Very nice, thanks!
---- Owen
I am an iPad, resistance is futile!
On Apr 9, 2010, at 2:43 PM, John Kennison <[email protected]> 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: [email protected] [[email protected]] On Behalf Of Owen
> Densmore [[email protected]]
> 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 <[email protected]> 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
