Re: abstraction

From: Noel Enyedy (
Date: Tue May 03 2005 - 15:07:58 PDT

Michael, you wrote:

> 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.

I think this is the story that Susan is telling in the article, except I
think that she is not anlayzing things on the level of the individual. I
see her "conjecturing" is the production of new, specific, fictive cases
(e.g. On page 292, 'THE' guppy population when it reaches [exactly] 6666).
Whereas her "linking" is the comparison and evaluation of these concrete
cases that sets the stage for "mathematical generalization". So, I take her
point to be NOT that abstractions are empty until filled with appropriate
content (as Mike said). But that abstractions don't get made until there are
enough concrete cases that are made to be appropriate. And that
generalizations once made, take a lot of work to make them do the work you
want them to do (which I guess is what Mike was saying about them being
empty with the twist that this is happening between folks not within them).

But I think I disagree when you say:

> Susan sketches the mathematical position on this, which is essentially
> Piagetian. Abstraction as stripping context.

It seems that in the examples she provides it is not a process of stripping
away of context but of selectively building up new contexts by highlighting
and elaborating certain aspects. In the talk itself there is nothing being
removed or asserted as irrelevant, which is what I would think of as
evidence for stripping away.

What I thought was interesting was that her examples of generalizing
occurred in the context of explaining a process of change (pg 279) or a
dynamic steady state (pg 292). I am sure this is due in part to the topic
and the tools that are mediating the activity (a simulation that produces
populations based on transformations of previous states), but it makes me
wonder which types of conversations lead to generalization and which ones
steer towards some other type of discursive move?

Noel Enyedy
Long time lurker, first time caller

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