Points

[StevenRoss]

A point does not have any dimension, right? If that is so, then I do not see how more than one point can line-up into a line segment or a line. It seems to me that even an infinite number of points together would have no length.

If a point has no dimension, I do not see how a point relates to anything. If a point does not relate to anything, I do not see how there can be more than one point.

Is there a correspondence to this in arithmetic? When I think of numbers, they appear to me to be discreet. For example, even 0.999 . . . (repeating infinitely) is the discrete number 1.

Thanks in advance for your help.

 

[yoscar04]

A point is a primitive notion or axiom in classical geometry. We can't define a point in terms of other defined objects. Now, many objects in Euclidean geometry consists of an infinite number of points. A point refers more to a specific location than anything else. A line segment, as you wrote, consists of an infinite number of points, its length would be equal to [infinite number of points]X[length of a point (which is 0)]=undetermined form that gives a finite number. There are definitions of dimension in terms of vector spaces, topology, etc. In all cases, the dimension of a point is considered to be 0. I am sure other members of the forum will have more to add. A friend of mine, once sent me the following way of looking at it:
To understand the dimensions of an object (actually a topological space), you can look at it in the following way:
"Consider an ant. Put your ant on the topological space you want to know the dimensions of, and ask yourself: “How many independent directions can that ant walk in on the space I’ve given it?”. Try to see what would be the dimension of a point, a line segment, a plane, space, etc.
Have you heard of the concept degrees of freedom?

 

[R.M.]

This isn't a formal mathematical example, but it helps my geometry students understand the basic idea:
In the solar system, a point can represent a planet or moon.
On the Earth, a point can represent a country.
In a country, a point can be represent a state.
In a state, a point can represent a city.
In a city, a point can represent a building.
In a building, a point can represent a particular room.
In a room, a point can represent the location of an object.
On an object, a point can represent a specific part of the object.
etc........
This can be continued infinitely many times.

Thinking of a point as an address in space, a segment is a straight object that connects 2 addresses together, and includes ALL addresses (points) between them, no matter how infinitely small they are.

 

[StevenRoss]

It is universally said that a geometric point is a location on a plane but I do not understand why that is necessarily the case. Suppose that there were one point and that is all, no other geometric space. Since there would be nothing to relate the point to other than itself, could you say that it has a location? Now, this point would have no dimension, so I do not see how you could even say that the point was here or there. I suppose that there are differences among an empty set, no set and an undefined set, although I am to sure of the differences. I suppose you would tell me that if there is one point and that is all, then there is a set of one point which has no dimension; it is not an empty set or an undefined set. There would be existence in a mathematic sense (although I am worried that I am talking nonsense) without any other attribute.

Also, is there a one-to-one correspondence with spacetime? Suppose there were a spacetime in which there were only one point. You would say that such spacetime is not empty and you would distinguish that spacetime from a spacetime without a point and from an undefined spacetime. Since you would say it is a point in spacetime, and since it would have no dimension, and there would be no spacetime other than this point in spacetime, how could you say that it has a location? So, if a point in spacetime is not necessarily defined by its location, how can there be two different points in spacetime? How can there be two singularities? I guess you would tell me that a spacetime with one point exists even though it has no dimension or location. The physicist Stephen Hawking once said that time is imaginary in both a mathematical and physical sense, but I do not see how a spacetime with only one point would be imaginary. Moreover, I have missed important meetings all my life, which destroyed friendships, relationships and business opportunities, because I have a very poor sense of time and arrive hours too late sometimes. I am working on not being egotistical but sometimes I miss appointments or a late because I forgot.

Is there a one-to-one correspondence between a geometric point and a number of arithmetic? When I think of numbers, they appear to me to be discreet. For example, even 0.999 . . . (repeating infinitely) is the discrete number 1. A point does not have any dimension so maybe a point corresponds to 0. But a point is distinguished from no point, so I do not see how 0 can be right. A point is defined, so a point does not correspond to when we say something in arithmetic is undefined, such as a fraction with 0 in the denominator.

 

[jonah2.0]

Beer soaked query follows.

A point is defined, so a point does not correspond to when we say something in arithmetic is undefined, such as a fraction with 0 in the denominator.

How exactly would you define a point?

 

[StevenRoss]

You asked the question which I have been asking. Many define a point as being a relative location on a plane and I am attempting to define it better than that, if possible.

 

[jonah2.0]

Beer soaked reply follows.

Many define a point as being a relative location on a plane and I am attempting to define it better than that, if possible.

Can you name those books/reference materials or links where these "many" defined a point?

 

[yoscar04]

From Wolfram:
A point is a 0-dimensional mathematical object which can be specified in

-dimensional space using an n-tuple (

,

, ...,

) consisting of

coordinates. In dimensions greater than or equal to two, points are sometimes considered synonymous with vectors and so points in n-dimensional space are sometimes called n-vectors. Although the notion of a point is intuitively rather clear, the mathematical machinery used to deal with points and point-like objects can be surprisingly slippery. This difficulty was encountered by none other than Euclid himself who, in his Elements, gave the vague definition of a point as "that which has no part."

The basic geometric structures of higher dimensional geometry--the line, plane, space, and hyperspace--are all built up of infinite numbers of points arranged in particular ways.

 

[tkhunny]

It would help if you let go of your Euclidean assumptions and visual reliance.

• "Point Geometries" are not new ideas.
• Perhaps it's just a member of a set.