Nick - OK . . . now that we recognize that terms like "point" are (should more properly be?) left intentionally undefined in the axiomatic systems, we can move to the next step . . .
A term like "point" (in an axiomatic theory) is a place where we can make a (temporary?) connection between the axiomatic system and a specific model. So, for example, in R^2 (the real plane), considered as a Euclidean space (a model where the axioms of Euclidean geometry hold), we can make the linkage: (Euclidean) point -> pair of real numbers (x, y) (with this linkage, we can say the R^2 is "a Euclidean space" -- i.e., with this linkage, the axioms of Euclidean geometry hold in R^2) Or, considering R (the real line) as the place where calculus happens, we can make the (different!) linkage: (calculus) point -> function from R to R ( x(t) ) With this "calculus" linkage, we can talk about the "position of the point x at the time t1" as x(t1). We can also talk about the "position of the point x at some other time t2" as x(t2). We can then sensibly say that the "position of the point x" changed from x(t1) to x(t2), and thus it also makes sense to talk about the "point moving from x(t1) to x(t2)" and we can talk about the "velocity of the point at time t1" as d(x(t))/dt evaluated at t1 (or, in other notation, x'(t1) . . .) Note that in this case, the (elaborated) technical term "position of the point x at time t" will have the technical "definition" x(t). The first step is to recognize that being too attached to "definitions" is the origin of suffering (to misquote someone or other :-) In order to do mathematics, we need to be ready to make and unmake attachments as needed . . . Euclid "made a mistake" in thinking that he needed to define terms like "point" and "line" and so, it turns out, he didn't really "define" them, he just left us a legacy of muddled language and ideas in that area . . . people like Hilbert made great progress in clearing up the meta-mathematical muddles that Euclid had left us . . . tom ============================================================ FRIAM Applied Complexity Group listserv Meets Fridays 9a-11:30 at cafe at St. John's College lectures, archives, unsubscribe, maps at http://www.friam.org