What is a Point? | Definition and Meaning

[zorro]

What is a point?

 

[Fredrik]

Members of arbitrary sets are sometimes referred to as points, but usually the term is used only for members of topological spaces (sets where we have specified which subsets to call "open sets").

Is that the kind of point you're interested in, or are you talking about points in geometry?

 

[zorro]

Point in Geometry.

 

[disregardthat]

The meaning of the word point depends on the context in which it is being used. In euclidean geometry it is taken as a primitive object for which we provide axioms. That is, it is not defined in terms of some previous concept. It is merely introduced by saying what properties it has. If you consider the plane as R^2 you can consider a point as an ordered pair of real numbers.

 

[Mentallic]

A point has no dimensions. It just just an arbitrary way to visualize a certain point (can't think of any other simile) in a single or multi-dimensional plane.

 

[zorro]

If a point is undefined, then there should not be any definition for a line either as line is a collection of points. Why do we define a line then?

 

[HallsofIvy]

Why do you think "there should not be any definition for a line either as line is a collection of points"? That certainly doesn't follow. I might point out that we could do it the other way- start with "line" as undefined, then define "point" as "the intersection of two lines". But, either way, starting with certain "primative notions" as Jarle says, then defining other things in terms of those is standard in mathematics.

 

[zorro]

A point has no dimensions. It just just an arbitrary way to visualize a certain point (can't think of any other simile) in a single or multi-dimensional plane.

If we draw two intersecting lines with a sketchpen, we get a big point.
If we do the same with a sharpened pencil, we get a small point.
I wonder if a point is really 'dimensionless'.

 

[Fredrik]

If we draw two intersecting lines with a sketchpen, we get a big point.
If we do the same with a sharpened pencil, we get a small point.
I wonder if a point is really 'dimensionless'.

Fortunately, lines in mathematics aren't defined using pens or pencils.

 

[zorro]

Fortunately, lines in mathematics aren't defined using pens or pencils.

Can you draw a line without a pen or pencil? There is something beyond just definition.

 

[ask_LXXXVI]

If we draw two intersecting lines with a sketchpen, we get a big point.
If we do the same with a sharpened pencil, we get a small point.
I wonder if a point is really 'dimensionless'.

The point you are talking about are simply approximations of abstract mathematical concepts.
What you draw using a sketchpen/pencil will appear different from different perspectives.

Points,circles,lines are definite mathematical concepts.They have an existence in an objective sense.Some prefer to say they exist in a Platonic world (the concept of Platonic world was envisaged by the Greek Philosopher Plato). Perfect circles,lines points may or may not be there in our physical universe.So don't try to "find" point in our universe.Think of a "point" as a mathematical concept.

 

[Borek]

As my math professor used to say - you can draw a line through any three points, assuming line is thick enough.

 

[ask_LXXXVI]

Can you draw a line without a pen or pencil? There is something beyond just definition.

like I said in previous post, the line which you draw using pen or pencil is just an approximation of the mathematical concept of a 'line'.In Euclidean geometry,it is a series of points that extends in 2 opposite directions without end.

In any mathematical system you need to have axioms. If you take 'point' to be axiom you can define 'line' out of it or vice-versa as a previous poster said.

 

[zorro]

Points,circles,lines are definite mathematical concepts.They have an existence in an objective sense.Some prefer to say they exist in a Platonic world (the concept of Platonic world was envisaged by the Greek Philosopher Plato). Perfect circles,lines points may or may not be there in our physical universe.So don't try to "find" point in our universe.Think of a "point" as a mathematical concept.

hmm... That means all the geometry work I do is an approximation.

 

[Borek]

No. Whatever you draw is an approximation. But if you calculate hypotenuse of right triangle with legs 3 & 4 to be 5, it is an exact result.

 

[Redbelly98]

If a point is undefined, then there should not be any definition for a line either as line is a collection of points. Why do we define a line then?

We don't. I just checked my high school geometry book, which says the terms "point", "line", and "plane" are accepted as intuitive concepts and not defined. They are used in the definitions of other terms, however.

You can't define every single term in math, not without getting into circular definitions. There has to be a starting point.

 

[zorro]

No. Whatever you draw is an approximation. But if you calculate hypotenuse of right triangle with legs 3 & 4 to be 5, it is an exact result.

How did you measure those legs?
By drawing them and using a scale?
1) Drawing them is an approximation.
2) Taking the reading is an approximation.

We can't do geometry without approximations!

 

[Fredrik]

You're missing the point. He's saying that if the two shorter sides of a right triangle* are 3 and 4 respectively, the longest side is 5. This is a theorem of geometry, and there's no fact about measurements on physical objects that can change that. The measurements you're talking about don't have anything to do with geometry.

*) Note that a triangle is a mathematical concept, not a physical object.

 

[G037H3]

A monad having position.

It's the concept of a position in space, a location.

 

[HallsofIvy]

A monad having position.

It's the concept of a position in space, a location.

And that definition will make sense if you first define "monad", "position", "space", and "location". Do you really consider those to be more fundamental notions than "point"?

You will also need to specify what discipline you are referring to- physics, mathematics, philosophy, ... ?

 

[G037H3]

And that definition will make sense if you first define "monad", "position", "space", and "location". Do you really consider those to be more fundamental notions than "point"?

You will also need to specify what discipline you are referring to- physics, mathematics, philosophy, ... ?

math+philosophy

I personally find monad with position to capture the essence of what a point is more than saying 'point'

Aristotle said that they have to be accepted as axioms, which is obviously true, but for the sake of illuminating the concept, monad with position is accurate

 

[Mentallic]

How did you measure those legs?
By drawing them and using a scale?
1) Drawing them is an approximation.
2) Taking the reading is an approximation.

We can't do geometry without approximations!

Most certainly not! If we try draw them and measure them that way, we will be making approximations and as such we won't be getting a perfect 3,4,5 side triangle.
Why do we label two identical angles as being the same? Couldn't we just see it? No, because drawings aren't always perfectly accurate. The idea that they are exactly equal is still valid and we make theories and assumptions on that idea.

A proof to show that the angles in any triangle add to 180^o is a definite requirement. We can't just draw up any triangle and measure the angles that way, we will get an approximation and this doesn't prove anything.

 

[epenguin]

You're missing the point.

That's what's worrying us all.

 

[zorro]

Most certainly not! If we try draw them and measure them that way, we will be making approximations and as such we won't be getting a perfect 3,4,5 side triangle.
Why do we label two identical angles as being the same? Couldn't we just see it? No, because drawings aren't always perfectly accurate. The idea that they are exactly equal is still valid and we make theories and assumptions on that idea.

A proof to show that the angles in any triangle add to 180^o is a definite requirement. We can't just draw up any triangle and measure the angles that way, we will get an approximation and this doesn't prove anything.

Nice point. I got it

 

[Diffy]

What is a point? That which has no part.

 

[Prpan]

Why should point have no definition? I guess it has a pretty good definition in integral calculus as a device for integration of 0-dimensional infinitesimally small quantities (points in algebraic and geometric sense) into multidimensional objects. 1-D integral will give a path, which is still an abstract object but integration over volume will give a real 3D object. Solving Zeno's paradox of the Tortoise and Achilles in integral calculus is a good example of linking the abstract concepts with the concepts of the real world.

 

[NotEnuffChars]

A point is something you make when debating a topic, eg debating about what a point is :D

Seriously though, I just define a point to be somewhere on a plane that I can describe using some sort of co-ordinate system, cartesian, polar and so forth. It seems to work for me.

 

[sk_saini]

What is a point?

A point is a geometrical figure which has existence with no dimensions.

 

[HallsofIvy]

math+philosophy

I personally find monad with position to capture the essence of what a point is more than saying 'point'

Aristotle said that they have to be accepted as axioms, which is obviously true, but for the sake of illuminating the concept, monad with position is accurate

Perhaps so, after you have defined "monad"!

What is a point? That which has no part.

Define "part".

Why should point have no definition? I guess it has a pretty good definition in integral calculus as a device for integration of 0-dimensional infinitesimally small quantities (points in algebraic and geometric sense) into multidimensional objects. 1-D integral will give a path, which is still an abstract object but integration over volume will give a real 3D object. Solving Zeno's paradox of the Tortoise and Achilles in integral calculus is a good example of linking the abstract concepts with the concepts of the real world.

And what definition is that?

A point is something you make when debating a topic, eg debating about what a point is :D

Seriously though, I just define a point to be somewhere on a plane that I can describe using some sort of co-ordinate system, cartesian, polar and so forth. It seems to work for me.

But there are many different things that can be given in terms of a co-ordinate system. Which of them is a "point"?

A point is a geometrical figure which has existence with no dimensions.

How are you defining "dimension"?

 

[Fredrik]

I'm laughing every time I open this thread, imagining people who haven't gotten to the pythagorean theorem yet being told that a point is a monad with position.

 

[zorro]

I'm laughing every time I open this thread, imagining people who haven't gotten to the pythagorean theorem yet being told that a point is a monad with position.

 

[G037H3]

I'm laughing every time I open this thread, imagining people who haven't gotten to the pythagorean theorem yet being told that a point is a monad with position.

Well, I study Heath's translation of the Elements. It introduces a lot of concepts to explain perspectives on what things are that are actually more difficult than the things they're describing. Many ways to define space and figures, etc.

 

[mathwonk]

As Hausdorff said, in reference to definitions of cardinal numbers as equivalence classes of sets under bijection, or some such nonsense, "For our purposes we do not need to know what numbers are, just how they behave".

Hence to understand points in geometry, it must be [and is] sufficient to know the axioms which describe their properties.

At least that is the axiomatic approach to geometry. if you prefer the model oriented approach, then you can define a Euclidean plane as the set of all ordered pairs of real numbers, and then in that model, a point is by definition an ordered pair of reals. E.g. in that model (1,2) is a point.

 

[NotEnuffChars]

But there are many different things that can be given in terms of a co-ordinate system. Which of them is a "point"?

I'm not sure what you're point is

Here's an example to qualify my statement.

A point in cartesian co-ordinates can be defined by x units in one direction, y units in a perpendicular direction and z units in the final perpendicular direction, usually denoted by (x,y,z). Similarly in spherical co-ordinates, (r, theta, phi) and so forth.

 

[Redbelly98]

I'm not sure what you're point is

Here's an example to qualify my statement.

A point in cartesian co-ordinates can be defined by x units in one direction, y units in a perpendicular direction and z units in the final perpendicular direction, usually denoted by (x,y,z). Similarly in spherical co-ordinates, (r, theta, phi) and so forth.

The point of HallsOfIvy's post was that, if you insist on a definition for everything, you'll then need to define all the terms used in those definitions. And the same for the definitions of those terms, etc. etc.

So, what are the definitions of "coordinate", "unit" and "direction"?

 

[NotEnuffChars]

The point of HallsOfIvy's post was that, if you insist on a definition for everything, you'll then need to define all the terms used in those definitions. And the same for the definitions of those terms, etc. etc.

So, what are the definitions of "coordinate", "unit" and "direction"?

Jarle answered that back on the first page. We need somewhere to start don't we?

Prior to the Hindu-Arabic mathematicians, calculations were made using Roman Numerals, which made arithmetic fairly difficult. Along comes these Hindu-Arabic mathematicians who "invented" the idea of a decimal point, which has since been widely used as a system to divide indivisible whole numbers. Clearly, it was a turning point in mathematics history.

That principle of a decimal point is like the starting point of a model that future mathematicians can use to define further principles, in my example, it gave rise to negative numbers or in the discussion example, a point, we can define a line and so forth.

 

[OmCheeto]

A point is a period ( . )

Only it has no radius.

So you only know it is there by defining it's coordinates.

 

[Redbelly98]

Jarle answered that back on the first page. We need somewhere to start don't we?

Yes, I agree completely:

You can't define every single term in math, not without getting into circular definitions. There has to be a starting point.

 

[chiro]

If we draw two intersecting lines with a sketchpen, we get a big point.
If we do the same with a sharpened pencil, we get a small point.
I wonder if a point is really 'dimensionless'.

If you're thinking in dimensions, a dimension in some system represents the number of independent variables needed to describe the system.

A point has zero dimensions because it needs no variables to describe it: it is just a constant definition.

A typical line (example y = x + 2) is one dimensional because it has one degree of freedom. If you supply a y (or an x) you get the other corresponding value which is dependent on the value you already supplied.

If I had an inequality like y <= x + 2 then we have a two dimensional system since y and x are independent of each other unless they lie on the line y = x + 2.

 

[Mu naught]

A point is the intersection of two lines

 

[zorro]

A point is a period ( . )

A point is not a period.
We denote it using a period, which is just an approximation.
A point can be just imagined - it has no radius. It can be defined as a circle with radius tending to 0 ( or more accurately equal to 0 ).

 

[OmCheeto]

A point is not a period.
We denote it using a period, which is just an approximation.
A point can be just imagined - it has no radius. It can be defined as a circle with radius tending to 0 ( or more accurately equal to 0 ).

I believe that if you'd read past my first sentence, you would see that we are in complete agreement.

 

[zorro]

I believe that if you'd read past my first sentence, you would see that we are in complete agreement.

We are in agreement only past your first sentence.
I was talking about first

 

[michaelc187]

How about a lack of pts elswhere

 

[Outlined]

It is an element of the set Rⁿ.