Unfortunately, I haven't found mathematicians in college to necessarily be any more punctilious than those in high school when it comes to universal agreement on terminology. The real source of punctiliousness is computer software, where the rubber really meets the road on issues like what is the "imaginary part". I don't think that mathematicians ever agreed on many of these definitions, and it was up to the people defining the software to come up with useful definitions. Once these definitions are embedded in many lines of software code, the mathematicians are usually then forced to be compatible. For example, the names of the trigonometric functions and their inverses didn't seem to stabilize until Fortran & other computer languages forced them to. In some cases, software has forced mathematicians to redefine some functions in a more useful manner -- e.g., atan2(y,x) = atan(y/x), except that it works even when x=0 (or sufficiently close to zero). Guy Steele worked very hard to rationalize these functions for Common Lisp, and one of the (good) legacies of Common Lisp is that these definitions have found their way into other languages. Unfortunately, even the Common Lisp number system has its quirks. For example, referential transparency is violated: (= x y) does not guarantee that (= (f x) (f y))! (Hint: consider the action of various functions on +0.0 and -0.0.) High school mathematics treats polynomials over the reals quite casually, often times ignoring the difference between (x^2-1)/(x+1) and (x-1). (So does Macsyma/Maxima, for that matter!) Those of us who have studied "typing systems" (ways of assigning "types" to language entities) know just how difficult it is to come up with an expressive type system that is also decidable, and also how incredibly pedantic it is to force people to exhaustively spell out all types & type conversions. This is one of the reasons for the "ML type system", which has a quite elegant type inferencing system for automatically deducing the correct types of things. The bottom line is that punctiliousness is important when one is first learning concepts, and when any sloppy usage can be a serious detriment to fully understanding a concept. But later, when the concept is fully understood, the punctiliousness becomes sheer pedantry, and gets in the way of efficient communication. At 09:13 AM 4/9/2006, James Propp wrote:
Has anyone written a good essay on the "two cultures" of high school mathematics and university mathematics?
The former is not just a subset of the latter. For one thing, I think high school teachers tend to be punctilious about some matters that university mathematicians are sloppy about. A case in point is the meaning of "imaginary part". Is the imaginary part of a+bi equal to b, or to bi? I don't think that I myself am consistent about this. I got the impression that Barry Mazur, in his book "Imagining Numbers", used a convention at odds with what's being taught in high schools these days, because one irate amazon reviewer of the book opined that Mazur was not qualified to write a book on advanced math because his misuse of the term "imaginary part" showed that he didn't even know basic high school math!
I also think that high school math culture enshrines some 19th century usages that have fallen into disfavor. For instance, the word "cone" in three-dimensional analytic geometry refers to a complete locus, not just the half that lies on one particular side of the vertex (and some high school teachers are rather pedantic about that); but from calculus onward, "cone" in university math means what high school teachers would call a half-cone.
Part of what's going on is a time-lag between developments in research and the training received by high school math teachers, and the nature of the courses that future high school teachers take. To give just one example: I've seen more than one course in "modern geometry for future high school teachers" in which the discussion of non-Euclidean geometry is extremely axiomatic and historical, and hyperbolic geometry is illustrated by way of a single picture (the pseudosphere), maybe supplemented by pictures of the Klein or Poincare models, but with no upper half-plane (and no Escher pictures!). Who would write such a book? Someone who had read such a book at a formative age, and who had a very static view of mathematics.
It would probably be easy to write a one-sided essay that simply contended "high school mathematics is bad mathematics" (cf. the essay on applied mathematics by Halmos). But I think there's more to it than that.
Comments?
Jim Propp