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
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