What is a Point? | Definition and Meaning

[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.