Date: Wed, 10 Apr 1996 13:36:47 -0300 (ADT) Subject: Re: Categories and Education at TSM (fwd) Date: Wed, 10 Apr 1996 08:42:59 -0400 (EDT) From: James Stasheff ---------- Forwarded message ---------- Date: Wed, 10 Apr 1996 07:40:02 -0500 From: Ed Dubinsky To: jds@math.unc.edu, mathed@math.sunysb.edu Cc: bbf@sage.cc.purdue.edu 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 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 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