Ricardo Ribeiro Gudwin wrote:
> Marshall Clemens wrote:
>
> > I don't beleive there is any way to
> > describe a whole without resorting to a reductionist explanation. If
> > you can think of an example please let me know.
>
> I believe I have an example: Imagine a circle in XY plan. I can make the
> projections of this circle in line X and line Y. But now, when we try to
> re-compose the circle, it is not a circle anymore, it is a square. So, a
> circle in 2D can not be decomposable into its 1D projections. We have
> to use the equation x2 + y2 = r2 as a holistic description, in order to
> understand it.
> OBSERVATION: This is only a (bad) analogy. To be more correct,
> I would have to talk about a function that can not be decomposable
> in terms of sums, squares and products. It is interesting as I can't
> describe such functions in an algebraic way, as the whole algebra
> is based on sums, products and other primitive (holistic) functions.
Yes, and if you move to direct products and direct sums via category theory you
get what you are looking for. one group od models which is reducable and
another which is not! Rosen does this with great care.
> BUT
> I can conceive the existence of such functions. A totally random function
> would be a function like that. I can imagine also other functions that are
> not random, but still are not "realizable" through mathematics. The most
> interesting is that they may be realizable through a physical device. For
> example: physical lens. With its imperfections and non-linearities. The
> input of this "function" would be the angle in which a light bean inputs the
> lens, and
> the result would be the angle it outputs the lens. Depending on the physical
>
> format of the lens, this function would not be "realizable" through
> mathematics.
> The maximum that mathematics can gives me is an APPROXIMATION of
> such functions. The question here is .... can we live only with an
> approximation,
> instead the real function ? If there is no chaotic behavior in the system,
> it's OK,
> but if there is some tendency to chaos, the fact we are using an
> approximation
> instead the real function would imply in unexpected behavior.
> Maybe Don can exemplify other problems that would appear due to the use
> of approximations instead of the real functions. This problem with chaos is
> one that
> occurs now on my mind. Maybe Rosen had catalogued others ?
Yes....this is the subject of a number of chapters in "Life Itself" and has its
basis in "Fundamentals of measurement". It is closely tied to the notions of
recursiveness mentioned by others. Direct sums give a reductionist
decomposition while dirct products do not. We can make models either way. In
the relational world, function is characterized by direct products while
material parts go together as direct sums. An essential ouitcome is that a
decompostion by function has no meaning out of contest....whereas a piece is a
piece in the system or out.
> Best regards,
> Ricardo
> --
> //\\\
> (o o)
> +-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-oOO--(_)--OOo-=-=-+
> \ Prof. Ricardo Ribeiro Gudwin /
> / Intelligent Systems Development Group \
> \ DCA - FEEC - UNICAMP | INTERNET /
> / Caixa Postal 6101 | gudwin@dca.fee.unicamp.br \
> \ 13081-970 Campinas, SP | gudwin@fee.unicamp.br /
> / BRAZIL | gudwin@correionet.com.br \
> +-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-+
> \ URL: http://www.dca.fee.unicamp.br/~gudwin/ /
> / Telephones: +55 (19) 788-3819 DCA/Unicamp (University) \
> \ +55 (19) 254-0184 Residencia (Home) /
> / FAX: +55 (19) 289-1395 \
> +-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-=-+
respectfully,
Don Mikulecky