From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7606 Path: news.gmane.org!not-for-mail From: Jocelyn Ireson-Paine Newsgroups: gmane.science.mathematics.categories Subject: Re: Categories in developmental psychology! Date: Fri, 15 Feb 2013 08:55:04 +0000 (GMT) Message-ID: References: Reply-To: Jocelyn Ireson-Paine NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: TEXT/PLAIN; format=flowed; charset=US-ASCII X-Trace: ger.gmane.org 1360936279 29064 80.91.229.3 (15 Feb 2013 13:51:19 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 15 Feb 2013 13:51:19 +0000 (UTC) To: Categories list , Jamie Vicary Original-X-From: majordomo@mlist.mta.ca Fri Feb 15 14:51:40 2013 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.32]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1U6LhS-0004gO-Um for gsmc-categories@m.gmane.org; Fri, 15 Feb 2013 14:51:39 +0100 Original-Received: from mlist.mta.ca ([138.73.1.63]:59718) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1U6Leg-0007K7-CS; Fri, 15 Feb 2013 09:48:46 -0400 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1U6Lef-00018z-8P for categories-list@mlist.mta.ca; Fri, 15 Feb 2013 09:48:45 -0400 In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7606 Archived-At: 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-and-transforming > > 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/ ]