It seems that the cunning policy: "It is better to be vague and partly
correct than to be precise and completely wrong" is applied
particularly to coordinate geometry form the very beginning. It seems
that even among mathematicians there is a difference of opinion on the
following fundamental issues: (1) Whether a point occupoies space or
does not occupy space. (2) Does a point specify only 'location' and
not a number? (3) Whether each point has to represent the same number
OR a point can represent any number from 0 to infinity, e.g. any
value of TanA. (4) Whether a line becomes contiguous/continuous if we
ADD points to a line one AFTER another (assign value of X, then
calculate the value of Y and finally place the point on its
appropriate position) or it requires a different condition.

On 3 Nov 2002, V.Gopal wrote:

(1) It occupies a pinpoint.
(2) It's there.
(3) Yes you can count points.
(4) Continuous lines have all the little porous pores filled.

"Philosophy, the science and art of antiquated quibbling."  -- WE

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  does not occupy space.

"space" is not math it's physics, in ph there is no point -> everything
occupies space

a one dimensional point is a number

I didn't really get the question. but two equal points represent the
same number ...

after all everything's a matter of definition

akalos

[snip]

You have made your point that you know nothing.  Rather than continue
eructating amidst thinking people who loathe you, why don't you infest
religious newsgroups and delite your peer group of loud morons spewing
revelation and inerrancy?  

--
Uncle Al
http://www.mazepath.com/uncleal/
 (Toxic URL! Unsafe for children and most mammals)
"Quis custodiet ipsos custodes?"  The Net!

In article <3DC552FD.9889B...@hate.spam.net>, Uncle Al

  ^^^^^^^^^^

Thanks again for the vocabulary building.

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In article <38af3945.0211030641.722c8...@posting.google.com>,

What is there about co-ordinate geometry that annoys you so?  Did you fail
a class in it or something?  Your crusade against it seems completely
unreasonable, but you must have your reasons.

Seems to whom?  Please cite a *single* bona-fide mathematician (not some
Usenet crank) who has the slightest problem with the foundations of
co-ordinate geometry.

(1) Whether a point occupoies space or does not occupy space.

This actually can be a meaningful question in point-set topology.  As
applied to Euclidean space, a point does NOT occupy space.  Do you have a
cite of some mathematician who claims that it does?!?

Huh?  In co-ordinate geometry, a location IS a tuple of numbers.  Do you
have an example of the distinction between the two that you are trying to
get at?

Since each point IS a tuple of numbers, why ask whether it can be several
different tuples?  Clearly it can not.

What *is* there about tan(A) that intrigues you so?  Is it the fact that
tan(pi/2) is infinite?  If so, this would also explain your fascination
with the hyperbola XY=1.

There is no disagreement among mathematicians about this; it does require
"a different condition" in the general case.  Clearly, a line with a
countable number of point holes in it can be filled in by the procedure
you describe.  But, as Cantor showed, the number of points in a line
segment (such as a gap in your discontinuous line) is vastly greater than
the number that can be filled in one-at-a-time.

HTH, really.

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In fact, in physics the word used for mass is 'point mass' or 'mass
point'.
The path along which it moves is a 'line'.
"A consistent field theory requires continuity of all elements of
theory, not only in time but also in space, and in all points in
space. Hence neither the concept of point mass nor that of motion can
play a part in a field theory." I am not talking of Field physics. I
am talking of coordinate geometry.
In geometry can we use a point to represent 0, one point to represet
10^-100, then one point each to represent 10^-10, 1, 10 etc? In this
context I am asking whether a point has to represent the same number?
Do you mean that points have size when you use the phrase "equal
points"?
I believe that each line has to represent one and the same number and
to represent another number we have to use another line and these
lines cannot intersect each other. If 0 to 1 is continuous then we
must have infinite number of parallel lines to cover all the numbers
between 0 and 1.
For those who bring in religion into geometry -  I cannot understand
in waht way religion is connected to geometry. In any case I do not
believe that God can perform miracles.

A point is a point is a point.

Is a mathematical point a physical point
or, to make my point, a philosphical point?

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news:20021103213013.F9434-100000@agora.rdrop.com...

    Yes, but physical points aren't mathematical points,
    since Einstein wannabees need to be reminded
    to use 12pt type in their writeups or they'll
    be mistaken for Heil Hilter wankers once
    again, and have to be Schroedinger's non-existent
    cat nuked again. Or even continously erased by
    the nonexistent continuum.

jim.hun...@jhuapl.edu

If you can't explain something simply, you don't know enough about it.
        -- Albert Einstein

 _     Yes, but physical points aren't mathematical points,

   As far as the laws of mathematics refer to reality, they are not
certain, and as far as they are certain, they do not refer to reality.
    -- Albert Einstein;  www.phnet.fi/public/mamaa1/einstein.htm

 _     since Einstein wannabees need to be reminded
 _     to use 12pt type in their writeups or they'll
 _     be mistaken for Heil Hilter wankers once
 _     again, and have to be Schroedinger's non-existent
 _     cat nuked again. Or even continously erased by
 _     the nonexistent continuum.

        Are you appealing, to some invocation
        That is revealing, a small revocation
        For my repealing, your incantation?

                Pointfullessness
Now changing the point, to make my point which is the point I deserve for
pointing out the most pointedly pointed point ever pointed, may I point
out to you, the point I wish you to appoint me without disappointment at
our next appointment?  -- the archives of Will's Will

----

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   As far as gravity refers to anything except self-reflecting
   morons, we'll assume that Einstone wasn't a complete dork,
   but merely a partial dork.

   God only plays with dice, he does not play with
   real number reality wankers, since reality is too
   complex for idiots like tensor goobs to
   even partially understand

Yaba Daba Doo!  Are you Flintstone?

Tensor goobs?  Some candy your mother gives you?

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   Moma always said that Tensors are the bubble gum of logic dude,
   since it's only rolling, bent-dick, religoid mathemawankers and their
   Jesuit stand-ins who believe in moron SPACE-TIME to begin with.

Hi,
where did you learn mathematics? It seems there must be something
non-standard about your education, for the confusions I find in your
postings are not the ones I am used to finding. Below you will find
geometry the way I like it most. Hopefully it will answer some of your
questions, and hopefully you and others will enjoy reading it.

Consider an n-dimensional Euclidean space S. In it we have (Euclidean)
subspaces of dimensions less than n. I will use the following
synonyms:
A point is a 0-space.
A line is a 1-space.
A plane is a 2-space.
I will assume in what follows that n > 2, so that I can illustrate
some things I want to illustrate.

We have S, and we know there are points, lines, planes, m-spaces (0 <=
m < n) on it, but we do not have any of theese points, lines or
planes; we cannot *point to* any point, line, or plane.

Suppose now that someone gives us a point P of S. We can now point to
P, and not only that: the fact that we are given P means that the
other points divide themselves according to how far away from P they
are. For along with a point we automatically have the (n-dimensional)
spheres centered at the point (though we cannot as yet point to any
particular (n-)sphere). In addition, lines, planes, and more generally
m-spaces, can be divided into those passing through P and those not
passing through P.

Let us go on and suppose that someone gives us another point Q. Among
the spheres we got when we got P, we can now distinguish the one on
which Q is located. Let us use s_AB to denote the sphere centered at A
that passes through B. Then we now have s_PQ. Also, among the lines
passing through P we now note the one that also passes through Q. By
line AB I will understand the line passing through points A and B; so
what we now have is the line PQ. We can also look at things from the
point of view of Q. Forgetting about P for a second, we then have the
(n-)spheres with center at Q, and for any m (> 0, < n) m-spaces are
divided into those that pass through Q and those that do not. With P
we then have the sphere at Q on which P lies, that is, s_QP. For any
m-space M we now have four possibilities:
~(P on M) & ~(Q on M)
~(P on M) & (Q on M)
(P on M) & ~(Q on M)
(P on M) & (Q on M)
("~" stands for "not," "&" stands for "and," and "M on N" means that
space M is part of space N. In particular, if dim(M)=1, and dim(N)=2,
then "M on N" means point M lies on line N.)
Actually, more can be defined. For example, there is the unique point
that lies precisely between P and Q, and there is a unique (n-1)-space
having the property that points on it are just as distant from P as
from Q.

We could go on this way and ask what we can point to when various
things are given to us (I recommend it as an exercise), but I will now
turn to something else. With all theese geometrical objects one can
ask the question: do they exist? My personal opinion is that geometry
should be understood in a way that makes this a meaningless question,
but normally another approach is taken, and as part of axiomatisations
of geometry one consequently has to state things like "there exists
n+1 points such that no (n-1)-space contains them all." With the
approach I prefer one speaks (or can speak, anyway) about *being
given* things, and also, no importance is put on the particular way
one describes what one has. An example of the latter is that I do not
have to speak about points and lines at all; any way of representing
what I have is fine.

More interestingly, we can speak about so-called points at infinity. A
point at infinity is where parallell (that is, non-intersecting) lines
intersect each other. For any collection of parallell lines there is
exactly one point at infinity through which they all pass. Thus, given
any two lines they intersect at exactly one point, either a normal one
or one at infinity. I hope you get the idea. More generally, it is
useful to consider m-spaces at infinity. For example, given two points
at infinity there is a unique line passing through both, and this line
is a line at infinity. Another example is that given two points at
infinity and one normal point there is a unique plane passing through
all three.

Through the approach I prefer, we can completely avoid questions like
whether points at infinity exist. The one who asks whether they exist
has introduced a new concept (that of existence) and the question
cannot be answered until the questioner specifies what she understands
by that concept. Not only do we avoid the question as to whether
points at infinity exist, we also avoid having to choose between
various continuity properties. An example of a continuity property
(one called the circular continuity principle) is "given two circles
c1 and c2 in a plane p that have points inside each other, c1 and c2
intersect at exactly two points." With the normal meaning of point
this is true, but with a non-standard meaning it could be false. The
important thing here is that given points in one sence we can
introduce points in another sence. Henceforth points will be
understood to have the same completeness proprty as the real numbers.
This can be stated thus: any Cauchy-sequence of points converges to a
point (if you do not know what this means it does not matter).

Points are distinguished from spaces of dimension > 0 in that they
"have no parts," but this does not mean we cannot introduce parts if
we like. Consider a line l with the two directions D1 and D2 (I hope
the meaning of a direction is intuitively clear), and consider a P of
l. We can think of P as a combination of "P from the direction D1" and
"P from the direction D2." We could also speak about things like
"approaching P at speed s" or "approaching P at speed s from the
direction D." The important thing is that having parts, like
existance, is a bit relative. For example, a line can be thought of as
the collection of all points on it, but it is equally possible to
think of a point as all the lines passing through it. That said, there
is an obvious sence in which a point has no parts and in which spaces
of dimension > 0 have parts. By now, everyone reading this posting
must have noted that I am not particularly "formal". Actually, my
philosophy is that it does not matter at all how we choose to convey
an idea. We can use diagrams, formal notation, informal English, or
anything we like.

Let S be a space. Let Congr(A, B, C, D) stand for "line segment AB is
congruent to line segment CD." In a sence, this relation is all there
is to geometry, for out of it everything else can be defined (it is
not the only relation having this property, though). Note that Congr
is given as soon as the space is. Congr is not particularly convenient
to use. It would be more practical to have a function dist(A, B) which
given points A and B gives us their distance. One could then define
Congr as
Congr(A, B, C, D) <-> dist(A,B) = dist(C, D),
and one could write easy-to-read things like
4 * dist(A, B) = 3 * dist(C, D).
It would be easy to say what properties distances must satisfy, and
obviously theese properties would be satisfied by real numbers. The
problem is that out of what we are given we cannot define a function
that given two points gives us a real number. Distances and angles are
different in this respect. So what are we to do? We may inelegantly
assume that two points determining a unit of length are given (this is
inelegant because the points must be arbitrarily chosen), but let us
opt for the elegant approach. This means we must think of distances as
something other than numbers. We could call them quantities. For the
moment I will assume we understand intuitively what distances are.

Now that we have distances, the only think we lack is the concept of
direction. It is intuitively clear what a direction is, so I need not
say much about it. For any direction D there is an opposite direction
-D, and {D, -D} (the class consisting of D and -D) corresponds
directly to a point at infinity. Directions will enter the geometry
when we consider motions in a given direction, which is what we will
now do.

For each combination of a direction D and a distance d there
corresponds what I will call an adder a = add_D_d, which given a point
P gives us the point a(P) that lies in the direction D relative to P
and lies at distance d from P. What I have named adders we could name
motions, but the name adder makes sence when what I call multipliers
are also considered. A multiplier could also be called a magnifier. It
takes an adder and magnifies or shrinks it. For example, there is an
identity multiplier m_id that leaves the adder it gets unchanged, and
there is a doubler that given an adder a1 gives us an adder a2 such
that
a2(P) = a1(a1(P))
Here it is appropriate to make some definitions:
a1 + a2 = a1a2 = the adder which for any point P gives a1(a2(P))
m1 * m2 = m1m2 = the multiplier which for any adder a gives m1(m2(a))
m1 + m2 = the multiplier which for any adder a gives m1(a)m2(a)
(m)(d) = the distance which is "m times as great as" d
Note that multipliers ...

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