slogans

slogans

[Al Vilcius]

As a bit of a fun project, I would like to put together a collection of
slogans, and am asking for contributions.  Such a collection may or may not
turn out to be "useful", but at least it will be amusing.

I'm thinking of a slogan (apart from being a Scottish Highland war-cry) as a
pithy little phrase or saying or motto or truism ..., just short of being
poetry, that can tweak the neural pathways to follow a previously traveled
path of understanding.

Slogans tend to be little gems, often found in the folklore, but sometimes
published, such as the 5 slogans in Jim Lambek & P. Scott's "Intro to higher
order categorical logic" - these are:
I: many objects of interest in mathematics congregate in concrete
categories.
II: many objects of interest to mathematicians are themselves small
categories.
III: many objects of interest to mathematicians may be viewed as functors
from small categories to Sets.
IV: many important concepts in mathematics arise as adjoints, right or left,
to previously known functors.
V: many equivalence and duality theorems in mathematics arise as an
equivalence of fixed subcategories induced by a pair of adjoint functors.

I think there are many more such gems; I like (and use) the one that says:
the arrows always go through the limit.
- I attribute this one to Armin Frei circa 1971 at UBC.

Another really good one is: adjoints are the unity and identity of
opposites.
- I attribute this one to Bill Lawvwere

Here are a couple more, from the physics of information:
* there is no information without representation.
* there is no processing without a process.
- attributed to Benjamin Schumacher of Kenyon College.

Each slogan can explode into rich and meaningful ideas, sort of the "url's
of the mind" (if you like that kind of cyber-geek talk).

Anyway, I hope the intention is clear, and that everyone will contribute
their favourites as well.  Please either post a response or send to me
directly (no flames please), along with appropriate credit if known; I will
assemble, and depending on how it goes, make the collection available.

Could be fun and interesting - I look forward to your responses.

Best regards to all .... Al

   /\      / Al R. Vilcius, Canada
  /  \    / mailto:al.r@vilcius.com
 /--->\  / Tel./FAX  (905) 854-3342 /3371
/      \/ web site: http://www.VILCIUS.com

Re: slogans

[Ronnie Brown]

How do you like:

Higher dimensional algebra supplies algebraic inverses to subdivision, for
application to local-to-global problems. 

(That has been the intention of our programme since the mid-1960's.) 

Ronnie Brown

On Wed, 26 Jan 2000, Al Vilcius wrote:

> As a bit of a fun project, I would like to put together a collection of
> slogans, and am asking for contributions.  Such a collection may or may not
> turn out to be "useful", but at least it will be amusing.
> 

New popularisation project

http://www.bangor.ac.uk/ma/CPM/rpamath/overall.htm

Help sought!


School of Informatics, Mathematics Division,
University of Wales, Bangor
Dean St., Bangor, Gwynedd LL57 1UT, United Kingdom
Tel. direct:+44 1248 382474|office:     382475
fax: +44 1248 361429
World Wide Web:
home page: http://www.bangor.ac.uk/~mas010/
(Links to survey articles:
Higher dimensional group theory
Groupoids and crossed objects in algebraic topology)

Symbolic Sculpture and Mathematics:
http://www.bangor.ac.uk/SculMath/
Mathematics and Knots:
http://www.bangor.ac.uk/ma/CPM/exhibit/welcome.htm

Re: slogans

[Giuseppe Longo]

... about adding slogans:
"there is no mathematics without structures"
and
"there are no structures without transformations (between structures)"

In stating this, I would go back to (Gauss and) Reimann (and Klein).  
 This was a turning point of last century mathematics.  Geometry is  
the analysis of (possibly) curved space and the unity of geometries  
is found on the notion of transformation (over manifolds, say:  
continuous, differentiable ...).
Mathematics is no more found (only) on "quantities", since ratios of  
length and of angles, at the heart of Euclidean geometry, are not  
preserved in non-euclidean frames (their group of automorphisms are  
not closed under omotheties).

Category Theory is the theory which inherited this fantastic  
broadening of perspective.


--Giuseppe Longo
Lab. "Jacques Herbrand"
CNRS et Ecole Normale Superieure
(Postal addr.:  LIENS
45, Rue D'Ulm
75005  Paris   (France) )

http://www.dmi.ens.fr/users/longo
e-mail: longo@di.ens.fr
(tel. ++33-1-4432-3328, FAX 4432-2080)


Upon kind permission of the M.I.T. Press, the book below is
currently downloadable from Longo's web page above (its n-th
edition is out of print...):

Andrea Asperti and Giuseppe Longo. Categories, Types and
Structures: an introduction to Category Theory for the working
computer scientist. M.I.T.- Press, 1991. (pp. 1--300).