Seminal Papers in Complexity
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At risk of sounding soft-brained, I'd recommend considering other domains as a source of inspiration for a new non-linear kind of math. Music in particular, and other so-called "creative" forms, offer insight into different kinds of problem-solving approaches, and quasi-logical expression.
And besides, Einstein liked playing his violin more than just about anything ;-) db ----- Original Message ----- From: Michael Agar To: The Friday Morning Applied Complexity Coffee Group Sent: Tuesday, June 19, 2007 7:41 AM Subject: Re: [FRIAM] Seminal Papers in Complexity This thread is sliding around some, but still I?d like to add this overlong comment in case it?s useful. The emails have been good brain food. The problem I keep worrying about in my own work is, I use many core concepts metaphorically because they work at the human organizational scale in powerful and useful ways that I believe respect their scientific origins but at the same time allow the human/social world to see and understand and act differently. But I also want to be clear on those origins, to know and describe when and where and how I?m stretching the concepts. The problem I have is, up close the conceptual basis of ?complexity? more often than not turns to mush. Mea culpa much of the time, I?m sure, but look what happened to reductionism in this thread. Even Wikipedia has several entries. I don?t know how much credence to give them, but here they are: 0.1 Varieties of reductionism 0.1.1 Ontological reductionism 0.1.2 Methodological reductionism 0.1.3 Methodological individualism 0.1.4 Theoretical reductionism 0.1.5 Scientific reductionism 0.1.6 Set-Theoretic Reductionism 0.1.7 Linguistic reductionism 0.1.8 Greedy reductionism 0.1.9 Eliminativism And now emergence. I?ve heard it used in several ways. Way back when, we used it in anthropology as a form of methodological defense against the usual social science model of everything planned in a modular way before the research started. Emergence was shorthand for ?I can?t tell you what I?m going to do until I get there and learn what?s worth learning and how to learn it.? Then it?s also used more generally as shorthand for ?surprise,? the presence and nature of which depends on perspective and prior knowledge of observer. Then it?s used for the end result of a deterministic process that has characteristics unlike the elements of that process, like water out of hydrogen and oxygen. Then it?s used for the need for different concepts and methods for different levels of a phenomenon, like phonology, morphology and syntax in linguistics. Then it?s used for unexpected evolutionary and historical transitions, like the Cambrian explosion. Probably many other uses if we sampled a lot of texts and conversations. Probably some of the sources cited already in the thread help with the problem. I need to read them. Maybe the field has outgrown the concepts that got it started. If true, that?s probably a good sign. So I think I?ll work on nonlinearity for awhile. Russell writes: ?most of my readers understand perfectly well what a linear function is: one that obeys f(a*x+b*y) = a*f(x)+b*f(y).? That?s clear, resembles the definition in the Wikipedia entry. But then he writes : ?If neither * or + are defined for your objects of discussion, you cannot talk about (non-)linearity.? That won?t do. I have to be able to talk about nonlinear effects of, say, mental health policy on local programs in a qualitative way. I know it makes sense to do so from experience. Problem is to make it clear what the term means in that context. If the math won?t do it, something else has to. I?ll puzzle over the NECSI definition and the opening pages of Strogatz? book for awhile. So maybe nonlinearity won?t be so easy either. There?s the famous Einstein quote for inspiration: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Maybe we need a new nonlinear kind of math. Maybe it exists. Enough already. Mike ------------------------------------------------------------------------------ ============================================================ 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 -------------- next part -------------- An HTML attachment was scrubbed... URL: http://redfish.com/pipermail/friam_redfish.com/attachments/20070619/c3880300/attachment.html |
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In reply to this post by Merle Lefkoff
On Jun 16, 2007, at 1:43 PM, Merle Lefkoff wrote:
> Owen, > > I took advantage of the CNLS printer to print LOTS of articles about > complexity that seemed to do more than just gestate in utero (let's > all > feminize seminal). Before I toss them all, I will pass on a few > suggestions for the list. Do you want the titles annotated? > > -Merle- Hi Merle. No need for annotations other than the titles/authors of the ones you really liked, unless you have time to comment on them. I would be interested in those you feel capture a topic well, especially if not in Your Basic Textbook On Mumble Science. I.e. things that are not in a single discipline. I'm reading the recent Scott Page & John Miller book, and one of their Themes is a nice on: Complexity often falls In Between disciplines. So within the summer school we are getting: - Computer Science and Statistical Mechanics - Finance and Dynamics - Sociology and Game Theory - Ecology and Graph Theory - Data Mining and Topology .. kinda interesting. The school site is open, I think: http://www.santafe.edu/events/workshops/index.php/CSSS_2007_Santa_Fe The readings are quite good and getting more added to each day. The use of a wiki has been a considerable win. -- Owen |
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In reply to this post by Phil Henshaw-2
Interesting .. when following these links I end up on one of my
favorite sites: http://cscs.umich.edu/~crshalizi/notebooks/ashby.html Cosma Shalizi is one of the more interesting mavericks of the SFI world, and taught at the SFI 2000 summer school. Definitely one of my favorite reads. -- Owen On Jun 16, 2007, at 9:35 AM, Phil Henshaw wrote: > One problem with the seminal papers on complexity is that they don't > connect. Take the foundational works of H.T. Odum, the systems > ecologist(1) or the cybernetic systems thinkers Ross Ashby (2) or > Norbert Wiener(3). It's hard to link them to other branches of > complex > systems study like Prigigene's 'Exploring Complexity' or Wolfram's > 'New > kind of Science' or Barabasi's 'Linked' (leaving out numerous > important > others). As a consequence few people are aware of the general > timeline > of complexity as a subject(4), and any timeline of the field is > bound to > be missing major contributions. > > The problem seems is partly that the study of complex systems is > interdisciplinary, because systems are, and what happens is each > discipline goes off on its own tangent and acts like it is trying to > take over the subject as a whole, each vying to erase each other > rather > than connect with each other. My work seems to be an example of an > attempt to link approaches, a new form of physics intended > expressly for > use by any discipline, and incorporating unique useful pieces of > what's > been developed from all the disciplines I've been exposed to. My work > may be 'odd' in more ways than that, but it's partly because I'm > trying > to write in a common language that makes it look 'foreign' to every > discipline, so no one'll publish it... Catch 22! :-) > > (1) Odum: 1994 'Ecological and General Systems' (see > http://www.eoearth.org/article/Odum,_Howard_T.) > (2) Ross Ashby's 1947 'Ecological and General Systems' or his 1956 > "Introduction to Cybernetics" (& see > http://en.wikipedia.org/wiki/W._Ross_Ashby) > (3) Weiner 1948 'Control and Communication in the Animal and the > Machine' > (3) complex systems thinking timeline from the cybernetics soc. > (http://www.asc-cybernetics.org/foundations/timeline.htm), > > > Phil Henshaw ????.?? ? `?.???? > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > 680 Ft. Washington Ave > NY NY 10040 > tel: 212-795-4844 > e-mail: pfh at synapse9.com > explorations: www.synapse9.com > > >> -----Original Message----- >> From: friam-bounces at redfish.com >> [mailto:friam-bounces at redfish.com] On Behalf Of Owen Densmore >> Sent: Friday, June 15, 2007 7:38 PM >> To: The Friday Morning Applied Complexity Coffee Group >> Subject: [FRIAM] Seminal Papers in Complexity >> >> >> Several of us have been attending the SFI Summer School this year. >> One thing that has stood out for me is that there are very few >> appropriate texts on the detailed, seminal ideas within complexity. >> Either the books are "popular" or they are technical/formal enough, >> but without broad view of complexity itself. Indeed, they may be >> *too* advanced in their speciality for the broad use complexity >> wishes to make. >> >> One example today was the intersection of computational theory and >> statistical mechanics given by Cris Moore: >> A Tale of Two Cultures: Phase Transitions in >> Physics and Computer Science >> Here are the slides: http://www.santafe.edu/~moore/Oxford.pdf >> You'd be unlikely to find a book bridging algorithms, computational >> complexity, and statistical mechanics. >> >> This leads me to believe that seminal papers are likely to be a good >> solution for bridging the various cultures, hopefully with some that >> *do* bridge gaps between specialties. >> >> Sooo -- gentle reader -- this brings me to a request: I'd like to >> start a collection of seminal papers who's goal is to bridge the gap >> between popular books and over-specialized texts, which are formal >> enough to be useful for multi-discipline complexity work. This may >> be daft, but I think not. >> >> As an example, I'd say Shannon's 1948 paper A Mathematical Theory of >> Communication would be good. >> >> -- Owen >> >> >> >> ============================================================ >> 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 Russell Standish
Nope, that's not at all what I meant. The centuries old qualitative/
quantitative issue, needs revisiting now, but that's a course of study, not an email. > > When I hear "nonlinear effects of mental health policy" I immediately > think of some variable (eg some measure of social good) that > depends on > some other variable (eg money) in a nonlinear way (eg social good > varies as the square of money spent). > > Whilst you may be using the term a little imprecisely by not being > quantitative, it is still a perfectly valid use of the term. > However, if > the above paragraph is not what you mean, then you've immediately lost > one of your readers. > > |
Seminal Papers in Complexity
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Then it would be an abuse of terminology. One that would immediately
mislead the reader, unless you very, very carefully explain that you are using nonlinear in an unconventional sense, every single time you use the term. BTW, what exactly do you mean by nonlinear, if not in the sense I suggested? Cheers On Wed, Jun 20, 2007 at 07:24:55AM -0600, Michael Agar wrote: > Nope, that's not at all what I meant. The centuries old qualitative/ > quantitative issue, needs revisiting now, but that's a course of > study, not an email. > > > > > > When I hear "nonlinear effects of mental health policy" I immediately > > think of some variable (eg some measure of social good) that > > depends on > > some other variable (eg money) in a nonlinear way (eg social good > > varies as the square of money spent). > > > > Whilst you may be using the term a little imprecisely by not being > > quantitative, it is still a perfectly valid use of the term. > > However, if > > the above paragraph is not what you mean, then you've immediately lost > > one of your readers. > > > > > > > ============================================================ > 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 -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpcoder at hpcoders.com.au Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- |
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In reply to this post by Phil Henshaw-2
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Hash: SHA1 Phil Henshaw wrote: > I think that's a very consistent argument, and very similar to the one > Bohr used as the basis for the Copenhagen convention and dumping > Einstein's idea of the physical world. As I recall, the argument was > that science is information and so nothing exists for science except > what exists as scientific information, and so uncertainties that define > a limit to scientific knowledge also define a limit to any meaningful > scientific reality. So as scientists, reality does not exist beyond > what is knowable. I think it was that slim logical thread that kept the > otherwise very unsatisfying assertion that phenomena are created by our > observations from being tossed out as ridiculous. Interesting. Your paraphrase certainly seems analogous to what I submitted if information and action are taken as analogous fulcra. But, I'm not sure the analogy is very robust. My argument hinges on action and unity (discretion) where "things" are only things, separate from all the other goo in which we're bathed, by virtue of their being acted upon or acting upon as a unit. Hence, an "emergent thing" is fictitious if it cannot be acted upon separately from the other things to which it's related, including its constituents. For example, an "emergent phenomenon", in order to be a real thing, would have to either act as a unit or be acted upon as a unit. (By "act", I don't mean "ascribed thing-hood by an observer". I mean "does physical work"... moves objects around, generates heat, etc.) Now, personally, I tend to agree with G?nther and I don't rely on the word "emergence" for conversations I regard as important. I have yet to see an emergent phenomenon.... and I doubt that I ever will. Of course, that doesn't mean they don't exist. It also doesn't mean that the concept is useless. Unicorns may not exist either; but, it is important that we can think about them and perform thought experiments with them. Given these details (especially the descent into discretion begged by the action requirement), it's hard for me to maintain the analogy between my argument and information as scientific currency. > For most people the question comes down to which way they *like* > thinking about the world, since either one can be made satisfying if > that's what you like... I prefer, and find more productive, thinking > that I'm exploring a world that exists without my knowledge of it, and > is built in such a complicated way that my descriptions will inevitably > be flawed. That's the 'bad' part of it I suppose. It also leaves me > always beginning my learning rather than trying to end it, and open to > being surprised. Yes! You've pointed out an extremely important part of the human condition (and science, as the search for truth). Stated preference is a symptom of the historical accretion of (often accidental) experiences each of us goes through. And so preference is a very important indicator that compresses (a lossy one) lots of information about a person's history into a digestible chunk. The same can be said of a person's actions. When presented with a situation and a suite of possible actions, those actions chosen by the subject are indicators of that person's historical accretion of experiences. Personally, I prefer to flip back and forth amongst various different points of view. I arrogantly think that I can do this purposefully; but, it's probably more accidental than anything else. If I can do it purposefully at all, it's probably more guided by intuition or instinct than anything conscious. Reductionism is useful in many situations. Concepts like "emergence" are useful in many situations. Internal loci of control are sometimes useful, likewise with external loci. Sometimes it's handy for me to think that I created the universe to entertain myself. [grin] And sometimes it's useful for me to think I'm an insignificant spec that can be faithfully modeled as an ideal gas molecule. Such flip-flopping often leads others to think I contradict myself. In reality, it is logically impossible for a person to contradict themselves because all their actions (including statements of preference) flow from their historically accreted experience. The perceived contradictions come because partial models (usually ideal or abstracted) based on one subset of actions often contradict partial models based on some other subset of actions. It's also important to remember that evidence taken via self-reporting is highly suspect (and usually misleading or flawed). So statements of preference are not to be trusted! Hence, though I may _say_ my preference is to flip-flop, it's probably not true. [grin] - -- glen e. p. ropella, 971-219-3846, http://tempusdictum.com Men who are unhappy, like men who sleep badly, are always proud of the fact. -- Bertrand Russell -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.6 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org iD8DBQFGeUnKZeB+vOTnLkoRAityAJ9RrKtQrNHD3kZ2FNd1hdtx1wN53wCfTF5X MOLPElbaRRFOGGDPa3tzhTQ= =Zbmx -----END PGP SIGNATURE----- |
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In reply to this post by Russell Standish
On Jun 20, 2007, at 6:53 AM, Russell Standish wrote: > > > BTW, what exactly do you mean by nonlinear, if not in the sense I > suggested? > As described in past posts, that's exactly what I'm trying to figure out--formal math definition doesn't help, metaphorical use too vague. Whatever the solution is, it's likely to be propositional/schematic rather than numeric and involve observer perspective/background knowledge. I'll write more to the list when I think I'm onto a solution. |
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In reply to this post by Owen Densmore
Glad cybernetics is 'back on the list' even if it's absense was only one
of being of more historic rather than of current interest. I do think the general phenomenon I pointed to is real, that each systems theory discipline has tried to write it's own self-sufficient whole theory rather than work by cross fertilizing. It's a little like the reason the popular press fails to tend toward concensus on things like global warming. The popular press is an entertainment medium with each contributor trying to distance themselves from all others to be entertaining, and does not run out of entertaining alternate points of view until they're exhausted, and so seeks confusion rather than self-critical consensus. I think if we looked carefully at cybernetics we'd find a very key, and very obvious fundamental principle of control entirely missing, that probably would have been found long ago if all the different systems theories didn't seem to focus on differentiating themselves from each other rather than building from each other. Phil Henshaw ????.?? ? `?.???? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: pfh at synapse9.com explorations: www.synapse9.com > -----Original Message----- > From: Owen Densmore [mailto:owen at backspaces.net] > Sent: Wednesday, June 20, 2007 12:15 AM > To: sy at synapse9.com; The Friday Morning Applied Complexity > Coffee Group > Subject: Re: [FRIAM] Seminal Papers in Complexity > > > Interesting .. when following these links I end up on one of my > favorite sites: > http://cscs.umich.edu/~crshalizi/notebooks/ashby.html > > Cosma Shalizi is one of the more interesting mavericks of the SFI > world, and taught at the SFI 2000 summer school. Definitely one of > my favorite reads. > > -- Owen > > > On Jun 16, 2007, at 9:35 AM, Phil Henshaw wrote: > > > One problem with the seminal papers on complexity is that > they don't > > connect. Take the foundational works of H.T. Odum, the systems > > ecologist(1) or the cybernetic systems thinkers Ross Ashby (2) or > > Norbert Wiener(3). It's hard to link them to other branches of > > complex > > systems study like Prigigene's 'Exploring Complexity' or Wolfram's > > 'New > > kind of Science' or Barabasi's 'Linked' (leaving out numerous > > important > > others). As a consequence few people are aware of the general > > timeline > > of complexity as a subject(4), and any timeline of the field is > > bound to > > be missing major contributions. > > > > The problem seems is partly that the study of complex systems is > > interdisciplinary, because systems are, and what happens is each > > discipline goes off on its own tangent and acts like it is > trying to > > take over the subject as a whole, each vying to erase each other > > rather > > than connect with each other. My work seems to be an example of an > > attempt to link approaches, a new form of physics intended > > expressly for > > use by any discipline, and incorporating unique useful pieces of > > what's > > been developed from all the disciplines I've been exposed > to. My work > > may be 'odd' in more ways than that, but it's partly because I'm > > trying > > to write in a common language that makes it look 'foreign' to every > > discipline, so no one'll publish it... Catch 22! :-) > > > > (1) Odum: 1994 'Ecological and General Systems' (see > > http://www.eoearth.org/article/Odum,_Howard_T.) > > (2) Ross Ashby's 1947 'Ecological and General Systems' or his 1956 > > "Introduction to Cybernetics" (& see > > http://en.wikipedia.org/wiki/W._Ross_Ashby) > > (3) Weiner 1948 'Control and Communication in the Animal and the > > Machine' > > (3) complex systems thinking timeline from the cybernetics soc. > > (http://www.asc-cybernetics.org/foundations/timeline.htm), > > > > > > Phil Henshaw ????.?? ? `?.???? > > ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > > 680 Ft. Washington Ave > > NY NY 10040 > > tel: 212-795-4844 > > e-mail: pfh at synapse9.com > > explorations: www.synapse9.com > > > > > >> -----Original Message----- > >> From: friam-bounces at redfish.com > [mailto:friam-bounces at redfish.com] On > >> Behalf Of Owen Densmore > >> Sent: Friday, June 15, 2007 7:38 PM > >> To: The Friday Morning Applied Complexity Coffee Group > >> Subject: [FRIAM] Seminal Papers in Complexity > >> > >> > >> Several of us have been attending the SFI Summer School this year. > >> One thing that has stood out for me is that there are very few > >> appropriate texts on the detailed, seminal ideas within > complexity. > >> Either the books are "popular" or they are > technical/formal enough, > >> but without broad view of complexity itself. Indeed, they may be > >> *too* advanced in their speciality for the broad use complexity > >> wishes to make. > >> > >> One example today was the intersection of computational theory and > >> statistical mechanics given by Cris Moore: > >> A Tale of Two Cultures: Phase Transitions in > >> Physics and Computer Science > >> Here are the slides: http://www.santafe.edu/~moore/Oxford.pdf > >> You'd be unlikely to find a book bridging algorithms, > computational > >> complexity, and statistical mechanics. > >> > >> This leads me to believe that seminal papers are likely to > be a good > >> solution for bridging the various cultures, hopefully with > some that > >> *do* bridge gaps between specialties. > >> > >> Sooo -- gentle reader -- this brings me to a request: I'd like to > >> start a collection of seminal papers who's goal is to > bridge the gap > >> between popular books and over-specialized texts, which are formal > >> enough to be useful for multi-discipline complexity work. > This may > >> be daft, but I think not. > >> > >> As an example, I'd say Shannon's 1948 paper A Mathematical > Theory of > >> Communication would be good. > >> > >> -- Owen > >> > >> > >> > >> ============================================================ > >> 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 Michael Agar
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Hash: SHA1 Michael Agar wrote: > As described in past posts, that's exactly what I'm trying to figure > out--formal math definition doesn't help, metaphorical use too vague. > Whatever the solution is, it's likely to be propositional/schematic > rather than numeric and involve observer perspective/background > knowledge. I'll write more to the list when I think I'm onto a solution. Formal math definitions do help. You just can't be myopic about it and restrict yourself to arithmetic. Open it up to higher math. It seems you want to generalize linearity to apply to _other_ composition functions. The typical definition of linearity applies only to addition, i.e. f(x+y) != f(x) + f(y). If you abstract up just a bit, linearity means "on the same line", which is a way of saying "in the same space" where the space is 1 dimensional. It's simply a closure under addition. But, there's no reason you couldn't define the same _type_ of thing with other composition operators. All you need to do to have an unambiguous definition of what you mean by "linearity" is to a) define the composition operator you're talking about and b) define the closure of that operator. Of course there are plenty of such constructs already, they just aren't referred to with the word "linearity". For example, there's a thing called an "affine plane": a set, P, of points, together with a set, L, of subsets of P called lines. The points and lines must satisfy the following incidence axioms. i) any two distinct points lie on exactly one line. ii) for each line l and point x not on l, there exists a unique line m containing x and not meeting l. iii) there exist 3 points not lying on a line. Basically, this is a topological construct wherein lines don't cross or jump out of the plane. It's a kind of closure in that sense. A composition where a line _did_ cross another or jump out of the plane would be non-affine-planar (similar to non-linear). - -- glen e. p. ropella, 971-219-3846, http://tempusdictum.com A day an hour of virtuous liberty is worth a whole eternity of bondage. - -- Cato -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.6 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org iD8DBQFGfAgQZeB+vOTnLkoRAvIVAKCbXSSEjvIwT3Ik6CXkXfigmGnuPQCgzKUW Nb8OAIqejb47DJdojESBYHA= =48OG -----END PGP SIGNATURE----- |
another idea for a generalized "nonlinearity" (was Re: Seminal Papers in Complexity)
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Hash: SHA1 I just realized there's another general sense of "linearity" that some non-mathematical descriptions target, that of "balance". The idea is that a system shows some sort of balance where no one component contributes more than any other component. Simple examples would be adding a nonlinear term to a previously linear equation: 1) z = a*x + b*y, changed to 2) z = a*x^2 + b*y Technically, (2) is linear because f(x,y) = f(x) + f(y) (note that just because the sets described are not planes doesn't mean the function is nonlinear). It is still describable as linear because one can cleanly separate out the co-domain (by definition) into X and Y. I.e. in the characterization of the co-domain, X and Y contribute equally, any point in that product space is fair game. But, if we were to bias it in some way, let's say we define functions as going from the positive reals (R+) crossed with the reals (f : R+ x R -> R). Then that may touch on someone's intuition of what "nonlinear" means. That sort of concept is captured in linear algebra by the concept of a "balanced set". E.g. R+ x R is not balanced because R+ is not balanced. The set described by (2) above is not balanced where (1) above _is_ balanced, even though both are linear functions. Of course, in order for one to have a sense of balance, one has to have a fulcrum about which to balance. And sometimes its useful to describe spaces that don't have such fulcrums (as in the affine plane described previously). So the linear algebra "balanced set" doesn't generalize very well, especially to vague descriptions of spaces and mappings between them. Glen E. P. Ropella wrote: > But, there's no reason you couldn't define the same _type_ of thing with > other composition operators. All you need to do to have an unambiguous > definition of what you mean by "linearity" is to a) define the > composition operator you're talking about and b) define the closure of > that operator. Of course there are plenty of such constructs already, > they just aren't referred to with the word "linearity". ============================================================ 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 - -- glen e. p. ropella, 971-219-3846, http://tempusdictum.com I have an existential map. It has 'You are here' written all over it. -- Steven Wright -----BEGIN PGP SIGNATURE----- Version: GnuPG v1.4.6 (GNU/Linux) Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org iD8DBQFGfBywZeB+vOTnLkoRAnM1AKDdMkLIf3LNW9pnhVA1M6wcoMQPMQCdERKI UthB//12Jk4flYLe0c+PJhU= =1Gja -----END PGP SIGNATURE----- |
Seminal Papers in Complexity
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In reply to this post by glen ep ropella
On Fri, Jun 22, 2007 at 10:34:09AM -0700, Glen E. P. Ropella wrote:
> -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > Michael Agar wrote: > > As described in past posts, that's exactly what I'm trying to figure > > out--formal math definition doesn't help, metaphorical use too vague. > > Whatever the solution is, it's likely to be propositional/schematic > > rather than numeric and involve observer perspective/background > > knowledge. I'll write more to the list when I think I'm onto a solution. > > Formal math definitions do help. You just can't be myopic about it and > restrict yourself to arithmetic. Open it up to higher math. > > It seems you want to generalize linearity to apply to _other_ > composition functions. The typical definition of linearity applies only > to addition, i.e. f(x+y) != f(x) + f(y). If you abstract up just a bit, > linearity means "on the same line", which is a way of saying "in the > same space" where the space is 1 dimensional. It's simply a closure > under addition. Not just addition, but also scalar multiplication by a member of a field. For any group G, one can consider the class of functions f:G->G satisfying f(x+y)=f(x)+f(y). This induces a linear-like property over N x G, ie for all a, b in N and for all x and y in G, f(ax+by) = af(x)+bf(y) where ax = \sum_i=0^a x However such objects are not linear functions, and don't appear to have a name. Perhaps they're not all that useful. -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpcoder at hpcoders.com.au Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- |
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In reply to this post by Michael Agar
Mike,
Non-linear does not need to be defined with equations. Non-linear can be a process having continuity. Any process that begins and ends with continuity (i.e. w/o discontinuity) is inherently non-linear because it requires finite periods of that have all derivatives all of the same sign. It 'only' requires is developing a calculus for physical system rates. Search for 'continuity' on my site for some things. It's a new non-linear kind of math. Phil Henshaw ????.?? ? `?.???? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: pfh at synapse9.com explorations: www.synapse9.com <http://www.synapse9.com/> -----Original Message----- From: [hidden email] [mailto:[hidden email]] On Behalf Of Michael Agar Sent: Tuesday, June 19, 2007 10:41 AM To: The Friday Morning Applied Complexity Coffee Group Subject: Re: [FRIAM] Seminal Papers in Complexity This thread is sliding around some, but still I?d like to add this overlong comment in case it?s useful. The emails have been good brain food. The problem I keep worrying about in my own work is, I use many core concepts metaphorically because they work at the human organizational scale in powerful and useful ways that I believe respect their scientific origins but at the same time allow the human/social world to see and understand and act differently. But I also want to be clear on those origins, to know and describe when and where and how I?m stretching the concepts. The problem I have is, up close the conceptual basis of ?complexity? more often than not turns to mush. Mea culpa much of the time, I?m sure, but look what happened to reductionism in this thread. Even Wikipedia has several entries. I don?t know how much credence to give them, but here they are: 0. <http://en.wikipedia.org/wiki/Reductionism#Varieties_of_reductionism> 1 Varieties of reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Ontological_reductionism> 1.1 Ontological reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Methodological_reductionism> 1.2 Methodological reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Methodological_individualism> 1.3 Methodological individualism 0. <http://en.wikipedia.org/wiki/Reductionism#Theoretical_reductionism> 1.4 Theoretical reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Scientific_reductionism> 1.5 Scientific reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Set-Theoretic_Reductionism> 1.6 Set-Theoretic Reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Linguistic_reductionism> 1.7 Linguistic reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Greedy_reductionism> 1.8 Greedy reductionism 0. <http://en.wikipedia.org/wiki/Reductionism#Eliminativism> 1.9 Eliminativism And now emergence. I?ve heard it used in several ways. Way back when, we used it in anthropology as a form of methodological defense against the usual social science model of everything planned in a modular way before the research started. Emergence was shorthand for ?I can?t tell you what I?m going to do until I get there and learn what?s worth learning and how to learn it.? Then it?s also used more generally as shorthand for ?surprise,? the presence and nature of which depends on perspective and prior knowledge of observer. Then it?s used for the end result of a deterministic process that has characteristics unlike the elements of that process, like water out of hydrogen and oxygen. Then it?s used for the need for different concepts and methods for different levels of a phenomenon, like phonology, morphology and syntax in linguistics. Then it?s used for unexpected evolutionary and historical transitions, like the Cambrian explosion. Probably many other uses if we sampled a lot of texts and conversations. Probably some of the sources cited already in the thread help with the problem. I need to read them. Maybe the field has outgrown the concepts that got it started. If true, that?s probably a good sign. So I think I?ll work on nonlinearity for awhile. Russell writes: ?most of my readers understand perfectly well what a linear function is: one that obeys f(a*x+b*y) = a*f(x)+b*f(y).? That?s clear, resembles the definition in the Wikipedia entry. But then he writes : ?If neither * or + are defined for your objects of discussion, you cannot talk about (non-)linearity.? That won?t do. I have to be able to talk about nonlinear effects of, say, mental health policy on local programs in a qualitative way. I know it makes sense to do so from experience. Problem is to make it clear what the term means in that context. If the math won?t do it, something else has to. I?ll puzzle over the NECSI definition and the opening pages of Strogatz? book for awhile. So maybe nonlinearity won?t be so easy either. There?s the famous Einstein quote for inspiration: As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. Maybe we need a new nonlinear kind of math. Maybe it exists. Enough already. Mike -------------- next part -------------- An HTML attachment was scrubbed... URL: http://redfish.com/pipermail/friam_redfish.com/attachments/20070623/ca55e5fe/attachment.html |
Seminal Papers in Complexity
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On Sat, Jun 23, 2007 at 12:03:42AM -0400, Phil Henshaw wrote:
> Mike, > Non-linear does not need to be defined with equations. Non-linear can > be a process having continuity. Any process that begins and ends with > continuity (i.e. w/o discontinuity) is inherently non-linear because it > requires finite periods of that have all derivatives all of the same > sign. It 'only' requires is developing a calculus for physical system > rates. Search for 'continuity' on my site for some things. It's a > new non-linear kind of math. > > No - wrong again. Linear processes are also continuous. You really need to learn some maths. Cheers ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpcoder at hpcoders.com.au Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- |
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In reply to this post by glen ep ropella
Glen,
Well, the 'fault' of considering things from multiple points of view is not contradiction, but confusing all those who don't! Speaking of your observation that "Internal loci of control are sometimes useful" wouldn't it be wise for us to switch the exponential growth of exploiting the earth to refining our uses of it before it's too late? I've been noticing mammoth errors seeming to stem from people using concepts of change that are several centuries out of date with respect to the changes in our real environment. Did you know that virtually all 'sustainable design' advocates thorough ally believe that continually doubling output with a 10%-20% reduction in waste reduces impacts on the earth if you just do it for the right purpose? It's stunning! They actually think what they're 'talking about' is what they're 'literally saying' when the two are whole worlds apart. On whether this confusion we all experience between information and action is robust or not, I certainly accept and observe some inconsistency, but think most people remain hamstrung by generally not knowing when. The simple case in point is how easily and confidently we understand some things going out of control, as with a singer's voice cracking, or a businessman not getting expert help until it's too late, or a party or a friendship erupting in thrill that changes to tragedy, and see nothing wrong at all with the speed, complexity and magnitude of unknown impacts of decision-making about our permanent life support system, doubling, regularly, forever. It's the inconsistency! All forms of excess look to be much the same problem to me, and can be read with the same metrics. It shouldn't be a tough problem, well except for confusing information and action. just an idle thought, of course... :,) Phil Henshaw ????.?? ? `?.???? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: pfh at synapse9.com explorations: www.synapse9.com > -----Original Message----- > From: friam-bounces at redfish.com > [mailto:friam-bounces at redfish.com] On Behalf Of Glen E. P. Ropella > Sent: Wednesday, June 20, 2007 11:38 AM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] reductionism > > > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > Phil Henshaw wrote: > > I think that's a very consistent argument, and very similar > to the one > > Bohr used as the basis for the Copenhagen convention and dumping > > Einstein's idea of the physical world. As I recall, the > argument was > > that science is information and so nothing exists for > science except > > what exists as scientific information, and so uncertainties that > > define a limit to scientific knowledge also define a limit to any > > meaningful scientific reality. So as scientists, reality does not > > exist beyond what is knowable. I think it was that slim logical > > thread that kept the otherwise very unsatisfying assertion that > > phenomena are created by our observations from being tossed out as > > ridiculous. > > Interesting. Your paraphrase certainly seems analogous to > what I submitted if information and action are taken as > analogous fulcra. But, I'm not sure the analogy is very > robust. My argument hinges on action and unity (discretion) > where "things" are only things, separate from all the other > goo in which we're bathed, by virtue of their being acted > upon or acting upon as a unit. > > Hence, an "emergent thing" is fictitious if it cannot be > acted upon separately from the other things to which it's > related, including its constituents. For example, an > "emergent phenomenon", in order to be a real thing, would > have to either act as a unit or be acted upon as a unit. (By > "act", I don't mean "ascribed thing-hood by an observer". I > mean "does physical work"... moves objects around, generates > heat, etc.) > > Now, personally, I tend to agree with G?nther and I don't > rely on the word "emergence" for conversations I regard as > important. I have yet to see an emergent phenomenon.... and > I doubt that I ever will. Of course, that doesn't mean they > don't exist. It also doesn't mean that the concept is > useless. Unicorns may not exist either; but, it is important > that we can think about them and perform thought experiments > with them. > > Given these details (especially the descent into discretion > begged by the action requirement), it's hard for me to > maintain the analogy between my argument and information as > scientific currency. > > > For most people the question comes down to which way they *like* > > thinking about the world, since either one can be made > satisfying if > > that's what you like... I prefer, and find more > productive, thinking > > that I'm exploring a world that exists without my knowledge > of it, and > > is built in such a complicated way that my descriptions > will inevitably > > be flawed. That's the 'bad' part of it I suppose. It > also leaves me > > always beginning my learning rather than trying to end it, > and open to > > being surprised. > > Yes! You've pointed out an extremely important part of the > human condition (and science, as the search for truth). > Stated preference is a symptom of the historical accretion of > (often accidental) experiences each of us goes through. And > so preference is a very important indicator that compresses > (a lossy one) lots of information about a person's history > into a digestible chunk. The same can be said of a person's > actions. When presented with a situation and a suite of > possible actions, those actions chosen by the subject are > indicators of that person's historical accretion of experiences. > > Personally, I prefer to flip back and forth amongst various > different points of view. I arrogantly think that I can do > this purposefully; but, it's probably more accidental than > anything else. If I can do it purposefully at all, it's > probably more guided by intuition or instinct than anything > conscious. Reductionism is useful in many situations. > Concepts like "emergence" are useful in many situations. > Internal loci of control are sometimes useful, likewise with > external loci. Sometimes it's handy for me to think that I > created the universe to entertain myself. [grin] And > sometimes it's useful for me to think I'm an insignificant > spec that can be faithfully modeled as an ideal gas molecule. > Such flip-flopping often leads others to think I contradict > myself. In reality, it is logically impossible for a person > to contradict themselves because all their actions (including > statements of > preference) flow from their historically accreted experience. > The perceived contradictions come because partial models > (usually ideal or > abstracted) based on one subset of actions often contradict > partial models based on some other subset of actions. > > It's also important to remember that evidence taken via > self-reporting is highly suspect (and usually misleading or > flawed). So statements of preference are not to be trusted! > Hence, though I may _say_ my preference is to flip-flop, it's > probably not true. [grin] > > - -- > glen e. p. ropella, 971-219-3846, http://tempusdictum.com > Men who are unhappy, like men who sleep badly, are always > proud of the fact. -- Bertrand Russell > > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1.4.6 (GNU/Linux) > Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org > > iD8DBQFGeUnKZeB+vOTnLkoRAityAJ9RrKtQrNHD3kZ2FNd1hdtx1wN53wCfTF5X > MOLPElbaRRFOGGDPa3tzhTQ= > =Zbmx > -----END PGP SIGNATURE----- > > ============================================================ > 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 > > |
another idea for a generalized "nonlinearity" (was Re: Seminal Papers in Complexity)
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In reply to this post by glen ep ropella
Just to rephrase, there's a great way to reapply all the basic theorems of calculus directly to real physical processes (skipping the interceding equations). Use data curves with an appropriate rule for determining a value and slope at any point by iteration. Works great and provides a crystal clear identification of the emergent non-linear phases of real processes. Like anything, you'd expect many questions, and slow beginning, then big strides. One of the hurdles is the software... As powerful as they are I hate R, and Excel, and AutoCad, though I have nothing else to use... Phil Henshaw ????.?? ? `?.???? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: pfh at synapse9.com explorations: www.synapse9.com > -----Original Message----- > From: friam-bounces at redfish.com > [mailto:friam-bounces at redfish.com] On Behalf Of Glen E. P. Ropella > Sent: Friday, June 22, 2007 3:02 PM > To: The Friday Morning Applied Complexity Coffee Group > Subject: [FRIAM] another idea for a generalized > "nonlinearity" (was Re: Seminal Papers in Complexity) > > > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > > I just realized there's another general sense of "linearity" > that some non-mathematical descriptions target, that of > "balance". The idea is that a system shows some sort of > balance where no one component contributes more than any > other component. Simple examples would be adding a nonlinear > term to a previously linear equation: > > 1) z = a*x + b*y, changed to > 2) z = a*x^2 + b*y > > Technically, (2) is linear because f(x,y) = f(x) + f(y) (note > that just because the sets described are not planes doesn't > mean the function is nonlinear). It is still describable as > linear because one can cleanly separate out the co-domain (by > definition) into X and Y. I.e. in the characterization of > the co-domain, X and Y contribute equally, any point in that > product space is fair game. > > But, if we were to bias it in some way, let's say we define > functions as going from the positive reals (R+) crossed with > the reals (f : R+ x R -> R). Then that may touch on > someone's intuition of what "nonlinear" means. > > That sort of concept is captured in linear algebra by the > concept of a "balanced set". E.g. R+ x R is not balanced > because R+ is not balanced. The set described by (2) above > is not balanced where (1) above _is_ balanced, even though > both are linear functions. Of course, in order for one to > have a sense of balance, one has to have a fulcrum about > which to balance. And sometimes its useful to describe > spaces that don't have such fulcrums (as in the affine plane > described previously). So the linear algebra "balanced set" > doesn't generalize very well, especially to vague > descriptions of spaces and mappings between them. > > Glen E. P. Ropella wrote: > > But, there's no reason you couldn't define the same _type_ of thing > > with other composition operators. All you need to do to have an > > unambiguous definition of what you mean by "linearity" is > to a) define > > the composition operator you're talking about and b) define the > > closure of that operator. Of course there are plenty of such > > constructs already, they just aren't referred to with the word > > "linearity". > > ============================================================ > 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 > > > > - -- > glen e. p. ropella, 971-219-3846, http://tempusdictum.com > I have an existential map. It has 'You are here' written all > over it. -- Steven Wright > > -----BEGIN PGP SIGNATURE----- > Version: GnuPG v1.4.6 (GNU/Linux) > Comment: Using GnuPG with Mozilla - http://enigmail.mozdev.org > > iD8DBQFGfBywZeB+vOTnLkoRAnM1AKDdMkLIf3LNW9pnhVA1M6wcoMQPMQCdERKI > UthB//12Jk4flYLe0c+PJhU= > =1Gja > -----END PGP SIGNATURE----- > > ============================================================ > 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 > > |
another idea for a generalized "nonlinearity" (was Re: Seminal Papers in Complexity)
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On Sat, Jun 23, 2007 at 02:09:31AM -0400, Phil Henshaw wrote:
> One of the hurdles is the software... As powerful as they > are I hate R, and Excel, and AutoCad, though I have nothing else to > use... > Why are you restricted to these packages? Are they just what you know, and you're not prepared to learn anything else, or does your company madate you use just these and nothing else, or is your computing platform just so weird nothing else will run on it. Can't be cost - there are just so many great open source packages for doing just about anything. Cheers -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpcoder at hpcoders.com.au Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- |
another idea for a generalized "nonlinearity" (was Re: Seminal Papers in Complexity)
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Hating R: is like hating the air you breathe.
On Jun 22, 2007, at 1:57 PM, Russell Standish wrote: > On Sat, Jun 23, 2007 at 02:09:31AM -0400, Phil Henshaw wrote: >> One of the hurdles is the software... As powerful as they >> are I hate R, and Excel, and AutoCad, though I have nothing else to >> use... >> > > Why are you restricted to these packages? Are they just what you know, > and you're not prepared to learn anything else, or does your company > madate you use just these and nothing else, or is your computing > platform just so weird nothing else will run on it. > > Can't be cost - there are just so many great open source packages for > doing just about anything. > > Cheers > > -- > > ---------------------------------------------------------------------- > ------ > A/Prof Russell Standish Phone 0425 253119 (mobile) > Mathematics > UNSW SYDNEY 2052 hpcoder at hpcoders.com.au > Australia http://www.hpcoders.com.au > ---------------------------------------------------------------------- > ------ > > ============================================================ > 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 Marko A. Rodriguez Los Alamos National Laboratory (P362-proto) Los Alamos, NM 87545 Phone +1 505 606 1691 http://www.soe.ucsc.edu/~okram -------------- next part -------------- An HTML attachment was scrubbed... URL: http://redfish.com/pipermail/friam_redfish.com/attachments/20070623/4362d774/attachment.html |
Seminal Papers in Complexity
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In reply to this post by Russell Standish
Exactly! Linear functions (or operators) can be broken into smaller,
summable pieces where all terms can be scaled simultaneously. You can factor common things out and operate on parts in a piecemeal manner; e.g. the Taylor or Fourier Series. When you get into non-linear stuff, you lose this fantastic mathematical tool of manipulation and simplification; i.e. factoring out (or distributing in) something common over a set of parts. Linear equations often give you the ability to look at one part of the equation in isolation from the rest; sort of like a spectral series. It's easier to understand because you don't have to look at everything at once. That's why non-linear functions (like some integral equations) are more often than not, very difficult to analyze, and often impossible to solve analytically. Our brains must think linearly. One can extend an imaginary line to any distance with perfect accuracy. But curve it in a non-trivial way, and the accuracy quickly attenuates. Robert Howard Phoenix, Arizona -----Original Message----- From: [hidden email] [mailto:[hidden email]] On Behalf Of Russell Standish Sent: Friday, June 22, 2007 12:52 PM To: The Friday Morning Applied Complexity Coffee Group Subject: Re: [FRIAM] Seminal Papers in Complexity On Fri, Jun 22, 2007 at 10:34:09AM -0700, Glen E. P. Ropella wrote: > -----BEGIN PGP SIGNED MESSAGE----- > Hash: SHA1 > > Michael Agar wrote: > > As described in past posts, that's exactly what I'm trying to figure > > out--formal math definition doesn't help, metaphorical use too vague. > > Whatever the solution is, it's likely to be propositional/schematic > > rather than numeric and involve observer perspective/background > > knowledge. I'll write more to the list when I think I'm onto a solution. > > Formal math definitions do help. You just can't be myopic about it and > restrict yourself to arithmetic. Open it up to higher math. > > It seems you want to generalize linearity to apply to _other_ > composition functions. The typical definition of linearity applies only > to addition, i.e. f(x+y) != f(x) + f(y). If you abstract up just a bit, > linearity means "on the same line", which is a way of saying "in the > same space" where the space is 1 dimensional. It's simply a closure > under addition. Not just addition, but also scalar multiplication by a member of a field. For any group G, one can consider the class of functions f:G->G satisfying f(x+y)=f(x)+f(y). This induces a linear-like property over N x G, ie for all a, b in N and for all x and y in G, f(ax+by) = af(x)+bf(y) where ax = \sum_i=0^a x However such objects are not linear functions, and don't appear to have a name. Perhaps they're not all that useful. -- ---------------------------------------------------------------------------- A/Prof Russell Standish Phone 0425 253119 (mobile) Mathematics UNSW SYDNEY 2052 hpcoder at hpcoders.com.au Australia http://www.hpcoders.com.au ---------------------------------------------------------------------------- ============================================================ 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 -------------- next part -------------- An HTML attachment was scrubbed... URL: http://redfish.com/pipermail/friam_redfish.com/attachments/20070625/fa4a2f06/attachment.html |
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In reply to this post by Russell Standish
Na, I think even the most sophisticated math misses all the truly supple
shape of natural form, and it it's of huge signifiance in our missunderstanding of natural phenomena. Phil Henshaw ????.?? ? `?.???? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: pfh at synapse9.com explorations: www.synapse9.com > -----Original Message----- > From: friam-bounces at redfish.com > [mailto:friam-bounces at redfish.com] On Behalf Of Russell Standish > Sent: Friday, June 22, 2007 3:52 PM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] Seminal Papers in Complexity > > > On Fri, Jun 22, 2007 at 10:34:09AM -0700, Glen E. P. Ropella wrote: > > -----BEGIN PGP SIGNED MESSAGE----- > > Hash: SHA1 > > > > Michael Agar wrote: > > > As described in past posts, that's exactly what I'm > trying to figure > > > out--formal math definition doesn't help, metaphorical > use too vague. > > > Whatever the solution is, it's likely to be > propositional/schematic > > > rather than numeric and involve observer perspective/background > > > knowledge. I'll write more to the list when I think I'm > onto a solution. > > > > Formal math definitions do help. You just can't be myopic about it > > and restrict yourself to arithmetic. Open it up to higher math. > > > > It seems you want to generalize linearity to apply to _other_ > > composition functions. The typical definition of linearity applies > > only to addition, i.e. f(x+y) != f(x) + f(y). If you > abstract up just > > a bit, linearity means "on the same line", which is a way of saying > > "in the same space" where the space is 1 dimensional. It's > simply a > > closure under addition. > > Not just addition, but also scalar multiplication by a member > of a field. > > For any group G, one can consider the class of functions > f:G->G satisfying f(x+y)=f(x)+f(y). This induces a > linear-like property over N x G, ie for all a, b in N and for > all x and y in G, > > f(ax+by) = af(x)+bf(y) > > where ax = \sum_i=0^a x > > However such objects are not linear functions, and don't > appear to have a name. Perhaps they're not all that useful. > > > -- > > -------------------------------------------------------------- > -------------- > A/Prof Russell Standish Phone 0425 253119 (mobile) > Mathematics > UNSW SYDNEY 2052 hpcoder at hpcoders.com.au > Australia http://www.hpcoders.com.au > -------------------------------------------------------------- > -------------- > > ============================================================ > 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 Russell Standish
Is a curve more constrained than what it represents? Why?
Why I bother to ask will probably be given away from my example of a new Carbon Offsets board game done by a Princeton environmental education program and sponsored by AAAS (as heard on NPR). It's really a wonderful educational tool, but with a surprise ending! The idea is that you need 7 carbon offset wedges (ramping up some choice of new technologies up over 50 years). Maybe you'd do two solar ones, three nukes, and two conservation ones, but their huge size, cost, impacts, and institutional & political implications, etc. makes them really tough choices. It's a simple model, and looks like a wonderful group process educational game. When I first described it to some folks I thought it was actually a board game, and that the idea was for the first move to lead to a second, but then I noticed that it's not quite like that. At the end of the very first had of play, time just sort of dribbles off all bye itself, whew.., gone, kaput, aaa..nd the-game-is-over! Hmmmm... from the shape of the triangle, can you predict the actual next hand in the game? http://www.aaas.org/news/press_room/climate_change/mtg_200702/wedge_conc ept_teacher_guide.pdf Phil Henshaw ????.?? ? `?.???? ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 680 Ft. Washington Ave NY NY 10040 tel: 212-795-4844 e-mail: pfh at synapse9.com explorations: www.synapse9.com -------------- next part -------------- A non-text attachment was scrubbed... Name: wdge_concept.jpg Type: image/jpeg Size: 70756 bytes Desc: not available Url : http://redfish.com/pipermail/friam_redfish.com/attachments/20070625/8c5c0b85/attachment.jpg |
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