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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