invitation + introduction

[Leigh Fanning]

First, the invitation:
On Thusday, the University of New Mexico Computer Science department will
hold it's annual student conference highlighting active research within
the department.  Dr. Melanie Mitchell will give the keynote address at
11:00 am.  

The conference is open with no admission fees, however we are not able to
provide you with lunch.  Proceedings hardcopy can be ordered for $10,
and will be available for free via download from the website shortly.
Details, and the keynote talk abstract, are below.

Next, the introduction:
By way of introduction, I am largely a FRIAM lurker, but have met a few
of you and in particular would like to further encourage Nick's suggestion
of a Robert Rosen reading group.  My PhD research area is molecular computing
and I am developing a formal system for reasoning about molecular computing
systems, specifically those composed of heterogeneous mixtures of DNA
oligonucleotides.  Milner's pi calculus, and Alur and Dill's timed automata
have been inspirational starting points.  Of course it's supremely simple
to find these inspirations, and attempt physics-style reductionist
techniques, in the engineering of synthetic biological systems.  However
one quickly determines that building even the simplest systems with a
biological basis must be done with a different approach.   The difficulty
in system calibration and readout, and the large number of tunable input
parameters, prevent breaking down molecular computing systems into neat
modules and demand study of how living systems execute their own
engineering and maintenance.    

My training, and I use this word with great trepidation following recent
discussion, is Engineering Physics, B.S. from CU-Boulder, Computer Science,
M.S. from UNM, and among other industry jobs, 7+ years doing Guidance,
Navigation and Control engineering for the Space Shuttle program in
the middle years when the fleet was "upgraded" to handle heavier weight
missions to the Space Station -- all old hat now and soon to retire, but
initially a load of interesting problems to work out.

Leigh Fanning

--------

The train between SF and ABQ works well, the bus system has a straight
shot up to the UNM campus from the depot.  Otherwise about an hour
of driving time is needed from Santa Fe, followed by some patience to work out
parking on campus.  There is a large parking garage by Popejoy hall
just off of Central Ave, or street parking just SW of campus can
work well sometimes.  If this works for your schedule, please come
and enjoy!

The schedule of talks is here:

http://cs.unm.edu/~csgsa/conference/

The location is the new Centennial Engineering building on the west
end of campus, bordering University Blvd, just north of Central Ave.

---

Melanie Mitchell, Portland State University and Santa Fe Institute                                                                                                                                                        
Thursday, 8 April, 2010
11 am - 12:00 pm
Centennial Engineering Center auditorium

Enabling computers to understand images remains one of the hardest
open problems in artificial intelligence.  No machine vision system
comes close to matching human ability at identifying the contents of
images or visual scenes or at recognizing similarity between different
scenes, even though such abilities pervade human cognition.  In this
talk I will describe research---currently in early stages---on
bridging the gap between low-level perception and higher-level image
understanding by integrating a cognitive model of perceptual
organization and analogy-making with a neural model of the visual
cortex.

Bio: Melanie Mitchell is Professor of Computer Science at Portland
State University and External Professor at the Santa Fe Institute.
She attended Brown University, where she majored in mathematics and
did research in astronomy, and the University of Michigan, where she
received a Ph.D. in computer science, working with her advisor Douglas
Hofstadter on the Copycat project, a computer program that makes
analogies.  She is the author or editor of five books and over 70
scholarly papers in in the fields of artificial intelligence,
cognitive science, and complex systems.  Her most recent book,
"Complexity: A Guided Tour", published in 2009 by Oxford University
Press, was named by Amazon.com as one of the ten best science books of
2009.

============================================================
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

[Grant Holland]

Leigh,

Great background! Looking forward to your reporting to us the many
interesting developments of your upcoming research activities.

Grant

Leigh Fanning wrote:

> First, the invitation:
> On Thusday, the University of New Mexico Computer Science department will
> hold it's annual student conference highlighting active research within
> the department.  Dr. Melanie Mitchell will give the keynote address at
> 11:00 am.  
>
> The conference is open with no admission fees, however we are not able to
> provide you with lunch.  Proceedings hardcopy can be ordered for $10,
> and will be available for free via download from the website shortly.
> Details, and the keynote talk abstract, are below.
>
> Next, the introduction:
> By way of introduction, I am largely a FRIAM lurker, but have met a few
> of you and in particular would like to further encourage Nick's suggestion
> of a Robert Rosen reading group.  My PhD research area is molecular computing
> and I am developing a formal system for reasoning about molecular computing
> systems, specifically those composed of heterogeneous mixtures of DNA
> oligonucleotides.  Milner's pi calculus, and Alur and Dill's timed automata
> have been inspirational starting points.  Of course it's supremely simple
> to find these inspirations, and attempt physics-style reductionist
> techniques, in the engineering of synthetic biological systems.  However
> one quickly determines that building even the simplest systems with a
> biological basis must be done with a different approach.   The difficulty
> in system calibration and readout, and the large number of tunable input
> parameters, prevent breaking down molecular computing systems into neat
> modules and demand study of how living systems execute their own
> engineering and maintenance.    
>
> My training, and I use this word with great trepidation following recent
> discussion, is Engineering Physics, B.S. from CU-Boulder, Computer Science,
> M.S. from UNM, and among other industry jobs, 7+ years doing Guidance,
> Navigation and Control engineering for the Space Shuttle program in
> the middle years when the fleet was "upgraded" to handle heavier weight
> missions to the Space Station -- all old hat now and soon to retire, but
> initially a load of interesting problems to work out.
>
> Leigh Fanning
>
> --------
>
> The train between SF and ABQ works well, the bus system has a straight
> shot up to the UNM campus from the depot.  Otherwise about an hour
> of driving time is needed from Santa Fe, followed by some patience to work out
> parking on campus.  There is a large parking garage by Popejoy hall
> just off of Central Ave, or street parking just SW of campus can
> work well sometimes.  If this works for your schedule, please come
> and enjoy!
>
> The schedule of talks is here:
>
> http://cs.unm.edu/~csgsa/conference/
>
> The location is the new Centennial Engineering building on the west
> end of campus, bordering University Blvd, just north of Central Ave.
>
> ---
>
> Melanie Mitchell, Portland State University and Santa Fe Institute                                                                                                                                                        
> Thursday, 8 April, 2010
> 11 am - 12:00 pm
> Centennial Engineering Center auditorium
>
> Enabling computers to understand images remains one of the hardest
> open problems in artificial intelligence.  No machine vision system
> comes close to matching human ability at identifying the contents of
> images or visual scenes or at recognizing similarity between different
> scenes, even though such abilities pervade human cognition.  In this
> talk I will describe research---currently in early stages---on
> bridging the gap between low-level perception and higher-level image
> understanding by integrating a cognitive model of perceptual
> organization and analogy-making with a neural model of the visual
> cortex.
>
> Bio: Melanie Mitchell is Professor of Computer Science at Portland
> State University and External Professor at the Santa Fe Institute.
> She attended Brown University, where she majored in mathematics and
> did research in astronomy, and the University of Michigan, where she
> received a Ph.D. in computer science, working with her advisor Douglas
> Hofstadter on the Copycat project, a computer program that makes
> analogies.  She is the author or editor of five books and over 70
> scholarly papers in in the fields of artificial intelligence,
> cognitive science, and complex systems.  Her most recent book,
> "Complexity: A Guided Tour", published in 2009 by Oxford University
> Press, was named by Amazon.com as one of the ten best science books of
> 2009.
>
> ============================================================
> 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

[John Kennison]

Hi Leigh,

I guess I’m a Friam lurker too. I’m a friend of Nick Thompson and a retired math professo. I like to read the Friam posts but I comment only occasionally. I’m currently working on dynamical systems and using category theory to break a system down into its cyclic components.

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.

For example, in “Life Itself” Rosen starts by talking about “Newtonian science” and the need to go beyond it, but then continues with a misunderstanding of Taylor’s theorem which, thankfully, is never really used in the rest of the book. Similarly, in some of his writings, Rosen talks about the insolubility of the three-body problem, about Godel’s theorem, about category theory, but never gets close to using any of this stuff.

Rosen’s definition of “component of a system” and his method of dealing with “non-recursiveness” are not just mathematically imprecise, they seem completely heavy-handed and insensitive to what the situation demands. In chapter 6, he gives a decomposition of a mathematical system into parts, but he claims the decomposition is unique and even gives a “proof” of this fact. The proof is bogus and it’s easy to find counter-examples to what he claimed he proved. (Rosen is aware of the problem since he casually notes that there are some exceptions to the theorem but these are not important for he wants to do. )
Perhaps one real difference is that Rosen is a scientist, an “inductivist” who generalizes from experiments and doesn’t worry if there are exceptions. I am a mathematician, a “deductivist” who can’t tolerate exceptions. But even an inductivist needs some mathematical skills and mathematical sensitivities, particularly when tackling such an ambitious project as life itself.

I’d be interested in hearing about your experience with reading Rosen.

Welcome to Friam, from one lurker to another.

________________________________________
From: [hidden email] [[hidden email]] On Behalf Of Leigh Fanning [[hidden email]]
Sent: Tuesday, April 06, 2010 9:19 PM
To: The Friday Morning Applied Complexity Coffee Group
Subject: [FRIAM] invitation + introduction

First, the invitation:
On Thusday, the University of New Mexico Computer Science department will
hold it's annual student conference highlighting active research within
the department.  Dr. Melanie Mitchell will give the keynote address at
11:00 am.

The conference is open with no admission fees, however we are not able to
provide you with lunch.  Proceedings hardcopy can be ordered for $10,
and will be available for free via download from the website shortly.
Details, and the keynote talk abstract, are below.

Next, the introduction:
By way of introduction, I am largely a FRIAM lurker, but have met a few
of you and in particular would like to further encourage Nick's suggestion
of a Robert Rosen reading group.  My PhD research area is molecular computing
and I am developing a formal system for reasoning about molecular computing
systems, specifically those composed of heterogeneous mixtures of DNA
oligonucleotides.  Milner's pi calculus, and Alur and Dill's timed automata
have been inspirational starting points.  Of course it's supremely simple
to find these inspirations, and attempt physics-style reductionist
techniques, in the engineering of synthetic biological systems.  However
one quickly determines that building even the simplest systems with a
biological basis must be done with a different approach.   The difficulty
in system calibration and readout, and the large number of tunable input
parameters, prevent breaking down molecular computing systems into neat
modules and demand study of how living systems execute their own
engineering and maintenance.

My training, and I use this word with great trepidation following recent
discussion, is Engineering Physics, B.S. from CU-Boulder, Computer Science,
M.S. from UNM, and among other industry jobs, 7+ years doing Guidance,
Navigation and Control engineering for the Space Shuttle program in
the middle years when the fleet was "upgraded" to handle heavier weight
missions to the Space Station -- all old hat now and soon to retire, but
initially a load of interesting problems to work out.

Leigh Fanning

--------

The train between SF and ABQ works well, the bus system has a straight
shot up to the UNM campus from the depot.  Otherwise about an hour
of driving time is needed from Santa Fe, followed by some patience to work out
parking on campus.  There is a large parking garage by Popejoy hall
just off of Central Ave, or street parking just SW of campus can
work well sometimes.  If this works for your schedule, please come
and enjoy!

The schedule of talks is here:

http://cs.unm.edu/~csgsa/conference/

The location is the new Centennial Engineering building on the west
end of campus, bordering University Blvd, just north of Central Ave.

---

Melanie Mitchell, Portland State University and Santa Fe Institute
Thursday, 8 April, 2010
11 am - 12:00 pm
Centennial Engineering Center auditorium

Enabling computers to understand images remains one of the hardest
open problems in artificial intelligence.  No machine vision system
comes close to matching human ability at identifying the contents of
images or visual scenes or at recognizing similarity between different
scenes, even though such abilities pervade human cognition.  In this
talk I will describe research---currently in early stages---on
bridging the gap between low-level perception and higher-level image
understanding by integrating a cognitive model of perceptual
organization and analogy-making with a neural model of the visual
cortex.

Bio: Melanie Mitchell is Professor of Computer Science at Portland
State University and External Professor at the Santa Fe Institute.
She attended Brown University, where she majored in mathematics and
did research in astronomy, and the University of Michigan, where she
received a Ph.D. in computer science, working with her advisor Douglas
Hofstadter on the Copycat project, a computer program that makes
analogies.  She is the author or editor of five books and over 70
scholarly papers in in the fields of artificial intelligence,
cognitive science, and complex systems.  Her most recent book,
"Complexity: A Guided Tour", published in 2009 by Oxford University
Press, was named by Amazon.com as one of the ten best science books of
2009.

============================================================
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

[Leigh Fanning]

Thank you for the welcome, and thank you also to any of you who attended
Melanie Mitchell's talk or any of the other talks today.

I don't have this particular book, however your examples are found in other
Rosen books such as the Dynamical System Theory in Biology volume from 1970.  
I'm surmising that the Life Itself book must be a summary of previous works.  

I cannot argue that there isn't a decidedly physics-based bent in construction
of various points.  A section on scaling in a different volume uses set
theoretic notation as a starting point which seems strange.  But, as in
picking stocks, the hindsight view is superior.  Rosen's companion volume
on Cellular Systems was published from the "Center for Theoretical
Biology," in the earlier 70s this must have been among the first such
institutions formed.  In contrast, nearly 20 years later, Manfred Schroeder's
1991 Fractals, Chaos, Power Laws book gives an excellent and thorough treatment
of scaling and similarity.  And you probably have your different favorite
as well.  

Having said that, I'm somewhat neutral because I haven't absorbed enough to make more
than observational rather than critical statements.  The initial things I have
read were intriguing, in particular one of the online essays details the idea
of anticipatory systems.  

Leigh  
 
On 07 Apr 2010 at 02:10 PM, John Kennison related

>
>
> Hi Leigh,
>
> I guess I?m a Friam lurker too. I?m a friend of Nick Thompson and a retired math professo. I like to read the Friam posts but I comment only occasionally. I?m currently working on dynamical systems and using category theory to break a system down into its cyclic components.
>
> 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.
>
> For example, in ?Life Itself? Rosen starts by talking about ?Newtonian science? and the need to go beyond it, but then continues with a misunderstanding of Taylor?s theorem which, thankfully, is never really used in the rest of the book. Similarly, in some of his writings, Rosen talks about the insolubility of the three-body problem, about Godel?s theorem, about category theory, but never gets close to using any of this stuff.
>
> Rosen?s definition of ?component of a system? and his method of dealing with ?non-recursiveness? are not just mathematically imprecise, they seem completely heavy-handed and insensitive to what the situation demands. In chapter 6, he gives a decomposition of a mathematical system into parts, but he claims the decomposition is unique and even gives a ?proof? of this fact. The proof is bogus and it?s easy to find counter-examples to what he claimed he proved. (Rosen is aware of the problem since he casually notes that there are some exceptions to the theorem but these are not important for he wants to do. )
> Perhaps one real difference is that Rosen is a scientist, an ?inductivist? who generalizes from experiments and doesn?t worry if there are exceptions. I am a mathematician, a ?deductivist? who can?t tolerate exceptions. But even an inductivist needs some mathematical skills and mathematical sensitivities, particularly when tackling such an ambitious project as life itself.
>
> I?d be interested in hearing about your experience with reading Rosen.
>
> Welcome to Friam, from one lurker to another.
>
> ________________________________________
> From: [hidden email] [[hidden email]] On Behalf Of Leigh Fanning [[hidden email]]
> Sent: Tuesday, April 06, 2010 9:19 PM
> To: The Friday Morning Applied Complexity Coffee Group
> Subject: [FRIAM] invitation + introduction
>
> First, the invitation:
> On Thusday, the University of New Mexico Computer Science department will
> hold it's annual student conference highlighting active research within
> the department.  Dr. Melanie Mitchell will give the keynote address at
> 11:00 am.
>
> The conference is open with no admission fees, however we are not able to
> provide you with lunch.  Proceedings hardcopy can be ordered for $10,
> and will be available for free via download from the website shortly.
> Details, and the keynote talk abstract, are below.
>
> Next, the introduction:
> By way of introduction, I am largely a FRIAM lurker, but have met a few
> of you and in particular would like to further encourage Nick's suggestion
> of a Robert Rosen reading group.  My PhD research area is molecular computing
> and I am developing a formal system for reasoning about molecular computing
> systems, specifically those composed of heterogeneous mixtures of DNA
> oligonucleotides.  Milner's pi calculus, and Alur and Dill's timed automata
> have been inspirational starting points.  Of course it's supremely simple
> to find these inspirations, and attempt physics-style reductionist
> techniques, in the engineering of synthetic biological systems.  However
> one quickly determines that building even the simplest systems with a
> biological basis must be done with a different approach.   The difficulty
> in system calibration and readout, and the large number of tunable input
> parameters, prevent breaking down molecular computing systems into neat
> modules and demand study of how living systems execute their own
> engineering and maintenance.
>
> My training, and I use this word with great trepidation following recent
> discussion, is Engineering Physics, B.S. from CU-Boulder, Computer Science,
> M.S. from UNM, and among other industry jobs, 7+ years doing Guidance,
> Navigation and Control engineering for the Space Shuttle program in
> the middle years when the fleet was "upgraded" to handle heavier weight
> missions to the Space Station -- all old hat now and soon to retire, but
> initially a load of interesting problems to work out.
>
> Leigh Fanning
>
> --------
>
> The train between SF and ABQ works well, the bus system has a straight
> shot up to the UNM campus from the depot.  Otherwise about an hour
> of driving time is needed from Santa Fe, followed by some patience to work out
> parking on campus.  There is a large parking garage by Popejoy hall
> just off of Central Ave, or street parking just SW of campus can
> work well sometimes.  If this works for your schedule, please come
> and enjoy!
>
> The schedule of talks is here:
>
> http://cs.unm.edu/~csgsa/conference/
>
> The location is the new Centennial Engineering building on the west
> end of campus, bordering University Blvd, just north of Central Ave.
>
> ---
>
> Melanie Mitchell, Portland State University and Santa Fe Institute
> Thursday, 8 April, 2010
> 11 am - 12:00 pm
> Centennial Engineering Center auditorium
>
> Enabling computers to understand images remains one of the hardest
> open problems in artificial intelligence.  No machine vision system
> comes close to matching human ability at identifying the contents of
> images or visual scenes or at recognizing similarity between different
> scenes, even though such abilities pervade human cognition.  In this
> talk I will describe research---currently in early stages---on
> bridging the gap between low-level perception and higher-level image
> understanding by integrating a cognitive model of perceptual
> organization and analogy-making with a neural model of the visual
> cortex.
>
> Bio: Melanie Mitchell is Professor of Computer Science at Portland
> State University and External Professor at the Santa Fe Institute.
> She attended Brown University, where she majored in mathematics and
> did research in astronomy, and the University of Michigan, where she
> received a Ph.D. in computer science, working with her advisor Douglas
> Hofstadter on the Copycat project, a computer program that makes
> analogies.  She is the author or editor of five books and over 70
> scholarly papers in in the fields of artificial intelligence,
> cognitive science, and complex systems.  Her most recent book,
> "Complexity: A Guided Tour", published in 2009 by Oxford University
> Press, was named by Amazon.com as one of the ten best science books of
> 2009.
>
> ============================================================
> 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

[Owen Densmore]

On Apr 7, 2010, at 12:10 PM, John Kennison <[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

[Nick Thompson]

Owen,

Sorry.  couldn't see how this was relevant.

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: Owen Densmore <[hidden email]>
> To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
> Date: 4/9/2010 10:50:06 AM
> Subject: Re: [FRIAM] invitation + introduction
>
> On Apr 7, 2010, at 12:10 PM, John Kennison <[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

[Grant Holland]

I second Owen's request. Could you discuss how it can be applied to CAS?

Thanks,
Grant

Owen Densmore wrote:

On Apr 7, 2010, at 12:10 PM, John Kennison [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

[John Kennison]

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] [[hidden email]] On Behalf Of Owen Densmore [[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]> 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

[John Kennison]

Leigh,

Is there a more complete title for the "Dynamical System Theory in Biology" volume from 1970? Is it a journal? or a series? It does sound interesting. Thanks.
________________________________________
From: [hidden email] [[hidden email]] On Behalf Of Leigh Fanning [[hidden email]]
Sent: Friday, April 09, 2010 1:35 AM
To: The Friday Morning Applied Complexity Coffee Group
Subject: Re: [FRIAM] invitation + introduction

Thank you for the welcome, and thank you also to any of you who attended
Melanie Mitchell's talk or any of the other talks today.

I don't have this particular book, however your examples are found in other
Rosen books such as the Dynamical System Theory in Biology volume from 1970.
I'm surmising that the Life Itself book must be a summary of previous works.

I cannot argue that there isn't a decidedly physics-based bent in construction
of various points.  A section on scaling in a different volume uses set
theoretic notation as a starting point which seems strange.  But, as in
picking stocks, the hindsight view is superior.  Rosen's companion volume
on Cellular Systems was published from the "Center for Theoretical
Biology," in the earlier 70s this must have been among the first such
institutions formed.  In contrast, nearly 20 years later, Manfred Schroeder's
1991 Fractals, Chaos, Power Laws book gives an excellent and thorough treatment
of scaling and similarity.  And you probably have your different favorite
as well.

Having said that, I'm somewhat neutral because I haven't absorbed enough to make more
than observational rather than critical statements.  The initial things I have
read were intriguing, in particular one of the online essays details the idea
of anticipatory systems.

Leigh

On 07 Apr 2010 at 02:10 PM, John Kennison related

>
>
> Hi Leigh,
>
> I guess I?m a Friam lurker too. I?m a friend of Nick Thompson and a retired math professo. I like to read the Friam posts but I comment only occasionally. I?m currently working on dynamical systems and using category theory to break a system down into its cyclic components.
>
> 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.
>
> For example, in ?Life Itself? Rosen starts by talking about ?Newtonian science? and the need to go beyond it, but then continues with a misunderstanding of Taylor?s theorem which, thankfully, is never really used in the rest of the book. Similarly, in some of his writings, Rosen talks about the insolubility of the three-body problem, about Godel?s theorem, about category theory, but never gets close to using any of this stuff.
>
> Rosen?s definition of ?component of a system? and his method of dealing with ?non-recursiveness? are not just mathematically imprecise, they seem completely heavy-handed and insensitive to what the situation demands. In chapter 6, he gives a decomposition of a mathematical system into parts, but he claims the decomposition is unique and even gives a ?proof? of this fact. The proof is bogus and it?s easy to find counter-examples to what he claimed he proved. (Rosen is aware of the problem since he casually notes that there are some exceptions to the theorem but these are not important for he wants to do. )
> Perhaps one real difference is that Rosen is a scientist, an ?inductivist? who generalizes from experiments and doesn?t worry if there are exceptions. I am a mathematician, a ?deductivist? who can?t tolerate exceptions. But even an inductivist needs some mathematical skills and mathematical sensitivities, particularly when tackling such an ambitious project as life itself.
>
> I?d be interested in hearing about your experience with reading Rosen.
>
> Welcome to Friam, from one lurker to another.
>
> ________________________________________
> From: [hidden email] [[hidden email]] On Behalf Of Leigh Fanning [[hidden email]]
> Sent: Tuesday, April 06, 2010 9:19 PM
> To: The Friday Morning Applied Complexity Coffee Group
> Subject: [FRIAM] invitation + introduction
>
> First, the invitation:
> On Thusday, the University of New Mexico Computer Science department will
> hold it's annual student conference highlighting active research within
> the department.  Dr. Melanie Mitchell will give the keynote address at
> 11:00 am.
>
> The conference is open with no admission fees, however we are not able to
> provide you with lunch.  Proceedings hardcopy can be ordered for $10,
> and will be available for free via download from the website shortly.
> Details, and the keynote talk abstract, are below.
>
> Next, the introduction:
> By way of introduction, I am largely a FRIAM lurker, but have met a few
> of you and in particular would like to further encourage Nick's suggestion
> of a Robert Rosen reading group.  My PhD research area is molecular computing
> and I am developing a formal system for reasoning about molecular computing
> systems, specifically those composed of heterogeneous mixtures of DNA
> oligonucleotides.  Milner's pi calculus, and Alur and Dill's timed automata
> have been inspirational starting points.  Of course it's supremely simple
> to find these inspirations, and attempt physics-style reductionist
> techniques, in the engineering of synthetic biological systems.  However
> one quickly determines that building even the simplest systems with a
> biological basis must be done with a different approach.   The difficulty
> in system calibration and readout, and the large number of tunable input
> parameters, prevent breaking down molecular computing systems into neat
> modules and demand study of how living systems execute their own
> engineering and maintenance.
>
> My training, and I use this word with great trepidation following recent
> discussion, is Engineering Physics, B.S. from CU-Boulder, Computer Science,
> M.S. from UNM, and among other industry jobs, 7+ years doing Guidance,
> Navigation and Control engineering for the Space Shuttle program in
> the middle years when the fleet was "upgraded" to handle heavier weight
> missions to the Space Station -- all old hat now and soon to retire, but
> initially a load of interesting problems to work out.
>
> Leigh Fanning
>
> --------
>
> The train between SF and ABQ works well, the bus system has a straight
> shot up to the UNM campus from the depot.  Otherwise about an hour
> of driving time is needed from Santa Fe, followed by some patience to work out
> parking on campus.  There is a large parking garage by Popejoy hall
> just off of Central Ave, or street parking just SW of campus can
> work well sometimes.  If this works for your schedule, please come
> and enjoy!
>
> The schedule of talks is here:
>
> http://cs.unm.edu/~csgsa/conference/
>
> The location is the new Centennial Engineering building on the west
> end of campus, bordering University Blvd, just north of Central Ave.
>
> ---
>
> Melanie Mitchell, Portland State University and Santa Fe Institute
> Thursday, 8 April, 2010
> 11 am - 12:00 pm
> Centennial Engineering Center auditorium
>
> Enabling computers to understand images remains one of the hardest
> open problems in artificial intelligence.  No machine vision system
> comes close to matching human ability at identifying the contents of
> images or visual scenes or at recognizing similarity between different
> scenes, even though such abilities pervade human cognition.  In this
> talk I will describe research---currently in early stages---on
> bridging the gap between low-level perception and higher-level image
> understanding by integrating a cognitive model of perceptual
> organization and analogy-making with a neural model of the visual
> cortex.
>
> Bio: Melanie Mitchell is Professor of Computer Science at Portland
> State University and External Professor at the Santa Fe Institute.
> She attended Brown University, where she majored in mathematics and
> did research in astronomy, and the University of Michigan, where she
> received a Ph.D. in computer science, working with her advisor Douglas
> Hofstadter on the Copycat project, a computer program that makes
> analogies.  She is the author or editor of five books and over 70
> scholarly papers in in the fields of artificial intelligence,
> cognitive science, and complex systems.  Her most recent book,
> "Complexity: A Guided Tour", published in 2009 by Oxford University
> Press, was named by Amazon.com as one of the ten best science books of
> 2009.
>
> ============================================================
> 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

[Grant Holland]

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] [[hidden email]] On Behalf Of Owen Densmore [[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] 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

[Owen Densmore]

Very nice, thanks!

    ---- Owen


I am an iPad, resistance is futile!

On Apr 9, 2010, at 2:43 PM, John Kennison <[hidden email]> 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] [[hidden email]] On Behalf Of Owen Densmore [[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]> 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

[John Kennison]

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] [[hidden email]] On Behalf Of Grant Holland [[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]> [[hidden email]<mailto:[hidden email]>] On Behalf Of Owen Densmore [[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]> 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

[Steve Smith]

John

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. 

  

Very good summary of Category Theory (CT)... very accessible and intuitive (for anyone who already knows what groups, rings, etc. are ;)

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).
  

I'm left wondering how you might think it applies to Complex Adaptive Systems (CAS)?

My colleagues,  Dr. Tom Caudell (UNM) and Dr. Michael Healy (UW emeritus) are working on a theory of Neural Architectures based on Category Theory
    http://portal.acm.org/citation.cfm?id=1568850
    http://portal.acm.org/citation.cfm?id=1704175.1704367 
    http://www.ece.unm.edu/~mjhealy/Healy-LOR-rev.pdf

that begins to encroach on the application of CT to CAS .

This fits with Leigh's announcement of Melanie's talk, or at least Melanie's  seminal work in "Analogy Making as Perception".    My interest as a Visualization Scientist (Trained in Physics/Math, practiced in CS/CE and focused mostly on the range of topics revolving around synthesized perceptual spaces for exploration, discovery and analysis of (possibly complex) phenomena) is in the formalization of Metaphor (Thus Analogy Making as Perception and Category Theory models of Cognition.)   I'm also convinced that it has application to Agency (what really good, deep, Agents  should have?)

I've read through Jocelyn Paine's compilation of her own exposition in this area:
    http://www.j-paine.org/why_be_interested_in_categories.html

and find it motivating but beyond my limited capabilities.   

However,  Jocelyn's expose on how Excel Spreadsheets have motivated her to investigate Category Theory and getting to that was worth the effort of reading it through to the end!

I look forward to more unfolding on the application of Category Theory to CAS here...  if it has legs anyway.

- Steve





============================================================
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

[Grant Holland]

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] [[hidden email]] On Behalf Of Grant Holland [[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]> [[hidden email]<mailto:[hidden email]>] On Behalf Of Owen Densmore [[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]> 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

[Nick Thompson]

John,

I am amazed.  I had no idea you were this deep down this rabbit-hole.  

Has anybody out there read Sommerhoff: (sp?) Analyical Biology? About 1950.
Is it relevant?  

 It was concerned with what I am going to call, for want of better terms,
diachronic teloi.  The self aiming gun.  It's diachronic because you get
your idea that the gun is self aiming from observing it over time.  Most of
my writing (and it has just been words, words, words,) has been about
synchronic teloi, synchronic because you get the idea that the birds at
your feeder are designed  from observing several different kinds of birds
at the same moment in [geologic] time.  In my work, I called both of these
forms of "natural design."  (objective teleology).  See also, Powers work
on Control Systems. Also, surprisingly, John Bowlby.  

I originally read Rosen because (1) Carl Tollander put me on to Category
Theory and I thought it might connect our worlds and (2) because I thought
Rosen might lead me to a mathematical characterization of "natural design."

I will be interested in how this conversation develops, even though it is
technologically beyond my ken.  

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: John Kennison <[hidden email]>
> To: The Friday Morning Applied Complexity Coffee Group <[hidden email]>
> Date: 4/10/2010 5:21:24 AM
> Subject: Re: [FRIAM] invitation + introduction
>
>
>
> 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] [[hidden email]] On Behalf Of
Grant Holland [[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]>
[[hidden email]<mailto:[hidden email]>] On Behalf Of
Owen Densmore [[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]> 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

[Nick Thompson]

Steve,

 

Oh, Wow! 

 

You wrote:

 

My interest as a Visualization Scientist (Trained in Physics/Math, practiced in CS/CE and focused mostly on the range of topics revolving around synthesized perceptual spaces for exploration, discovery and analysis of (possibly complex) phenomena) is in the formalization of Metaphor (Thus Analogy Making as Perception and Category Theory models of Cognition.)   I'm also convinced that it has application to Agency (what really good, deep, Agents  should have?)

We (Eric Charles Lee Rudlolph,  and collaborators) are trying to resubmit (favorable review first time around but not quite favorable enough)  on the mathematical characterization (via "configuration spaces") of emotional behavior ("expression", if you must).  The idea is to mathematically describe visual displays that people respond to emotionally and then tweak the maths to produce what we ethologists call "super normal stimuli"

 

Would you have any interest in this project?  We had thought perhaps to run some part of it through the Complex.

 

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: [hidden email]

To: [hidden email]

Sent: 4/10/2010 8:35:24 AM

Subject: Re: [FRIAM] invitation + introduction

John

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. 

  

Very good summary of Category Theory (CT)... very accessible and intuitive (for anyone who already knows what groups, rings, etc. are ;)

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).
  

I'm left wondering how you might think it applies to Complex Adaptive Systems (CAS)?

My colleagues,  Dr. Tom Caudell (UNM) and Dr. Michael Healy (UW emeritus) are working on a theory of Neural Architectures based on Category Theory
    http://portal.acm.org/citation.cfm?id=1568850
    http://portal.acm.org/citation.cfm?id=1704175.1704367 
    http://www.ece.unm.edu/~mjhealy/Healy-LOR-rev.pdf

that begins to encroach on the application of CT to CAS .

This fits with Leigh's announcement of Melanie's talk, or at least Melanie's  seminal work in "Analogy Making as Perception".    My interest as a Visualization Scientist (Trained in Physics/Math, practiced in CS/CE and focused mostly on the range of topics revolving around synthesized perceptual spaces for exploration, discovery and analysis of (possibly complex) phenomena) is in the formalization of Metaphor (Thus Analogy Making as Perception and Category Theory models of Cognition.)   I'm also convinced that it has application to Agency (what really good, deep, Agents  should have?)

I've read through Jocelyn Paine's compilation of her own exposition in this area:
    http://www.j-paine.org/why_be_interested_in_categories.html

and find it motivating but beyond my limited capabilities.   

However,  Jocelyn's expose on how Excel Spreadsheets have motivated her to investigate Category Theory and getting to that was worth the effort of reading it through to the end!

I look forward to more unfolding on the application of Category Theory to CAS here...  if it has legs anyway.

- Steve

============================================================
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

[Douglas Roberts-2]

Nick,

It sounds like your proposal aims to digitize a completely non-linear analog process.  Is there any hope that a mathematical characterization such as you describe has a chance of being robust enough to produce anything more than a "cartoon" representation of emotional behavior?

--Doug

On Sat, Apr 10, 2010 at 10:47 AM, Nicholas Thompson <[hidden email]> wrote:

Steve,

 

Oh, Wow! 

 

You wrote:

 

My interest as a Visualization Scientist (Trained in Physics/Math, practiced in CS/CE and focused mostly on the range of topics revolving around synthesized perceptual spaces for exploration, discovery and analysis of (possibly complex) phenomena) is in the formalization of Metaphor (Thus Analogy Making as Perception and Category Theory models of Cognition.)   I'm also convinced that it has application to Agency (what really good, deep, Agents  should have?)

We (Eric Charles Lee Rudlolph,  and collaborators) are trying to resubmit (favorable review first time around but not quite favorable enough)  on the mathematical characterization (via "configuration spaces") of emotional behavior ("expression", if you must).  The idea is to mathematically describe visual displays that people respond to emotionally and then tweak the maths to produce what we ethologists call "super normal stimuli"

 

Would you have any interest in this project?  We had thought perhaps to run some part of it through the Complex.

 

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: [hidden email]

To: [hidden email]

Sent: 4/10/2010 8:35:24 AM

Subject: Re: [FRIAM] invitation + introduction

John

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. 

  

Very good summary of Category Theory (CT)... very accessible and intuitive (for anyone who already knows what groups, rings, etc. are ;)

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).
  

I'm left wondering how you might think it applies to Complex Adaptive Systems (CAS)?

My colleagues,  Dr. Tom Caudell (UNM) and Dr. Michael Healy (UW emeritus) are working on a theory of Neural Architectures based on Category Theory
    http://portal.acm.org/citation.cfm?id=1568850
    http://portal.acm.org/citation.cfm?id=1704175.1704367 
    http://www.ece.unm.edu/~mjhealy/Healy-LOR-rev.pdf

that begins to encroach on the application of CT to CAS .

This fits with Leigh's announcement of Melanie's talk, or at least Melanie's  seminal work in "Analogy Making as Perception".    My interest as a Visualization Scientist (Trained in Physics/Math, practiced in CS/CE and focused mostly on the range of topics revolving around synthesized perceptual spaces for exploration, discovery and analysis of (possibly complex) phenomena) is in the formalization of Metaphor (Thus Analogy Making as Perception and Category Theory models of Cognition.)   I'm also convinced that it has application to Agency (what really good, deep, Agents  should have?)

I've read through Jocelyn Paine's compilation of her own exposition in this area:
    http://www.j-paine.org/why_be_interested_in_categories.html

and find it motivating but beyond my limited capabilities.   

However,  Jocelyn's expose on how Excel Spreadsheets have motivated her to investigate Category Theory and getting to that was worth the effort of reading it through to the end!

I look forward to more unfolding on the application of Category Theory to CAS here...  if it has legs anyway.

- Steve

============================================================
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

[Nick Thompson]

This fits with Leigh's announcement of Melanie's talk, or at least Melanie's  seminal work in "Analogy Making as Perception".  

 

YES!!!!!!!!!!!!!!!  Does anybody have links to such work!!!!!

 

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: [hidden email]

To: [hidden email]

Sent: 4/10/2010 8:35:24 AM

Subject: Re: [FRIAM] invitation + introduction

John

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. 

  

Very good summary of Category Theory (CT)... very accessible and intuitive (for anyone who already knows what groups, rings, etc. are ;)

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).
  

I'm left wondering how you might think it applies to Complex Adaptive Systems (CAS)?

My colleagues,  Dr. Tom Caudell (UNM) and Dr. Michael Healy (UW emeritus) are working on a theory of Neural Architectures based on Category Theory
    http://portal.acm.org/citation.cfm?id=1568850
    http://portal.acm.org/citation.cfm?id=1704175.1704367 
    http://www.ece.unm.edu/~mjhealy/Healy-LOR-rev.pdf

that begins to encroach on the application of CT to CAS .

This fits with Leigh's announcement of Melanie's talk, or at least Melanie's  seminal work in "Analogy Making as Perception".    My interest as a Visualization Scientist (Trained in Physics/Math, practiced in CS/CE and focused mostly on the range of topics revolving around synthesized perceptual spaces for exploration, discovery and analysis of (possibly complex) phenomena) is in the formalization of Metaphor (Thus Analogy Making as Perception and Category Theory models of Cognition.)   I'm also convinced that it has application to Agency (what really good, deep, Agents  should have?)

I've read through Jocelyn Paine's compilation of her own exposition in this area:
    http://www.j-paine.org/why_be_interested_in_categories.html

and find it motivating but beyond my limited capabilities.   

However,  Jocelyn's expose on how Excel Spreadsheets have motivated her to investigate Category Theory and getting to that was worth the effort of reading it through to the end!

I look forward to more unfolding on the application of Category Theory to CAS here...  if it has legs anyway.

- Steve

============================================================
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

[John Kennison]

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] [[hidden email]] On Behalf Of Grant Holland [[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] [[hidden email]] On Behalf Of Grant Holland [[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]> [[hidden email]<mailto:[hidden email]>] On Behalf Of Owen Densmore [[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]> 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

[Grant Holland]

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] [[hidden email]] On Behalf Of Grant Holland [[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] [[hidden email]] On Behalf Of Grant Holland [[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][hidden email] [[hidden email][hidden email]] On Behalf Of Owen Densmore [[hidden email][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][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