Subject: categories: Michael Healy's question on math and AI

From: "F. William Lawvere" <[email-masked]>
Date: Wed, 24 Jan 2001 23:26:27

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

---

From: Todd Wilson <[email-masked]>
Date: Thu, 25 Jan 2001 19:08:26 -0800 (PST)

On Wed, 24 Jan 2001, F. William Lawvere wrote:
> cohesive and variable sets (or `spaces') and their relationships.  The ZFvN 
> and analysts who pay lip service to it as a `foundation' never in practice 
> use its formalism.
> 
> (B) [...]
> 
> 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

---

From: Michael Barr <[email-masked]>
Date: Fri, 26 Jan 2001 13:14:14 -0500 (EST)

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

>   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.

>   (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.

>  (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.

> 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