What is a Line? A Definition and Explanation

[TheAlkemist]

Can someone give me the definition of a line?

 

[G037H3]

the result of the movement of a point

if it's a straight line, it's the shortest distance between two points

Greek philosophers commented that because humans can conceive of perfectly straight lines even though they don't exist in material form, that humans must have knowledge that cannot be classified as material

 

[Mark44]

the result of the movement of a point

Since a point can be free to move in any direction, the path it traces out can be any curve, and not necessarily a line.

if it's a straight line, it's the shortest distance between two points

The terms line and straight line are usually taken to be synonyms.

Greek philosophers commented that because humans can conceive of perfectly straight lines even though they don't exist in material form, that humans must have knowledge that cannot be classified as material

Wikipedia gives a reasonable definition: an infinitely-extending one-dimensional figure that has no curvature.

 

[arildno]

With Hilbert, it is more fruitful to define what relations the "line" has with other basic notions like "the point" and "the plane".

Thus, these three terms are essentially undefined terms in the sense that they do not have any intrinsic meaning beyond that meaning they bear within the relationships with each other.

 

[G037H3]

Since a point can be free to move in any direction, the path it traces out can be any curve, and not necessarily a line.

A line is a straight curve. :D

 

[arildno]

A line is a straight curve. :D

Nope. It is a bent parabola.

 

[TheAlkemist]

so i take it that there'sno objective definition huh

 

[CRGreathouse]

so i take it that there'sno objective definition huh

See post #4. When you understand it you will have achieved enlightenment.

 

[TheAlkemist]

See post #4. When you understand it you will have achieved enlightenment.

LOL
I'll go with this one:
an infinitely-extending one-dimensional figure that has no curvature.
Which would be the scientific definition.

The mathematical definition would be:
the result of the movement of a point

Since math only deals with dynamical and not static concepts

 

[HallsofIvy]

it depends upon your starting point. In basic geometry, "line", as well as "point", "plane", etc. are taking as "undefined terms".

In, say, calculus III, a line, in three dimensions, can be defined as the trace of a linear function of one paratmeter: x= at+ b, y= ct+ d, z= et+ f.

In the differential of curved surfaces, there may be no "straight" lines so we typically use "line" to mean a "geodesic", a curve that is, locally, the shortest distance between two points.

 

[trambolin]

Motion of a point defines a continuum (in which you have to define what a point is anyway), but in set theoretical mathematics, the line is a set of points, which is visualized as a line. That does not make those equal unless you say motion of a point which jumps from one real number to another real number .

So to be uber-rigorous, you cannot use concepts from geometry and set theory at same time unless you state explicitly the corresponding mapping between them e.g. a line is a set. But a line is not a set in geometrical terms.

Another example: in geometry (and maybe physics?) the space is the encapsulation of everything possible, I mean everything is in the space, this is how you visualize. But in set theory it is the collection of points. In other words, space is made up of its elements contrary to the first case. There is a difference and unfortunately a very important difference which I become aware of very late to understand mathematics properly in the first place. So many things i had to unlearn due to this misconception of spaces.

A final question, how do you visualize a set of continuous functions defined on [0,1]. Do you see graphs of functions on the cartesian plane hanged in the voidness? Is it natural to think it as such when the space is discrete or any other esoteric space of functions? That is exactly the problem of epsilon-delta proofs since they are given with a geometric intuition which is directly violated by any slightly complicated space metric.

See my point?

 

[TheAlkemist]

it depends upon your starting point. In basic geometry, "line", as well as "point", "plane", etc. are taking as "undefined terms".

In, say, calculus III, a line, in three dimensions, can be defined as the trace of a linear function of one paratmeter: x= at+ b, y= ct+ d, z= et+ f.

In the differential of curved surfaces, there may be no "straight" lines so we typically use "line" to mean a "geodesic", a curve that is, locally, the shortest distance between two points.

undefined terms? how can something exist if it has no definition? Although something can have a definition and not exist. Like Bugs Bunny for example.

Motion of a point defines a continuum (in which you have to define what a point is anyway), but in set theoretical mathematics, the line is a set of points, which is visualized as a line. That does not make those equal unless you say motion of a point which jumps from one real number to another real number .

so the act of visualization/looking transforms a bunch of points into a line?

So to be uber-rigorous, you cannot use concepts from geometry and set theory at same time unless you state explicitly the corresponding mapping between them e.g. a line is a set. But a line is not a set in geometrical terms.

so which is it? Or are you saying the definition is contextual?

Another example: in geometry (and maybe physics?) the space is the encapsulation of everything possible, I mean everything is in the space, this is how you visualize. But in set theory it is the collection of points. In other words, space is made up of its elements contrary to the first case. There is a difference and unfortunately a very important difference which I become aware of very late to understand mathematics properly in the first place. So many things i had to unlearn due to this misconception of spaces.

The first definition you have of space makes more sense. In science, space is described as boundless, infinite and without form. We perceive/experience structures in space with respect to 3 dimensions. The second definition, the one you call the set theory definition, where space in MADE from stuff, not so much.

A final question, how do you visualize a set of continuous functions defined on [0,1]. Do you see graphs of functions on the cartesian plane hanged in the voidness? Is it natural to think it as such when the space is discrete or any other esoteric space of functions? That is exactly the problem of epsilon-delta proofs since they are given with a geometric intuition which is directly violated by any slightly complicated space metric.
See my point?

I'm not a mathematician so please forgive my ignorance as I attempt to explain what I understand about the epsilon-delta thing. It's basically asking when does a set of discrete things become continuous?
Space isn't discrete. Stuff IN space is.

 

[trambolin]

undefined terms? how can something exist if it has no definition? Although something can have a definition and not exist. Like Bugs Bunny for example.

My point is if you want to define anything you have to define it in one domain. You might not define Bugs Bunny in terms of evolution theory right, just because it looks like a bunny?

so the act of visualization/looking transforms a bunch of points into a line?

Yes, using facts about arithmetic such as one number is larger than the other etc.

so which is it? Or are you saying the definition is contextual?

Definition is always contextual in mathematics. http://en.wikipedia.org/wiki/Cone"

The first definition you have of space makes more sense. In science, space is described as boundless, infinite and without form. We perceive/experience structures in space with respect to 3 dimensions. The second definition, the one you call the set theory definition, where space in MADE from stuff, not so much.

Though it does not make sense at first sight, it does and that is how mathematics describe things. The natural intuitive way is left the stage to discretization program starting from Weierstrass and many others in the end of 1800s.

I'm not a mathematician so please forgive my ignorance as I attempt to explain what I understand about the epsilon-delta thing. It's basically asking when does a set of discrete things become continuous?
Space isn't discrete. Stuff IN space is.

Actually it is not related to what you say but let me give an example. Read the definition of the http://en.wikipedia.org/wiki/Contin..._.28epsilon-delta.29_of_continuous_functions". Do you get a geometric feeling for it?

And interestingly, no the spaces that most of the mathematics define are discrete. That is the subtlety that I am trying to draw your attention to.

 

[TheAlkemist]

My point is if you want to define anything you have to define it in one domain. You might not define Bugs Bunny in terms of evolution theory right, just because it looks like a bunny?

OK. But that definition has to be consistent within that domain. Do you agree?
And evolution theory describes changes in living things. Let's go with biology. In biology you actually have to physically analyze and characterize something in order to describe or define it. In order to do that the thing actually has to exist. Bugs Bunny doesn't exist.

Yes, using facts about arithmetic such as one number is larger than the other etc.

I don't get it. Please explain.

Definition is always contextual in mathematics. http://en.wikipedia.org/wiki/Cone"

But I noticed in that example, cone is either used as an adjective or the noun an adjective is qualifying. Regardless, if this is the case, then i'd have to conclude that mathematics is not science nor is it even the so-called "language of science". And I don't mean this in a condescending way. I just can't see how this contextual definition thing would work in science. I'm an organic chemist and a carbon atom is a carbon atom regardless of what molecule or polymer it's a part of or what branch of chemistry it was being studied in. This is just my opinion by the way.

Though it does not make sense at first sight, it does and that is how mathematics describe things. The natural intuitive way is left the stage to discretization program starting from Weierstrass and many others in the end of 1800s.

Hmm. Interesting. Hey, could you elaborate more on what you said earlier:
"There is a difference and unfortunately a very important difference which I become aware of very late to understand mathematics properly in the first place. So many things i had to unlearn due to this misconception of spaces."

Actually it is not related to what you say but let me give an example. Read the definition of the http://en.wikipedia.org/wiki/Contin..._.28epsilon-delta.29_of_continuous_functions". Do you get a geometric feeling for it?

I read it. And yes I do now. Thanks.

And interestingly, no the spaces that most of the mathematics define are discrete. That is the subtlety that I am trying to draw your attention to.

I see. Now could this be that mathematics really only studies and describes dynamical systems and concepts (motion, change) through the relations between discrete entities while static concepts are only understood with this idea of discrete space from which geometric objects are constructed or generated? For example, geometry studies shape and size as properties of connected discrete space while fields like topology an differential geometry study the dynamic properties of shapes using other dynamic mathematical methods like calculus.
Am I making any sense here? Feel free to let me know if I'm not LOL.

 

[trambolin]

OK. But that definition has to be consistent within that domain. Do you agree?
And evolution theory describes changes in living things. Let's go with biology. In biology you actually have to physically analyze and characterize something in order to describe or define it. In order to do that the thing actually has to exist. Bugs Bunny doesn't exist.

Yes but Bugs Bunny exists if you define the living things to be the things that move and sound on TV. This is where all the mathematician jokes are starting. Assume that ...

I don't get it. Please explain.

I mean you can order the numbers. they can follow a queue. Just like a line or a circle like the ones on the clock.

But I noticed in that example, cone is either used as an adjective or the noun an adjective is qualifying. Regardless, if this is the case, then i'd have to conclude that mathematics is not science nor is it even the so-called "language of science". And I don't mean this in a condescending way. I just can't see how this contextual definition thing would work in science. I'm an organic chemist and a carbon atom is a carbon atom regardless of what molecule or polymer it's a part of or what branch of chemistry it was being studied in. This is just my opinion by the way.

Yes, but imagine there is a mathematical framework that can model your very complicated stuff in a couple of equations, wouldn't you be willing to use it? Suppose you used this framework and obtained incredible results. Now is it the science or the math that led you to these results? My opinion is the both, it takes equally sharp-witted individuals both to produce the math and also to identify its use in mind-boggling places.

Hmm. Interesting. Hey, could you elaborate more on what you said earlier:
"There is a difference and unfortunately a very important difference which I become aware of very late to understand mathematics properly in the first place. So many things i had to unlearn due to this misconception of spaces."

What I am trying to say is that, sometimes, for the sake of clarity and other unknown reasons, geometric analogies and similar shortcuts are used to teach one particular subject. But as many pointed out over the years that this is deteriorating the math education and leading to a drastic percentage of confused students and math-haters. It is the stupid math snobbery and misunderstanding of the fundamental ideas of the metaphors such as the number line, sets etc. that leads to this uncomfortable situation.

I see. Now could this be that mathematics really only studies and describes dynamical systems and concepts (motion, change) through the relations between discrete entities while static concepts are only understood with this idea of discrete space from which geometric objects are constructed or generated? For example, geometry studies shape and size as properties of connected discrete space while fields like topology an differential geometry study the dynamic properties of shapes using other dynamic mathematical methods like calculus.
Am I making any sense here? Feel free to let me know if I'm not LOL.

Well I would not classify the math with respect to the things that they study. You can very well study the strength of materials, dislocation theory etc. with calculus which is supposedly studying motion. It is the modeling part that chooses which mathematical domain is relevant for that particular problem. For example, how on Earth would you be aware of the fact that http://en.wikipedia.org/wiki/Penrose_tiling" which have a particular pattern that is also ubiquitous in medieval buildings in Middle-East? Now which one is the real science?

 

[TheAlkemist]

Yes but Bugs Bunny exists if you define the living things to be the things that move and sound on TV. This is where all the mathematician jokes are starting. Assume that ...

In essence, "based on this/these axiom(s) ..."
But you can have a logically valid argument whose premise is false can't you? Of course science isn't exempt from this too. But the science checks for this (or at least good science attempts to) using methods like falsification of hypothesis and Bayesian Inference.

I mean you can order the numbers. they can follow a queue. Just like a line or a circle like the ones on the clock.

Using the same "assume that ..." ?

Yes, but imagine there is a mathematical framework that can model your very complicated stuff in a couple of equations, wouldn't you be willing to use it? Suppose you used this framework and obtained incredible results. Now is it the science or the math that led you to these results? My opinion is the both, it takes equally sharp-witted individuals both to produce the math and also to identify its use in mind-boggling places.

Of course i'd use it. And I use them routinely. But I'm not and have never questioned the utility of mathematics as a method of modeling the real world. Indeed math has been quite successful in this regard. And I agree with you--both science and math lead to results. As for the credibility of the results, well that's usually based on what's anticipated. And in the case of science, also, who and how many people have been able to repeat it.

What I am trying to say is that, sometimes, for the sake of clarity and other unknown reasons, geometric analogies and similar shortcuts are used to teach one particular subject. But as many pointed out over the years that this is deteriorating the math education and leading to a drastic percentage of confused students and math-haters. It is the stupid math snobbery and misunderstanding of the fundamental ideas of the metaphors such as the number line, sets etc. that leads to this uncomfortable situation.

OK.

Well I would not classify the math with respect to the things that they study. You can very well study the strength of materials, dislocation theory etc. with calculus which is supposedly studying motion. It is the modeling part that chooses which mathematical domain is relevant for that particular problem. For example, how on Earth would you be aware of the fact that http://en.wikipedia.org/wiki/Penrose_tiling" which have a particular pattern that is also ubiquitous in medieval buildings in Middle-East? Now which one is the real science?

But I'm not trying to instigate a battle between math and science. OK. so math 'models' physical problems in the domain of scientific inquiry. Fair enough. I believe humans can encounter the patterns in nature by various modes of inquiry and expression; physics, maths, art, music, metaphysics, etc. So to me, Roger Penrose not being aware that his discovery of the periodicity and symmetry of an array of shapes being connected to crystal structures isn't a question of which method of inquiry supersedes the other but simply what knowledge came first.
But this still doesn't answer my question about what kinds of systems maths models.

 

[trambolin]

maybe I don't understand your question then. Care to repeat it?

By the way, I am nothing close to being a math ambassador, so don't expect too much from me philosophically. I am just pointing out to an important detail. Mathematical definitions and intuitive analogies in general describe different things no matter how close they might seem.

 

[TheAlkemist]

maybe I don't understand your question then. Care to repeat it?

By the way, I am nothing close to being a math ambassador, so don't expect too much from me philosophically. I am just pointing out to an important detail. Mathematical definitions and intuitive analogies in general describe different things no matter how close they might seem.

I slightly edited my last post -- the end of it. And by the way, thanks for tolerating me in this discussion. I rarely get a chance to have these kinds of conversations with mathematicians so I appreciate it. And I'm far from a science ambassador too.

As far as the relationship between math definition and analogies, I get that. I'm cool with your explanation.

My last question was basically this: does maths model, exclusively, dynamic systems/concepts? By dynamic systems/concepts I mean moving particles/objects in space or that make up space.

 

[trambolin]

Well, in control theory for example, this is exclusively studied, in which physical systems are modeled by differential equations and stability is analyzed. But this does not mean that actual trajectories of pistons, particles, distillation columns are being exactly modeled as moving dynamic objects (it is sometimes approximated and even neglected). Their particular dynamics are described by mathematical objects namely differential equations. But in the end, the stability is characterized by the eigenvalues of a matrix. So imagine, stability of all the possible infinitely many outcome possibility of a system is determined by the eigenvalues of a matrix of its model whose size is mostly less than 200.

You might wonder if this is a coincidence to have differential equations for motion or not. Well, I think yes and no. Since you can also use the very same framework to model the finance or stock markets where money seems to be the moving particle. Or predator prey population dynamics. It is quite interesting to see that the population, namely the NUMBER of individuals in a COLLECTION , and its CHANGE over TIME are all concepts (capitalized) that we have defined on top of these creatures. But we are all treating these notions as separate things since we are used to them as different scientific concepts and naming them as (MATH, PHYSICS, CHEMISTRY etc.). In actuality we are trying to make a consistent story out of it.

 

[TheAlkemist]

Well, in control theory for example, this is exclusively studied, in which physical systems are modeled by differential equations and stability is analyzed. But this does not mean that actual trajectories of pistons, particles, distillation columns are being exactly modeled as moving dynamic objects (it is sometimes approximated and even neglected). Their particular dynamics are described by mathematical objects namely differential equations. But in the end, the stability is characterized by the eigenvalues of a matrix. So imagine, stability of all the possible infinitely many outcome possibility of a system is determined by the eigenvalues of a matrix of its model whose size is mostly less than 200.

You might wonder if this is a coincidence to have differential equations for motion or not. Well, I think yes and no. Since you can also use the very same framework to model the finance or stock markets where money seems to be the moving particle. Or predator prey population dynamics. It is quite interesting to see that the population, namely the NUMBER of individuals in a COLLECTION , and its CHANGE over TIME are all concepts (capitalized) that we have defined on top of these creatures. But we are all treating these notions as separate things since we are used to them as different scientific concepts and naming them as (MATH, PHYSICS, CHEMISTRY etc.). In actuality we are trying to make a consistent story out of it.

So you're saying yes? That mathematics models moving particles, events, sequences, itineraries, etc?

 

[trambolin]

Of course yes, but in a particular way. The topic is getting out of hand. I hope I don't offend you if I stop the discussion here on my side and recommend you a book that explains everything we have discussed here and quite more. Because as I said there are quite subtle constructions that we use when we discuss these issues and one reply window is quite limited.

I have recommended this book many many times on this forum. (Did not get any feedback though.) It is from Lakoff, Nuñez "Where Mathematics Come From" 2000, Basic Books. Received quite some negative criticism by the way which is almost always unavoidable anyhow.

The title is resembles a popular science book but it is far from it. Very boring and repetitive in the beginning but pays off quite nicely in the remaining. I would be more than happy to discuss about the findings of the book.

 

[mathwonk]

I depends on the context. In carpentry, it means the edge of a ruler or the blue mark left by a taut string.] dusted with chalk. In celestial physics it may mean the path of a light beam.

In geometry, there is the desire to have the word refer to many different examples. This leads to the approach of not choosing one example i.e. not "defining" it precisely, but merely saying what properties it should have, possibly in relation to some other undefined objects, like "points". There should also be an undefined relation called "incidence" or lies on.

E.g. one may want to have two types of objects, "lines" and "points, such that two points lie on one line, and two lines have only one point lying on both.

This undefined approach allows the conclusions that are arrived at to apply to many different examples of "lines" and points".

I also found this very confusing for a long time. Until I realized the benefit of no0t saying exactly what a line is allows it to be many different things in different settings.

However once the particular setting is chosen, one should say what the lines are in the case. E.g. Halls above said that if the setting is the Cartesian plane R^2, then the lines are usually chosen to be solutions of linear equations.

but if the setting is a class of functions that solve a given linear differential equation, then lines are sets of functions of form f+cg where f,g are different solutions and c is an arbitrary constant.


On a curved space with a distance, a "line" may be a path that has minimal length in some sense.

Intuitively lines have two geometric properties: "one dimensional", and "straight".

Those notions also can have different meanings in different settings.

good question.

 

[TheAlkemist]

Of course yes, but in a particular way. The topic is getting out of hand. I hope I don't offend you if I stop the discussion here on my side and recommend you a book that explains everything we have discussed here and quite more. Because as I said there are quite subtle constructions that we use when we discuss these issues and one reply window is quite limited.

I have recommended this book many many times on this forum. (Did not get any feedback though.) It is from Lakoff, Nuñez "Where Mathematics Come From" 2000, Basic Books. Received quite some negative criticism by the way which is almost always unavoidable anyhow.

The title is resembles a popular science book but it is far from it. Very boring and repetitive in the beginning but pays off quite nicely in the remaining. I would be more than happy to discuss about the findings of the book.

thanks. i'll check it out.

I depends on the context.

However once the particular setting is chosen, one should say what the lines are in the case.

...and stick to it. yes, i agree. but there are many examples where this ins't the case.

 

[HallsofIvy]

undefined terms? how can something exist if it has no definition? Although something can have a definition and not exist. Like Bugs Bunny for example.

The same way you can have a generic "template" with blank spaces to be filled in for a particular purpose.

"Undefined terms" are absolutely essential to mathematics! They are what gives mathematics it great generality and applicability. In order to use mathematics to solve a given problem, in physics, economic, or whatever, I have to find a mathematical model which means we must find a mathematical structure in which we interpret the undefined terms as concepts in the application. As long as the "axioms" and "postulates" are true statements about the particular application, then you know that all theorems and methods proved from those axioms and postulates are true.

so the act of visualization/looking transforms a bunch of points into a line?


so which is it? Or are you saying the definition is contextual?



The first definition you have of space makes more sense. In science, space is described as boundless, infinite and without form.

This was posted under "General Mathematics", not physics or any other "science".

We perceive/experience structures in space with respect to 3 dimensions. The second definition, the one you call the set theory definition, where space in MADE from stuff, not so much.


I'm not a mathematician so please forgive my ignorance as I attempt to explain what I understand about the epsilon-delta thing. It's basically asking when does a set of discrete things become continuous?
Space isn't discrete. Stuff IN space is.

 

[daniel rey m.]

I was surprised to see that nobody gave the simple, clear, elementary definitions that marveled me when I came upon them in my venerable Webster's New School & Office Dictionary (1946):

Point - that which has position but no magnitude

Line - length without breadth

Oddly, when it comes to defining "plane" it fails to follow through and it says uninspiringly "in geometry, an even superficies", when it should've completed the glorious series by saying "length and breadth without depth".

 

[trambolin]

Because they are limited and context dependent

Point - that which has position but no magnitude

For example a function can be seen as a point belonging to a space of, say continuous functions. Why should a function have a magnitude? This is only a geometrical insight and does not help with understanding mathematics in general.

Line - length without breadth

Same here... also why should anything be compatible with the ordinary 3D dimensions? Why don't you define a line as saying a line = breadth without depth. Why do you start from length and depth comes always the third place? That's why we spent two pages just to come to half conclusion.

Oddly, when it comes to defining "plane" it fails to follow through and it says uninspiringly "in geometry, an even superficies", when it should've completed the glorious series by saying "length and breadth without depth".

Because of the problems I have tried to mention above. It can help you up to a point. Then you have to leave the safety of "seeing things" and move to "handling things".

 

[D H]

undefined terms? how can something exist if it has no definition?

http://userpages.umbc.edu/~rcampbel/Math306/Axioms/Hilbert.html

 

[equilibrum]

Would it be appropriate to say that a line is the direct displacement between two points?

 

[HallsofIvy]

Would it be appropriate to say that a line is the direct displacement between two points?

Only if you first defined "direct displacement"!

 

[daniel rey m.]

"Only if you first defined 'direct displacement'!" - HallsofIvy

"Movement on a flat surface along the shortest path" (i.e., no beating around the bush, no detours, no bypasses, but linger along the way and watch the landscape if you like)

"(…) why should anything be compatible with the ordinary 3D dimensions? Why don't you define a line as saying a line = breadth without depth? Why do you start from length and depth comes always the third place?" - trambolin

One couldn't possibly define a line in such a way because breadth implies at least two dimensions. "Breadth" means "the measure of any surface from side to side". Also, "depth" means "the state or degree of being deep", and neither lines nor planes are deep, which is why they are to be seen in the imaginary two-dimensional world someone thought up and called Flatland many years ago. Consequently, as one progresses from the definition of the point, to that of the line and, finally, to that of the plane, one must start out and then proceed as indicated.

What do you mean, "always the third place"? It has to be mentioned in the third case for the very first time because it must be denied when defining a plane.

"That's why we spent two pages just to come to half conclusions." - trambolin

The two pages were ploughed through as a first step so as not to make any comments "a priori" and the warnings concerning context were duly assimilated. The definitions taken from a general-purpose dictionary were offered on the understanding that they are a first approximation to the matter.

 

[trambolin]

"Only if you first defined 'direct displacement'!" - HallsofIvy

"Movement on a flat surface along the shortest path" (i.e., no beating around the bush, no detours, no bypasses, but linger along the way and watch the landscape if you like)

I gave you an example, how do you define a line in the set of continuous functions? You are only giving us vague geometric analogies but not actually defining something.

"(…) why should anything be compatible with the ordinary 3D dimensions? Why don't you define a line as saying a line = breadth without depth? Why do you start from length and depth comes always the third place?" - trambolin

One couldn't possibly define a line in such a way because breadth implies at least two dimensions. "Breadth" means "the measure of any surface from side to side". Also, "depth" means "the state or degree of being deep", and neither lines nor planes are deep, which is why they are to be seen in the imaginary two-dimensional world someone thought up and called Flatland many years ago. Consequently, as one progresses from the definition of the point, to that of the line and, finally, to that of the plane, one must start out and then proceed as indicated.

So, a line namely the z-axis itself cannot be defined, if I am looking towards the x-y plane because it has no length but depth. See our point? It immediately creates confusion on the simplest examples if you insist on having physical values in the definition. We are creating an unnecessary expectation by using physical analogies which we must avoid when we are trying to communicate with these ideas. We can not and should not shape the idea before we convey to others because every individual has a different imaging device in the brain.

What do you mean, "always the third place"? It has to be mentioned in the third case for the very first time because it must be denied when defining a plane.

We have an underlying habit of generalizing two dimensions to three i.e starting from length then width and depth etc. We are used to look at the flatland from the top. But we can also look at it from the +ve x-axis and see things becoming bigger or smaller and also moving up and down. There is absolutely no constraint for doing not so. But it is just not the habit.

"That's why we spent two pages just to come to half conclusions." - trambolin

The two pages were ploughed through as a first step so as not to make any comments "a priori" and the warnings concerning context were duly assimilated. The definitions taken from a general-purpose dictionary were offered on the understanding that they are a first approximation to the matter.

Sorry I don't understand what you say here.

 

[equilibrum]

Only if you first defined "direct displacement"!

daniel rey m. 's explanation is similar to what i was trying to convey but really,anything goes. A line is an axiom anyway,no?

 

[daniel rey m.]

"A line is an axiom anyway,no?" -- equilibrium

No, not according to Hilbert's axioms for geometry. D H, our friendly neighborhood PF Mentor here, has furnished us a link to them, and Hilbert says that "point", "line" and "plane" are "undefined terms".

" So, a line namely the z-axis itself cannot be defined, if I am looking towards the x-y plane because it has no length but depth. See our point?" -- trambolin

That is so in a 3-D context, but not when defining a line as an ideal, isolated entity, which is how it's done when teaching math on a first, elementary approach.

" We are used to look at the flatland from the top. But we can also look at it from the +ve x-axis and see things becoming bigger or smaller and also moving up and down." -- trambolin

I'm still trying to understand that. Do you mean to say look at it edge-on, as when you place a rigid piece of cardboard horizontally before your eyes and all you see is a line? In that case all you'll see of Flatland will also be a line, not things going up and down, since in that case they'd be jumping out of Flatland and falling back into it. From no point of view would you see anything there moving up and down. From above you'd see them moving within its two dimensions, on the same plane, somewhat like an airplane passenger looking down.

"Sorry I don't understand what you say here." -- trambolin

I couldn't've expressed myself more clearly in that last paragraph. It's a paragon of conciseness and clarity! Maybe the expression in Latin is causing the confusion? Meditate on its opposite --"a posteriori"-- and then maybe you'll grasp the idea in all its shining glory. Sorry, I refuse to be your virtual dictionary.

 

[trambolin]

"Sorry I don't understand what you say here." -- trambolin

I couldn't've expressed myself more clearly in that last paragraph. It's a paragon of conciseness and clarity! Maybe the expression in Latin is causing the confusion? Meditate on its opposite --"a posteriori"-- and then maybe you'll grasp the idea in all its shining glory. Sorry, I refuse to be your virtual dictionary.

Thanks, that's very kind of you.

 

[HallsofIvy]

A line is an axiom anyway,no?

I'm not sure what that even means. A "line" is a geometric object. An axiom is a statement. How can an object be a statement? I think what you intend is what I said earlier- that "line" is an undefined term in geometry.