I think Thurston gives a great example of ambiguity in his paper "On Proof and Progress in Mathematics," where he lists 7+1 ways of understanding the derivative. Infinitesimal, Symbolic, Logical, Geometric, Rate, Approximation, Microscopic + "... Lagrangian section of the cotangent bundle ...."
-Roger Frye On Dec 29, 2009, at 12:09 PM, ERIC P. CHARLES wrote: > Well, of course, all of this (Glen and Nick's posts) is ignoring the obvious > fact that ambiguity is the antithesis of mathematics. Of course (?!?), there > is a nuanced resolution of this tension, having something to do with a > difference in worlds between the lofty professor and the practical man, but > I'm not sure what it is. > > When a teacher asks a student what 2+2 is (hint: 4), the length of the area > of a circle with radius 1 (hint: pie), what the integral of a given function > is, whether a given number is prime or not, etc. etc. etc., the student > doesn't get full credit for saying "Its ambiguous, and the world is better > that way!" I doubt anyone would argue that students and lower-level teachers > of mathematics are completely wrong in their view that these questions have > unambiguous answers. (Though surely some will claim the problems are not > adequately specified. For example, is the circle in euclidean space?) So, how > do we reconcile claims that ambiguity is at the heart of mathematics with the > obvious truth that mathematicians really like producing, teaching, and > preaching about unambiguous things? > > Also, re Glen's post specifically, I think there is value in discriminating > between accidental and intentional ambiguity. Not all claims of ambiguity is > are claims of ignorance, sometimes situations are actually ambiguous and > therefore claims of ambiguity are claims of knowledge. For an example of the > former, I may claim that the pitter patter on my roof "May be acorns falling > or it may be rain, its ambiguous". In that case, we all agree that it either > IS acorns OR rain (while retaining the chance it is both), and it is clear > that I am stating my ignorance as to which it is. For an example of the > latter, we might ask whether George W.'s "Free Speech Zones" were protecting > people's freedom of speech. One possible answer to that question, one that > expresses a good understanding of the situation, NOT severe ignorance, might > be "In some ways it technically was, but in other ways it severely undermined > freedom speech, so the situation is ambiguous." On a lighter note, many jokes > an innuendo take advantage of ambiguity, and if you don't think the situation > is ambiguous, you won't get it. For example, I once shot an elephant in my > pajamas..... what he was doing in my bedroom I'll never know. > > Eric > > > On Tue, Dec 29, 2009 01:21 PM, "glen e. p. ropella" > <g...@agent-based-modeling.com> wrote: > 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 > > > Eric Charles > > Professional Student and > Assistant Professor of Psychology > Penn State University > Altoona, PA 16601 > > > ============================================================ > 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 Roger Frye, 505-670-8840 Qforma, Inc. 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