Michael Healy's question on math and AI

Michael Healy's question on math and AI

[F. William Lawvere]

Re : Michael Healy’s question on math and AI


This is to answer Mike and also several other people who have contacted me 
recently asking how I would respond to queries about

(1)	Artificial Intelligence, cognitive science, linguistic engineering, 
knowledge representation, and related attempts at creating modern subjects, 
and

(2)	the relevance of category theory and of mathematics in general to these.

     My basic response is strong advice to actually learn some category 
theory, rather than resting content with slinging back and forth ill-defined 
epithets like “set theory”, “contingency”, etc..

So much confusion has been accumulated that an opposition of the form
“set-theoretical versus non-set-theoretical” has at least seven wholly 
distinct meanings, hence billions of electrons and drops of ink can be 
spilled by surreptitiously identifying any two of these.  For example, the 
opposition can concern whether or not large cardinal assumptions are needed 
for a certain result, which is mathematically meaningful and hence 
independent of whether or not the ZFvN rigidification of Cantor is being 
used as a framework.  Another example is the opposition habitually used in 
geometry between properties of spaces which can be explained in terms of 
arbitrary mappings versus those which depend on the cohesion being studied 
(e.g. “the underlying abstract group vs. the Lie group”). Obviously these 
two oppositions are not the same although they may be related.

One of the oppositions which I have emphasized since 1964 is
    the ZFvN rigid hierarchy based on galactically “meaningful”       
inclusion, requiring the totally arbitrary “singleton” operation of Peano 
with the resulting chains of mathematically spurious rigidified membership, 
on the one hand,
                             versus
    the category of abstract sets, involving many potential universes of
discourse and arbitrary specific relations between them, on the other hand.
(Abstract sets can CARRY structures of mathematical interest, but precisely 
because of the need of flexibility in the latter, they themselves have only 
very few properties, unlike the ZFvN “sets”).

Within Cantor’s original conception itself there is a fundamentally 
important opposition: the abstract sets, which he called “Kardinalzahlen”, 
versus the cohesive and variable sets which he called “Mengen”.  (An 
additional confusion stems from the use, by nearly all of Cantor’s 
followers, of the term “cardinal number” to mean (not a 
Kardinalzahl=abstract set, but) a label for an isomorphism class of abstract 
sets, an invariant which Cantor of course also studied, but which is too 
abstract to support the specific relations between abstract sets themselves, 
the mappings, and hence cannot carry the needed mathematical structures).

(A)    The real issue is that for purposes of pure AND applied mathematics, 
we need to be able to represent (without spurious ingredients) these 
cohesive and variable sets (or “spaces”) and their relationships.  The ZFvN 
rigidification fails so miserably in doing this that even those geometers 
and analysts who pay lip service to it as a “foundation” never in practice 
use its formalism.

(B)   Category theory made explicit some universal features of the 
relationship between quantity and quality whose fundamental importance had 
been forced into consciousness by the work of Volterra and Hurewicz (both of 
whom made basic contributions to both functional analysis and algebraic 
topology) and of many others. This relationship between quantitative and 
qualitative aspects concerns cohesive and variable sets and structures built 
on such spaces.  For example, Volterra already recognized that spaces have 
“elements” other than points, and Hurewicz recognized the need for 
cartesian-closed categories (even before the lambda-calculus formalism, or 
category theory, was devised); moreover, the original fiber bundles were 
explicitly modeling dynamical situations, etc.

Many people working in the new fields, striving to realize the dream of a 
theoretical computer science, do not seem to be aware of points like  (A) 
and (B). It would certainly be a bad strategy for the advancement of science 
to “hide” the fact that category theory belongs to the background of a new 
result and thus to help perpetuate that sort of ignorance.

The role of mathematics in general (not only of category theory) also
seems to be widely misunderstood, even in those fields which definitely need 
more mathematics in order to mature and make a real contribution.  For 
example, some say that logic is more general than mathematics, partly 
because of ignoring the strongly qualitative aspect of modern mathematics 
and partly because of the philosophical tradition of hiding the fact that no 
logic other than mathematical logic has had any significant real-world 
applications. Because of the minimal
mathematical education required of students of philosophy, the claim is too 
easily accepted in many philosophical circles that “mathematics is 
unsuitable” for some given issue of conceptual analysis; this conclusion 
seems to be based on the syllogism:
        mathematics is set theory (a misconception which the philosophers
themselves have done much to disseminate),
        set theory is clearly not suitable (actually because of the defects 
of the ZFvN rigidification, which make it ill-suited for mathematics as 
well)
        hence ......
This syllogism serves as an excuse to indefinitely postpone learning 
mathematics (and category theory in particular).

An older sort of excuse is the assertion that the proposed science should 
concern the REAL WORLD, not pure mathematics. This superficially appealing 
truism has frequently been used to mask the fact that comparing reality with 
existing concepts does not alone suffice to produce the level of 
understanding required to change the world; a capacity for constructing 
flexible yet reliable SYSTEMS of
concepts is needed to guide the process. This realization (not Platonism) 
was the basis of the supreme respect for mathematics expressed by champions 
of reality like Galileo, Maxwell, and Heaviside. For example, the 
differential calculus provides the capacity to construct systems descriptive 
of celestial motions, fluid interactions, electromagnetic radiation fields, 
etc., and therefore engineers have learned it. The functorial calculus helps 
to provide a similar capacity adequate to the requirements, not only of the 
older sciences,
but of the newer would-be sciences as well. Hence my response.

                 Bill Lawvere

Re: Michael Healy's question on math and AI

[Todd Wilson]

On Wed, 24 Jan 2001, F. William Lawvere wrote:
> (A)    The real issue is that for purposes of pure AND applied mathematics, 
> we need to be able to represent (without spurious ingredients) these 
> cohesive and variable sets (or `spaces') and their relationships.  The ZFvN 
> rigidification fails so miserably in doing this that even those geometers 
> and analysts who pay lip service to it as a `foundation' never in practice 
> use its formalism.
> 
> (B) [...]
> 
> Many people working in the new fields, striving to realize the dream of a 
> theoretical computer science, do not seem to be aware of points like  (A) 
> and (B). 

As someone who is "striving to realize the dream of a theoretical
computer science", I would better like to understand the point that
Lawvere is making here.  Am I right in assuming that, in using terms
such as "spurious ingredients" and "rigidification", Lawvere is
referring to the fact that (to use some computer science terminology)
set theory is too much implementation and not enough specification?
That the rigid epsilon-structure of set theory cannot represent
abstract mathematical structure faithfully, without introducing
unwanted details?

If so, can category theory really do better?  Can we give some
concrete examples in both "pure AND applied mathematics" that really
make the difference in representational ability clear?  (These
questions, like the ones below, are not rhetorical or deprecatory; I'd
really like to know some answers.)

To take the first example that comes to mind, consider the cartesian
product of two objects A and B.  The "implementation" of this in set
theory as a set of ordered pairs (which are themselves specific
doubleton sets) certainly introduces some "spurious ingredients", but
the category-theoretic version has its own idiosynchrasies as well:

- Although we constantly speak of "the" product, we really only have
  "a" product (at least if we take the category-theoretic perspective
  seriously).  What is really involved, formally, in making the move
  from "a" to "the"?  A formal language translation scheme?  Coherence
  theorems?  How much technical work is really involved here?

- Related to this, what about the fact that if

      (pi0: A x B -> A, pi1: A x B -> B)

  is a product, then so is (pi1, pi0), indistinguishable categorically
  from the other product?  Does the arbitrary choice between one of
  these products introduce a "spurious ingredient"?  If we find this
  particular "implementation detail" aesthetically displeasing, can we
  abstract away from it by defining an "unordered cartesian product"?
  (I couldn't see how to do it.)

- Is there anything to be made of the fact that the set-theoretic
  cartesian product is a local construction, involving only the sets A
  and B and certain small sets made up of their elements, whereas
  a/the category-theoretic product depends on the whole category
 (because of the quantification in the universal property)?

And if these idiosynchracies do carry any weight (and I'm not claiming
that they do), why are they "better" idiosynchracies than those of the
set-theoretic cartesian product?  And, finally, shouldn't "better"
really be "better for what"?  In other words, aren't the two
communities really just arguing past one another, like people arguing
over types of automobile?  What really is the issue here?

Sorry for all the questions (and all the "really"s).

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

Re: Michael Healy's question on math and AI

[Michael Barr]

At the risk of offering my head on a tray, I will make a stab at some of
this.

On Thu, 25 Jan 2001, Todd Wilson wrote:

> On Wed, 24 Jan 2001, F. William Lawvere wrote:
> > (A)    The real issue is that for purposes of pure AND applied mathematics, 
> > we need to be able to represent (without spurious ingredients) these 
> > cohesive and variable sets (or `spaces') and their relationships.  The ZFvN 
> > rigidification fails so miserably in doing this that even those geometers 
> > and analysts who pay lip service to it as a `foundation' never in practice 
> > use its formalism.
> > 
> > (B) [...]
> > 
> > Many people working in the new fields, striving to realize the dream of a 
> > theoretical computer science, do not seem to be aware of points like  (A) 
> > and (B). 
> 
> As someone who is "striving to realize the dream of a theoretical
> computer science", I would better like to understand the point that
> Lawvere is making here.  Am I right in assuming that, in using terms
> such as "spurious ingredients" and "rigidification", Lawvere is
> referring to the fact that (to use some computer science terminology)
> set theory is too much implementation and not enough specification?
> That the rigid epsilon-structure of set theory cannot represent
> abstract mathematical structure faithfully, without introducing
> unwanted details?
> 
> If so, can category theory really do better?  Can we give some
> concrete examples in both "pure AND applied mathematics" that really
> make the difference in representational ability clear?  (These
> questions, like the ones below, are not rhetorical or deprecatory; I'd
> really like to know some answers.)
> 
> To take the first example that comes to mind, consider the cartesian
> product of two objects A and B.  The "implementation" of this in set
> theory as a set of ordered pairs (which are themselves specific
> doubleton sets) certainly introduces some "spurious ingredients", but
> the category-theoretic version has its own idiosynchrasies as well:
> 
> - Although we constantly speak of "the" product, we really only have
>   "a" product (at least if we take the category-theoretic perspective
>   seriously).  What is really involved, formally, in making the move
>   from "a" to "the"?  A formal language translation scheme?  Coherence
>   theorems?  How much technical work is really involved here?
> 

Any two products are uniquely isomorphic in a way that preserves the
projections in the obvious way.  That is all that need be said.  There are
coherence statements that can be made, but they follow from the above and
are unnecessary.


> - Related to this, what about the fact that if
> 
>       (pi0: A x B -> A, pi1: A x B -> B)
> 
>   is a product, then so is (pi1, pi0), indistinguishable categorically
>   from the other product?  Does the arbitrary choice between one of
>   these products introduce a "spurious ingredient"?  If we find this
>   particular "implementation detail" aesthetically displeasing, can we
>   abstract away from it by defining an "unordered cartesian product"?
>   (I couldn't see how to do it.)
> 

The product is the product of the set {A,B}, which is equal to the set
{B,A}, but our orthography forces us to write one or the other.  Of
course, a product is really defined for {A_i|i in I} and is inherently
unordered.  In set theory, the usual A x B is quite a different set from B
x A and in category theory they are indistinguishable.  You seem to
consider that a disadvantage to category theory, but I consider it an
advantage.

> - Is there anything to be made of the fact that the set-theoretic
>   cartesian product is a local construction, involving only the sets A
>   and B and certain small sets made up of their elements, whereas
>   a/the category-theoretic product depends on the whole category
>  (because of the quantification in the universal property)?
> 

Our familiar categories have regular generators and for them the product
condition can be reduced to the universal mapping condition when the
domain is/are the generator(s), which is local.  On the other hand, check
out the product in the category of affine schemes that is really
comprehensible only in terms of the categorical definition.

> And if these idiosynchracies do carry any weight (and I'm not claiming
> that they do), why are they "better" idiosynchracies than those of the
> set-theoretic cartesian product?  And, finally, shouldn't "better"
> really be "better for what"?  In other words, aren't the two
> communities really just arguing past one another, like people arguing
> over types of automobile?  What really is the issue here?
> 
> Sorry for all the questions (and all the "really"s).
> 


For another example, consider the traditional definition of Z as the set
{0,{0},{0,{0}},{0,{0}{0,{0}}},...}
and contrast that to the categorical specification.

Michael Barr

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

Re: Michael Healy's question on math and AI

[zdiskin]

 It seems that a discussion "CT vs. ST " (despite some silent resistance 
at the beginning :) has nevertheless started in this list too, and is 
going in a few different directions ranging from rather technical thru 
methodological to philosophical aspects. Probably it's unavoidable (and
useful) when we talk about math in general and its applications but, as 
Bill Lawvere pointed, too much quite different stuff is packed into the 
same title of CTvsST and the type mismatch in the very discussion and 
its comprehension by the (rather diverse as I could guess) audience is
very possible. (Under type mismatch I mean something like this. Suppose 
we discuss the result of  multiplication 3 x 3. If the alternatives  are 
8,9 and, say, 12, we have good chances for a reasonable discussion, and 
if even the alternatives are 12, 37 and 111, type correctness is still 
respected and we have chances to achieve useful results, but if the 
alternatives to be discussed are 8, 9, that triangle and that <favorite 
keyword>, the situation is hopeless). So, some methodological 
arrangement and figuring out explicitly the relevant meanings of the 
CTvsST-problem would be useful.
 Here are a few contexts already touched in the discussion, where we 
deal with methodologically quite different problems whose merging under 
the same title CTvsST may be misleading.

 a). Ways of setting/defining math structures.

 The actual meaning of CTvsST here is the opposition between the 
following two ways (described by Steve Vickers yesterday, I'll just 
rephrase slightly his presentation).

--(1)  Math structure  Carrying set (of abstract elements) + 
Structure over it defined in terms of the elements (so, structure 
resides in elements of the carrier).

--(2)  Math structure  Category (collection of abstract objects and 
their morphisms) + Structure over it defined in terms of the morphisms 
(and then structure on a carrier object  resides in morphisms adjoint to 
it).

 The title CTvsST may be misleading here since in the both ways we use 
sets, their elements, mappings between them. To name the two ways 
somehow, I'd propose to call the 1st  one Boubakian (since this way of 
setting math structures got its classical completion in Bourbaki's 
volumes), and the 2nd one categorical. So, the actual opposition here is 
CatStr vs. BrbStr (but, of course, a category with structure is itself a 
Bourbakian math structure)

 This opposition is of extreme great relevance for software engineering, 
business modeling, knowledge representation ... but even brief outline 
of the reasons needs a more detailed description of the issue. Let's 
postpone that for a next posting. 

 b). Modeling (formalization) of set theory underlying the setting for 
math structures and reasoning about them (usually referred to as 
foundations).

 The actual meaning of CTvsST here is "categorical set theory" vs. 
"formal set theory(ies)" (ZF, NBG,...), or CatST vs. FrmST.

 This context and meaning are totally irrelevant to applications i 
question and asking about what is better, CT or ST, for applications in 
AI,SE, ... is very much like asking what is better for applications in 
mechanical engineering: the differential/integral calculi or ST;  or  
what is better for general theory of relativity, the tensor calculus or 
ST etc. 

 c). Modeling (formalization) of reasoning about math structures (often 
called metamathematics)

 Of course, what we have here depends on which way of setting math 
structures we consider: Bourbakian or categorical. Nevertheless, it's 
possible and make sense to treat reasoning about Bourbakian  structures 
categorically and, say, to treat reasoning about categories in a 
elementwise fashion.

 We again have two different approaches. Historically first (originated 
by Tarsky and Mal'cev) was formalization of first order logic (FOL, 
including syntax and semantics) in a quite immediate way now well known 
to a wide audience of quite different backgrounds including computer 
scientists and philosophers. This way is often referred to as "Tarskian 
formalization of the notion of truth", and let's call this entire style 
Tarskian MetaMathematics, TarMMath.

 The other approach was developed in CT and is usually called  
categorical logic, CatLog. So, the opposition we actually have here is 
CatLog vs. TarMMath, or, if you prefer, CatMMath vs. TarMMath.

 This oppostion, though of course connected with that in (a), CatStr vs. 
BrbStr, has its own peculiarities which are of extreme high relevance 
for knowledge representation, business modeling and similar domains. So, 
details here would be very useful but I'd again postpone them for a next
posting.

 So, as it was said in Lawvere's note, there are a few aspects of CTvsST 
(understood in the wide sense), some of them may be not relevant for 
applications (for example, b) while others are of great importance and 
deserve more detailed exposition (a,c).  On the other hand, all the 
three context we have considered are special cases of a quite general 
intellectual activity usually referred to as modeling. Mathematics is 
itself a special (very refined) discipline of modeling, and foundations 
and meta-mathematics are nothing but modeling math by math means (well, 
there are other means, say, philosophical :). It seems that to make 
these contexts more understandable for a wider audience (and for myself 
:), not only more details should be provided but some sketch of what 
modeling in general and math modeling in particular are, would be also 
useful. So, I'd ventured to sketch some general tutorial on applying 
math to engineering domains and to itself but certainly it's for a next 
posting.

 --Zinovy Diskin

Re: Michael Healy's question on math and AI

[John Duskin]

At the risk of putting my head on a platter (as Mike Barr 
said).....If you take the point of view of ``representable functors", 
an object P, together with an orderded pair of arrows (pr_A:P --->A, 
pr_B:P---->B) is a product of A with B iff for all objects T, the 
mapping Hom(T,P)--->Hom(T,A)xHom(T,B) defined by 
f|---->(f.pr_A,f.pr_B) is a bijection. If you compose this map with 
the bijection
(a,b)|--->(b,a): Hom(T,A)xHom(T,B)--->Hom(T,B)xHom(T,A), you get that 
P, together with the ordered pair of arrows ( pr_B:P---->B, pr_A:P 
--->A) represents a "product of B with A". and this is different even 
if we are talking about a product of A with itself. In other words, 
the "product we usually are thinking about"  is universal for ordered 
pairs of arrows (a:T--->A,b:T--->B). If this means that "ordered 
pair" needs to be made a primitive notion within the underlying logic 
(as the Bourbakists did), so be it, because the simple translation of 
the above statement out of "set theoretic" terms" into "category 
theoretic" terms (no Hom-sets allowed) needs the notion of "ordered 
pair" in order to state it independently of any overarching "theory 
of sets".

Re: Michael Healy's question on math and AI

[S.J.Vickers]

Todd Wilson asks:
> As someone who is "striving to realize the dream of a theoretical
> computer science", I would better like to understand the point that
> Lawvere is making here.  Am I right in assuming that, in using terms
> such as "spurious ingredients" and "rigidification", Lawvere is
> referring to the fact that (to use some computer science terminology)
> set theory is too much implementation and not enough specification?
> That the rigid epsilon-structure of set theory cannot represent
> abstract mathematical structure faithfully, without introducing
> unwanted details?

Executive summary of a long reply
-------------------------------------------------

* Spaces - for instance topological spaces - cannot always be analysed as
set + structure.
* Category theory enables structure to be found amongst the
morphisms.(generalized elements) instead of within the objects (amongst
concrete elements in a set-theoretic analysis).
* Variation of kind of space bears some relation to variation of logic.
* The logical questions are important in computer science, even if topology
and geometry at first appear not to be.
------------------------------------------------------------------


The discussion of Cartesian product does illustrate set theory vs.
categories as implementation vs. specification, and, sure, there's a lot to
be said for specifying categorically before you implement set-theoretically.

However, Bill's point is much deeper than that. He is saying that set theory
is actually inadequate as an implementation language in many contexts.

I'll set out my own understanding of this, but I hope I'm not
misrepresenting Bill if I draw on his note.

I think within his note you already see two criticisms of set theory.

The less deep of the two is of the "rigidity" of the epsilon-structure. Most
mathematicians would probably agree that it doesn't make much sense to
investigate the elements of pi (say), yet still view set theory as a
remarkably successful encoding of mathematics into an epsilon-structure,
successful enough to be used as foundations.

The deeper criticism is that even for things that are unambiguously
collections of some kind, so they really do have elements, the set-theoretic
approach is inadequate. Bill refers to "cohesive and variable sets (or
`spaces')", and I take it that an example of "cohesive" set is a topological
space. There it is not enough to know merely what the elements are. They
also "cohere", sit in rather subtle relationship with each other. Maps
should respect the coherence, i.e. be continuous.

One way to view this is that in order to know the coherence in a space X you
need to know not just its "point-like" elements but also its line-like,
plane-like or any-shape-you-choose elements - as Bill said, "Volterra
already recognized that spaces have elements other than points". These are
the "generalized elements", the maps from other spaces to X. Hence in the
category of spaces the structure of X is seen not within the object but
amongst the morphisms.

Correspondingly, we get two approaches to topological spaces.

1) Set theory - structure within objects. Space = set + topology. The
topology provides criteria so that out of all the functions, mostly
discontinuous, between the sets of points, some can be identified as
continuous.

2) Category theory - structure amongst morphisms. We are now free to
describe the objects in point-free ways, for instance the frames of locale
theory, the schemes of algebraic geometry, or Grothendieck toposes as
generalized spaces, and still hope to get (via the generalized elements) a
good spatial feel for the objects.

Given a choice, why use the point-free ways at all?

But in fact there's less of a choice than one might think, as the geometers
discovered.

Approach (1) does not generalize well to situations where you want to vary
the set theory. This want arises naturally, for concrete mathematical
reasons, when you start working with sheaves, since the class of sheaves
over a space provides a non-standard set theory, usually non-classical, and
one in which rigid "element of" or "subset of" are not useful. As you vary
the set theory there are ways of transforming sets in one to sets in
another, but it turns out that sets of points of spaces cannot be
transformed in the same way: the space and its set of points part company.
[See Appendix for a technical discussion.]

Examples thus show that the decomposition "space = set + topology" is not
stable under change of set theory.

The inadequacies of point-set topology are well known in constructive
mathematics, where the point-set statements of important results such as the
Heine-Borel theorem turn out to be false while point-free versions still
hold.

Note that category theory in itself doesn't tell you what category to use
(sheaves? locales? toposes? schemes?). But it does help you to see what you
are looking for in a category of spaces that goes beyond space =
set+structure.

Computer science
---------------------------

That was a long preamble to say Beware! Spaces might not be sets!

But why should workers on Knowledge Interchange Formats worry about this?
Why can't they just use first-order logic, assume a classical set-theoretic
semantics and keep well away from sheaves?

One reason lies in the classical first-order logic itself. It works in a
blandly uniform way on its formulae that ignores any difference in status as
knowledge. For instance, maybe a knowledge database should distinguish
between formulae that represent observed facts and those that represent
background assumptions or scientific hypotheses. It could make a difference
to what negation means or how negation behaves. But such nuances are not
present in classical first-order logic.

These logical questions could be a natural part of the theory of knowledge
representation.

The interesting thing is that the variability of logic can be connected to
the possible un-set-likeness of spaces. When you replace set theory's
elements by category theory's generalized elements, you sometimes get a
calculus that's almost like set theory but with different logic. For
instance, intuitionistic logic (internal logic of toposes) for sheaves,
geometric logic for spaces. The effect is of enabling the point-free spaces
and maps to be handled just like sets and functions - describe a space by
its points, define a map without needing to prove continuity - but with a
funny logic.

Hence although Knowledge Interchange Formats might seem a million miles away
from topology or algebraic geometry, subtle logical questions might create
direct links. (My particular favourite is geometric logic, whose distinction
between formulae and sentences actually looks quite like that between
observed facts and background assumptions.)

Steve Vickers.

Appendix
-------------
Let f: X -> Y be a map between spaces and let f^* be the inverse image
functor on sheaves. It is the way of transforming "sets over Y" to "sets
over X". If p: Z -> Y is a local homeomorphism - one way of describing a
sheaf - then its inverse image f^*(p): f^*(Z) -> X is just the pullback of p
along f. Just from the categorical property of pullback one can say that the
(generalized) points of f^*(Z) are the pairs (x,z), x a point of X and z a
point of Z, such that f(x) = p(z).

Now it makes good sense to say that a space over Y is just an arbitrary map
(not necessarily a local homeomorphism) g: W -> Y and again it is natural to
use pullback to transform it to a space g': Z' -> X over X. Any such space
over Y has a corresponding sheaf over Y, its "set of points". However, f^*
of the set of points is not in general the set of points of the pullback
space.

Re: Michael Healy's question on math and AI

[Michael Barr]

I mean the initial algebra for the theory with a nullary operation 0:1 -->
N and a unary operation s:N --> N.  1 stands for the terminal object
(empty product).

On Sun, 28 Jan 2001, Bill Halchin wrote:

> 
> 
> >For another example, consider the traditional definition of Z as the set
> >{0,{0},{0,{0}},{0,{0}{0,{0}}},...}
> >and contrast that to the categorical specification.
>      ^^^ Michael, to make things more explicit I take you
>       mean the category with one object N and two arrows, 0 & S,
>    such that
> 
>        0:N->N and s:N->N
> 
>    Yes?
> 
>    Regards,
> 
>    Bill Halchin

Re: Michael Healy's question on math and AI

[Colin McLarty]

Todd Wilson <twilson@csufresno.edu> wrote: 

>- Although we constantly speak of "the" product, we really only have
>  "a" product (at least if we take the category-theoretic perspective
>  seriously).  What is really involved, formally, in making the move
>  from "a" to "the"?  A formal language translation scheme?  Coherence
>  theorems?  How much technical work is really involved here?

	Actually, nothing is involved if we introduce a product operator.
That is, we take operators _x_, p0_,_ p1_,_ and say:

For any objects (or types) A and B the object AxB has arrows

p0A,B:AxB --> A   and p1A,B:AxB -->B with the properties of a categorical
product.

	Notice that the categorical property is exactly what you want in a
product type: From a record of type AxB you can recover the A entry, and
the B entry, via the projections. And whenever you have a pair of values,
one in A and one in B, there is a correspondng single value in AxB.
Notice, the "values" may be parametrized, so we are actually dealing with
operations f:T-->A and g:T-->B and the resulting (f,g):T-->AxB.

	Then AxB will generally not be the only product of A and B in the
category, but it will be one, and that is what we need. Coherence theorems
indeed are important. But they are provable from the above. So there is no
need to give them as part of specifying the product--the same as a computer
need not have the Chinese remainder theorem programmed into it, to
implement arithmetic.  

>
>- Related to this, what about the fact that if
>
>      (pi0: A x B -> A, pi1: A x B -> B)
>
>  is a product, then so is (pi1, pi0), indistinguishable categorically
>  from the other product?  

	Sadly, that is not a fact. The pair (pi0,pi1) gives a product of A
and B, while (pi1,pi0) gives a product of B and A. This is revealed
categorically by the fact that the codomain of pi0 is A, while the
codomain of pi1 is B.

	In programming terms, a data record of <your age in years, your
height in inches> is different from a record of <your height in inches,
your age in years>. It is quite important practically, as well as
theoretically, to distinguish the product of A and B from that of B and A.

	Even in a product BxB we need to keep the projections in order. For
example, that is how we distinguish between pairs <x,y> of reals with x
less than y, and pairs <x,y> of reals with y less than x. An important
distinction. This is why a correct specification of the categorical product
specifies the projection arrows as p0_,_ p1_,_, or in words:

	"projection to the first of the following two objects"

and	"projection to the second of the following two objects" 
	 
>- Is there anything to be made of the fact that the set-theoretic
>  cartesian product is a local construction, involving only the sets A
>  and B and certain small sets made up of their elements, whereas
>  a/the category-theoretic product depends on the whole category
> (because of the quantification in the universal property)?

	This is one reason why a computer implementation of the categorical
product, in any reasonably rich environment, will be incomplete. But it
pales beside other reasons why computer implementations of any reasonably
strong construction are incomplete. The ZF set-theoretic product will also
be incomplete in any computer implementation

	Compare the way Goedel's theorem shows that computer implementations of
arithmetic will all be incomplete. It pales beside the fact that normal
implementations don't try to implement induction at all. 

>And, finally, shouldn't "better"
>really be "better for what"?  In other words, aren't the two
>communities really just arguing past one another, like people arguing
>over types of automobile?  What really is the issue here?

	There are many different issues, and correspondingly different
arguments. Which one did you mean to address?

Re: Michael Healy's question on math and AI

[Peter McBurney]

Professor Lawvere's message included the following sentence:

On Wednesday, January 24, 2001 11:26 PM, F. William Lawvere 
[SMTP:wlawvere@hotmail.com] wrote:

>For > example, some say that logic is more general than mathematics, 
partly
> because of ignoring the strongly qualitative aspect of modern mathematics 
> and partly because of the philosophical tradition of hiding the fact that 
no
> logic other than mathematical logic has had any significant real-world
> applications.


It's not entirely clear to me what is being asserted in the second part of 
this sentence.  If what is being asserted includes the statement that the 
only logic which has had any significant real-world applications is 
mathematical logic, then this assertion is incorrect.  To give just one 
example, over the last decade the Advanced Computation Laboratory of the 
Imperial Cancer Research Fund (ICRF) in London, UK, has built intelligent 
computer decision-support systems for medical applications using logics of 
argumentation.  These logics typically use non-deductive modes of 
reasoning, and are based on the work of philosophers of argumentation 
dating from the 1950s; this work in philosophy was undertaken outside, and 
in strong opposition to, the tradition of mathematical logic.  A 
category-theoretic semantics has been provided for some of these logics of 
argumentation.    The resulting decision-support systems have found 
real-world application in cancer treatment advice, in drug prescription and 
in the automated assessment of chemical properties, such as toxicity and 
carcinogenicity.   Moreover, current research in Artificial Intelligence is 
developing the use of non-deductive argumentation formalisms for automated 
dialogues between autonomous software agents in multi-agent systems, work 
that is likely to form the basis of next-generation e-commerce systems.






Peter McBurney

************************************************************************  
**********
Peter McBurney
Agent Applications, Research and Technologies (Agent ART) Group
Department of Computer Science
University of Liverpool
Liverpool L69 7ZF
U. K.

Email:  p.j.mcburney@csc.liv.ac.uk
Web-page:  www.csc.liv.ac.uk/~peter

************************************************************************  
**********




-----Original Message-----
From:	F. William Lawvere [SMTP:wlawvere@hotmail.com]
Sent:	Wednesday, January 24, 2001 11:26 PM
To:	categories@mta.ca
Subject:	categories: Michael Healy's question on math and AI

Re : Michael Healy's question on math and AI


This is to answer Mike and also several other people who have contacted me
recently asking how I would respond to queries about

(1)	Artificial Intelligence, cognitive science, linguistic engineering,
knowledge representation, and related attempts at creating modern subjects, 
and

(2)	the relevance of category theory and of mathematics in general to 
these.

     My basic response is strong advice to actually learn some category
theory, rather than resting content with slinging back and forth 
ill-defined
epithets like "set theory", "contingency", etc..

So much confusion has been accumulated that an opposition of the form
"set-theoretical versus non-set-theoretical" has at least seven wholly
distinct meanings, hence billions of electrons and drops of ink can be
spilled by surreptitiously identifying any two of these.  For example, the
opposition can concern whether or not large cardinal assumptions are needed 
for a certain result, which is mathematically meaningful and hence
independent of whether or not the ZFvN rigidification of Cantor is being
used as a framework.  Another example is the opposition habitually used in
geometry between properties of spaces which can be explained in terms of
arbitrary mappings versus those which depend on the cohesion being studied
(e.g. "the underlying abstract group vs. the Lie group"). Obviously these
two oppositions are not the same although they may be related.

One of the oppositions which I have emphasized since 1964 is
    the ZFvN rigid hierarchy based on galactically "meaningful"
inclusion, requiring the totally arbitrary "singleton" operation of Peano
with the resulting chains of mathematically spurious rigidified membership, 
on the one hand,
                             versus
    the category of abstract sets, involving many potential universes of
discourse and arbitrary specific relations between them, on the other hand.
(Abstract sets can CARRY structures of mathematical interest, but precisely 
because of the need of flexibility in the latter, they themselves have only 
very few properties, unlike the ZFvN "sets").

Within Cantor's original conception itself there is a fundamentally
important opposition: the abstract sets, which he called "Kardinalzahlen",
versus the cohesive and variable sets which he called "Mengen".  (An
additional confusion stems from the use, by nearly all of Cantor's
followers, of the term "cardinal number" to mean (not a
Kardinalzahl=abstract set, but) a label for an isomorphism class of 
abstract
sets, an invariant which Cantor of course also studied, but which is too
abstract to support the specific relations between abstract sets 
themselves,
the mappings, and hence cannot carry the needed mathematical structures).

(A)    The real issue is that for purposes of pure AND applied mathematics, 
we need to be able to represent (without spurious ingredients) these
cohesive and variable sets (or "spaces") and their relationships.  The ZFvN 
rigidification fails so miserably in doing this that even those geometers
and analysts who pay lip service to it as a "foundation" never in practice
use its formalism.

(B)   Category theory made explicit some universal features of the
relationship between quantity and quality whose fundamental importance had
been forced into consciousness by the work of Volterra and Hurewicz (both 
of
whom made basic contributions to both functional analysis and algebraic
topology) and of many others. This relationship between quantitative and
qualitative aspects concerns cohesive and variable sets and structures 
built
on such spaces.  For example, Volterra already recognized that spaces have
"elements" other than points, and Hurewicz recognized the need for
cartesian-closed categories (even before the lambda-calculus formalism, or
category theory, was devised); moreover, the original fiber bundles were
explicitly modeling dynamical situations, etc.

Many people working in the new fields, striving to realize the dream of a
theoretical computer science, do not seem to be aware of points like  (A)
and (B). It would certainly be a bad strategy for the advancement of 
science
to "hide" the fact that category theory belongs to the background of a new
result and thus to help perpetuate that sort of ignorance.

The role of mathematics in general (not only of category theory) also
seems to be widely misunderstood, even in those fields which definitely 
need
more mathematics in order to mature and make a real contribution.  For
example, some say that logic is more general than mathematics, partly
because of ignoring the strongly qualitative aspect of modern mathematics
and partly because of the philosophical tradition of hiding the fact that 
no
logic other than mathematical logic has had any significant real-world
applications. Because of the minimal
mathematical education required of students of philosophy, the claim is too 
easily accepted in many philosophical circles that "mathematics is
unsuitable" for some given issue of conceptual analysis; this conclusion
seems to be based on the syllogism:
        mathematics is set theory (a misconception which the philosophers
themselves have done much to disseminate),
        set theory is clearly not suitable (actually because of the defects 
of the ZFvN rigidification, which make it ill-suited for mathematics as
well)
        hence ......
This syllogism serves as an excuse to indefinitely postpone learning
mathematics (and category theory in particular).

An older sort of excuse is the assertion that the proposed science should
concern the REAL WORLD, not pure mathematics. This superficially appealing
truism has frequently been used to mask the fact that comparing reality 
with
existing concepts does not alone suffice to produce the level of
understanding required to change the world; a capacity for constructing
flexible yet reliable SYSTEMS of
concepts is needed to guide the process. This realization (not Platonism)
was the basis of the supreme respect for mathematics expressed by champions 
of reality like Galileo, Maxwell, and Heaviside. For example, the
differential calculus provides the capacity to construct systems 
descriptive
of celestial motions, fluid interactions, electromagnetic radiation fields, 
etc., and therefore engineers have learned it. The functorial calculus 
helps
to provide a similar capacity adequate to the requirements, not only of the 
older sciences,
but of the newer would-be sciences as well. Hence my response.

                 Bill Lawvere