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Re: [xmca] Numbers - Natural or Real?
Andy:
In the realm of numbers, Hegel's observation that what is real is rational and what is rational is real is simply not true.
The square root of two is a real number. But it's not rational; the decimal expansion of the square root of two has no end. Similarly, the naturals are rational, but the rationals are not necessarily natural (e.g. negative numbers and zero).
The historical progression, from naturals to rationals to reals is a very good example of what Anna was talking about, the tendency to reify a whole system as merely one special case of a larger metadiscourse about numbers.
That's why I suggested that the notion of quantity doesn't really exhaust the idea of numbers: quantities (in the sense of taking a set of seven oranges and then mentally taking away the oranges) are really only one part of numbers. The "three" in "two thirds" is not really a quantity of oranges minus oranges, but rather a procedure, and this is certainly true of the 3 in f(x) = 3x. .
Anna likes to consider the HISTORICAL progression (from naturals to rationals to reals to complex numbers) as the basis of numerical instruction (and so does Piaget!). I find Vygotsky's own account a much more linguistic one. Based on Thinking and Speech, especially Chapter Five (but also his distinction between indicating, naming, and signifying), I would say it looks something like this:
a) POINTING: Pointing to three oranges. Counting by pointing. Number is a concrete group of objects.
b) NAMING: Saying “three oranges”. Counting by wording. Number is a concrete group of objects called by a name.
c) GENERALIZING: Learning that “three oranges” and “three apples” are both kinds of “three”. “Number” is a quality of a group of objects we call “quantity”.
d) ABSTRACTING: Thinking “three oranges” WITHOUT THE ORANGES, as a potential but not necessarily real quantity of objects.
e) CONCEPTUALIZING: Thinking of “three” WITHOUT THE QUANTITY, e.g. y = 3x, where neither side of the equation actually equals three, where three is simply a relationship between y and x and not an actual quantity at all.
David Kellogg
Seoul National University of Education
--- On Tue, 7/5/11, Andy Blunden <ablunden@mira.net> wrote:
From: Andy Blunden <ablunden@mira.net>
Subject: Re: [xmca] Numbers - Natural or Real?
To: annasfar@math.msu.edu, "eXtended Mind, Culture, Activity" <xmca@weber.ucsd.edu>
Date: Tuesday, July 5, 2011, 7:58 AM
Can I just add a minor observation to this discussion, as to why I used the term "rational" rather than "real" in contadistinction to "natural"?
It seemed intuitively impelling that the correct contrast to counting numbers [natural] was numbers expressing the continuum we imagine when we set out to measure something in the real world. "Rational number" evokes concepts like 3/4 or 1.76 and so on, and looks like a discontinuous series. But in fact, the rational numbers constitute a continuum, just as must as the real numbers do: between any pair of rational numbers, there is an infinity of other rational numbers.
The technical point, the reason for saying "rational" and not "real" is that "real numbers" include numbers like the circumference of a circle of unit diameter, i.e., pi, and such numbers can never be the result of a measurement, they are hypothetic extensions of the concept of measuring.
So, though "real" sounds right, and "rational" sounds wrong, I think it has to be "Natural or Rational"
THis is just an afterthought and I am not trying to make any particular point here. :)
Andy
anna sfard wrote:
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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*Andy Blunden*
Joint Editor MCA: http://www.informaworld.com/smpp/title~db=all~content=g932564744
Home Page: http://home.mira.net/~andy/
Book: http://www.brill.nl/default.aspx?partid=227&pid=34857
MIA: http://www.marxists.org
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