SUO: IFF Thread 1: Why Category Theory?
All,
Many of the 32 messages from our recent first round of comments on the IFF
starter documents dealt with the topic "Why Category Theory" for the IFF.
Please consider this new thread a vehicle for continuing a focused
discussion on that topic.
I am particularly interested in whether those who raised and reinforced
this issue have received satisfactory responses. My suspicion is that
they have not. I hope we can use this thread to get this issue dealt with
satisfactorily for all.
I include below 5 sections, extracted from different messages. Each section
includes the relevant SUO message numbers. I am not necessarily
suggesting that we continue to lug all this text around in replies
to this thread. I include it here primarily for consolidated reference.
In fact, it would actually help the editing process if you can include just
those pieces to which your response pertains. Use your best judgment.
If I have made errors, please let me know off list. I will correct
them and report back to the list.
You may wish to refer to certain subsets of comments available from
http://suo.ieee.org/IFF/comments/cindex.html
Thank you for your input,
Jim Farrugia
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Here we go:
SECTION 1 is a from exchanges between Pat Hayes and Jon Awbrey
(with two other comments tucked into the top part.)
SECTION 2 is from exchanges between Josiane Caron and Chris Menzel.
SECTION 3 is from exchanges between Josian Caron and Pat Hayes.
SECTION 4 is from Matthew West.
SECTION 5 is from exchanges between Leo Obrst and Pat Hayes
Name abbreviations are as follows:
JA = Jon Awbrey
PH = Pat Hayes
CM = Chris Menzel
JC = Josiane Caron
LO = Leo Obrst
MW = Matthew West
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SECTION 1 - all comments from http://suo.ieee.org/email/msg06691.html
except where otherwise indicated
PH: OK, I have a few.
PH: First, I fail to see the utility of the emphasis placed on category theory.
This is not motivated anywhere, but it badly needs to be motivated if you
expect anyone to take it seriously enough to even read the sources to
find out what you are talking about.
CM: I heartily agree. I've never heard an adequate answer to this question.
( http://suo.ieee.org/email/msg06636.html )
LO: I agree the motivation is missing from the document. I do think there
is a motivation. I hope to add to it incrementally.
...
I intend to discuss your issues a bit further (and later), since I don't
see any incongruity between the "logic" view and the "category theory"
view. The category theory view is simply more general.
( http://suo.ieee.org/email/msg06704.html )
JA: I think that the following is a fair statement:
A modest amount of category theory, along with
a modest amount of set theory, is indispensable
to understanding what mathematics is about and how
mathematics is done today. This is important, not
just for representing the ontology of mathematical
objects, structures, and systems, but further, and
more importantly for applications, because these
objects, structures, and systems are used in
modeling most other objects, processes, and
situations of any complexity that anyone
might happen to care about.
PH: Well, maybe. I would still like to see a bit more detail, however.
JA: Be careful what you (pretend to?) wish for.
PH: No pretence, I assure you. Notice that I am not asking for a tutorial
on category theory (not do I propose to give anyone else such a tutorial,
though I will hold people to their promises on foundational issues), but
a reasoned account (if only brief) of what its perceived relevance is to
the overall topic/project of the SUO.
JA: I think that I already said this in my initial statement,
and I limited myself to facts that are fairly obvious to
anyone who looks around at what tools are actually being
used, where and for what. I am trying to get out of the
habit of repeating the obvious to an unneccesary degree.
PH: Just being nifty mathematics that all good men should
know isn't good enough (and one example of one person's
work isn't good enough, either, Jon, just to save you
the cutting and pasting.)
JA: This is hardly a matter of a few people's eccentricities.
PH: This response really amounts to saying that Category Theory is a Good Thing.
JA: I am saying that a modest amount of category theory is indispensable
to several of our objectives here, which, though qualified, is still
a slightly stronger statement.
PH: Well, again, I'd like to see a case made. The case may be there,
but it isn't obvious; it needs to be actually stated. A little
brisk canter through the foothills of Birkhoff and McLane might be
good mental exercise, but I need a lot more convincing to base my
entire ontological metatheory on foundational distinctions which
are only meaningful in circles that are even more restricted than
the FOM mailing list.
JA: What you say here does not give an accurate picture of the situation in
the field. Category theory is a basic tool in widespread use, for basic
algebra and analysis, discrete math and computer science, automata and
formal languages, modeling and simulation, yea, verily, unto all the
sinful vales of impure math and engineering. In many applications it is
preferred to set theory for its utilities in getting the subject off the
ground far more quickly than would otherwise be possible.
PH: In fact, however, I think that its influence has almost entirely been
in pure mathematics (where indeed it is part of the general competence
expected of a professional mathematician these days) but hardly at all
outside pure mathematics (and even within large parts of mathematics,
it really amounts to little more than a style of terminological usage.)
JA: Yes, there are people who do category theory for its own sake, just
as there are people who do descriptive set theory for its own sake,
but the word "modest" was meant to set aside those further reaches
of both subjects. I'm talking about the part of category theory
that is a standard working tool in almost every branch of math
that I can remember taking. It ain't all that much, but it's
"indispensable".
PH: It hardly involves getting all concerned with distinctions like
proper class versus set, however. These are distinctions in the
foundations of topos theory, not the usual fodder of the working
mathematician.
JA: I tend to agree with this. It has always been my sense that this
distinction is more of a non-small deal to the set theorist, though,
and that the working class of category-theory-users bother with it only
so that set theorists will continue taliking to them. Just my sense of
the sociodynamics -- I am not up to talking topoi/toposes just yet. For
my part, I use the distinction mostly to mark off the more overweening
stuff that I do not care all that much about.
PH: Most ontological modelling is not mathematical modelling,
and category theory plays virtually no significant role
in mathematical modelling in any case. Fractal theory
would be far more germane, for example.
JA: By "mathematical model" I meant not just the sort of elaborate model that
usually comes to mind, but the sort of thing that we are doing as soon as
we use "abstract objects" like numbers and sets as "intermediary fictions",
if you prefer to think of them that way, to describe a world that is made
of more concrete things like numbers of apples, sets of oranges, and even
on upward to the hydrodynamic flow of oil through a pipeline from Seville.
In my bemused observations of the discussions of elementary set theory
that have belabored this group for more than a year now, I sense that
a due appreciation of difference between the territory being modeled
and the mathematical objects forming the model, even in such a basic
form of modeling activity as counting and "setting" them, might just
be, well, indispensable to our progress here.
PH: I would agree that particular 'due appreciation' is rather important, and
that it isn't found widespread in nature. But that has nothing specially
to do with topos theory. If anything, topos theory is rather cavalier
about the the map/territory distinction, like most of mathematics, since
it is almost entirely concerned with the maps (in the case of topos theory,
literally so); and one certainly does not *need* topos theory to understand
these distinctions or to state them properly. In fact, I find the topos
METAtheory to be quite opaque, c.f. my comments on the interpretational
issues arising from first-order axiomatizations of topos theory. I
certainly can see no good argument for adopting such strongly-worded
slogans as the 'categorial principle', or for assuming that our first
business should be to give a foundational ontology for categories.
JA: I did not address the topic of topos theory in my initial remark.
Some people include it in their modest set, some do not. Given
the number of times in my life that I have had to readjust my
sense of modesty, I think that I will bow out on this for now.
But a modicum of category theory has been found by very many people
to be of utility in organizing their studies of modeling relations --
this is only natural when you consider the fact that a morphism is
just a mathematical metaphor. Of course, people are free to avoid
the use of any tool that they are dead-set against -- the meaning
of the word "indispensable" is not designed to cover outliers like
that -- but if the tool is really "necessary" in a practical sense
the demands of the problem domain itself will usually require them
to re-create it , however obscurely transmongrelfied.
My own crisis of foundations is decades behind me now -- from the pragmatic
perspective that I take today I tend to think of the whole thing as having
been yet another undergraduate hangover of post-cartesian
dis-illusionment -- and now there is just too much of a practical nature
that can be done and demands to be done while yet I have the time to do
it. I am not all that concerned with telling people where they should
live, foundation-wise, or even how they should talk -- human nature is
just plain against it -- but if I can string a few lines of communication
between diverse perspectives, I think that it'd be worth the trouble to
do so, and non-obtrusive enough for almost anybody -- though not for the
Shakers, bless their souls.
PH: BTW, perhaps it would aid communication if I said why I tend to
be rather cynical about basing ontology activity in topos theory,
or indeed almost any other piece of 'neat' mathematics. This cynicism
comes from many years trying to apply elegant mathematics to real-world
ontologizing, and finding again and again that the structures one wants
most to capture are precisely the ones that do not fit well into the
mathematical theories. Reasoning with notions of approximation would
be easy if approximations were equivalence relations, or metric open
spheres or closed algebras of one kind or another; but they aren't,
which is where the ontology gets interesting. Real-world 'fractal'
spaces arent quite genuinely fractal: they typically are only in
some ad-hoc ranges of scale. Ontological spatial reasoning isnt
supported well by conventional topologies (which are too 'rubbery')
or by conventional metric-space geometry (which is too 'stiff');
and so on. Topos theory is of such wonderful utility in mathematics
precisely because it is a general framework for all kinds of otherwise
disparate mathematical structures (and so it greatly simplifies and
rationalizes the process of proving theorems about them), but these
structural families all have the characteristic closure properties
that realistic spaces, of the kind that physical ontologies must
often try to tackle, usually conspicuously lack. Mathematics
studies the universals; ontology is often trying to do
a better job on a smaller territory.
JA: I think that different people have different ideas about the scope of
"neat" mathematics. Most of the areas that I consider the "neatest"
were not on your list. Closure properties are a convenience appropriate
to maps -- how else would you fold them? -- and were never meant to be
taken as properties of the landscape, nor indeed of mathematics itself.
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SECTION 2
JC: I do not yet know if IFF is able to work for every problem I have
already list. But I am sure I can start with.
( http://suo.ieee.org/email/msg06656.html )
CM: Have you already attempted to approach your problems via the
rather simpler and more familiar framework of first-order logic and
model theory? If not, why start with category theory? If so, could you
detail exactly the shortcomings you found in the first-order approach?
( http://suo.ieee.org/email/msg06659.html )
JC: Nice to answer me, but I am not sure to understand what you are
referring to with the terminology ' first order logic and model theory' ?
are you referring to the different indications you gave in your preceding
mails about the set/class distinctions, and that I had commented? In this
case it would be easy for me to explain what I understood, viz how to use
some of the elements or functions to modelize several accounts and notably
what I called 'locations' in my verbal protocol analysis.
and I would be very glad to have your comments.
( http://suo.ieee.org/email/msg06664.html )
CM: Yes, now that I look back, pretty much, though the point there was
to lay out the standard first-order approach to *sorted* logic.
...
I'd be happy to see it -- though my interest is not so much in the
specific content of your work (interesting as I'm sure it is!) but in any
*shortcomings* of the first-order framework that you believe are addressed
by IFF. I am at this point completely unconvinced that IFF offers any
insight or utility over first-order logic vis-a-vis building ontologies
(or even building the theoretical underpinnings of ontology generally), so
would interested in seeing any evidence to the contrary.
( http://suo.ieee.org/email/msg06668.html )
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SECTION 3
JC: But most of current work on problem solving consider only success
or failure of experts' strategies. They do not consider the
strategies of novices, notably at the the begining of the solving
process for two reasons: one is that they do not know how to do;
the second reason is that it is too much difficult and that it needs
too much work. It is not that this kind of research would not
interest them. If the work to know how to do was easier they will do.
( http://suo.ieee.org/email/msg06656.html )
PH: I'm not sure I follow your point. Are you meaning to suggest that
topos theory is likely to provide a good foundation for a
psychological theory of performance of novices? That is a remarkable
idea. You may be right, but (after trying to teach topos theory to
undergraduates) I find it so wildly unlikely that I would like to see
some case made for it.
( http://suo.ieee.org/email/msg06674.html )
JC: No I do not suggest that topos is a good foundation. I said that in Kent's
version it did not bother, disturb me because I was able to understand.
But I did not say that it was the part that I like and see how to use.
The other times I read or hear work on topos I understood nothing. I
prefer more simple vocabulary.
( http://suo.ieee.org/email/msg06687.html )
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SECTION 4
MW: One of the things I have been finding unsatisifactory is the
concept of a relation, and over the last few years I've been looking
at this with others to try to understand what we mean when we use that
construct. Interestingly, when I now start to read about Category
Theory, I find a very good correspondance with where that was going.
So I think it likely that Category Theory has a proper place.
( http://suo.ieee.org/email/msg06742.html )
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SECTION 5
LO: However, we do need to build a framework which enables
us to cobble together different ontologies (theories) in a reasonable
fashion and perhaps even a framework to enable us to "compose" theories,
i.e., to be able to "place" theories in an overarching framework and
then link or (God help us) even "project" those theories'
"intersections". If we don't do this, who will? Personally, I think
category theory will help.
( http://suo.ieee.org/email/msg06704.html )
PH: Ah, now THAT does make sense. BUt this seems to be quite a different
theme than the one in the IFF document. What you are saying here is
that the category theory can provide a framework for a kind of
structural meta-theory of ontologies; for talking about structural
relationships between ontologies, as is done in the SpecWare system
also. That isn't what the IFF seems to be saying at all, though: it
is talking about categories being the *subject-matter* of ontologies.
For example, your picture of categories in the structural meta-theory
is quite compatible with the ontologies themselves being written in,
say, DAML and being about, say, states of viscosity of different
grades of crude oil.
( http://suo.ieee.org/email/msg06712.html )
LO: I intend to discuss your issues a bit further (and later), since I don't
see any incongruity between the "logic" view and the "category theory"
view. The category theory view is simply more general.
( http://suo.ieee.org/email/msg06704.html )
PH: Well, if so, I'm tempted to ask, why is FOL offered as the basic
axiomatization of topos the
( http://suo.ieee.org/email/msg06712.html )
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