Re: A well kept secret?

[Ronnie Brown <ronnie.profbrown@btinternet.com>]

 From teaching first year analysis I saw  that we need rules for
constructing continuous (and then differentiable) functions (as the
texts do, of course).
I guess this led me later to emphasise constructions of continuous
functions in topology, and this leads naturally in many cases to
universal properties. (Are categorical methods relevant to functions of
bounded variation?) This is the conceptual revolution in which of course
a particular construction is defined by its relation to all other
objects of the `category of discourse'. This can be related to
programming; given any input of the required type, the output is a
function or morphism. It also emphasises properties rather than mode of
construction.

So categorical methods can be used without explicitly saying at the
first instance that one is doing `category theory'; I was also an
advocate of set notation
in calculus, for example to name the domains of functions defined by
formulae, without introducing `set theory' as a `big deal'.

Ronnie Brown







Ellis D. Cooper wrote:
> At 11:09 PM 12/17/2009, John Baez wrote:
>> I think it's premature to introduce category theory in the undergrad
>> curriculum.
>
> I think there are enough very interesting simple examples of
> categories that the language and diagrams could be introduced
> to  high school students. For example, lists are terrific examples
> for discussion of the free monoid functor, its unit, and counit, but
> they don't have to be called by their official names. And tables give
> a 2-dimensional version of that discussion, with an exchange law that
> is simple but interesting. Kinship trees or the trees used in high
> school probability class can be used to talk about partially ordered
> sets, but they don't have to be called that. The idea would be to get
> diagrams into the student consciousness, so they learn about
> connecting the dots. Advanced high school students know about
> multiplication of matrices, so they could learn something about
> arrows standing for linear transformations, and composition of arrows
> corresponding to matrix multiplication. The slogan is, "algebra is
> the geometry of notation," and high school students can learn to look
> at and play with diagrams. I bet some kind of school-yard game could
> be based on diagram chasing.
>
> As I see it the greater problem is that high school mathematics
> teachers need more education. Therefore, I am preparing a book with
> no calculus beyond the AP level, but using Robinson infinitesimals,
> Kolmogorov probability spaces, and Eilenberg-Mac Lane categories
> wherever these things come up simply and naturally in a certain
> context to do with biology. It has been announced for pre-order at
> amazon.com by World Scientific and should be available in April,
> 2010. The work-in-progress is available for examination (and feedback
> to me!) upon request.
>
> Ellis D. Cooper
>

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