RE: ONT Re: Higher Order Categorical Logic -- Discussion

Dear Jon,

First, thank you for this excellent tutorial. Even I can
more or less follow you.

Can I chip in a question here. You talk about structure
preserving functions below. Is that the same as an
isomorphism? If not what is the difference?


Matthew West
Principal Consultant
Shell Information Technology International Limited
Shell Centre, London SE1 7NA, United Kingdom

Tel: +44 20 7934 4490 Mobile: +44 7796 336538
Email: [email protected]
Internet: http://www.shell.com
http://www.matthew-west.org.uk


> -----Original Message-----
> From: Jon Awbrey [mailto:[email protected]]
> Sent: 07 January 2004 22:36
> To: Inquiry; Ontology
> Cc: Jack Park
> Subject: ONT Re: Higher Order Categorical Logic -- Discussion
> 
> 
> 
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> 
> HOC.  Discussion Note 6
> 
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> 
> JA: The next examples are sets plus "structure",
>     in these cases, something like a "sums table",
>     a "times table", or a "less than" relation is
>     defined on the sets of the category and also
>     preserved by the arrows of the category.
> 
> L&S: | Example C2.  The category of 'monoids'.  Its objects are
>      | monoids, that is, semigroups with unity element, and its
>      | morphisms are homomorphisms, that is, mappings which
>      | preserve multiplication (the semigroup operation)
>      | and the unity element.
> 
> MA: The unity element being "="?  What does the
>     phrase "preserve multiplication" mean?
> 
> JA: A "semigroup" is a set with a 2-ary operation (*)
>     subject to an associative law, a*(b*c) = (a*b)*c.
>     Sometimes you think of (*) as multiplication,
>     and write a*b as ab.  Other times you think
>     of (*) as addition, and write a*b as a+b.
> 
> JA: The "unity element" is just another name
>     for the identity element in the system.
>     If you are thinking of (*) on analogy
>     with multiplication, you will use "1"
>     and write 1*a = a = a*1.  If you are
>     thinking of (*) on analogy with sum,
>     you use "0" and write 0+a = a = a+0.
> 
> JA: A classic example would be the logarithm mapping from
>     a domain (D, *) of real numbers under multiplication (*)
>     to a domain (E, +) of real numbers under addition (+).
> 
>     1.  log : (D, *) -> (E, +).
> 
>         The log function maps the object D to the object E,
>         mapping the structure of (*) to the structure of (+).
> 
>     2.  log(1) = 0.
> 
>         The log function maps the multiplicative identity 1
>         to the additive identity 0.
> 
>     3.  log(x * y) = log(x) + log(y)
> 
>         One says:  "the image of the product is the sum of 
> the images".
>         this describes a form of analogy or metaphor between 
> (*) and (+).
> 
> MA: I *think* I understand this.
> 
> That figure of speech -- called "chiasma" or "chiasmus" in 
> literary circles,
> is one of the recognizable signatures by which you may know 
> that a morphism
> has set its hand to the work.  Here, the word "image" refers 
> to the morphic
> image, that is, the functional value of the 
> structure-preserving function.
> 
> Let's try to get at the notion of morphisms as 
> "structure-preserving maps".
> suppose we have two structured sets (X, L) and (Y, M) and a 
> map f : X -> Y.
> What does it mean that f maps the structure L on X to the 
> structure M on Y?
> 
> The use of the word "preserve" for a correspondence 
> established between two
> structures will make more sense if you remember that the 
> paradigmatic case
> is one where both L and M are thought of under the same name, 
> say (*), (+),
> (=<), etc., even if that is strictly speaking an act of great 
> abstraction
> wrapped in a figure of hardly heard homophony.
> 
> The ingredients of a potential morphism are as follows:
> 
>    1.  We have a set X with a certain "structure" L that is 
> defined on it.
>        It could a 2-adic relation L c X x X that has the 
> properties of an
>        order relation, or it could be a 3-adic relation L c X 
> x X x X that
>        is associated with a 2-ary operation like addition, 
> multiplication,
>        or any one of several 2-ary logical connectives.
> 
>    2.  We have a set Y with a comparable structure M that is 
> defined on it.
> 
> For the sake of a concrete example, let's say that both L and 
> M are 3-adic
> relations of the kind that are associated with 2-ary 
> operations.  Thus, we
> can write (X, L) = (X, *) and (Y, M) = (Y, +), where L c X^3 
> and M c Y^3.
> As a generic name for the result of an operation, I'll use 
> "resultant".
> 
>    3.  We are given a mapping f : X -> Y, and we would like 
> to test whether
>        f maps the structure L on X to the structure M on Y, 
> in which case
>        we will bow to tradition and say that f preserves the 
> respective
>        attachments of structure in the passage from X and Y.
> 
> Here is one way to formulate the property that we need to test.
> In order to say that f : X -> Y preserves the form of L in the
> form of M, the following equation must hold for all u, v in X.
> 
>    f(u * v)  =  f(u) + f(v)
> 
> In the idiom that is commonly used, we are asking whether the
> following parable, properly interpreted, is a constant truth:
> 
>    The image of the resultant is the resultant of the images.
> 
> In order to read this right, you have to keep in mind that
> "image of" means "f evaluated at", the first "resultant"
> refers to L or (*) evaluated on a pair u, v in X, and
> the second "resultant" refers to M or (+) evaluated
> on the corresponding pair f(u), f(v) in Y.
> 
> Saved by the dinner bell ...
> I will use the interval
> to rustle up some kinds
> of pictures that might
> help with this mess.
> 
> Jon Awbrey
> 
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> http://www.cs.bsu.edu/homepages/mighty/history.html
> o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o~~~~~~~~~o
> 
>