Categories in developmental psychology!
Dear all, I recently came across an intriguing volume entitled "Morphisms and Categories: Comparing and Transforming". The lead author is Piaget, one of the foremost names in developmental psychology, who died in 1980. The volume is a collection of papers, many of which seem to enthusiastically apply ideas of category theory to developmental psychology. I haven't seen the book itself, but this link lets you read the full introduction and the first page of each chapter: http://www.questia.com/library/6364330/morphisms-and-categories-comparing-an... Reading the introduction, it's clear that Piaget took very seriously the idea that category theory could provide a formal foundation for psychology. Does this perspective survive in the modern psychology literature? Does anybody know how Piaget came to be acquainted with these categorical ideas in the first place? Is there anything here that could be of interest to modern category theorists? Best wishes, Jamie. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
Hello, some comments are below. On Tue, Feb 12, 2013 at 9:12 AM, Jamie Vicary <jamie.vicary@cs.ox.ac.uk>wrote:
Dear all,
I recently came across an intriguing volume entitled "Morphisms and Categories: Comparing and Transforming". The lead author is Piaget, one of the foremost names in developmental psychology, who died in 1980. The volume is a collection of papers, many of which seem to enthusiastically apply ideas of category theory to developmental psychology.
I tried reading the book some time ago: it was interesting but not an easy read. What is left in my head are vague memories and a rough schema of Piaget's view on intellectual development in children: 2-7 year olds: pre-operational thinking, when a child concentrates on objects rather than operations with them (recall Euclid's definitions of point and line); 7-11(?) years: operational thinking (related to Bourbaki structures: I do not care what points and lines are, but here are operations I can perform with them); 11-... years: comparative and transformational thinking (structures could be thought of and operated as integral entities, e.g., be compared and transformed, hence, category theory). The papers in the book describe several series of experiments --- some very clever --- that reveal these qualities. As with any other experiments in psychology, whether these experiments really prove what they claim is a non-trivial question.
Reading the introduction, it's clear that Piaget took very seriously the idea that category theory could provide a formal foundation for psychology.
I'd not say it's about a formal foundation for psychology. Rather, it's about patterns of categorical thinking as a natural part of human intellectual development.
Does this perspective survive in the modern psychology literature?
It's probably not widely known, but some people are well aware of it -- Marian Petre from the Open University, for example. Perhaps she can provide references.
Does anybody know how Piaget came to be acquainted with these categorical ideas in the first place?
If I remember correctly, the preface to the book describes Piaget's mathematical evolution, and that his coming to category theory was a natural step in his epistemological studies. Not in the technical sense of CT being a formal foundation for psychology, but in the qualitative sense of the three-stage schema above.
Is there anything here that could be of interest to modern category theorists?
That categorical thinking is quite natural for a human being (but is suppressed by math education in school). Best regards, Zinovy
Best wishes, Jamie.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I guess these observations should be interpreted with circumspection, in the light of Lacan's usage of topological notions in psychoanalysis, see e.g. Sokal-Bricmont, Impostures intellectuelles p. 55. The term category goes back to Aristotle; the word (kategoria) in inself is older but it seems Aristotle attributed to it the kind of meaning we are talking about here. The term functor derives from the latin verb fungi (deponens). It seems Eilenberg-Mac Lane learnt the term functor from Carnap. Best Johannes On Tue, 12 Feb 2013, Jamie Vicary wrote:
Dear all,
I recently came across an intriguing volume entitled "Morphisms and Categories: Comparing and Transforming". The lead author is Piaget, one of the foremost names in developmental psychology, who died in 1980. The volume is a collection of papers, many of which seem to enthusiastically apply ideas of category theory to developmental psychology.
I haven't seen the book itself, but this link lets you read the full introduction and the first page of each chapter: http://www.questia.com/library/6364330/morphisms-and-categories-comparing-an...
Reading the introduction, it's clear that Piaget took very seriously the idea that category theory could provide a formal foundation for psychology. Does this perspective survive in the modern psychology literature? Does anybody know how Piaget came to be acquainted with these categorical ideas in the first place? Is there anything here that could be of interest to modern category theorists?
Best wishes, Jamie.
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Thu, 14 Feb 2013 09:01:33 +0100 (CET), Johannes Huebschmann wrote:
in the light of Lacan's usage of topological notions in psychoanalysis, see e.g. Sokal-Bricmont, Impostures intellectuelles p. 55.
The title reflects its subject faithfully but its object may well be the authors themselves... Plenty of responses have been given to S&B, including by mathematicians (e.g. J. Goguen), but they are almost never cited along the reference to S&B, so one may wonder if people are looking for a better understanding of, or whether they just want to settle scores with social & human sciences. Interestingly, this question which relates to the status of scientific research, the way we represent ourselves, the way pwer is structured in the society (incl. the scientific society) is precisely a topic addessed by Latour and Foucault for instance. Considering Lacan, his usage of topology came from work with mathematicians and certainly did not aim at doing mathematics but rather at translating a certain situation or operation (necessarily expressed in natural language) in more formal, less ambiguous terms. In order to facilitate communication and comparison with other works and to minimize interpretation by the audience (BTW, this is why Lacan says that this is not a metaphor). The fact that there may be, sometimes, some mistakes on the mathematical side is certainly irritating to the mathematical eyes but is without interest w.r.t. Lacan's objective. Finally, a more constructive reaction for mathematicians may be to rejoice that some people are trying to formalise their research; the former could even help the latter do so, actually. dc -- David Chemouil Onera - DTIM Tel./fax: +33 (0) 5 6225 2936 / 2593 <http://www.onera.fr/staff/david-chemouil> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
On Tue, 12 Feb 2013, Jamie Vicary wrote:
Dear all,
I recently came across an intriguing volume entitled "Morphisms and Categories: Comparing and Transforming". The lead author is Piaget, one of the foremost names in developmental psychology, who died in 1980. The volume is a collection of papers, many of which seem to enthusiastically apply ideas of category theory to developmental psychology.
I haven't seen the book itself, but this link lets you read the full introduction and the first page of each chapter: http://www.questia.com/library/6364330/morphisms-and-categories-comparing-an...
Reading the introduction, it's clear that Piaget took very seriously the idea that category theory could provide a formal foundation for psychology. Does this perspective survive in the modern psychology literature? Does anybody know how Piaget came to be acquainted with these categorical ideas in the first place? Is there anything here that could be of interest to modern category theorists?
Hi Jamie, I used to teach Artficial Intelligence to psychologists, and I came across category theory very rarely in the literature. I don't remember seeing it ever mentioned in the journal "Cognitive Science", for example. Only one of the psychologists I talked to in the department at Oxford had ever heard of category theory. So the perspective does not survive, at least in my experience. I also note "A Category Theory Approach to Cognitive Development" by Graeme S. Halford and William H. Wilson (1980) ( http://www.cse.unsw.edu.au/~billw/reprints/halwil1980.pdf ) who say "Since category theory does not appear to have been used in psychology except for some discussions by Piaget (1970) and Piaget, Grize, Szeminska, and Vinh-Bang (1968) ...." Granted, they did write that in 1980. This neglect surprises me. To me, it seems obvious that category theory might be very useful when mapping between different representations. For example, I'd have expected the connectionists to have used it in reconciling their representations with "symbolic" logic-based representations. There was a lot of interest in that reconciliation during the mid-1980s when connectionism became popular, but none of it (that I know of) used category theory. I have come across some psychology-related category-theory stuff, but it all feels like outliers. For example, in a posting on 2012-10-19 to this list, in the "Algorithms arising from category theory" thread, I cited a paper by Michael Healy on combining neural networks categorically; and one by Healy, Thomas Caudell, and Timothy Goldsmith that suggests compound concepts can be represented as categorical diagrams, and compared by comparing these diagrams. This seems close in spirit to Piaget's use of morphisms. I also cited Goguen's use of colimits and 3/2-colimits to model conceptual blending and the interpretation of metaphors. Also, there is a book "The Logical Foundations of Cognition" edited by John Macnamara and Gonzalo E. Reyes (1994) (http://books.google.co.uk/books?id=Outqx9VKqIMC&pg=PA57&lpg=PA57&dq=reyes+category+theory&source=bl&ots=JKg1Qm9Y-x&sig=x6u0aIwOGxzqbOVhXmTU_HIndOs&hl=en&sa=X&ei=3OcdUar3KcOb1AWuy4CIDQ&ved=0CDUQ6AEwAQ#v=onepage&q=reyes%20category%20theory&f=false ). Google have blocked access to most of the text - at one time, I was able to view it all - but the introduction is still accessible. In it, Macnamara and Reyes mention three current approaches to psychology - psychometric testing, experiments on mental operations, and computer modelling - and contrast with a fourth, which is the topic of the papers in the book. That fourth approach uses category theory to formalise the notion of reference. The authors say on page 4 about this: "in fact we see the relation between categorical logic and cognition as parallel to that between calculus and dynamics." Amongst the papers is one by Lawvere, "Closed Categories and Toposes" which includes a categorical formulation of the part-whole relationship and of modelling the evolution of systems with parts. Lawvere ends by saying, "Despite some simplifications in the above, needed for rapid description, I hope that I have made clear that there is a great deal of useful precision lying behind my illustrations, and a great deal to be developed on the same basis. Thus I believe to have demonstrated the plausibility of my thesis that category theory will be a necessary tool in the construction of an adequately explicit science of knowing". This book seems like another outlier, though: I've not seen any sign of the ideas entering the mainstream journals. As far as Jamie's question about how Piaget came to be acquainted with these categorical ideas: I did some Googling, and the results suggest he did so late. "The foundation of Piaget's theories: mental and physical action" by H. Beilin and G. Fireman G (1999) ( http://www.ncbi.nlm.nih.gov/pubmed/10884847 ) says in its abstract: "... This view of the child's logical development, which relied heavily on truth-table (extensional) logic, underwent a number of changes. First from the addition of other logics: category theory and the theory of functions among them. ...". Which suggests Piaget was extending his knowledge of logic, searching for new logical tools to model development with. And "Piaget: Or, The Advance of Knowledge" by Jacques Montangero and Danielle Maurice-Naville (1997) ( http://books.google.co.uk/books?id=ztHtYviIG7EC&pg=PA86&lpg=PA86&dq=piaget+morphisms&source=bl&ots=NJ4_8XbVqC&sig=Srya9Iu-m-OqFTCCRmquoWOGE0k&hl=en&sa=X&ei=Qt0dUeuDFIml0AXBxYDIAg&ved=0CFsQ6AEwBQ#v=onepage&q=piaget%20morphisms&f=false ) says this on page 86. Note the "towards the end of his life": Correspondences or morphisms are instruments of knowledge and as such fill one of the two main functions that Piaget (towards the end of his life) attributed to reason - that of comparing objects, states, or transformations. The other major function - that of transforming - depends on the mental operations and actions from which they are derived. In Piaget's last works, morphisms and correspondences are distinguished, therefore, from operatory structures that alone incarnated reason in previous works. Piaget suggested using the term "correspondence" to designate the most general aspect of comparing, including its most elementary forms. Morphisms are correspondences that account for the structure of the systems that are being compared. Elementary correspndences are sometines described as premorphisms. Nevertheless, in several passages, Piaget used morphisms and correspondences as though they were synonymous." I've also found, but not yet read, "Development of Morphisms: An Initial Investigation and Its Link with Piaget's New Theory of Concrete Operations" by Philip M. Davidson (1986) ( http://www.eric.ed.gov/ERICWebPortal/search/detailmini.jsp?_nfpb=true&_&ERICExtSearch_SearchValue_0=ED272317&ERICExtSearch_SearchType_0=no&accno=ED272317 ). This is the abstract: "This paper examines the category-theoretic formulation of cognitive development introduced by Piaget in the late 1960's and elaborated during the 1970's. The new theory is interpreted as the focal point of Piaget's investigations into topics such as function, correspondences, and commutability. Hypotheses arising from Piaget's new model were investigated in a developmental study of the morphism concept, which is central to the category-theoretic formulation. Subjects, 36 boys and 36 girls ranging in age from 5 to 7 years, were tested individually on three morphism problems. Each problem consisted of a game board divided into colored sections, a number of colored wooden tiles to be mapped to these sections, and a picture representing an operation or relation to be preserved while conducting the mapping. Results, as hypothesized, indicated that cognition of morphisms accompanies or precedes operational reasoning. Discussion focuses on several implications of further developing Piaget's category-theoretic model. Directions such development might take are pointed out. It is concluded that the new formalism offers the potential for an integrated model of cognitive process and cognitive structure, and is therefore a significant advance in constructivist theory." I wondered whether Piaget was influenced by Michael Arbib's ( http://nlab.mathforge.org/nlab/show/Michael+Arbib ) work on categories and cybernetics. I've not found any evidence for that. I also wondered whether he came to it via structuralism, and the (by then very old) ideas of the Erlangen Program. I suppose that must have influenced sructuralism too. I do notice in the Wikipedia article on the Erlangen program this quote: In his book Structuralism (1970) Jean Piaget says, "In the eyes of contemporary structuralist mathematicians, like Bourbaki, the Erlangen Program amounts to only a partial victory for structuralism, since they want to subordinate all mathematics, not just geometry, to the idea of structure." Maybe that book would be a good place to start looking at his intellectual influences.
Best wishes, Jamie.
Cheers, Jocelyn Ireson-Paine http://www.j-paine.org +44 (0)7768 534 091 Jocelyn's Cartoons: http://www.j-paine.org/blog/jocelyns_cartoons/ [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
I'm very happy to see others writing about Lacan's use of mathematics, e.g. his use of knot theory in Encore XX, something that I've always meant to return to and make an honest and careful study of, "when I have time." About categories in Aristotle--this summer I read Aristotle's work "Categories" and for a little while was very excited about the category-theoretic possibilities in it. Part of what he does in "Categories" is this: regard all "things" (leaving aside for the moment the question of defining a "thing"!) as having a partial-ordering in which x \leq y iff y is predicated of x. Again, Aristotle doesn't define "predicated," but from context he seems to only want to include propositions which are true or meaningful or something like this in his notion of predication, so that this partial-ordering tells you some useful things. At first read, this seems very exciting--since there are many possible ways in which y can be predicated of x, this should be a category in which all those propositions which predicate y of x are the hom-sets. But reading further in the Organon and then looking at a bit of other people's commentaries on "Categories," it seems that Aristotle only wants to consider the copula, "y is x," as his only form of predication. So his category really indeed is just a partially-ordered set. (And he can say some interesting things about it--e.g. existents, really-existing material things in the world, are just minimal elements in this partially-ordered set.) So my sense of it is that Aristotle did not use the word "kategoria" in a way like how we use it, which is a bit disappointing. But there is probably a good opportunity here for someone to come along and give a properly categorical generalization of what Aristotle does in "Categories" and see what you can do with it. Sorry if it's too off-topic, Andrew S. On Thu, 14 Feb 2013, Johannes Huebschmann wrote:
I guess these observations should be interpreted with circumspection, in the light of Lacan's usage of topological notions in psychoanalysis, see e.g. Sokal-Bricmont, Impostures intellectuelles p. 55.
The term category goes back to Aristotle; the word (kategoria) in inself is older but it seems Aristotle attributed to it the kind of meaning we are talking about here.
The term functor derives from the latin verb fungi (deponens). It seems Eilenberg-Mac Lane learnt the term functor from Carnap.
Best
Johannes
[For admin and other information see: http://www.mta.ca/~cat-dist/ ]
If this is not too much off topic: To make it clear: The classical Greek word "kategoria" has a somewhat different meaning: A quote from the internet: Recently, Kevin Stein (2008) added a third element???antapologia???to the speech act. According to Stein???s description, an offended party issues an accusation (kategoria). Aristotle uses the plural form "kategoriai". Best Johannes [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (6)
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Andrew Salch -
David Chemouil -
Jamie Vicary -
Jocelyn Ireson-Paine -
Johannes Huebschmann -
Zinovy Diskin