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
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
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
For > example, some say that logic is more general than mathematics,
Professor Lawvere's message included the following sentence: On Wednesday, January 24, 2001 11:26 PM, F. William Lawvere [SMTP:wlawvere@hotmail.com] wrote: 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
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?
participants (5)
-
Colin McLarty -
F. William Lawvere -
Michael Barr -
Peter McBurney -
Todd Wilson