Re: Categories ridiculously abstract

Re: Categories ridiculously abstract

[Vaughan Pratt]

1 ab.stract \ab-'strakt, 'ab-,\ adj (15c)
 [ML abstractus, fr. L, pp. of abstrahere to draw away, fr. abs-, ab- + 
trahere to draw -- more at DRAW]
    1a: disassociated from any specific instance <abstract entity> 
    1b: difficult to understand: ABSTRUSE <abstract problems> 
    1c: IDEAL <abstract justice> 
    1d: insufficiently factual: FORMAL <possessed only an abstract right> 
    
    2: expressing a quality apart from an object <the word poem is concrete, 
    poetry is abstract> 
    3a: dealing with a subject in its abstract aspects: THEORETICAL <abstract 
    science> 
    3b: IMPERSONAL, DETACHED <the abstract compassion of a surgeon --Time> 
    
    4: having only intrinsic form with little or no attempt at pictorial 
    representation or narrative content <abstract painting> -- ab.stract.ly 
    \ab-'strak-(t)l<e^->, 'ab-,\ adv -- ab.stract.ness \ab-'strak(t)-n<e>s, 
    'ab-,\ n


1a: Sets and categories as mathematical abstractions are equally
disassociated from specific instances.

1b: For almost every interesting known theorem of category theory there
is a harder interesting known theorem of set theory, and vice versa.
It is plausible that the exceptions from set theory outnumber those
from category theory, but it is equally plausible that a majority of
mathematical literates judge category theory harder than set theory.
No clear winner here.

1c: Sets and categories are both ideal entities.

1d: Set theory and category theory are equally factual, and equally
formal.

2: In this sense set theory and category theory are both abstract while
sets and categories are objects and so not abstract.

3a: Set theory and category theory deal equally with the abstract aspects
of their respective subjects.

3b: The FOM mailing list tends to get worked up much more often and
rather more heatedly about the set-vs-category debate than does the
categories mailing list.

4.  Categories lend themselves better to diagrams than do sets.


Conclusions (organized by dictionary meaning of "abstract"):

	1 to 3a: No difference.

	3b:      Category theorists are more abstract than set theorists.

	4:       Sets are more abstract than categories.


--
Vaughan Pratt                             O res ridicula! immensa stultitia.
                                          --Chorus of Old Men, Catulli Carmina

Re: Categories ridiculously abstract

[Tom Leinster]

Michael Barr wrote:
> 
> And here is a question: are categories more abstract or less abstract than
> sets? 

A higher-dimensional category theorist's answer:
"Neither - a set is merely a 0-category, and a category a 1-category."

There's a more serious thought behind this.  Sometimes I've wondered, in a
vague way, whether the much-discussed hierarchy

0-categories (sets) form a (1-)category, 
(1-)categories form a 2-category, 
...

has a role to play in foundations.  After all, set-theorists seek to found
mathematics on the theory of 0-categories; category-theorists sometimes talk
about founding mathematics on the theory of 1-categories and providing a
(Lawverian) axiomatization of the 1-category of 0-categories; you might ask
"what next"?  Axiomatize the 2-category of (1-)categories?  Or the
(n+1)-category of n-categories?  Could it even be, I ask with tongue in cheek
and head in clouds, that general n-categories provide a more natural
foundation than either 0-categories or 1-categories?


Tom

Re: Categories ridiculously abstract

[Michael MAKKAI]

In "Towards a categorical foundation of mathematics" (Logic Colloquium
'95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic
no.11, 1998; pp.153-190) and in subsequent work, I am proposing an
approach to a foundation whose universe consists of the weak n-categories
and whatever things are needed to connect them. This is done on the basis
of a general point of view concerning the role of identity of mathematical
objects. Readers of said paper who have followed developments on weak
higher dimensional categories will note that much has been done since
towards fleshing out the program.

Michael Makkai


On Thu, 30 Nov 2000, Tom Leinster wrote:

> 
> Michael Barr wrote:
> > 
> > And here is a question: are categories more abstract or less abstract than
> > sets? 
> 
> A higher-dimensional category theorist's answer:
> "Neither - a set is merely a 0-category, and a category a 1-category."
> 
> There's a more serious thought behind this.  Sometimes I've wondered, in a
> vague way, whether the much-discussed hierarchy
> 
> 0-categories (sets) form a (1-)category, 
> (1-)categories form a 2-category, 
> ...
> 
> has a role to play in foundations.  After all, set-theorists seek to found
> mathematics on the theory of 0-categories; category-theorists sometimes talk
> about founding mathematics on the theory of 1-categories and providing a
> (Lawverian) axiomatization of the 1-category of 0-categories; you might ask
> "what next"?  Axiomatize the 2-category of (1-)categories?  Or the
> (n+1)-category of n-categories?  Could it even be, I ask with tongue in cheek
> and head in clouds, that general n-categories provide a more natural
> foundation than either 0-categories or 1-categories?
> 
> 
> Tom
> 

Re: Categories ridiculously abstract

[DR Mawanda]

My understanding of the relation between  category theory and  set  theory
is that category theory  is a formal theory built on abstract concepts
(objects and morphisms). The way of defining category theory need a
metalanguage which is closed to the logic of set theory language (a
particular case of what is called boolean logic). There is a sort of
dichothomy between logic behind the two theories. This dichothomy come from
our limitation of talking about category theory. We use  already two-valued
logic (true and false)  which we cannot avoid if we need to talk about
identity of objects and morphisms. Now a kind of Godel's arguments about
natural numbers (If N is consistent, then there is no proof of its
consistency by method formalizable within the theory ) is  what is going on.
This doesn't stop the category theory 'game'. When you give birth to a child
you will never know in advance if the child will be an honest person or a
criminal. Category theory have generated many structures which can help us
to understand why many mathematicians have work differently to describe a
same mathematical concept in different ways. As an example we know, from
category theory,  that Cauchy and Dedekind were defining real numbers from
rational numbers but the two definitions are not saying the same thing.

----- Original Message -----
From: "Michael MAKKAI" <makkai@scylla.math.mcgill.ca>
To: <categories@mta.ca>
Sent: Saturday, December 02, 2000 12:19 AM
Subject: categories: Re: Categories ridiculously abstract


> In "Towards a categorical foundation of mathematics" (Logic Colloquium
> '95, ed's: J. A. Makowsky and E. V. Ravve, Springer Lecture Notes in Logic
> no.11, 1998; pp.153-190) and in subsequent work, I am proposing an
> approach to a foundation whose universe consists of the weak n-categories
> and whatever things are needed to connect them. This is done on the basis
> of a general point of view concerning the role of identity of mathematical
> objects. Readers of said paper who have followed developments on weak
> higher dimensional categories will note that much has been done since
> towards fleshing out the program.
>
> Michael Makkai
>
>
> On Thu, 30 Nov 2000, Tom Leinster wrote:
>
> >
> > Michael Barr wrote:
> > >
> > > And here is a question: are categories more abstract or less abstract
than
> > > sets?
> >
> > A higher-dimensional category theorist's answer:
> > "Neither - a set is merely a 0-category, and a category a 1-category."
> >
> > There's a more serious thought behind this.  Sometimes I've wondered, in
a
> > vague way, whether the much-discussed hierarchy
> >
> > 0-categories (sets) form a (1-)category,
> > (1-)categories form a 2-category,
> > ...
> >
> > has a role to play in foundations.  After all, set-theorists seek to
found
> > mathematics on the theory of 0-categories; category-theorists sometimes
talk
> > about founding mathematics on the theory of 1-categories and providing a
> > (Lawverian) axiomatization of the 1-category of 0-categories; you might
ask
> > "what next"?  Axiomatize the 2-category of (1-)categories?  Or the
> > (n+1)-category of n-categories?  Could it even be, I ask with tongue in
cheek
> > and head in clouds, that general n-categories provide a more natural
> > foundation than either 0-categories or 1-categories?
> >
> >
> > Tom
> >
>

Re: Categories ridiculously abstract

[Robert J. MacG. Dawson]

Tom Leinster wrote:
> 
> Michael Barr wrote:
> >
> > And here is a question: are categories more abstract or less abstract than
> > sets?
> 
> A higher-dimensional category theorist's answer:
> "Neither - a set is merely a 0-category, and a category a 1-category."
> 
> There's a more serious thought behind this.  Sometimes I've wondered, in a
> vague way, whether the much-discussed hierarchy
> 
> 0-categories (sets) form a (1-)category,
> (1-)categories form a 2-category,
> ...
> 
> has a role to play in foundations.  After all, set-theorists seek to found
> mathematics on the theory of 0-categories; category-theorists sometimes talk
> about founding mathematics on the theory of 1-categories and providing a
> (Lawverian) axiomatization of the 1-category of 0-categories; you might ask
> "what next"?  Axiomatize the 2-category of (1-)categories?  Or the
> (n+1)-category of n-categories?  

	Surely we should start with the set of (-1)-categories? <gd&r>

	-Robert Dawson

Re: Categories ridiculously abstract

[John Duskin]

Readers of the cat list may be interested in the one meaningful post 
to Slate's "The Fray" in reply to Holt's MSRI "Diary" article. It was 
made  by David Yetter:

                                              
                                               category theory
                                               David Yetter
                                               28 Nov 2000 20:29


It is sad more than a decade on since the
proof of the remarkable categorical coherence
theorem of Shum that mathematicians can
continue to view category theory as a mere
linguistic convention or useless abstraction.

Shum's theorem shows that axioms
completely natural from the internal dynamic of
category theory completely characterize
framed tangles, relative versions of the framed
knots and links which are central to
smooth topology in 3 and 4 dimensions (notice
the dimensionality of space and of
space-time: hardly divorced from meaning.) Other
categories satisfying the same axioms
include the categories of representations of
quantum groups, physically motivated
algebraic structures which have become central
objects of study for mathematicians from
many old branches of mathematics.

Indeed, Shum's theorem, a theorem of
category theory, is the only really satisfying
explanation for the intimate connection
between quantum groups and low-dimensional
topology.

Re: Categories ridiculously abstract

[Michael Barr]

I don't think one should blame the guy whose remarks Peter quoted.  He is
not a mathematician and presumably knows nothing more than some college
level mathematics.  He has picked up that attitude from the high-powered
mathematicians that inhabit places like MSRI (and the CRM, Fields Inst.,
and PIMS in Canada).  Ignoring the fact that category theory was fathered
by two of the most eminent mathematicians of the last century and
god-fathered by arguably the very greatest, they still go around saying
that it is without content and nothing but meaningless abstraction.  I was
unaware of what David Yetter mentioned, but I am certainly aware of the
crucial role categories had in proving the Weil conjectures and the fact
that people like John Baez seem to believe that higher dimensional
categories will be important in physics.  I might also point out that
categories were the right framework for Kaplansky's very elegant proof of
the Auslander-Buchsbaum theorem.  And here is a question: are categories
more abstract or less abstract than sets?

Michael

Re: Categories ridiculously abstract

[Todd Wilson]

On Wed, 29 Nov 2000, Michael Barr wrote:
> And here is a question: are categories more abstract or less
> abstract than sets?

There is a trap lurking in this question, and it has to do with
understanding the term "abstract":  different notions of "abstract"
can lead to different answers to the question.  In the case of sets
and categories, since these are of different similarity types,
something other than inclusion of classes of models is meant.  For
example "abstract", applied to sets and categories, might mean:

1. Having wider applicability.  In this case, we can observe that the
   theorems of category theory (e.g., products are unique up to unique
   isomorphism) are generally more widely applicable than theorems of
   set theory (e.g., the powerset of a set has greater cardinality
   than the set itself), and so we would be inclined to say that
   categories are more abstract than sets on this criterion.

2. Having more general conditions for being an instance.  In order to
   specify a set, we need only give (list, characterize) its members.
   To specify a category we need to do the same thing for both the
   collection of objects and the collection of arrows, and then we
   need to specify the composition law.  (Even in an arrows-only
   formulation of category theory, we still need to specify both the
   collection of arrows and the composition law.)  So, on this
   criterion, sets come out as more abstract.

Some time ago, on the Foundations of Mathematics mailing list (FOM),
there was a long and sometimes heated debate on alternative
foundations of mathematics (where alternative meant non-set-theoretic)
-- in particular on the viability of some kind of category-theoretic
foundation for mathematics (e.g., elementary topos theory + some
additional axioms) -- and the majority view seemed to be that

- Set theory is more all-encompassing.  The standard arguments about
  the bi-interpretability of category theory and set theory were met
  with the challenge (unanswered, as far as I know) to produce, in a
  category-theoretic foundation, a natural linearly-ordered sequence
  of axioms of higher infinity that can be used to "calibrate" the
  existential commitments of extensions to the basic axioms comparable
  to the large cardinal axioms of set theory, where the naturality
  requirement supposedly precludes the slavish translation of these
  large cardinal axioms into the language of category theory.  (Recall
  that all known large cardinal axioms for set theory fall into a very
  nice linear hierarchy that can be used to gauge the consistency
  strength of a theory.)

- Set theory is conceptually simpler.  Set theory axiomatizes a
  single, very basic concept (membership), expressed using a single
  binary relation, and posits a natural set of axioms for this
  relation that are (more or less) neatly justified in terms of a
  fairly (some would say perfectly) clear semantic conception, the
  cumulative hierarchy.  Category theory, the view goes, could only
  approach the scope of set theory, if at all, by adding many axioms
  that are unnatural and quite complicated to state and work with
  without the aid of multiple layers of definitions and definitional
  theorems (for products, exponentials, power-objects/subobject
  classifier, higher replacement-like closure conditions on the
  category, etc.).

The arguments put forward in support of these views were very similar
to those that are implicit in the labeling of category theory as
"ridiculously abstract", and there are no doubt many readers of this
list who would disagree with part or all of these views (me, for one).
However, my intention in reporting them here is *not* to start another
set-theory vs category theory thread, but rather to point out that,
although category theorists have yet to make a convincing case -- at
least I haven't seen one -- that category theory is more fundamental
or foundational in any important sense (sorry, Paul), recent research
in cognitive science on the embodied and metaphorical nature of our
thinking indicates that category theory may well be able to make such
a claim after all.  See the books

    G. Lakoff and M. Johnson.  Philosophy in the Flesh: The Embodied
    Mind and Its Challenge to Western Thought.  Basic Books, 1999.

    G. Lakoff and R. Nuñez.  Where Mathematics Comes From: How the
    Embodied Mind Brings Mathematics into Being.  Basic Books, 2000.

for a popular account of this research.  I should mention, of course,
that, closer to home, the book

    F.W. Lawvere and S.H. Schanuel, Conceptual Mathematics:  A First
    Introduction to Category Theory.  Cambridge University Press, 1997. 

is certainly a step in this direction.

-- 
Todd Wilson                               A smile is not an individual
Computer Science Department               product; it is a co-product.
California State University, Fresno                 -- Thich Nhat Hanh