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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 __________________________________________ _____ xmca mailing list xmca@weber.ucsd.edu http://dss.ucsd.edu/mailman/listinfo/xmca --
*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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