Subject: Re: Categories and Education at TSM (fwd)
Date: Wed, 10 Apr 1996 13:36:47 -0300 (ADT)

Date: Wed, 10 Apr 1996 08:42:59 -0400 (EDT)
From: James Stasheff <[email-masked]>

---------- Forwarded message ----------
Date: Wed, 10 Apr 1996 07:40:02 -0500
From: Ed Dubinsky <[email-masked]>
To: [email-masked], [email-masked]
Cc: [email-masked]
Subject: Re: Categories and Education at TSM (fwd)

People might be interested to know that towards the end of his life, Piaget
was trying to see how categroy theory might be used to help understand
children's thinking.  I think he was particularly interested in the move
from thinking about a category as a class of sets with certain permissible
functions between them, to the more axiomatic description in terms of
morphisms.  He wrote one book and at least one paper on the subject.

ed dubinsky

Date: Sat, 13 Apr 1996 08:41:56 -0300 (ADT)
Subject: Re: Categories and Education at TSM

Date: Sat, 13 Apr 1996 11:38:14 +1000 (EST)
From: Ronnie BROWN <[email-masked]>

Tim Porter and I wrote an article for the general rerader `The
methodology of mathematics' published
in Math. Gazette July, 1995, in which we emphasise the question as to what
are the objects of study of mathematics, what does it try to find out
about these objects,  and what are the tools with
which these objects are studied. These would seem likely to be  basic
questions for the beginners in mathematics.

The paper can be downloaded from
http://www.bangor.ac.uk/~mas010/papers/methmatg.ps

Answers could be of the type:

Mathematics is the art and science of patterns and structures, and their
growth and change. This is why mathematics underlies most of the
other sciences, and gives a language for description, deduction and
calculation with strange new objects which arise in everyday life, and in
science, technology, and social sciences.

In order to study structures, you need a mathematics of structures, and
in the current age this seems to be category theory, par excellence.

The interesting question is how much can people who are used to maths as
pluses and minuses actually appreciate of this. It is well worth trying
to put it over, because if people do not have some feeling for these
basic points,
then the subject will be regarded with a mystique, and something not for
them.

Another advantage of this metalevel discussion is that it enables the
possibility
of some critique of category theory as practised. I am not trying to make
this critique, but suggesting that we need a context in which discussion
can take place, to allow an analysis of future directions!

The point also is that we need an argument for mathematics (and in
particular, for category theory), which can
be put to the general public and to government. The current situation is
that the applicable parts of maths are tapped off for the development of
other subjects, where they can  often get notable funding, none of which
goes to the source of these ideas. Indeed, they are no longer
`mathematics'. How long can this source last, without proper support and
nourishment, and an understanding of its achievements?

Ronnie Brown

Date: Sun, 14 Apr 1996 08:47:35 -0300 (ADT)
Subject: Piaget

Date: Sat, 13 Apr 1996 15:05:29 +0100
From: Alberto Peruzzi <[email-masked]>

The use made of categories by Piaget was unsatisfactory in many respects,
although it is an undoubtful merit to have recognised the relevance of
categories in dealing with the architecture of mathematical structures in
relationship with the cognitive development. The main defect was that also
in 1974 he didn't realise the role of adjoints: the lack of consideration
for adjoints prevented him from having a precise model of the growth in
structural complexity (as when he tried to explain the passage from a
monoid of "operations" to its "completion" into a group). This general
remark can be easily substantiated by those who know Piaget's theory of
cognitive stages. I argued for it already in a paper going back to 1978,
"unfortunately" written in Italian. (I can mail the paper, or some related
publications in English, to those who are interested.)

Alberto Peruzzi