> Meantime, I have printed out the Hegel and appreciate greatly the
> associated discussion, even as I await further discussion of the issue
> of generalization in mathematics.
Susan sketches the mathematical position on this, which is essentially
Piagetian. Abstraction as stripping context.
You may find it interesting that scientists often cannot interpret
graphs in introductory textbooks of their own discipline, exactly
because the context (how data were collected, how data are processed
and transformed, the details of the object [system, animal] under
study, etc.) On the other hand, when you ask scientists to articulate
and explicate a graph from their own or related work, they ALWAYS begin
with thick descriptions of the system under study, then they explain
data collection and instrumentation, then the transformation, then
perhaps some of the scientific background concepts . . . until finally,
after a long account, they come to talk about the graph itself:
Roth, W.-M. (2003). Toward an anthropology of graphing. Dordrecht, The
Netherlands: Kluwer Academic Publishing.
Back to "mathematical generalization". What appears to occur is that
the individual needs to pursue the study of concrete cases, see how the
object behaves as a result of the various actions and operations
applied. Once you know many concrete cases, you may begin to see
invariances, and it is only after this process of ascension to the
concrete that you can engage in mathematical abstraction, that is, see
the invariances across many concrete cases.
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