Re: A well kept secret?

[Zinovy Diskin <zdiskin@gmail.com>]

>>  I think there are enough very interesting simple examples of categories
>> that the language and diagrams could be introduced to high school students.
>>

I've heard that Piaget experimented, successfully, with teaching
category theory to 12-year-old children (but I do not have any
references).

>
>
> Math trickles down.  Right now we need more category theory taught at the
> graduate level, so someday enough professors will understand it well enough
> to teach it at the undergrad level, so that eventually enough high school
> teachers will know enough to teach it at the high school level.
>
> If this seems overly optimistic, it's worth thinking about calculus, which
> in Newton's day was regarded as comprehensible only by a few experts.
>

For calculus, the transformation of an esoteric into a basic
discipline was largely  driven by engineering applications. After
mathematicians demonstrated that calculus could be applied to
practical engineering problems, and developed a methodology for such
applications, engineers recognized that calculus should be taught at
the then-undergrad level. This created a demand in professors capable
of teaching calculus to engineers, and further along the chain, as
John described.  This mechanism should work for category theory as
well: software engineering is saturated with problems to which
categories have something essential to offer. The situation is even
more favorable because software engineers themselves reinvent
categorical constructs (more accurately, their inventions can be seen
as a reinvention of categorical constructs). I believe that software
engineering is ready (theoretically :) to accept categorical methods.

Of course, much needs to be done to adapt category theory as a basic
mathematical discipline for software engineering but it would not be a
waste of time and effort. This work should be profitable for
categories in two ways:
1) Public appreciation, funding etc.
2) Engineering applications are a source of interesting problems and
interpretations that may be mathematically fruitful.

Focusing on engineering allows treating "the opprobrium issue" in a
different way (my apologies if it is too vulgar).  Category theory
provides methods and tools, and there are other tools on the market.
At least part of the attempts to sell category theory to a general
mathematical public is like selling it to a competing vendor, and
hence doomed to fail from the very beginning. It's more fruitful to
sell (whatever that means) to prospective users/customers. Working
mathematics, physics, computer science are such users, and they do
appreciate category theory. However, these groups of customers are not
particularly numerous. A very promising prospective user is software
engineering: it's massive, dynamic, and eager (as any other
engineering)  to adapt any widget helpful to do the job, be it
calculus, vector algebra or abstract nonsense. Having such a customer
would dramatically change the market situation for categories
similarly to the case of mechanical engineering-calculus.

Z.


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