invitation + introduction
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I'm fascinated by the abstracts of the papers mentioned on this topic. I feel I have a lot of work to do. The attached file should give an intuitive overview of what I have been doing. John ________________________________________ From: [hidden email] [[hidden email]] On Behalf Of Grant Holland [[hidden email]] Sent: Saturday, April 10, 2010 2:38 PM To: The Friday Morning Applied Complexity Coffee Group Subject: Re: [FRIAM] invitation + introduction Thanks, John. I'll take a look. Grant John Kennison wrote: 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: [hidden email]<mailto:[hidden email]> [[hidden email]<mailto:[hidden email]>] On Behalf Of Grant Holland [[hidden email]<mailto:[hidden email]>] 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: [hidden email]<mailto:[hidden email]> [[hidden email]<mailto:[hidden email]>] On Behalf Of Grant Holland [[hidden email]<mailto:[hidden email]>] 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: [hidden email]<mailto:[hidden email]><mailto:[hidden email]><mailto:[hidden email]> [[hidden email]<mailto:[hidden email]><mailto:[hidden email]><mailto:[hidden email]>] On Behalf Of Owen Densmore [[hidden email]<mailto:[hidden email]><mailto:[hidden email]><mailto:[hidden email]>] 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 <[hidden email]><mailto:[hidden email]><mailto:[hidden email]><mailto:[hidden email]> 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 ============================================================ 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 |
Dynamical Systems and Toposes.pdf (73K)
Re: invitation + introduction
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Re: invitation + introduction
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