ambiguity and mathematics
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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 ([hidden email])
http://www.cusf.org [City University of Santa Fe]
============================================================ 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 |
Re: ambiguity and mathematics
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Beautiful and useful quotes. Thanks, Nick.
Tory On Dec 28, 2009, at 11:33 PM, Nicholas Thompson wrote:
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Re: ambiguity and mathematics
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In reply to this post by Nick Thompson
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 |
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In reply to this post by Nick Thompson
Glen,
Rosen, again, eh? A couple of years ago I tried to get us goiing on Rosen and it fell flat. Are there Roseners and non Roseners? What exactly IS a rosener. Who of you out there ON THE LIST would say that YOU are a Rosener? I still have to go to Houston. N Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([hidden email]) http://home.earthlink.net/~nickthompson/naturaldesigns/ http://www.cusf.org [City University of Santa Fe] > [Original Message] > From: glen e. p. ropella <[hidden email]> > To: The Friday Morning Applied Complexity Coffee Group <[hidden email]> > Date: 12/29/2009 11:22:33 AM > Subject: Re: [FRIAM] ambiguity and mathematics > > > 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, > > > > 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 ============================================================ 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 |
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In reply to this post by glen e. p. ropella-2
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" <[hidden email]> wrote: Eric CharlesThis perspective is the essential gist of Robert Rosen's message, if you 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 |
Re: ambiguity and mathematics
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In reply to this post by Nick Thompson
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:
============================================================ 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 |
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In reply to this post by Eric Charles
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" <[hidden email]> 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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In reply to this post by Nick Thompson
All,
Make sure that you all process Byers's definition of ambiguity before you crank up your rhetorical engines too high. Unfortuately the definition was "below the fold" in my original message. Again, it is, "...a single situation or idea that is perceived in two self-consistent but mutually incompatible frames of reference." These are pretty demanding conditions. Ambiguity a la Byers is not just any old confusion. It is a highly elaborated contradiction. You have to do a lot of careful work to achieve ambiguity in Byers's sense. Nick Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([hidden email]) http://home.earthlink.net/~nickthompson/naturaldesigns/ http://www.cusf.org [City University of Santa Fe] > [Original Message] > From: Roger Frye <[hidden email]> > To: The Friday Morning Applied Complexity Coffee Group <[hidden email]> > Cc: <[hidden email]> > Date: 12/29/2009 2:31:38 PM > Subject: Re: [FRIAM] ambiguity and mathematics > > 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" <[hidden email]> 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. > > 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. (formerly CommodiCast) > INNOVATE, NAVIGATE, SURPASS > Confidentiality Notice: This e-mail transmission may contain confidential or entity named in the e-mail address. If you are not the intended recipient, you are hereby notified that any disclosure, copying, distribution, or reliance upon the contents of this e-mail is strictly prohibited. If you have received this e-mail transmission in error, please reply to the sender, so that Qforma can arrange for proper delivery, and then please delete the message from your inbox. Thank you. > > > > > ============================================================ > 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 ============================================================ 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 |
Re: ambiguity and mathematics
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In reply to this post by Nick Thompson
Quoting Nicholas Thompson circa 09-12-29 10:44 AM:
> Rosen, again, eh? A couple of years ago I tried to get us goiing on Rosen > and it fell flat. Are there Roseners and non Roseners? What exactly IS a > rosener. Who of you out there ON THE LIST would say that YOU are a > Rosener? I am not a Rosenite; but I'm fond of Rosenites. I have lots of sympathy for them. It falls flat, in my opinion, because of the sophistry surrounding Rosen's works. As with all sophistry, it's difficult to find any fundamental flaws; but the flaws are there. My main objection lies in the assumption that there is an accurate fine-grained mapping between what goes on "in here" (inferential entailment) and what goes on "out there" (causal entailment). The relationship between the two is, to me, merely analogical. -- 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 |
Re: ambiguity and mathematics
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In reply to this post by Eric Charles
Quoting ERIC P. CHARLES circa 09-12-29 11:09 AM:
> Well, of course, all of this (Glen and Nick's posts) is ignoring the obvious > fact that ambiguity is the antithesis of mathematics. That's just not the case. Mathematics easily captures the concept of ambiguity. Hence, ambiguity can't be the antithesis of math. Math is a language. Languages describe. All languages can be reflective (circular). In English, we can describe circularity with sentences like: "This sentence is false." In math, we can describe circularity with sentences like: "Let X = {a,b,X}." Ambiguity is, formally, just a type of circularity. In generic circularity, the evaluation of some predicates on X is undefined. In ambiguous cases, the evaluation of those predicates simply have multiple values. I.e. the mapping is 1 to many. This isn't, in any interpretation, the antithesis of math. It is well described by math, less well described by English. > 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? Because math is a _means_ not an _end_. Ambiguity is at the heart of math because math is our attempt to disambiguate the ambiguous ... to refine what is coarse ... to peek into the little nooks and crannies created by our prior theorems. > 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. Again, I have to disagree. All claims of ambiguity are statements of ignorance. Granted, we can whittle away at the ignorance and refine the ambiguity to a very fine point (which is what Rosen does). But in the end, ambiguity ... as Byer's and the rest of us use the term [grin] ... means "We don't know what makes it take value X as opposed to value Y in this evaluation." A great example is the square root of 4. There are 2 answers: 2 and -2. In order to know whether the answer is 2 or -2, we need more information. I.e. we're too ignorant to answer that question. > 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. It's still a statement of ignorance, though perhaps not of _severe_ ignorance. As I said before, the use of the word "ambiguity" may well be taken as a methodological point, not an ontological or epistemological one. Whether "Free Speech Zone" evaluates as "protect" xor "undermine" is definitely an expression of ignorance. Ultimately, due to our lack of understanding of the social processes involved, we can't state for sure whether it is one or the other. Further, we can't even state that the two are disjoint! Perhaps one MUST undermine freedom of speech in order to protect freedom of speech, in some circumstances? The point is that we don't know. We are ignorant. In the case of shooting an elephant in pajamas, again, it's a statement of ignorance. We can resolve it by 1 question: "Was the elephant wearing your pajamas?" Prior to the answer, it's ambiguous. Post answer, it's not. -- 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 |
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In reply to this post by Nick Thompson
Nick writes:
> Make sure that you all process Byers's definition of > ambiguity before you crank up your rhetorical engines > too high. Unfortuately the definition was > "below the fold" in my original message. Again, it is, > > "...a single situation or idea that is perceived in > two self-consistent but > mutually incompatible frames of reference." As you may recall, Nick, I have been working for several years now to elaborate a mathematical model of ambiguity for applications in psychology (etc.), largely inspired by various goings-on in the Kitchen Seminar and SEC Forum, and in particular with our various discussions about schematization and emergence, Jaan's notion (in a paper with Emily Abbey) of "emergence of meanings through ambivalence" (where he, inadvisedly, uses "ambivalence" to mean "ambiguity"...damned Estonophones), and your and Jim's repeated pep-talks on "levels of organization". I hope to have a manuscript ready by January 25 (the deadline for submission to a conference for which the subject *might* be appropriate), and will circulate it to these lists when it's finished or on January 26, whichever comes first. The working title is "Stratified manifolds, finite topological spaces, posets, and (in)decision trees: Ambiguity as a mathematical foundation for schematization in robotics" (hey, it's a robotics conference, okay?). As the title sort of gives away (but only to initiates), one of my concerns is how it is that continuous (and, incidentally, infinite) "manifolds" of data/stimuli/worldstuff/what-have-you become "schematized" by human consciousness (and language). I can assimilate Byers's definition (as just quoted) to my general ideas by rephrasing it like this: "ambiguity consists in subsuming two (or more) self-consistent but mutually (more or less) incompatible situations or ideas into a single situation or idea". That is, "an ambiguity between (or resolvable to) A and B (and C...)" is a higher-level 'thing' than A and B. (Notice that I have committed a rhetorical move by suddenly introducing the notion of "an" ambiguity. That's actually what I prefer to talk about, rather than "ambiguity" as a general ... what? process?) I admit the possibility of an ambiguity existing between *any* two 'things' (for a given person at a given time). What's important to me (and what I'm trying to write this paper in such a way as to promote) is the idea that an ambiguity between A and B is a first-class object in its own right, not some sort of derivative construction. That's why I've (I think) coined the phrase "indecision tree", by which I mean nothing other than a "decision tree" in the usual sense (a finite acyclic digraph [whose underlying graph *may* be cyclic {i.e., in Jaan's language, there *may* be--and often is--equifinality: different paths to the same endpoint}]), but viewed in such a way that the non-terminal nodes are conceived of as "states of indecision" (or ambiguity) and treated as first-class objects. (Fence-sitters of the world unite!) Why in the world are you going to Houston? The Austin Lounge Lizards have a fabulous song, "I'm Going Back to Dallas, Texas (To See If Anything Could Be Worse Than Losing You)"; surely Houston *is* worse than even Dallas. Lee P.S. I see that I haven't said anything about ambiguity *of* mathematics. That's because I can't make much sense of Bohm's or Byers's comments as quoted. Maybe I need more context. ============================================================ 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 |
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In reply to this post by Nick Thompson
In any discussion such as this one, lest the discussion just spin out of
control (which gives everybody a giddy sense of whizzing around but eventually gets nowhere) we have to understand which definition of ambiguity we are working with. I suggested that we work with Byers's. The is nothing coarse about Byers ambiguity. To be ambiguous in Byers sense, a situation must include two well articulated ideas that are mutually antogonistic but bound together in the same well articulated system of thought. To have achieved Byers-ambiguity is to have clarified a lot. N Nicholas S. Thompson Emeritus Professor of Psychology and Ethology, Clark University ([hidden email]) http://home.earthlink.net/~nickthompson/naturaldesigns/ http://www.cusf.org [City University of Santa Fe] > [Original Message] > From: glen e. p. ropella <[hidden email]> > To: The Friday Morning Applied Complexity Coffee Group <[hidden email]> > Date: 12/29/2009 4:49:08 PM > Subject: Re: [FRIAM] ambiguity and mathematics > > Quoting ERIC P. CHARLES circa 09-12-29 11:09 AM: > > Well, of course, all of this (Glen and Nick's posts) is ignoring the obvious > > fact that ambiguity is the antithesis of mathematics. > > That's just not the case. Mathematics easily captures the concept of > ambiguity. Hence, ambiguity can't be the antithesis of math. Math is a > language. Languages describe. All languages can be reflective > (circular). In English, we can describe circularity with sentences > like: "This sentence is false." In math, we can describe circularity > with sentences like: "Let X = {a,b,X}." Ambiguity is, formally, just a > type of circularity. In generic circularity, the evaluation of some > predicates on X is undefined. In ambiguous cases, the evaluation of > those predicates simply have multiple values. I.e. the mapping is 1 to > many. > > This isn't, in any interpretation, the antithesis of math. It is well > described by math, less well described by English. > > > So, how do we > > reconcile claims that ambiguity is at the heart of mathematics with the > > truth that mathematicians really like producing, teaching, and preaching about > > unambiguous things? > > Because math is a _means_ not an _end_. Ambiguity is at the heart of > math because math is our attempt to disambiguate the ambiguous ... to > refine what is coarse ... to peek into the little nooks and crannies > created by our prior theorems. > > > 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. > > Again, I have to disagree. All claims of ambiguity are statements of > ignorance. Granted, we can whittle away at the ignorance and refine the > ambiguity to a very fine point (which is what Rosen does). But in the > end, ambiguity ... as Byer's and the rest of us use the term [grin] ... > means "We don't know what makes it take value X as opposed to value Y in > this evaluation." > > A great example is the square root of 4. There are 2 answers: 2 and -2. > In order to know whether the answer is 2 or -2, we need more > information. I.e. we're too ignorant to answer that question. > > > For an example of the latter, we > > might ask whether George W.'s "Free Speech Zones" were protecting > > 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. > > It's still a statement of ignorance, though perhaps not of _severe_ > ignorance. As I said before, the use of the word "ambiguity" may well > be taken as a methodological point, not an ontological or > epistemological one. Whether "Free Speech Zone" evaluates as "protect" > xor "undermine" is definitely an expression of ignorance. Ultimately, > due to our lack of understanding of the social processes involved, we > can't state for sure whether it is one or the other. Further, we can't > even state that the two are disjoint! Perhaps one MUST undermine > freedom of speech in order to protect freedom of speech, in some > circumstances? The point is that we don't know. We are ignorant. > > In the case of shooting an elephant in pajamas, again, it's a statement > of ignorance. We can resolve it by 1 question: "Was the elephant > wearing your pajamas?" Prior to the answer, it's ambiguous. Post > answer, it's not. > > -- > 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 ============================================================ 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 |
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In reply to this post by Nick Thompson
Well, I can see that goals are not explicitly mentioned. But let's see what happens if we say that the heart of the enterprise, the thing that drives it forward, is not its "goal" Littering is the 'heart" of sanitation? The "signs" are all wrong.
Nick
Nicholas S. Thompson
Emeritus Professor of Psychology and Ethology,
Clark University ([hidden email])
http://www.cusf.org [City University of Santa Fe]
============================================================ 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 |
Re: ambiguity and mathematics
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In reply to this post by Nick Thompson
Quoting Nicholas Thompson circa 09-12-29 05:02 PM:
> In any discussion such as this one, lest the discussion just spin out of > control (which gives everybody a giddy sense of whizzing around but > eventually gets nowhere) we have to understand which definition of > ambiguity we are working with. > > I suggested that we work with Byers's. The is nothing coarse about Byers > ambiguity. To be ambiguous in Byers sense, a situation must include > two well articulated ideas that are mutually antogonistic but bound > together in the same well articulated system of thought. > > To have achieved Byers-ambiguity is to have clarified a lot. Under Byers' use of the term ("a single situation that is perceived in two mutually incompatible frames of reference"), ambiguity can be thought of as a METHOD. Byers is talking about ambiguity as an attribute of a function/process that takes a single input and produces multiple outputs. The perception/interpretation according to distinct and mutually incompatible frames of reference is just an elaborate way to say that there are details about the evaluation that are obscure, as in the case of the square root function and the sign of the result. When the same situation can be evaluated to two distinct results, it helps to do as Lee suggests and formulate the ambiguity as an explicit part of a larger evaluation. For example, if the square root of 4 can be 2 or -2, we need some larger unifying context within which to reconcile the two answers. I.e. There is some data missing. If we had the extra data, we would know whether the √4 evaluates to +2 or -2. For example, perhaps the "4" represents the height reached when you throw a baseball up in the air. The answers +2 vs. -2 then mean either the place you stood when you threw it or the place it landed. Obviously, where you're standing and where the ball lands are mutually incompatible answers to the same (ambiguous) question. But they fit into a larger, unifying whole, which is what Byers' talks about on those pages surrounding his definition. Add more attributes to the evaluation and the ambiguity disappears. Hence, as I said, ambiguity is used as a method for refining (from coarse to fine) a question. If you prefer Lee's "an ambiguity", then you can say it as: The explicit expression of an ambiguity is a method for refining a question. It shows us exactly why, where, and when we need new data. For reference, here's what I said before... to help those who don't read carefully. [grin] >> [Original Message] >> From: glen e. p. ropella <[hidden email]> >> [...] >> Because math is a _means_ not an _end_. Ambiguity is at the heart of >> math because math is our attempt to disambiguate the ambiguous ... to >> refine what is coarse ... to peek into the little nooks and crannies >> created by our prior theorems. -- 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 |
Re: ambiguity and mathematics
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In reply to this post by lrudolph
Quoting [hidden email] circa 09-12-29 04:18 PM:
> P.S. I see that I haven't said anything about > ambiguity *of* mathematics. That's because I > can't make much sense of Bohm's or Byers's > comments as quoted. Maybe I need more context. Byers and Bohm are merely making the point that the type of thing you are trying to say in your paper is pervasive in mathematics. In any situation where we're trying to be maximally explicit in formulating our questions, math is a tool that we use to lay out what we do and do not know. Ambiguity is just one type of statement of ignorance. There are others. In your case, you're simply inverting the focus so we can pay direct attention to the _hole_ in what we know, the ignorance. It's a fantastic idea. So many people are so boggled by all the bricks in the wall of knowledge that they don't notice the little holes of ignorance between the bricks. My favorite example of when this sort of "necker cube" focus inversion is useful is the use of positively charged holes going forward through an electronic circuit, rather than focusing on negatively charged electrons going backward through the circuit. That little inversion helped me get through my "electronic properties of materials" class in college. [grin] Byers and Bohm are saying that this sort of "ignorance highlighting" is the heart of math (in Byers case) and science (in Bohm's case). -- 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 |
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