Categories in developmental psychology!

Categories in developmental psychology!

[Jamie Vicary]

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?

Best wishes,
Jamie.


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Re: Categories in developmental psychology!

[Zinovy Diskin]

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.
>

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Re: Categories in developmental psychology!

[Johannes Huebschmann]

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-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?
>
> 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/ ]

Re: Categories in developmental psychology!

[Jocelyn Ireson-Paine]

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/



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