Beautiful and useful quotes. Thanks, Nick.
Tory
On Dec 28, 2009, at 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])
http://home.earthlink.net/~nickthompson/naturaldesigns/
http://www.cusf.org [City University of Santa Fe]
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============================================================
FRIAM Applied Complexity Group listserv
Meets Fridays 9a-11:30 at cafe at St. John's College
lectures, archives, unsubscribe, maps at http://www.friam.org
