A question about straight lines

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A question about straight lines

Neighbor
I should very much like to know whether a straight-line has infinite points on it or a finite number of points on it?
For those who answer I'd like to have an explanation of what they say too.
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Re: A question about straight lines

kirby urner-4

My answer would be "that depends on what the straight-line is defined to represent".  You can't tell just by looking at the line.  You need more context.

When the "set of real numbers" is what's considered to be represented, then the answer is there's an infinity of points between any two.  But that's not the only set people care about and you can draw lines and not mean "real numbers" by them.

So don't let lines by themselves dictate.  If the context is missing i.e. the set is unspecified, then take the liberty of applying your own meaning.  I personally don't care for the "real numbers" a whole lot, so if a loophole is left open, I might well take advantage and go with some other set, even the complex numbers.

Kirby



On Sun, Jan 26, 2014 at 1:36 PM, Neighbor <[hidden email]> wrote:
I should very much like to know whether a straight-line has infinite points on it or a finite number of points on it?
For those who answer I'd like to have an explanation of what they say too.

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Re: A question about straight lines

Neighbor
In reply to this post by Neighbor
I mean to say "a line without breadth with points on it" and in the Euclid's Elements this particular kind of lines (i.e. straight-lines) is defined as one which lies evenly with points on itself and a point is that of which there is no part, and my question is: Are there infinite points or not on them?
If there is an infinite number of points on them and each point is dimensionless (i.e. nothing, having neither breadth nor width) then the sum of infinite points which are nothing will have as sum also nothing, but if we admit that a point has only breadth (a very small amount but not 0), then the sum of infinite points will also have breadth, and since they are infinite the straight-line they made up is also with infinite breadth (i.e. an infinite straight-line). Therefore, if we suppose that straight-lines have infinite points on them, then all the straight-lines will be either infinite or nothing (depending how we define a point). On the other hand as you know, there are also finite straight lines, how can we admit their existence if they are made up of infinite points?
Now, if we suppose that straight-lines have a finite number of points on them, and a point is defined as having only breadth, the smallest possible and not having parts, then a finite number of such points will make a finite straight-line and an infinite number of them, a infinite straight-line.
As you can see this is the only case which seems to be true, but if so, the divisibility to infinity of a finite straight-line would also be impossible, for dividing this we can arrive to the single point which have no parts, as an atom defined by Democritus. Maybe I've been talking nonsense, 'cause all these concepts puzzle me, and if you can show something better, please, I should very much like to know what you think.
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Re: A question about straight lines

Robert Hansen
Your notion of "infinite" is the issue. Between any two points, there is another point, halfway between them. This continues without end.

Bob Hansen

> On Jan 27, 2014, at 1:44 AM, Neighbor <[hidden email]> wrote:
>
> If there is an infinite number of points on them and each point is dimensionless (i.e. nothing, having neither breadth nor width) then the sum of infinite points which are nothing will have as sum also nothing

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Re: A question about straight lines

Neighbor
In reply to this post by Neighbor
> Your notion of "infinite" is the issue. Between any
> two points, there is another point, halfway between
> them. This continues without end.
>
> Bob Hansen

But if between any two points there is another, I really cannot understand how a finite straight-line can be made up of infinite points. I cannot help thinking that the sum of infinite points is either nothing or an infinite straight-line. At any rate, when we have two finite straight-lines, and comparing them (by applying one on the other), if the first exceeds the other, then we say that the first is greater than the second, but if they are both made up of infinite points how can one of them be greater than the other? Would you not rather say that the first being greater is made up of more points? But if it's made up of more points, then what follows is that they are both made up of a finite number of points.
I'm really perplexed, and perhaps I'm talking nonsense, so I'd like to have a clear explanation.
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Re: A question about straight lines

Robert Hansen
A “point” is a place on a line. Do you ever run out of “places” on a line? That is what infinite means.

Bob Hansen

On Jan 27, 2014, at 10:16 AM, Neighbor <[hidden email]> wrote:

>  I really cannot understand how a finite straight-line can be made up of infinite points.
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Re: A question about straight lines

Neighbor
In reply to this post by Neighbor
> A “point” is a place on a line. Do you ever run out
> of “places” on a line? That is what infinite means.
>
> Bob Hansen
Maybe you misunderstood me, what I mean to say is not a point of infinite (endless) dimensions on a line, but an INFINITE NUMBER of points, (a number that has no limits, endless) for if between each two points there is another, and this process is endless, then the number of points between two any points (like the endpoints of a straight-line) is also endless. This is my meaning.
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Re: A question about straight lines

Neighbor
In reply to this post by Neighbor
To avoid misunderstandings, every time I've wrote "Infinite points" I've meant "an infinite number of points".
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Re: A question about straight lines

Bishop, Wayne
In reply to this post by Neighbor
Your real problem is interpreting Euclid's axioms and definitions as completely meaningful from a modern perspective.  Even "straight" has problems; what does it mean?  Most of us take a "line" to be a subset of the point set (the "plane") that separates the plane into 3 mutually exclusive convex sets, one of which is the line itself.  How much more you need to identify the traditional Euclidean axioms requires more.  It takes at least a quarter, better all year, to even get started in a very formal context and is a topic for a few experts to indulge in while the rest of we peons proceed in a much more sensible manner.  We include a Ruler Postulate and a Protractor Postulate to take care of lots of stuff that has to follow from "measure-free" traditional geometry.  Google  Pappus Desargues  field

Wayne

At 10:44 PM 1/26/2014, Neighbor wrote:
I mean to say "a line without breadth with points on it" and in the Euclid's Elements this particular kind of lines (i.e. straight-lines) is defined as one which lies evenly with points on itself and a point is that of which there is no part, and my question is: Are there infinite points or not on them?
If there is an infinite number of points on them and each point is dimensionless (i.e. nothing, having neither breadth nor width) then the sum of infinite points which are nothing will have as sum also nothing, but if we admit that a point has only breadth (a very small amount but not 0), then the sum of infinite points will also have breadth, and since they are infinite the straight-line they made up is also with infinite breadth (i.e. an infinite straight-line). Therefore, if we suppose that straight-lines have infinite points on them, then all the straight-lines will be either infinite or nothing (depending how we define a point). On the other hand as you know, there are also finite straight lines, how can we admit their existence if they are made up of infinite points?
Now, if we suppose that straight-lines have a finite number of points on them, and a point is defined as having only breadth, the smallest possible and not having parts, then a finite number of such points will make a finite straight-line and an infinite number of them, a infinite straight-line.
As you can see this is the only case which seems to be true, but if so, the divisibility to infinity of a finite straight-line would also be impossible, for dividing this we can arrive to the single point which have no parts, as an atom defined by Democritus. Maybe I've been talking nonsense, 'cause all these concepts puzzle me, and if you can show something better, please, I should very much like to know what you think.

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Re: A question about straight lines

kirby urner-4
In reply to this post by Neighbor

Your questions have come up for students for centuries.  Since we usually get to them when they're still young, fully mature debate is rarely engaged in, and since Euclid's Axioms are defined as "given" (as in "lets accept that...") there's no obligation to offer "proofs" about any of this low-level stuff.  We're at the level of definition.

Just so you know you're not alone in imagining alternatives, a dimension theorist named Karl Menger proposed branching away from Euclideanism precisely around these definitions, suggesting a "geometry of lumps" we might define instead, one wherein points, lines, planes and polyhedrons aren't distinguished by "dimension numbers" (0,1,2,3).

http://coffeeshopsnet.blogspot.com/2009/03/res-extensa.html

Your discomfort with points of no size making lines would be replicated at the next levels, where lines of no breadth make "rafts" (planes) and cuts from planes of no thickness stack up to make whatever "3D printer"-like shapes.  How could that be?  How do we get "new dimensions" by packing and stacking what were ultimately no-dimensional points?  In Menger's "geometry of lumps" maybe we don't need to think that way. [1]

Spooky Greek metaphysics (Hellenism some call it) is part of your cultural heritage and "infinity" is going to keep dogging you, so might as well make friends and learn it the way they teach it.  But don't stop yourself from realizing that there are other axiomatic paradigms out there and Euclideanism is itself a finitude among other finitudes. [2]

Mathematics is about both/and thinking more than either/or thinking, though at any given time, you may need to wear blinders, just to keep those other ways of thinking from spoiling your concentration in the moment.

Kirby

[1] I used Menger's essay as an underpinning for a later-developed finite / discrete geometry that likewise diverges from standard "dimension talk", Bucky Fuller's polymorphic system, wherein points are more lump-like (published in the late 1970s, early 1980s by Macmillan).

[2] http://www.thefreedictionary.com/finitude



On Sun, Jan 26, 2014 at 10:44 PM, Neighbor <[hidden email]> wrote:
I mean to say "a line without breadth with points on it" and in the Euclid's Elements this particular kind of lines (i.e. straight-lines) is defined as one which lies evenly with points on itself and a point is that of which there is no part, and my question is: Are there infinite points or not on them?
If there is an infinite number of points on them and each point is dimensionless (i.e. nothing, having neither breadth nor width) then the sum of infinite points which are nothing will have as sum also nothing, but if we admit that a point has only breadth (a very small amount but not 0), then the sum of infinite points will also have breadth, and since they are infinite the straight-line they made up is also with infinite breadth (i.e. an infinite straight-line). Therefore, if we suppose that straight-lines have infinite points on them, then all the straight-lines will be either infinite or nothing (depending how we define a point). On the other hand as you know, there are also finite straight lines, how can we admit their existence if they are made up of infinite points?
Now, if we suppose that straight-lines have a finite number of points on them, and a point is defined as having only breadth, the smallest possible and not having parts, then a finite number of such points will make a finite straight-line and an infinite number of them, a infinite straight-line.
As you can see this is the only case which seems to be true, but if so, the divisibility to infinity of a finite straight-line would also be impossible, for dividing this we can arrive to the single point which have no parts, as an atom defined by Democritus. Maybe I've been talking nonsense, 'cause all these concepts puzzle me, and if you can show something better, please, I should very much like to know what you think.

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Re: A question about straight lines

Neighbor
In reply to this post by Neighbor
At any rate, I'd like to ask another question, which may seem to you really trivial:
What is that which divides a line into two lines? Is not that a point?
Having a straight-line AB we can cut this into two straight-lines AG and GB by means of a point (let it be G).
Now, suppose I ask you, when we separate the two lines, on what straight-line the point G will be? And since this is one point (which is initially on AB), it will either be on AG or GB, for one point cannot be on both straight-lines, BEING ONE. Let it remain on AG, then what will be the first endpoint of GB? Will it not be the next point to G? But we cannot find this point if we say that between any points there is another one ever, shall GB have only one end point (i.e. B)? What should we say?
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Re: A question about straight lines

Robert Hansen

On Jan 28, 2014, at 11:18 AM, Neighbor <[hidden email]> wrote:

> Having a straight-line AB we can cut this into two straight-lines AG and GB by means of a point (let it be G).
> Now, suppose I ask you, when we separate the two lines, on what straight-line the point G will be? And since this is one point (which is initially on AB), it will either be on AG or GB, for one point cannot be on both straight-lines, BEING ONE. Let it remain on AG, then what will be the first endpoint of GB? Will it not be the next point to G? But we cannot find this point if we say that between any points there is another one ever, shall GB have only one end point (i.e. B)? What should we say?

AB is a line (it extends forever in both directions).

AG is a ray (it has a beginning and extends forever in one direction).

What is left (what you are calling GB) is not a line, ray or segment. It might be called an “open ray” due to it being an open set of points. An open set of points does not include its boundary. A closed set of points does. A line, ray and segment are all closed.

http://en.wikipedia.org/wiki/Boundary_(topology)

Bob Hansen
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Re: A question about straight lines

Joe Niederberger
In reply to this post by Neighbor
>What is that which divides a line into two lines? Is not that a point?

That's kind of cute. Its very similar to the brilliant idea of Dedekind's from which came one of the first modern developments of real numbers.

The standard technical answer to the rest of your line here is that the "ends" of a line segment are of two types: open or closed. Did you really never hear that before? An open interval has no (least,greatest) point by the usual ordering. Its hard to imagine such, and it may not even exist in physical reality, but their are ways of grappling with it successfully for the purposes of mathematics.

More fundamentally, there are some long standing philosophical questions at the heart of the issues you raise.

#1. What is the relationship between mathematics and physical reality?

#2. How do mathematicians proceed in their work these days so as to not have to bump up against #1 too often?
 
I have my own partial answer to #1 that the margins are too small to contain.

The answer to #2 is that they have adopted the notion of building their theories up from axioms and definitions, which more or less remove them from worrying too much about #1 and instead forces the issue to be one of the rigid "mechanistic" relationships of a formal theory. When some questions cannot be answered within such a theory there are various ways of enlarging or bifurcating the theory.

Cheers,
Joe N
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Re: A question about straight lines

Robert Hansen

On Jan 28, 2014, at 12:17 PM, Joe Niederberger <[hidden email]> wrote:

> #1. What is the relationship between mathematics and physical reality?

Can you explain this in more detail? I don’t recall the original poster ever referencing anything “physical”, though they may have been thinking “physical”.

Bob Hansen
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Re: A question about straight lines

Neighbor
In reply to this post by Neighbor
I didn't say that AB is an endless line in both directions, I said that AB is a finite straight-line (having the endpoints A and B), therefore AG and GB in which AB has been divided are also finite straight-lines.
About the "open ray": how can a finite straight-line have no boundary???? Being made of points, there must be two boundaries in both directions, which we call endpoints, for if there is no boundary this is not a finite straight-line but an infinite one.
Therefore I don't understand your notion of "open ray", But if you will be willing to explain me this concept carefully and make me understand, I will be grateful.

>
> AB is a line (it extends forever in both directions).
>
> AG is a ray (it has a beginning and extends forever
> in one direction).
>
> What is left (what you are calling GB) is not a line,
> ray or segment. It might be called an “open ray” due
> to it being an open set of points. An open set of
> points does not include its boundary. A closed set of
> points does. A line, ray and segment are all closed.
>
> http://en.wikipedia.org/wiki/Boundary_(topology)
>
> Bob Hansen
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Re: A question about straight lines

Joe Niederberger
In reply to this post by Neighbor
>though they may have been thinking ?physical?.

Yes, its my inference. Either "Neighbor" is speaking naively, in which physical and mathematical notions are often confused, or he is purposely doing so, to make some point that is not clear.

For example, Neighbor says:
- --------------------
If there is an infinite number of points on them and each point is dimensionless (i.e. nothing, having neither breadth nor width) then the sum of infinite points which are nothing will have as sum also nothing, but if we admit that a point has only breadth (a very small amount but not 0), then the sum of infinite points will also have breadth, and since they are infinite the straight-line they made up is also with infinite breadth (i.e. an infinite straight-line).
- ----------------------

I think its clear he is drawing on physical intuition here at least partially, and its either because he doesn't know how the real number system is treated formally or he intentionally doesn't care.

Cheers,
Joe N
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Re: A question about straight lines

Robert Hansen
In reply to this post by Neighbor

On Jan 28, 2014, at 1:24 PM, Neighbor <[hidden email]> wrote:

> About the "open ray": how can a finite straight-line have no boundary????

All sets of points have boundaries. The question is whether the boundary is part of the set of points or not.

Think about a number line.
Now think about the set of all numbers greater than 4. (all the points to the right of 4 on the number line)
Now tell me what is the smallest number in that set? (the point closest to 4 on the number line)

You can’t answer the question. Even though the set of all numbers greater than 4 has a lower boundary, which is 4, it isn’t in the set itself.

You defined such a set of points by saying that G is part of AG and that BG is all the points left over. BG has a boundary (G) but the boundary itself isn’t part of BG (by your choice). Thus, BG is an open segment at G. At B it is a closed segment because B is part of BG.

You have actually used examples of both types of sets (sets that include their boundary and sets that do not), but you haven’t realized that this is what you are doing. When you say that BG doesn’t include G (because you used that with AG), you have basically defined a set (BG) that doesn’t include its boundary. Such a set is called an “open” set.

Bob Hansen

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Re: A question about straight lines

Joe Niederberger
In reply to this post by Neighbor
Neighbor says:
>About the "open ray": how can a finite straight-line have no boundary???? Being made of points, there must be two boundaries in both directions, which we call endpoints, for if there is no boundary this is not a finite straight-line but an infinite one.

You are correct that the "boundary" is the same point for both AG and GB, but that boundary point "G" can be considered either included (the "closed" situation) or excluded ("open") in either AG or GB. Being a boundary and belonging to the set to which it is a boundary are separate notions, and one does not follow logically from the other.

If you want the two segments to overlap at G, then you set theoretically consider it "included" in both AG & GB. (There is no logical conflict in including the number "2" in two distinct sets, say { 1, 2 } and { 1, 2, 3 } - same here.) If you want the two segments to be disjoint, then one or the other or both need to be open. If you want to recover the segment AB by "unioning" the two fragments set theoretically and also keep AG & GB disjoint, then one has to be open and the other closed. This is all elementary set theory applied to line segments considered as sets of point.

Cheers,
Joe N
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Re: A question about straight lines

Neighbor
In reply to this post by Neighbor
> All sets of points have boundaries. The question is
> whether the boundary is part of the set of points or
> not.
>
> Think about a number line.
> Now think about the set of all numbers greater than
> 4. (all the points to the right of 4 on the number
> line)
> Now tell me what is the smallest number in that set?
> (the point closest to 4 on the number line)
> You can’t answer the question. Even though the set of
> all numbers greater than 4 has a lower boundary,
> which is 4, it isn’t in the set itself.

If you consider only natural numbers, then next to four
there will be 5. But my question is not about numbers and their proprieties, but about straight-lines in Euclid's geometry with points on them.

> You defined such a set of points by saying that G is
> part of AG and that BG is all the points left over.
> BG has a boundary (G) but the boundary itself isn’t
> part of BG (by your choice). Thus, BG is an open
> segment at G. At B it is a closed segment because B
> is part of BG.

I will show you in the following example what I can't really understand about division of a straight-line:
Let's have a finite straight-line AB, and we want to cut it in half. If I'm not mistaken we do so by cutting AB with a point on it(let the point be M) into two equal straight-lines AM and MB (AM coming before M, and MB coming after). Now suppose that I want to add AM to a straight-line, and BM to another, therefore I need to separate AB into AM and MB, and being M one, it will be either on AM, MB, or neither. If we leave M and take A(M) and (M)B without it, then the two parts are not really halves of AB, for the sum of the two halves in which one thing is divided is the thing itself, but A(M)+(M)B will be less than AB, because AB = A(M)+M+(MB). Now if we take M on either AM or MB, let it be taken on AM, then when we separate AB we will have straight-lines AM and (M)B, but they will be unequal, inasmuch as A(M)=(M)B ==> AM>(M)B.
The inference seems to be that is not a point to divide a straight-line in half but something in between, I mean to say that when we want to divide AB in half, we should divide it in AM and NB, with M and N two consecutive points, and this won't cause troubles in the separation. Therefore what cuts AB in half is something between the two consecutive points M and N. But how two points can be consecutive if between them, as you say, there is another point ever.
And this is the very thing concerning division in half of a straight-line that I cannot understand.
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Re: A question about straight lines

Joe Niederberger
In reply to this post by Neighbor
Neighbor says:
>I mean to say that when we want to divide AB in half, we should divide it in AM and NB, with M and N two consecutive points

OK, for a while I thought maybe you were putting us on (what with your fake moniker and all) but I'm guessing now you are sincere. "Two consecutive points" has no meaning in the real number based continuum, or even in the set of rational numbers with usual ordering. So, if you cannot accept that between two real (or rational) numbers there is always another (which means that "two consecutive" simply does not exist in these games) then you will never accept the more or less accepted answers to your questions.  

So, lets simplify a bit: can you accept that between two points their is always another? Why or why not? If you do accept that, then what possible meaning does "two consecutive points" have? Any two you pick, trying to get that elusive consecutive pair, will always have another between them. Therefore they are not consecutive.  

Cheers,
Joe N
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