Re: SUO: Manifolds Of Sensuous Impressions (MOSI's)
Jon,
Your excerpt illustrates two points: (1) that the theories
of manifolds and categories are very powerful and general,
and (2) that they are terrifying to nonmathematicians.
>| Serge Lang, 'Differential & Riemannian Manifolds',
>| Springer-Verlag, New York, NY, 1995.
>
>In presenting this text I am obligated to change
>many Greek characters into Latin letters, and so
>by way of a slightly skewed form of compensation,
>I will convert Roman numerals to Arabic decimals.
>Notes from the translator [me] will be placed in
>square brackets, to ease the transits to English.
>
>¤~~~~~~~~~¤~~~~~~~~~¤~RECITATION~¤~~~~~~~~~¤~~~~~~~~~¤
>
>Chapt 2. Manifolds
>
>Starting with open subsets of Banach spaces [think R^n for the moment],
>one can glue them together with 'C^p'-isomorphisms [bijective mappings
>that are continuously differentiable up to at least as far as order p].
>The result is called a manifold. We begin by giving the formal definition.
>We then make manifolds into a category, and discuss special types of morphisms.
>We define the tangent space at each point, and apply the criteria following
>the inverse function theorem to get a local splitting of a manifold when
>the tangent space splits at a point.
>
>We shall wait until the next chapter to give a manifold structure
>to the union of all the tangent spaces.
I agree with you about point (1), but I was trying to make
some points that are quite different from (2):
1. Mathematicians are generally lazy, and they want to avoid
as much tedious writing and computation as possible.
2. Their powerful formalisms actually enable axioms to be
written in a form that make them easier to understand
and actually much closer to what many people would call
"common sense".
3. Unfortunately, when mathematicians talk to other
mathematicians, they feel no obligation to make their
points "obvious" to anyone who doesn't immediately see
that what they are saying is "obvious".
4. What I was trying to say is that this high-powered stuff
can make the axioms very much simpler, very much more
general, and very much easier to explain to the "average
man in the street" -- provided that you don't use typical
mathamtical jargon to explain your formalism.
Bottom line: Category theory and differentiable manifolds are
very powerful, they can make life a lot easier for everybody,
but some work is needed on the human factors.
John Sowa