This perspective is the essential gist of Robert Rosen's message, if you carve off all the surrounding sophistry. Ambiguity is the essence of life. If we specialize down into mathematicians, we can say that ambiguity is the essence of mathematics, as practiced by the animals we call mathematicians.
To some extent, this may seem to trivialize what Byers and Bohm are saying; but I don't think it does. It just places it in a larger context. But the paradox Nick points out extends beyond the "mathematics itself" question, in tact, up to the "life itself" question. And that brings me to my current comment: Asserting that ambiguity is the heart of _anything_ is, essentially, "begging the question" or petitio principii. Ambiguity is just multi-valued-ness, the ability of a [im]predicate [grin] to take on one value when evaluated in one context and another value when evaluated in another context. Hence, ambiguity is (like randomness) a statement of ignorance. So, there are 2 ways to parse the situation (and the quote from Byers) as a statement of ignorance: 1) Saying "ambiguity is the heart of math" is saying "we really don't understand what we're doing when we do math", or 2) Saying "ambiguity is the heart of math" is an expression that math is a _method_, not knowledge ... an approach, not a thing to be approached. Both are compatible with the "mechanism" that Rosen rails about. But (2) allows us to put off the controversy and continue working together as holists and reductionists. ... or not. ;-) Quoting Nicholas Thompson circa 09-12-28 10:33 PM: > 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. -- glen e. p. ropella, 971-222-9095, http://agent-based-modeling.com ============================================================ 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
