trying to find something new to say about
the subject, faced with all the prior work, working in a niche that's
obtuse to almost all of us. Then there's the teacher of mathematics who
insists on our learning quite unambiguous facts about the subject. Then
there are the applied mathematicians, practitioners, like engineers, for
whom ambiguity is the enemy.
Getting into the mind of the formost is probably quite hard for any but
his/her immediate colleagues. But I also sense 'ambiguity' is being
used as a synonym for the 'unknown' or 'not understood'. Try
transposing 'the ambiguous' for 'the misunderstood' and 'the
unambiguous' with 'the understood', etc., in the quotes and see how
they read.
Robert C
On 12/28/09 11:33 PM, Nicholas Thompson wrote:
Hi, everybody,
The most important part of this message is the first few paragraphs,
don't not read it because it is long.
THE TEXT:
Here are two stimulating quotes from William Byers, /How
Mathematicians Think/. You will find them on pp 23-25, which happen
to be up on Amazon's page for the book.
Last paragraph of the intro, page 24:
/The power of ideas resides in their ambiguity. Thus, any project
that would eliminate ambiguity from mathematics would
destroy mathematics. It is true that mathematicians are motivated to
understand, that is, to move toward clarity, but if they wish to be
creative then they must continually go back to the ambiguous, to the
unclear, to the problematic, that is where new mathematics comes
from. Thus, ambiguity, contradiction and their consequences
--conflict, crises, and the problematic-cannot be excised from
mathematics. They are its living heart./
Epigraph from chapter 1, page 25:
/"I think people get it upside down when they say the unambiguous is
the reality and the ambiguous merely uncertainty about what is really
unambiguous. Let's turn it around the other way: the ambiguous is the
reality and the unambiguous is merely a special case of it, where we
finally/ /manage to pin down some very special aspect. /
//
/David Bohm"/
A few pages later, Byers defines ambiguity as involving
/"...a single situation or idea that is perceived in two
self-consistent but mutually incompatible frames of reference."/
//
THE SERMON:
Now on the one hand, these passages filled me with joy, because a
little appreciated psychologist of great perspicacity once wrote:
/"The insight that science arises from contradiction among concepts is
a useful one for explaining characteristic patterns of birth, growth,
and decay in the sciences. Initially, a phenomenon is brought sharply
into focus by its relationship to a conceptual problem. A first
generation of imaginative investigators is attracted to the phenomenon
in the hope of casting light on the related conceptual issue. These
investigators generate a lot of argument, a little progress, and a lot
of publicity. Then a second generation of scientists attracted, who
are drawn to the problem more by the sound of battle than by any
genuine interest in the original issue. By then, the conceptual issue
has been straightened out, the good people have left, and those who
remain devote their time to swirling in ever tighter eddies of
technological perfection.// " (/Thompson, 1976, My Descent from the
Monkey, In P.P.G. Bateson and P.H. Klopfer (Eds.), /Perspectives in
Ethology/, /2,/ 221-230.
//
On the other hand, to call ambiguity the living heart of mathematics
seems a little like calling "mess-making" the living heart of cleaning
a house, or littering the living heart of public sanitation.
It is characteristic of all goals that, if they are achieved, the
activity associated with them ceases. Therefore, for goal directed
activity to continue, it must fail to achieve it's end. But that
hardly makes failure the goal of the activity.
I suspect that Byers may clear this up in subsequent pages, but I
thought it was interesting enough to put it before the group. One way
out of the paradox, lies in Byers's definition's insistence that
ambiguity defined by a contradiction between two */clear /*concepts
bound within the same system. If we understood mathematicians as
clarifying the concepts that are bound within a frame work until their
contradiction becomes evident, then the perhaps the specter of making
ambiguity the heart of mathematics becomes less horrifying.
Now, I have to go to Houston.
All the best,
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([email protected] <mailto:[email protected]>)
http://home.earthlink.net/~nickthompson/naturaldesigns/
<http://home.earthlink.net/%7Enickthompson/naturaldesigns/>
http://www.cusf.org [City University of Santa Fe]
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Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
