RE: [xmca] Numbers - Natural or Real?

[anna sfard <annasfar@math.msu.edu>]

Hi Bill, David, and Larry,

 

Just quickly.

 

*Bill*: 

My yesterday piece on natural and rational (or rather real) numbers was
supposed to be a commentary/footnote to Devlin's writings - didn't it show?
My note was an elaboration and explication of what Devlin, following
Davydov, meant by "beginning from rational numbers", and it was continuation
of and  support for Delving/Davydov's ideas (some of which you quote).  As
an aside, to understand these ideas better one should read at least two
notes from his MAA columns  - the one you're talking about and the one that
precedes it.

 

*David*:

Re your question 

"My question is whether there is a non-numerical "posthistory" to
mathematics in, say, algebraic relations (which are independent of specific
quantities) and imaginary numbers (which seem to me to be almost entirely
independent of any conceivable quantity at all)."

I'm not sure how you divide discourses into mathematical and not, or more
specifically, pre-mathematical, mathematical and post-mathematical, but for
me, all the discourses you mention are developmentally inter-related. This
is how I see it: in most general terms, mathematics expands by the
systematic annexation of its own meta-discourses, that is, by turning the
talk *about* mathematics into a part of mathematics itself. Thus, for
example, elementary algebra, the one learned in school, is a formalized
meta-arithmetic - a formalized discourse *about* arithmetic (it begins when
the child starts talking about numerical patterns and about unknown
quantities that produced a certain result). Similarly, the discourse on
complex numbers (once known as imaginary) is a kind of formalized
meta-discourse on the discourse on real numbers. Confused? Sorry, this is
the best I can do right now. I've written about all this extensively in my
book Thinking as Communication, though, and I hope it is written clearly
enough to be accessible also to interested non-mathematical readers.. 

 

*Larry*:

Thank you. Yes, like you, I believe communicating - the actual talk - should
be emphasized also in math classroom. This principle is explicitly present
in current policy documents, such as US Core Standards for teaching and
learning math. Whether and how this recommendation is implemented is a
different story. 

 

Happy 4 July to all the American xmca-ers,

anna

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