Grant,
My papers are found at http://www.tac.mta.ca/tac/ in Vol 22 [2009] No 14 and Vol 16 [2006] No 17 and Vol 10 [2002] No 15. (There are also some other papers I wrote with Mike Barr and Bob Raphael and some on DE's but these are not on dynamical systems.) I'm working on the PDF paper. --John ________________________________________ From: friam-boun...@redfish.com [friam-boun...@redfish.com] On Behalf Of Grant Holland [grant.holland...@gmail.com] Sent: Saturday, April 10, 2010 10:46 AM To: The Friday Morning Applied Complexity Coffee Group Subject: Re: [FRIAM] invitation + introduction John, Sounds very pertinent, and applicable to my current research - which I call "Organic Complex Systems". Looking forward to your PDF. Pls send links to, or copies of, your 3 pubs if you will. Thanks, Grant John Kennison wrote: > Thanks, Grant and Owen, for the votes of confidence. Concerning complex > adaptive systems, I would have to define a CAS in such a way that it can be > interpreted in any topos --then see if we can analyze CAS's by working in a > topos. > > Currently I am working on finding cycles. The idea is that we have a system > which can be in different states. Let S be the "set of all states that the > system can be in". Let t:S to S be a "transition function" so that if the > system is now in state x, then, in the next time unit, it will be in state > t(x). I then look for cycles (such as t(a) = b, t(b) = c, t(c) = a, so that > t^3(a)=t(t(t(a)))=a --or, more generally, states x for which t^n(x)=x for > some n, where t^n(x) = t(t(t( t(x)))))) iterating t n times. Then I can > map the system in "the best possible way" into a topos where it becomes > cyclic, meaning that for every x there is some n with t^n(x)=x. So n would be > a whole number in the topos, but whole numbers can jump around and be 3 in > some places and 5 in other places, etc. > > Just exploring this set-up has occupied me since 2001, and I have published 3 > papers on it in the TAC (a web-based journal). > > I'll say more and put it in a PDF file, so I can arrows and exponents and > keep tabbing and spacing the way I intended it. > > ---John > > > > ________________________________________ > From: friam-boun...@redfish.com [friam-boun...@redfish.com] On Behalf Of > Grant Holland [grant.holland...@gmail.com] > Sent: Friday, April 09, 2010 5:29 PM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] invitation + introduction > > John, > > I love such clarity - as expressed in your explanation of category theory. My > reaction is "Oh, so THAT's what category theory is!" Thanks for taking the > time to explain. > > Grant > > John Kennison wrote: > > Owen > Thanks for asking the question. In my answer, below, I describe the technical > terminology impressionistically. If you want more precision, the Wikipedia > articles are usually pretty good at giving precise definitions, along with > some sense of the underlying ideas. > > Category theory claims to be a formalization of how mathematics actually > works. For example, consider the following mathematical structures, which > have been defined in the 19th and 20th centuries: > Groups = “sets with a notion of multiplication” > Rings = “sets with notions of both > multiplication and addition” > Linear Spaces = “sets in which vector operations can be defined” > Topological Spaces = “sets with a notion of limit” > Each structure has a corresponding notion of a structure-preserving function: > Group homomorphism = “function f for which f(xy) = f(x)f(y)” > Ring homomorphism = “function f for which f(xy)=f(x)f(y) and > f(x+y)=f(x)+f(y)” > Linear map = “function preserving operations such as scalar mult: > f(kv)=kf(v)” > Continuous function = “function f for which f(Lim x_n) = Lim(f(x_n)” > > A category consists of a class of objects, together with a notion of > “homomorphism” or “map” or “morphism” between these objects. The main > operation in a category is that morphisms compose (given a morphism from X to > Y and another from Y to Z, there is then a composite morphism from X to Z). > Examples of catgeories: > Objects = Groups; Morphisms = > Group Homomorphisms > Objects = Rings; Morphisms = > Ring Homomorphisms > Objects = Linear spaces; Morphisms = Linear > maps > Objects = Top’l spaces; Morphisms = Cont. > functions > Objects = Sets; Morphisms = > Functions > (The above examples are respectively called the categories of groups, of > rings, of linear spaces, of top’l spaces, and of sets.) > > The claimed advantages of using categories are: > (1) The important and natural questions that mathematicians ask are > categorical in nature –that is they depend not on operations such as group > multiplication, but strictly on how the morphisms compose. (that is, the > objects are like black boxes, we don't see the limits or multiplication > inside the box, we only see arrows, representing morphisms going from one box > to another.) > (2) Looking at a subject from a category-theoretic point of view sheds > light on what is really happening and suggests new research areas. > (3) Proving a theorem about an arbitrary category can have applications > to all of the traditional categories mentioned above. > (4) As would be expected, there are suitable mappings between categories, > called functors, which enable us to compare and relate different parts of > mathematics. > I work in topos theory which ambitiously proposes to study where logic comes > from. We start by noting that many ideas in logic are closely tied to the > category of sets. > For example the sentence “x > 3” is true for some values of x and not > for others (if we assume, for example, that x is a real number) The compound > sentence “x > 3 and 3x = 12” is true on the intersection of the set where the > x > 3 with the set where 3x = 12. > On the other hand, “x >7 or x < 1” in true on a union. Of course “x > not equal to 3” is true on the complement of where “x = 3”. > Much of our assumptions about how the logical connectives “and”, > “or”, “not” are closely connected to how intersections, unions and > complements work in sets. But intersections, unions and (weak) complements > can be defined in categorical terms and then they may behave differently (for > example, categories need not obey the “law” of the excluded middle). A topos > is a category that resembles the category of Sets in some formal ways, but > which may lead to non-standard logics. One example of a topos can be thought > of as a category of sets in which the elements can change over time, such as > the set of all states in the US. Note that the element called Virginia splits > into 2 elements, West Virginia and Virginia, and, according to some views, > elements like Georgia were not in the set of US states during the Civil War. > The set of US states also has structure, such as the boundaries of the > states, which can change over time. > The advantage of uses toposes is that a traditional mathematical object can > be mapped, using a suitable functor, to a non-standard world (i.e. to a > related object in a topos) and this can reveal some of the inner structure of > the object. For example, an evolving system might be best viewed in a world > where elements can change over time. > > ________________________________________ > From: friam-boun...@redfish.com<mailto:friam-boun...@redfish.com> > [friam-boun...@redfish.com<mailto:friam-boun...@redfish.com>] On Behalf Of > Owen Densmore [o...@backspaces.net<mailto:o...@backspaces.net>] > Sent: Friday, April 09, 2010 12:50 PM > To: The Friday Morning Applied Complexity Coffee Group > Subject: Re: [FRIAM] invitation + introduction > > On Apr 7, 2010, at 12:10 PM, John Kennison > <jkenni...@clarku.edu><mailto:jkenni...@clarku.edu> wrote: > > > > Hi Leigh, > > <snip> > Nick introduced me to Rosen’s “Life Itself” and I have skimmed some articles > by Rosen. I am both fascinated and disappointed by Rosen’s work. Fascinated > by what Rosen says about the need to develop radically different kinds of > models to deal with biological phenomena and disappointed by Rosen’s > heavy-handed stabs at developing such models. And yet still stimulated > because I have enough ego to believe that with my mathematical and > category-theoretic background, I might succeed where Rosen failed. > > > > Category theory has been mentioned several times, especially in the early > days of friam. Could you help us out and discuss how it could be applied > here? CT certainly looks fascinating but thus far I've failed to grasp it. > I'd love a concrete example (like how to address Rosen's world) of it's use, > and possibly a good introduction (book, article). > > ---- Owen > > > I am an iPad, resistance is futile! > ============================================================ > 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 > > > > ============================================================ > 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 ============================================================ 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
