The Virgin Birth of Points
Lester Zick
The Virgin Birth of Points
~v~~
The Jesuit heresy maintains points have zero length but are not of
zero length and if you don't believe that you haven't examined the
argument closely enough.
~v~~
William Elliot
Clearly points don't have zero length, they have a positive infinitesimal
length for which zero is just the closest real approximation.
Robert J. Kolker
In Euclidean space a set which has exactly one pont as a member has
measure zero. But you can take the union of an uncountable set of such
singleton sets and get a set with non-zero measure.
Bob Kolker
Robert J. Kolker
>
> Clearly points don't have zero length, they have a positive infinitesimal
> length for which zero is just the closest real approximation.
You don't need to resort to non-standard analysis. Within the realm of
standard real numbers, the matter is settle using measure (either Borel
or Lebesque)
Bob Kolker
David C. Ullrich
Erm, no. Points (or rather singletons) have zero length.
************************
David C. Ullrich
Venkat Reddy
I agree. Also, like I said in the other post, points can only exist as
boundaries of higher dimensional regions. Lines, surfaces, solids etc
can exist as regions in their own world and as boundaries in higher
dimensions. When they are in the role of a boundary they are not part
of any regions (of higher dimension).
We can't observe life of a point as a region in its own dimensional
space.
- venkat
Hero
What measure will give a non-zero number/value?
With friendly greetings
Hero
John Jones
Points have zero length when construed as lying in a spatial
framework. However, points have no length because points are not
objects that arise in a spatial framework. Positions, not points,
arise in the spatial framework, and positions are always
constructions.
I conclude that the question about points cannot be a logical inquiry
or someone here would have been able to sort it out...
Randy Poe
The Lebesgue measure of the interval [0,1] is 1. The
Lebesgue measure of every finite and countable subset
of that interval is 0. The Lebesgue measure of the Cantor
set, which is uncountable, is also 0.
Is that what you were asking?
- Randy
Dave Seaman
Lebesgue measure will do so, not for all possible uncountable sets, but
for some. For example, the Lebesgue measure of an interval [a,b] is its
length, b-a.
--
Dave Seaman
Oral Arguments in Mumia Abu-Jamal Case heard May 17
U.S. Court of Appeals, Third Circuit
<http://www.abu-jamal-news.com/>
John Jones
An interval [a,b] is composed of positions, not points. But even
positions are constructions, and it is not appropriate to analyse a
construction in spatial terms.
Lester Zick
wrote:
Except the main purpose of this thread is less to discuss the zero
length of points than the heresy of maintaining self contradictory
predicates, as in "has zero length" and "is not of zero length".
~v~~
Lester Zick
<ma...@hevanet.remove.com> wrote:
I don't see how points "clearly" have infinitesimal length unless
they're infinitesimal to begin with. Clearly Newton didn't think
points were infinitesimal nor did Leibniz or they wouldn't have
drafted the notations they used. And Newton's calculations of tangents
were only defined at one point not at one infinitesimal.
We have a certain arithmetic notation such as 5-5=0 in which the
difference between 5 and itself is not infinitesimal but zero. Nor is
the difference between a line and itself infinitesimal but zero.
~v~~
Lester Zick
I wouldn't call the calculus non standard analysis.
~v~~
Lester Zick
Then I'm curious about this unionizing of points people talk about.
~v~~
Lester Zick
So if you unionize an infinite number of points, would the converse
operation be decertification of the union and wouldn't that constitute
division by zero?
~v~~
Robert J. Kolker
>
> Except the main purpose of this thread is less to discuss the zero
> length of points than the heresy of maintaining self contradictory
> predicates, as in "has zero length" and "is not of zero length".
Points do not have a length (0 or not). Some -sets- of points have
-measure-. In particular a set consisting of a single point has measure 0.
You have manage to confuse an object with a set whose element is that
object.
Bob Kolker
Dave Seaman
I think you need to learn some measure theory. This is a question about
mathematics, by the way, not philosophy.
Robert J. Kolker
>
> I wouldn't call the calculus non standard analysis.
Integrals are done over sets of points, not idividual points. Learn to
distinguish between sets and the elements of the sets.
Bob Kolker
Robert J. Kolker
>
>
> Then I'm curious about this unionizing of points people talk about.
You are curious about sets (and no wonder, you know nothing about them).
Bob Kolker
Lester Zick
wrote:
>On Nov 11, 9:40?pm, Lester Zick <dontbot...@nowhere.net> wrote:
>> The Virgin Birth of Points
>> ~v~~
>>
>> The Jesuit heresy maintains points have zero length but are not of
>> zero length and if you don't believe that you haven't examined the
>> argument closely enough.
>>
>> ~v~~
>
>Points have zero length when construed as lying in a spatial
>framework. However, points have no length because points are not
>objects that arise in a spatial framework. Positions, not points,
>arise in the spatial framework, and positions are always
>constructions.
So the intersections of lines are not points? Dearie, me.
>I conclude that the question about points cannot be a logical inquiry
>or someone here would have been able to sort it out...
Logically or illogically?
~v~~
Robert J. Kolker
>
> So if you unionize an infinite number of points, would the converse
> operation be decertification of the union and wouldn't that constitute
> division by zero?
You are making no sense here. Division is the inverse operation to
multiplication.
Bob Kolker
Igor
> On Mon, 12 Nov 2007 05:13:10 -0800, Venkat Reddy <vred...@gmail.com>
No. Your main purpose in this thread is the same as in any other of
your threads. And that is the intentional obfuscation of established
mathematical concepts.
Robert J. Kolker
>
>
> I think you need to learn some measure theory. This is a question about
> mathematics, by the way, not philosophy.
Zick is totally incapable of understanding either mathematics or
physics. Robert Heinlein had some clever things to say about people who
cannot cope with mathematics. Heinlein said they are subhuman but
capable of wearing shoes and keeping clean.
Bob Kolker
>
>
>
Robert J. Kolker
>
> So the intersections of lines are not points? Dearie, me.
Lines (which are sets of points) sometimes have a non-zero set
intersection which consists of a single point.
Once again you do not distinguish between objects and the sets of which
the objects are elements. Another evidence that you cannot cope with
mathematics.
Bob Kolker
Robert J. Kolker
>
>
> No. Your main purpose in this thread is the same as in any other of
> your threads. And that is the intentional obfuscation of established
> mathematical concepts.
>
>
He is unable to do otherwise. He cannot comprehend standard mathematical
concepts. Zick cannot cope with mathematics. Robert Heinlein had some
interesting things to say about people like Zick.
Bob Kolker
>
>
>
Hero
> Hero wrote:
> > Robert wrote:
> > > Lester Zick wrote:
> > > > The Virgin Birth of Points
> > > > ~v~~
>
> > > > The Jesuit heresy maintains points have zero length but are not of
> > > > zero length and if you don't believe that you haven't examined the
> > > > argument closely enough.
.
> > > In Euclidean space a set which has exactly one pont as a member has
> > > measure zero. But you can take the union of an uncountable set of such
> > > singleton sets and get a set with non-zero measure.
.
> > What measure will give a non-zero number/value?
> The Lebesgue measure of the interval [0,1] is 1. The
> Lebesgue measure of every finite and countable subset
> of that interval is 0. The Lebesgue measure of the Cantor
> set, which is uncountable, is also 0.
>
> Is that what you were asking?
Yes.Thanks, Randy.
And thanks to Dave, John and Robert too.
Now we have all we need to point to the important fact:
It is measuring sets of points, not measuring points or measuring many
points.
A point A is different from the set { A }.
And a hint to the standard topology of the real number line, which
according to Kuratowski, gives to every two points the intervall in
between, a set of points.
With friendly greetings
Hero
Dave Seaman
Zick is in my killfile. He is not the person I was responding to.
Randy Poe
> On Mon, 12 Nov 2007 06:48:11 -0500, "Robert J. Kolker"
>
> >William Elliot wrote:
>
> >> Clearly points don't have zero length, they have a positive infinitesimal
> >> length for which zero is just the closest real approximation.
>
> >You don't need to resort to non-standard analysis. Within the realm of
> >standard real numbers, the matter is settle using measure (either Borel
> >or Lebesque)
>
> I wouldn't call the calculus non standard analysis.
Calculus does not require infinitesimals or NSA. I
believe that was Leibniz's method, but we mostly
follow Newton's development which only requires
a theory of limits.
- Randy
John Jones
>
> - Show quoted text -
I think you need to distinguish between a position and a point before
wildly conflating them in both a philosophical and mathematical
confusion.
John Jones
> On Mon, 12 Nov 2007 07:40:03 -0800, John Jones <jonescard...@aol.com>
The intersections of lines are positions, not points. There is no
precedent for creating a new metaphysical entity from the arbitrary
arrangement of lines. I would have thought it obvious. But plainly I
was mistaken.
John Jones
A line is not a set of points because sets are indifferent to order.
However, if you care to order points we still do not have a minimal
definition of a line.
Hero
> Randy wrote:
.
> > Is that what you were asking?
.
> Yes.Thanks, Randy.
> And thanks to Dave, John and Robert too.
>
> Now we have all we need to point to the important fact:
> It is measuring sets of points, not measuring points or measuring many
> points.
> A point A is different from the set { A }.
>
> And a hint to the standard topology of the real number line, which
> according to Kuratowski, gives to every two points the intervall in
> between, a set of points.
>
I just see, that Robert wrote more or less the same, posted in the
same hour too, i didn't know about.
But anyhow, the truth can be expressed twice, without getting twisted.
With friendly greetings
Hero
Uncle Al
>
> The Virgin Birth of Points
> ~v~~
>
> The Jesuit heresy
Idiot. The Giant Flying Spaghetti Monster heresy. The thinner thighs
in 30 days heresy.
1) Who gives shit one what god thinks?
2) Who gives shit one what borign idiot Lester Zick spews?
Uncle Al sold his soul to Satan in a simple contract: Post mortality,
for all of remaining eternity, Uncle Al will torture meat puppets like
you, Zick, in a choice part of Hell. Uncle Al gets ice in his drink
and a fine party into eternity.
--
Uncle Al
http://www.mazepath.com/uncleal/
(Toxic URL! Unsafe for children and most mammals)
http://www.mazepath.com/uncleal/lajos.htm#a2
Dave Seaman
> On Nov 12, 6:05?pm, Dave Seaman <dsea...@no.such.host> wrote:
>> On Mon, 12 Nov 2007 07:50:52 -0800, John Jones wrote:
>> > On Nov 12, 3:42?pm, Dave Seaman <dsea...@no.such.host> wrote:
>> >> On Mon, 12 Nov 2007 07:06:39 -0800, Hero wrote:
>> >> > What measure will give a non-zero number/value?
>>
>> >> Lebesgue measure will do so, not for all possible uncountable sets, but
>> >> for some. For example, the Lebesgue measure of an interval [a,b] is its
>> >> length, b-a.
>>
>> > An interval [a,b] is composed of positions, not points. But even
>> > positions are constructions, and it is not appropriate to analyse a
>> > construction in spatial terms.
>>
>> I think you need to learn some measure theory. This is a question about
>> mathematics, by the way, not philosophy.
>>
>> - Show quoted text -
> I think you need to distinguish between a position and a point before
> wildly conflating them in both a philosophical and mathematical
> confusion.
In my statement that you quoted, I used neither of the terms "position"
or "point". I mentioned only Lebesgue measure, uncountable sets, and
intervals. Exactly what is your, er, point? Why do I need to distinguish
between terms that I didn't use?
Neither of those is a precise mathematical term, by the way. The meaning
depends on context, but to me a "point" is a member of some abstract
space (possibly a vector space, or a topological space, or a metric
space, or a measure space, or a Banach space, or whatever). A
"position", on the other hand, suggests a point that is given in some
coordinate system. That doesn't always apply. Lots of times we talk
about points in situations where there are no coordinates in sight.
I consider "position" to be too limited a term for that reason.
lwa...@lausd.net
> On Nov 12, 6:21?pm, "Robert J. Kolker" <bobkol...@comcast.net> wrote:
> > Once again you do not distinguish between objects and the sets of which
> > the objects are elements. Another evidence that you cannot cope with
> > mathematics.
> However, if you care to order points we still do not have a minimal
> definition of a line.
I've been thinking about the links to Euclid's and Hilbert's
axioms presented in some of the other geometry threads:
http://en.wikipedia.org/wiki/Hilbert%27s_axioms
These last few posts are posing the question, is a
point an _element_ of a line, or is a point a
_subset_ of a line?
The correct answer is neither. For let us review
Hilbert's axioms again:
"The undefined primitives are: point, line, plane.
There are three primitive relations:
"Betweenness, a ternary relation linking points;
Containment, three binary relations, one linking
points and lines, one linking points and planes,
and one linking lines and planes;
Congruence, two binary relations, one linking line
segments and one linking angles."
So we see that line is an undefined _primitive_,
and that there is a _primitive_ to be known as
"containment," so that a line may be said to
"contain" points.
Notice that the primitive "contain" has _nothing_
to do with the membership primitive of a set
theory such as ZFC. Why? Because this is a
geometric theory that is not even written in
the _language_ of ZFC.
So both "a point is an element of a line" and "a
point is a subset of a line" are incorrect.
The other question concerns what the intersection
of two lines is. Well, first we must define
"intersection" -- in terms of our _primitives_,
of course -- before we can answer. And the only
answer we can possibly give is in terms of
_containment_: the intersection of two lines a,b
is a point A such that a contains A and b contains
A as well, provided that such a point exists. We
can't call it a "position," since "position" is
not a _primitive_ of our theory.
So now all we have to do is prove that if such a
point exists, it must be unique. But this follows
directly from Axiom I.1. For if there were two
points of intersection A,B, then I.1 tells us that
two points determine a line, so that AB = a and
AB = b as well, therefore a = b. So if a,b are
distinct and intersect, then they intersect in a
unique point of intersection.
Kenneth Doyle
> unionizing of points
Wasn't that a mafia racket that started on the waterfront? Oh, hang on,
that was the unionizing of punts. Never mind.
Lester Zick
wrote:
So differentials are points?
~v~~
Lester Zick
wrote:
Actually in the New Yorker, the unionizing of puns.
~v~~
Lester Zick
Unlike yourself, Bobby.
~v~~
Robert J. Kolker
> Uncle Al sold his soul to Satan in a simple contract: Post mortality,
> for all of remaining eternity, Uncle Al will torture meat puppets like
> you, Zick, in a choice part of Hell. Uncle Al gets ice in his drink
> and a fine party into eternity.
And a hummer from his Lady.
Bob Kolker
>
Robert J. Kolker
> A line is not a set of points because sets are indifferent to order.
> However, if you care to order points we still do not have a minimal
> definition of a line.
Consider E2, the set of number pairs (x,y) x,y real taken as points.
Along with the pythagorian metric and the obvious definition of lines
(sets of (x,y) which satisfy a*x + b*y = c for some constants a,b,c) you
get a structure that satistfies Hilberts postulates for plane geometric
space. Since the axioms are categorica, all instances of Euclidea plane
geometry (as axiomatized by Hilbert) are isometric. So a model where
lines consist of points yields an instance of the geometry.
Since the line can be parametrized by a single variable it can be easily
ordered.
Where did you get you degree? I need to know, so I won't send my kids
there.
Bob Kolker
>
Robert J. Kolker
>
> The intersections of lines are positions, not points. There is no
> precedent for creating a new metaphysical entity from the arbitrary
> arrangement of lines. I would have thought it obvious. But plainly I
> was mistaken.
Take a pair of linear equations in two variables each of which define a
line. If the equations are not linearly dependent they determine a
unique solution (x,y) which is --- aha!---- the point of intersection.
In a Euclidean Plane lines when they intersect at all, have a unique
point of intersection. And mathematical objects are not metaphysical
entities. They are brain farts the blow about in our skulls.
Bob Kolker
>
Lester Zick
<bobk...@comcast.net> wrote:
>Lester Zick wrote:
>
>>
>> So the intersections of lines are not points? Dearie, me.
>
>Lines (which are sets of points) sometimes have a non-zero set
>intersection which consists of a single point.
And apparently sometimes they don't? So what is it exactly you're
sometimes saying, Bobby?
>Once again you do not distinguish between objects and the sets of which
>the objects are elements. Another evidence that you cannot cope with
>mathematics.
But I can certainly cope with the likes of homo habilis.
~v~~
John Jones
Lester Zick
wrote:
>On Nov 12, 6:07?pm, Lester Zick <dontbot...@nowhere.net> wrote:
Plainly you were, are, and will be for if a point is a metaphysical
entity then so I suggest are lines which I should have thought was
obvious.
~v~~
John Jones
A position may well not be a primitive, but the intersections of lines
construct positions, not points. Primitives are incommensurables.
Points, lines, planes, etc are incommensurables which do not contain
the properties of one within the other. Their 'synthesis' is not a
synthesis of properties or objects, but of the frameworks that
establish objects and properties (see Kant).
Lester Zick
wrote:
>Lester Zick wrote:
>>
>> The Virgin Birth of Points
>> ~v~~
>>
>> The Jesuit heresy
>[snip crap]
>
>Idiot. The Giant Flying Spaghetti Monster heresy. The thinner thighs
>in 30 days heresy.
>
> 1) Who gives shit one what god thinks?
Plainly Jesuits do.
> 2) Who gives shit one what borign idiot Lester Zick spews?
"borign"? I'm not familiar with that word. Pray tell, is it a learned
borrowing from the Jesuit?
>Uncle Al sold his soul to Satan in a simple contract: Post mortality,
>for all of remaining eternity, Uncle Al will torture meat puppets like
>you, Zick, in a choice part of Hell. Uncle Al gets ice in his drink
>and a fine party into eternity.
Al, look, you're just raving now. Obviously you have even less
coherence than usual. Tell you what, why not take a couple steps back
from the subject and go fuck yourself.
~v~~
Robert J. Kolker
> And apparently sometimes they don't? So what is it exactly you're
> sometimes saying, Bobby?
Parellel lines have no points of intersection. Next question?
Bob Kolker
Schlock
<dse...@no.such.host> wrote:
>On Mon, 12 Nov 2007 13:20:13 -0500, Robert J. Kolker wrote:
>> Dave Seaman wrote:
>>>
>>>
>>> I think you need to learn some measure theory. This is a question about
>>> mathematics, by the way, not philosophy.
>
>> Zick is totally incapable of understanding either mathematics or
>> physics. Robert Heinlein had some clever things to say about people who
>> cannot cope with mathematics. Heinlein said they are subhuman but
>> capable of wearing shoes and keeping clean.
>
>Zick is in my killfile. He is not the person I was responding to.
Aw c'mon, Davey. Everybody knows a Seaman just loves semen.
lwa...@lausd.net
> > So now all we have to do is prove that if such a
> > point exists, it must be unique. But this follows
> > directly from Axiom I.1. For if there were two
> > points of intersection A,B, then I.1 tells us that
> > two points determine a line, so that AB = a and
> > AB = b as well, therefore a = b. So if a,b are
> > distinct and intersect, then they intersect in a
> > unique point of intersection.
> A position may well not be a primitive, but the intersections of lines
> construct positions, not points. Primitives are incommensurables.
> Points, lines, planes, etc are incommensurables which do not contain
> the properties of one within the other. Their 'synthesis' is not a
> synthesis of properties or objects, but of the frameworks that
> establish objects and properties (see Kant).
I'm sorry, but I must concur with Mr. Kolker here. The
intersection of two lines is a point.
One way to see what's going on here is to consider
the standard model of Hilbert, namely R^3. (Kolker
uses the notation E2 for 2D Hilbert, but let us
consider the third dimension now as well.)
Now we can determine what this model happens to
map the primitives to. As Kolker has said, "line"
is mapped to the set of ordered triples (x,y,z)
satisfying a linear relation.
But what about "point"? Is "point" mapped to a
triple itself (an element of a line), or is it a
singleton whose sole element is an ordered
triple (a subset of a line)?
This is, of course, closely related to what the
primitive "containment" is mapped to. It could
be membership (point = ordered triple) or
inclusion (point = singleton of ordered triple).
I believe that mapping containment to membership
will be awkward. Let us recall what Hilbert
wrote about containment:
"Containment, three binary relations, one linking
points and lines, one linking points and planes,
and one linking lines and planes."
So we see that lines contain points, planes
contain points, and planes contain lines.
And here lies the problem. If we let containment
be mapped to membership, then planes would have
both points and lines as distinct elements. And
even if we only allowed planes to have lines as
elements, which lines would be the elements of
the plane anyway. For the plane z = 0, for
example, are x = constant the elements of the
plane, or y = constant, or all of them?
So it makes much more sense to map "containment"
to "inclusion." Thus points are singletons and
subsets of the lines and planes that happen to
"contain" them. And therefore the intersection
of two lines is the set intersection -- which
is exactly the "point."
Of course, what about the ordered triples --
the elements of points, lines -- themselves? We
may call them "positions," if we want. So the
single element of a point is a "position," and
the elements of a line are "positions." And so
answering the OP's question, positions don't
have a measure, but "points" do -- at least,
in the standard model R^3 of Hilbert, where
subsets in R^3 have a Lebesgue measure.
Of course, this is all only in the standard
model of Hilbert. In other models, "point,"
"line," may be mapped to something other
than sets, so we can't always refer to the
element of a point as a "position," because
"point" may be mapped to something that
doesn't have an element.
To see what I mean, let us take a page from
Han de Brujin's book and come up with a new
model of some subset of Hilbert's axioms. (If
you don't know who HdB is, it's not that
important for this example.)
Consider Hilbert's Axioms of Incidence only --
the ones labeled I.1 to I.7. Notice one can't
prove from these axioms alone that more than
finitely many points exist. Indeed, we
observe I.7:
"I.7: Upon every straight line there exist at
least two points, in every plane at least
three points not lying in the same straight
line, and in space there exist at least four
points not lying in a plane."
Apparently, by I.1 through I.7, we can't even
prove the existence of more than _four_ points,
and indeed, we can construct a model of I.1
through I.7 in which only four points exist.
Now in this model, we will map our four points
to natural numbers -- in particular, the
natural numbers 1, 2, 4, and 8. (Those familiar
with HdB should know by now where I am heading
with this.) Lines contain exactly two points --
mapped to the sum of the two points that lie
on them. Planes contain exactly three points --
mapped to the sum of the three points that lie
on them. So we have:
1. point (1)
2. point (2)
3. line (1+2)
4. point (4)
5. line (1+4)
6. line (2+4)
7. plane (1+2+4)
8. point (8)
9. line (1+8)
10. line (2+8)
11. plane (1+2+8)
12. line (4+8)
13. plane (1+4+8)
14. plane (2+4+8)
15. space (1+2+4+8)
In this model, containment is mapped to a
more complicated matter -- one can try
bitwise AND (or OR) to come up with the
relation onto which containment is mapped.
The important part is that this points
don't have "positions" at all.
Lester Zick
<bobk...@comcast.net> wrote:
>Lester Zick wrote:
>
>>
>> Except the main purpose of this thread is less to discuss the zero
>> length of points than the heresy of maintaining self contradictory
>> predicates, as in "has zero length" and "is not of zero length".
>
>Points do not have a length (0 or not). Some -sets- of points have
>-measure-. In particular a set consisting of a single point has measure 0.
>
>You have manage to confuse an object with a set whose element is that
>object.
Hey it's not my problem, Bobby. I'm not the one who claims points have
zero length but are not of zero length.Modern mathematics is a heresy.
~v~~
Lester Zick
>On Nov 12, 12:08 pm, Lester Zick <dontbot...@nowhere.net> wrote:
>> On Mon, 12 Nov 2007 05:13:10 -0800, Venkat Reddy <vred...@gmail.com>
>> wrote:
>>
>>
>>
>>
>>
>> >On Nov 12, 5:02 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>> >> On Sun, 11 Nov 2007 20:57:47 -0800, William Elliot
>>
>> >> <ma...@hevanet.remove.com> wrote:
>> >> >On Sun, 11 Nov 2007, Lester Zick wrote:
>>
>> >> >> The Virgin Birth of Points
>> >> >> ~v~~
>>
>> >> >> The Jesuit heresy maintains points have zero length but are not of
>> >> >> zero length and if you don't believe that you haven't examined the
>> >> >> argument closely enough.
>>
>> >> >Clearly points don't have zero length, they have a positive infinitesimal
>> >> >length for which zero is just the closest real approximation.
>>
>> >> Erm, no. Points (or rather singletons) have zero length.
>>
>> >I agree. Also, like I said in the other post, points can only exist as
>> >boundaries of higher dimensional regions. Lines, surfaces, solids etc
>> >can exist as regions in their own world and as boundaries in higher
>> >dimensions. When they are in the role of a boundary they are not part
>> >of any regions (of higher dimension).
>>
>> >We can't observe life of a point as a region in its own dimensional
>> >space.
>>
>> Except the main purpose of this thread is less to discuss the zero
>> length of points than the heresy of maintaining self contradictory
>> predicates, as in "has zero length" and "is not of zero length".
>
>
>No. Your main purpose in this thread is the same as in any other of
>your threads. And that is the intentional obfuscation of established
>mathematical concepts.
I know, Igor. I'm just making an exception in this particular case.
~v~~
Lester Zick
<bobk...@comcast.net> wrote:
>Igor wrote:
>
>>
>>
>> No. Your main purpose in this thread is the same as in any other of
>> your threads. And that is the intentional obfuscation of established
>> mathematical concepts.
>>
>>
>He is unable to do otherwise. He cannot comprehend standard mathematical
>concepts. Zick cannot cope with mathematics. Robert Heinlein had some
>interesting things to say about people like Zick.
Yes, I know, Bobby. He said they don't suffer heresies like modern
mathematics, relativity, and quantum theory gladly.
~v~~
Lester Zick
<bobk...@comcast.net> wrote:
>Lester Zick wrote:
>>
>>
>> Then I'm curious about this unionizing of points people talk about.
>
>You are curious about sets (and no wonder, you know nothing about them).
Yes but on the plus side I suffer fools gladly.
~v~~
Lester Zick
<bobk...@comcast.net> wrote:
>> I think you need to learn some measure theory. This is a question about
>> mathematics, by the way, not philosophy.
>
>Zick is totally incapable of understanding either mathematics or
>physics.
Well I know enough of mathematics to have convinced you there is no
real number line.
~v~~
Lester Zick
<bobk...@comcast.net> wrote:
>Lester Zick wrote:
>>
>> So if you unionize an infinite number of points, would the converse
>> operation be decertification of the union and wouldn't that constitute
>> division by zero?
>
>You are making no sense here. Division is the inverse operation to
>multiplication.
So? I'm just asking what the inverse operation of the unionization of
points is.
~v~~
Lester Zick
>> On Nov 12, 6:21?pm, "Robert J. Kolker" <bobkol...@comcast.net> wrote:
>> > Once again you do not distinguish between objects and the sets of which
>> > the objects are elements. Another evidence that you cannot cope with
>> > mathematics.
>> A line is not a set of points because sets are indifferent to order.
>> However, if you care to order points we still do not have a minimal
>> definition of a line.
>
>I've been thinking about the links to Euclid's and Hilbert's
>axioms presented in some of the other geometry threads:
Guesswork gives me a headache. Please spare us undemonstrated
assumptions of truth.
~v~~
Robert J. Kolker
>
> So? I'm just asking what the inverse operation of the unionization of
> points is.
There is none. THe set operations do not form a group. But, of course,
you knew that. The set operations constitute a lattice.
Bob Kolker
Robert J. Kolker
> Well I know enough of mathematics to have convinced you there is no
> real number line.
So what. The theory of real numbers can and is developed without any
geometric content of all. Any geometrical associations with real numbers
are merely aids to intuition, not logical necessity.
In the nineteenth century a purely analytic foundations for the theory
of real and complex variables was developed. Geometry was purged as a
logical necessity. Of course, geometry can be very helpful for the
right-brain operations associated with discovering new theorems to prove
or new mathematical systems.
Bob Kolker
Robert J. Kolker
>
>
> Hey it's not my problem, Bobby. I'm not the one who claims points have
> zero length but are not of zero length.Modern mathematics is a heresy.
Neither does any one else. You have created a straw man here.
Measure is associated with certain -sets of points-, not the points
themselves.
Bob Kolker
Virgil
>
>
> Hey it's not my problem, Bobby. I'm not the one who claims points have
> zero length but are not of zero length.Modern mathematics is a heresy.
Ultra-heretic Zick accusing others of his own sin?
It is to laugh!
Amicus Briefs
wrote:
>A position may well not be a primitive, but the intersections of lines
>construct positions, not points
So if we change the name of "points" to "positions" we'll solve the
problem?
Lester Zick
<bobk...@comcast.net> wrote:
>Lester Zick wrote:>
>> And apparently sometimes they don't? So what is it exactly you're
>> sometimes saying, Bobby?
>On Mon, 12 Nov 2007 13:21:56 -0500, "Robert J. Kolker"
><bobk...@comcast.net> wrote:
>>Lester Zick wrote:
>>
>>>
>>> So the intersections of lines are not points? Dearie, me.
>>
>>Lines (which are sets of points) sometimes have a non-zero set
>>intersection which consists of a single point.
>Parellel lines have no points of intersection. Next question?
So parallel lines (which are sets of points) sometimes have a non-zero
set intersection which consists of a single point but have no points
of intersection, oh homo habilis? No wonder the Jesuits got involved.
~v~~
Lester Zick
Speak for thyself.
~v~~
Amicus Briefs
Ah, Virgil. We feared lest you had gone astray teaching the trivium of
truth to schoolchildren at Our Lady of Perpetual Education. It's good
to know our fears were as groundless as your faith.
David C. Ullrich
wrote:
>On Nov 12, 5:02 pm, David C. Ullrich <ullr...@math.okstate.edu> wrote:
>> On Sun, 11 Nov 2007 20:57:47 -0800, William Elliot
>>
>> <ma...@hevanet.remove.com> wrote:
>> >On Sun, 11 Nov 2007, Lester Zick wrote:
>>
>> >> The Virgin Birth of Points
>> >> ~v~~
>>
>> >> The Jesuit heresy maintains points have zero length but are not of
>> >> zero length and if you don't believe that you haven't examined the
>> >> argument closely enough.
>>
>> >Clearly points don't have zero length, they have a positive infinitesimal
>> >length for which zero is just the closest real approximation.
>>
>> Erm, no. Points (or rather singletons) have zero length.
>>
>
>I agree.
Good for you.
>Also, like I said in the other post, points can only exist as
>boundaries of higher dimensional regions. Lines, surfaces, solids etc
>can exist as regions in their own world and as boundaries in higher
>dimensions. When they are in the role of a boundary they are not part
>of any regions (of higher dimension).
>
>We can't observe life of a point as a region in its own dimensional
>space.
Uh, no. The reason a set consisting of a single point has zero
length is that a - a = 0.
>- venkat
************************
David C. Ullrich
Lester Zick
<bobk...@comcast.net> wrote:
>Lester Zick wrote:
>>
>>
>> Hey it's not my problem, Bobby. I'm not the one who claims points have
>> zero length but are not of zero length.Modern mathematics is a heresy.
>
>Neither does any one else. You have created a straw man here.
Horseshit, Bobby. I didn't create the straw man. I can cite chapter
and verse.
>Measure is associated with certain -sets of points-,
But only rational/irrational measure and not real measure.
> not the points
>themselves.
They're not interested in measure but in points. Without them they
can't get anywhere beyond.
~v~~
Lester Zick
A lattice? And how is a lattice not a geometric form? And what lies in
the interstices of the lattice? Nothing? Something?
~v~~
Lester Zick
<bobk...@comcast.net> wrote:
>Lester Zick wrote:
>
>> Well I know enough of mathematics to have convinced you there is no
>> real number line.
>
>So what. The theory of real numbers can and is developed without any
>geometric content of all. Any geometrical associations with real numbers
>are merely aids to intuition, not logical necessity.
Sure, sure, Bobby. That's why the expression "real number line" pops
up all over the place. That's why you talk incessantly about lines,
points, and lattices etc.
>In the nineteenth century a purely analytic foundations for the theory
>of real and complex variables was developed. Geometry was purged as a
>logical necessity. Of course, geometry can be very helpful for the
>right-brain operations associated with discovering new theorems to prove
>or new mathematical systems.
Of course geometry can be very helpful because there is a geometry of
arithmetic but no arithmetic of geometry. The problem is there is no
set of real numbers. If there were you could dance on geometry. But
you can't define any set of real numbers because there is no single
modality for real numbers for the definition of any single set.Pi lies
on circular arcs as Archimedes showed and doesn't lie anywhere else.
~v~~
Robert J. Kolker
Better still call them potatoes. Two potatoes determine a kugel. Two
kugels intesect on a potatoe.
Bob Kolker
Venkat Reddy
> On Sun, 11 Nov 2007 20:57:47 -0800, William Elliot
>
> <ma...@hevanet.remove.com> wrote:
> >On Sun, 11 Nov 2007, Lester Zick wrote:
>
> >> The Virgin Birth of Points
> >> ~v~~
>
> >> The Jesuit heresy maintains points have zero length but are not of
> >> zero length and if you don't believe that you haven't examined the
> >> argument closely enough.
>
> >Clearly points don't have zero length, they have a positive infinitesimal
> >length for which zero is just the closest real approximation.
>
> they're infinitesimal to begin with. Clearly Newton didn't think
> points were infinitesimal nor did Leibniz or they wouldn't have
> drafted the notations they used. And Newton's calculations of tangents
> were only defined at one point not at one infinitesimal.
>
> We have a certain arithmetic notation such as 5-5=0 in which the
> difference between 5 and itself is not infinitesimal but zero. Nor is
> the difference between a line and itself infinitesimal but zero.
>
Lester Sir, but I thought you just said that the main purpose of this
thread is less to discuss the zero length of points than the heresy of
maintaining self contradictory
predicates, or to continue some of your silly fights with other
poster. Anyways, happy to see you discussing same ideas from my other
thread regarding the extent of points.
- venkat
Venkat Reddy
> On Nov 12, 6:05?pm, Dave Seaman <dsea...@no.such.host> wrote:
>
>
>
> > On Mon, 12 Nov 2007 07:50:52 -0800, John Jones wrote:
> > > On Nov 12, 3:42?pm, Dave Seaman <dsea...@no.such.host> wrote:
> > >> On Mon, 12 Nov 2007 07:06:39 -0800, Hero wrote:
> > >> > Robert wrote:
> > >> >> Lester Zick wrote:
> > >> >> > The Virgin Birth of Points
> > >> >> > ~v~~
>
> > >> >> > The Jesuit heresy maintains points have zero length but are not of
> > >> >> > zero length and if you don't believe that you haven't examined the
> > >> >> > argument closely enough.
>
> > >> >> measure zero. But you can take the union of an uncountable set of such
> > >> >> singleton sets and get a set with non-zero measure.
>
> > >> > What measure will give a non-zero number/value?
>
> > >> Lebesgue measure will do so, not for all possible uncountable sets, but
> > >> for some. For example, the Lebesgue measure of an interval [a,b] is its
> > >> length, b-a.
>
> > >> --
> > >> Dave Seaman
> > >> Oral Arguments in Mumia Abu-Jamal Case heard May 17
> > >> U.S. Court of Appeals, Third Circuit
> > >> <http://www.abu-jamal-news.com/>
> > > An interval [a,b] is composed of positions, not points. But even
> > > positions are constructions, and it is not appropriate to analyse a
> > > construction in spatial terms.
>
> > I think you need to learn some measure theory. This is a question about
> > mathematics, by the way, not philosophy.
>
> > Dave Seaman
> > Oral Arguments in Mumia Abu-Jamal Case heard May 17
> > U.S. Court of Appeals, Third Circuit
> > <http://www.abu-jamal-news.com/>- Hide quoted text -
>
> > - Show quoted text -
>
> I think you need to distinguish between a position and a point before
> wildly conflating them in both a philosophical and mathematical
> confusion.
The position of a point is relative to the reference coordinate
system. So, position is an attribute on a point to locate it with
reference to the given coordinate system.
Does it make some sense?
- venkat
Randy Poe
> On Mon, 12 Nov 2007 20:57:25 -0500, "Robert J. Kolker"
>
> >Lester Zick wrote:
>
> >> Hey it's not my problem, Bobby. I'm not the one who claims points have
> >> zero length but are not of zero length.Modern mathematics is a heresy.
>
> >Neither does any one else. You have created a straw man here.
>
> Horseshit, Bobby. I didn't create the straw man. I can cite chapter
> and verse.
Lester Zick citing a reference other than himself?
I'd like to see that. Exact quote, please, that says
points have zero length but not zero length.
- Randy
Venkat Reddy
He must be referring to the post by William in the thread "Lines
composed of points?".
- venkat
Venkat Reddy
> On Mon, 12 Nov 2007 05:13:10 -0800, Venkat Reddy <vred...@gmail.com>
I thought a set contains zero or more elements, and the size
(cardinality) of the set is the number of its elements. Whats the
"length" of a set? Why is it zero when the set contains a single
point? And, to which of my statement did you negate when you said
"no"?
- venkat
Wolf Kirchmeir
What about the fries? Do they form a Mandelbrot set?
Venkat Reddy
> On Mon, 12 Nov 2007 20:56:04 -0500, "Robert J. Kolker"
>
Values, quantities do not need any geometrical entities to lie in or
lie at. I can count 5 apples without using any geometrical space. So
where does the value "5" lie?
We should not try to model the continuum with the patch work of
different kinds of numbers and keep on worrying about the holes they
leave. Basically we faltered somewhere while moving from counting
numbers to real numbers and then applying them to continuum.
- venkat
Randy Poe
That doesn't have a meaning for general sets.
> Why is it zero when the set contains a single
> point?
The special kind of set called a "closed interval in R"
is defined by two endpoints, a and b with a <= b. And the length
of that kind of set can be defined as b - a.
So it's zero when b = a.
Measure theory generalizes the idea of length to handle
arbitrary sets of points. "Measure" corresponds to length
in the case of the special kind of set called a "closed
interval in R" but again you'd have a hard time defining
length for general point sets.
> And, to which of my statement did you negate when you said
> "no"?
Probably that "points only exist as boundaries of higher
dimensional regions".
- Randy
Venkat Reddy
> On Nov 12, 11:37 am, John Jones <jonescard...@aol.com> wrote:
>
> > On Nov 12, 6:21?pm, "Robert J. Kolker" <bobkol...@comcast.net> wrote:
> > > Once again you do not distinguish between objects and the sets of which
> > > the objects are elements. Another evidence that you cannot cope with
> > > mathematics.
> > A line is not a set of points because sets are indifferent to order.
> > However, if you care to order points we still do not have a minimal
> > definition of a line.
>
> I've been thinking about the links to Euclid's and Hilbert's
> axioms presented in some of the other geometry threads:
>
>
> These last few posts are posing the question, is a
> point an _element_ of a line, or is a point a
> _subset_ of a line?
>
> The correct answer is neither. For let us review
> Hilbert's axioms again:
>
> "The undefined primitives are: point, line, plane.
> There are three primitive relations:
>
> "Betweenness, a ternary relation linking points;
> Containment, three binary relations, one linking
> points and lines, one linking points and planes,
> and one linking lines and planes;
> Congruence, two binary relations, one linking line
> segments and one linking angles."
>
> So we see that line is an undefined _primitive_,
> and that there is a _primitive_ to be known as
> "containment," so that a line may be said to
> "contain" points.
>
> Notice that the primitive "contain" has _nothing_
> to do with the membership primitive of a set
> theory such as ZFC. Why? Because this is a
> geometric theory that is not even written in
> the _language_ of ZFC.
>
> So both "a point is an element of a line" and "a
> point is a subset of a line" are incorrect.
Excellent! This settles my question in the main thread.
>
> The other question concerns what the intersection
> of two lines is. Well, first we must define
> "intersection" -- in terms of our _primitives_,
> of course -- before we can answer. And the only
> answer we can possibly give is in terms of
> _containment_: the intersection of two lines a,b
> is a point A such that a contains A and b contains
> A as well, provided that such a point exists. We
> can't call it a "position," since "position" is
> not a _primitive_ of our theory.
>
> So now all we have to do is prove that if such a
> point exists, it must be unique. But this follows
> directly from Axiom I.1. For if there were two
> points of intersection A,B, then I.1 tells us that
> two points determine a line, so that AB = a and
> AB = b as well, therefore a = b. So if a,b are
> distinct and intersect, then they intersect in a
> unique point of intersection.
Venkat Reddy
Thanks for educating me a little on the sets.
>
> > And, to which of my statement did you negate when you said
> > "no"?
>
> Probably that "points only exist as boundaries of higher
> dimensional regions".
Yes, because if we consider 1D space, points can't exist without lines
and lines can't exist without points. Points have to be defined in
terms of line segments and lines in terms of points.
Similarly, the 2D regions are defined by the boundary consisting of
lines and points. These boundaries are defined by the regions and
regions are defined by the boundaries.
So points exist as regions in zero dimension and as boundaries in
higher dimensions.
I'm no expert in this, but just presenting my thoughts.
- venkat
William Hughes
Hardly. There is more than one way of defining lines
and points. Certainly you can take "line"
be a primative. In this case
a line is not composed of anything, it just is.
However, your argument is not only "it is possible
to define a line as not being composed of points",
but also that "it is impossible to define a line
as being composed of points". The latter statement
is false.
It is certainly possible to define points without
reference to lines and then to define a line
as a particular set of points. If we do this
then we need a definition of "extent" for a set
of points. Note there can be more than
one "size" for a set. The cardinality is one
size but the cardinality does not have the
properties we want. However, the Lebesque measure
does. If we define the extent of a set
to be the Lebesque measure, then the extent of
a point (formally the extent of the singleton
containing the point) is 0, but the extent of a set of
points may not be 0. The fact that you do
not like this will not make it go away.
- William Hughes
Robert J. Kolker
> What about the fries? Do they form a Mandelbrot set?
Only if mixed with almonds.
Bob Kolker
Dave Seaman
> On Nov 13, 12:31 am, John Jones <jonescard...@aol.com> wrote:
>> On Nov 12, 6:05?pm, Dave Seaman <dsea...@no.such.host> wrote:
>>
>>
>>
>> > On Mon, 12 Nov 2007 07:50:52 -0800, John Jones wrote:
>> > > On Nov 12, 3:42?pm, Dave Seaman <dsea...@no.such.host> wrote:
>> > >> On Mon, 12 Nov 2007 07:06:39 -0800, Hero wrote:
>> > >> > Robert wrote:
>>
>> > >> >> In Euclidean space a set which has exactly one pont as a member has
>> > >> >> measure zero. But you can take the union of an uncountable set of such
>> > >> >> singleton sets and get a set with non-zero measure.
>>
>> > >> > What measure will give a non-zero number/value?
>>
>> > >> Lebesgue measure will do so, not for all possible uncountable sets, but
>> > >> for some. For example, the Lebesgue measure of an interval [a,b] is its
>> > >> length, b-a.
>>
>> > > positions are constructions, and it is not appropriate to analyse a
>> > > construction in spatial terms.
>>
>> > I think you need to learn some measure theory. This is a question about
>> > mathematics, by the way, not philosophy.
>>
>> wildly conflating them in both a philosophical and mathematical
>> confusion.
> The position of a point is relative to the reference coordinate
> system. So, position is an attribute on a point to locate it with
> reference to the given coordinate system.
> Does it make some sense?
What if no coordinate system is specified? The definition of a measure
space says nothing about a coordinate system. For that matter, the
important elements of a measure space are not the points (elements of the
space itself), but rather the measurable sets (members of the specified
signma-algebra).
Robert J. Kolker
> I thought a set contains zero or more elements, and the size
> (cardinality) of the set is the number of its elements. Whats the
> "length" of a set? Why is it zero when the set contains a single
> point? And, to which of my statement did you negate when you said
> "no"?
Do not confuse cardinality with measure. They are quite distinct.
Bob Kolker
Robert J. Kolker
> The position of a point is relative to the reference coordinate
> system. So, position is an attribute on a point to locate it with
> reference to the given coordinate system.
>
> Does it make some sense?
In a way. Position is a name we give to points to identify them uniquely.
Bob Kolker
Lester Zick
wrote:
>On Nov 13, 6:44 pm, Randy Poe <poespam-t...@yahoo.com> wrote:
>> On Nov 13, 6:31 am, Lester Zick <dontbot...@nowhere.net> wrote:
>>
>> > On Mon, 12 Nov 2007 20:57:25 -0500, "Robert J. Kolker"
>>
>> > <bobkol...@comcast.net> wrote:
>> > >Lester Zick wrote:
>>
>> > >> Hey it's not my problem, Bobby. I'm not the one who claims points have
>> > >> zero length but are not of zero length.Modern mathematics is a heresy.
>>
>> > >Neither does any one else. You have created a straw man here.
>>
>> > Horseshit, Bobby. I didn't create the straw man. I can cite chapter
>> > and verse.
>>
>> Lester Zick citing a reference other than himself?
Only for the purpose of ridicule. I cite you all the time.
>> I'd like to see that. Exact quote, please, that says
>> points have zero length but not zero length.
>>
>
>He must be referring to the post by William in the thread "Lines
>composed of points?".
Sure. And when I inquired into the contradiction of predicates
involved I received the reply from him that it was obvious and he
couldn't discuss the subject with anyone who didn't find it obvious.
~v~~
Randy Poe
> On Tue, 13 Nov 2007 05:48:58 -0800, Venkat Reddy <vred...@gmail.com>
> wrote:
>
>
>
> >On Nov 13, 6:44 pm, Randy Poe <poespam-t...@yahoo.com> wrote:
> >> On Nov 13, 6:31 am, Lester Zick <dontbot...@nowhere.net> wrote:
>
> >> > On Mon, 12 Nov 2007 20:57:25 -0500, "Robert J. Kolker"
>
> >> > <bobkol...@comcast.net> wrote:
> >> > >Lester Zick wrote:
>
> >> > >> Hey it's not my problem, Bobby. I'm not the one who claims points have
> >> > >> zero length but are not of zero length.Modern mathematics is a heresy.
>
> >> > >Neither does any one else. You have created a straw man here.
>
> >> > Horseshit, Bobby. I didn't create the straw man. I can cite chapter
> >> > and verse.
>
> >> Lester Zick citing a reference other than himself?
>
> Only for the purpose of ridicule. I cite you all the time.
OK, then. Your offer to "cite chapter and verse" was
not a serious one. Thought not.
- Randy
Venkat Reddy
My humble thoughts - Given a pair of end points, a line segment is
uniquely defined in 1D. Conversely, given a line segment its two end
points are uniquely defined. There is no other way to find a point
except as an end point of a line segment. Generalizing, if possible, a
region is defined by its boundary and a boundary is defined by its
region. Within a fixed boundary, the region could be curved into
higher dimensional spaces, and the extent of the region could vary
based on its curvature. But we do not consider it while observing from
within the same dimensional space. For example, a sphere bounded by a
given surface could have varying volume based the curvature of its 3D
region in higher dimensions.
If I can imagine wild, a point could have finite or infinite extent as
a region in zero dimensional space and have zero extent in higher
dimensional spaces. We could apply Same rules for n-D regions. A line
has zero extent in 2D and a surface has zero extent in 3D.
> If we do this
> then we need a definition of "extent" for a set
> of points. Note there can be more than
> one "size" for a set. The cardinality is one
> size but the cardinality does not have the
> properties we want. However, the Lebesque measure
> does. If we define the extent of a set
> to be the Lebesque measure, then the extent of
> a point (formally the extent of the singleton
> containing the point) is 0, but the extent of a set of
> points may not be 0. The fact that you do
> not like this will not make it go away.
Well, my like and dislike will have infinitesimal impact on the tons
of math literature existing out there. I'm just trying to learn by
questioning. Thanks for helping me.
- venkat
John Jones
> On Nov 12, 2:45 pm, John Jones <jonescard...@aol.com> wrote:
>
> > > So now all we have to do is prove that if such a
> > > point exists, it must be unique. But this follows
> > > directly from Axiom I.1. For if there were two
> > > points of intersection A,B, then I.1 tells us that
> > > two points determine a line, so that AB = a and
> > > AB = b as well, therefore a = b. So if a,b are
> > > distinct and intersect, then they intersect in a
> > > unique point of intersection.
> > Points, lines, planes, etc are incommensurables which do not contain
> > the properties of one within the other. Their 'synthesis' is not a
> > synthesis of properties or objects, but of the frameworks that
> > establish objects and properties (see Kant).
>
> I'm sorry, but I must concur with Mr. Kolker here. The
> intersection of two lines is a point.
>
> One way to see what's going on here is to consider
> the standard model of Hilbert, namely R^3. (Kolker
> uses the notation E2 for 2D Hilbert, but let us
> consider the third dimension now as well.)
>
> Now we can determine what this model happens to
> map the primitives to. As Kolker has said, "line"
> is mapped to the set of ordered triples (x,y,z)
> satisfying a linear relation.
>
> But what about "point"? Is "point" mapped to a
> triple itself (an element of a line), or is it a
> singleton whose sole element is an ordered
> triple (a subset of a line)?
>
> This is, of course, closely related to what the
> primitive "containment" is mapped to. It could
> be membership (point = ordered triple) or
> inclusion (point = singleton of ordered triple).
>
> I believe that mapping containment to membership
> will be awkward. Let us recall what Hilbert
> wrote about containment:
>
> "Containment, three binary relations, one linking
> points and lines, one linking points and planes,
>
> So we see that lines contain points, planes
> contain points, and planes contain lines.
>
> And here lies the problem. If we let containment
> be mapped to membership, then planes would have
> both points and lines as distinct elements. And
> even if we only allowed planes to have lines as
> elements, which lines would be the elements of
> the plane anyway. For the plane z = 0, for
> example, are x = constant the elements of the
> plane, or y = constant, or all of them?
>
> So it makes much more sense to map "containment"
> to "inclusion." Thus points are singletons and
> subsets of the lines and planes that happen to
> "contain" them. And therefore the intersection
> of two lines is the set intersection -- which
> is exactly the "point."
>
> Of course, what about the ordered triples --
> the elements of points, lines -- themselves? We
> may call them "positions," if we want. So the
> single element of a point is a "position," and
> the elements of a line are "positions." And so
> answering the OP's question, positions don't
> have a measure, but "points" do -- at least,
> in the standard model R^3 of Hilbert, where
> subsets in R^3 have a Lebesgue measure.
>
> Of course, this is all only in the standard
> model of Hilbert. In other models, "point,"
> "line," may be mapped to something other
> than sets, so we can't always refer to the
> element of a point as a "position," because
> "point" may be mapped to something that
> doesn't have an element.
>
> To see what I mean, let us take a page from
> Han de Brujin's book and come up with a new
> model of some subset of Hilbert's axioms. (If
> you don't know who HdB is, it's not that
> important for this example.)
>
> Consider Hilbert's Axioms of Incidence only --
> the ones labeled I.1 to I.7. Notice one can't
> prove from these axioms alone that more than
> finitely many points exist. Indeed, we
> observe I.7:
>
> "I.7: Upon every straight line there exist at
> least two points, in every plane at least
> three points not lying in the same straight
> line, and in space there exist at least four
> points not lying in a plane."
>
> Apparently, by I.1 through I.7, we can't even
> prove the existence of more than _four_ points,
> and indeed, we can construct a model of I.1
> through I.7 in which only four points exist.
>
> Now in this model, we will map our four points
> to natural numbers -- in particular, the
> natural numbers 1, 2, 4, and 8. (Those familiar
> with HdB should know by now where I am heading
> with this.) Lines contain exactly two points --
> mapped to the sum of the two points that lie
> on them. Planes contain exactly three points --
> mapped to the sum of the three points that lie
> on them. So we have:
>
> 1. point (1)
> 2. point (2)
> 3. line (1+2)
> 4. point (4)
> 5. line (1+4)
> 6. line (2+4)
> 7. plane (1+2+4)
> 8. point (8)
> 9. line (1+8)
> 10. line (2+8)
> 11. plane (1+2+8)
> 12. line (4+8)
> 13. plane (1+4+8)
> 14. plane (2+4+8)
> 15. space (1+2+4+8)
>
> In this model, containment is mapped to a
> more complicated matter -- one can try
> bitwise AND (or OR) to come up with the
> relation onto which containment is mapped.
>
> The important part is that this points
> don't have "positions" at all.
If points are not positioned, as you say, then the line that has the
minimal requirement of two points by virtue of 1.7, is a concept, or
if you like, a set. The set has fractured and unfractured lines as its
members. The set has one of its line members fractured when two points
fall on the same position. Other fractured lines arise when 'before'
and 'after' points fall contrary to their expected positions.
Interesting.
John Jones
> On Mon, 12 Nov 2007 14:45:25 -0800, John Jones <jonescard...@aol.com>
> wrote:
>
> >A position may well not be a primitive, but the intersections of lines
>
> So if we change the name of "points" to "positions" we'll solve the
> problem?
I think many problems could be solved by not conflating point and
position. A position is not a point, nor a point a position. A
position is an object in a framework or construction; while a point,
like a line, is a framework for the construction of objects such as
positions. Frameworks are incommensurables while objects (positions)
are not. So the rules for frameworks are quite different to that of
the rules for objects. Perhaps Hibert rudimentally envisaged this when
he described points and lines as 'primitives'.
Lester Zick
wrote:
>On Nov 12, 10:15 pm, Lester Zick <dontbot...@nowhere.net> wrote:
>> On Sun, 11 Nov 2007 20:57:47 -0800, William Elliot
>>
>> <ma...@hevanet.remove.com> wrote:
>> >On Sun, 11 Nov 2007, Lester Zick wrote:
>>
>> >> The Virgin Birth of Points
>> >> ~v~~
>>
>> >> The Jesuit heresy maintains points have zero length but are not of
>> >> zero length and if you don't believe that you haven't examined the
>> >> argument closely enough.
>>
>> >Clearly points don't have zero length, they have a positive infinitesimal
>> >length for which zero is just the closest real approximation.
>>
>> I don't see how points "clearly" have infinitesimal length unless
>> they're infinitesimal to begin with. Clearly Newton didn't think
>> points were infinitesimal nor did Leibniz or they wouldn't have
>> drafted the notations they used. And Newton's calculations of tangents
>> were only defined at one point not at one infinitesimal.
>>
>> We have a certain arithmetic notation such as 5-5=0 in which the
>> difference between 5 and itself is not infinitesimal but zero. Nor is
>> the difference between a line and itself infinitesimal but zero.
>>
>
>Lester Sir,
It's just Lester, venkat. Mostly we're just on a first name basis with
the guys on these groups unless we use terms that would make a whore
blush with anticipation.
> but I thought you just said that the main purpose of this
>thread is less to discuss the zero length of points than the heresy of
>maintaining self contradictory
>predicates, or to continue some of your silly fights with other
>poster. Anyways, happy to see you discussing same ideas from my other
>thread regarding the extent of points.
Well of course I had a dual purpose in mind. The first was to discuss
the heresy of maintaining self contradictions in mathematics and
science generally. But I also wanted to spread the discussion to other
groups where I didn't feel comfortable changing your address list to
accommodate. In any event I certainly don't mind discussing your
original subject since it coincides nicely with threads I've posted in
the past. As for the silly fights, just ignore them. They're nothing
more than byplay.
~v~~
Lester Zick
wrote:
>On Nov 13, 4:41 pm, Lester Zick <dontbot...@nowhere.net> wrote:
Well the problem here, venkat, is that we can model arithmetic exactly
in terms of the difference between rational numbers.But we can't model
transcendental numbers using the same technique because the only
things we can model "between" rationals are straight line segments and
square roots of rationals.
Where geometry becomes essential to the grasp of mathematics lies in
the comprehension of curves. For example pi lies on circular curves as
shown by Archimedes and not on any straight line segment. And such
curves are what correspond to transcendental numbers. And without them
all we can do is arithmetic. Consequently what we find is geometry is
the basic progenitor to arithmetic.
~v~~
Amicus Briefs
<dse...@no.such.host> wrote:
>> The position of a point is relative to the reference coordinate
>> system. So, position is an attribute on a point to locate it with
>> reference to the given coordinate system.
>
>> Does it make some sense?
>
>What if no coordinate system is specified? The definition of a measure
>space says nothing about a coordinate system.
Well then there is no measure space to measure against.
> For that matter, the
>important elements of a measure space are not the points (elements of the
>space itself), but rather the measurable sets (members of the specified
>signma-algebra).
"Signma-algebra"? I must have missed that one in ninth grade algebra.
Lester Zick
Identify them uniquely? Why would we want to identify them uniquely?
Why not just wing it and unionize them instead?
~v~~
Schlock
Well that's okay as long as we're dealing with straight kugels. Curved
kugels are a whole nuther kettle of fish however.
Lester Zick
How's about pi kugels?
~v~~
Lester Zick
<wpih...@hotmail.com> wrote:
>> > So both "a point is an element of a line" and "a
>> > point is a subset of a line" are incorrect.
>>
>> Excellent! This settles my question in the main thread.
>Hardly. There is more than one way of defining lines
>and points. Certainly you can take "line"
>be a primative. In this case
>a line is not composed of anything, it just is.
>
>However, your argument is not only "it is possible
>to define a line as not being composed of points",
>but also that "it is impossible to define a line
>as being composed of points". The latter statement
>is false.
Well that certainly clears that up.
>It is certainly possible to define points without
>reference to lines and then to define a line
>as a particular set of points.
I knew it. If we define something without being able to define
something we thereby reach Jesuit comprehension of points.
> If we do this
>then we need a definition of "extent" for a set
>of points.
Why, since we only have your word for it that we can define something
we can't define?
> Note there can be more than
>one "size" for a set. The cardinality is one
>size but the cardinality does not have the
>properties we want. However, the Lebesque measure
>does. If we define the extent of a set
>to be the Lebesque measure, then the extent of
>a point (formally the extent of the singleton
>containing the point) is 0, but the extent of a set of
>points may not be 0. The fact that you do
>not like this will not make it go away.
~v~~
Schlock
wrote:
>A position may well not be a primitive, but the intersections of lines
>construct positions, not points.
Yes, well, the jury is still out on the difference between them.
> Primitives are incommensurables.
>Points, lines, planes, etc are incommensurables which do not contain
>the properties of one within the other. Their 'synthesis' is not a
>synthesis of properties or objects, but of the frameworks that
>establish objects and properties (see Kant).
I think we'll let you see Kant and please don't come back until you
do.
Lester Zick
Everything you say is quite distinct. It hasn't stopped you from
confusing them. In fact the one thing that stands out clearly in all
this nonsense is that one can't get a straight word out of you guys.
Points are this, measures are that, and positions are something else.
Modern mathematikers are all just a bunch of prancing princesses
preening terminology they can't even agree on.
~v~~