What is exactly a “Point”?

Question

I read somewhere that a line is made up of infinite points. Between any two points on that line, there are another infinite points. and between any two points BETWEEN those 2 points there are another infinite points.

Is this true? If so , HOW?

Also, i have read in physics that the smallest possible length is the “Planck’s Length” . Nothing can be smaller than it. So is the point even smaller than the Planck’s length?

Is a point spherical in shape? Rectangular? Square? Or it does not has any shape? If so, it must not have any length, that means a point does not exist. Is this correct?

Answer 1

First of all and most importantly: Mathematical existence has nothing whatsoever to do with physical existence. Just because something does not "exist physically" does not mean it cannot exist mathematically. Mathematics was partly created to model physics, but this does not mean that mathematics cannot make simplifications (like believing there are points). But even then, applications of mathematics go way beyond "just" physics. You can also think of it this way: Mathematics is so amazing, besides the physics you are used to it may explain any kind of physics you can come up with in your head.

A mathematical point (in let's say two dimensions) is just a pair $(x_0,y_0)$ consisting of real numbers $x_0,y_0$. If you have another point $(x_1,y_1)$, you get a line through these two points. It's the set of all points $(x,y)$ with $y=ax+b$, where $a = \frac{y_0-y_1}{x_0-x_1}$ and $b = y_0 - mx_0$. Of course, there are infinitely many points $(x,y)$ on said line (take any $x$ and calculate the corresponding $y$). But even between $(x_0,y_0)$ and $(x_1,y_1)$, there are infinitely many points. Take any $\lambda\in [0,1]$, then $(\lambda_0 + (1-\lambda)x_1, \lambda y_0 + (1-\lambda) y_1)$ will be a point "between" $(x_0,y_0)$ and $(x_1,y_1)$.

The concept of a point can be generalized to abritary dimensions ($\mathbb{R}^n)$, other rings ($\mathbb{Z}^n$, $\mathbb{Q}^n$, $\mathbb{C}^n$, ...), later to points in a metric space and "finally" to points in a topological space. Then points need not actually represent anything physical. For example you could take the vertices of a graph (that is a thing with vertices and edges between those vertices) to be your points and measure distance between these vertices (how many steps do I need to get from one vertix to another vertix?). This yields a metric space, which does not have anything directly to do with physics, but which is still very interesting and important.

There are also other approaches to formalizing the concept of 'point', for example incidence geometry: It abstractly deals with points and lines, and may be used to formalize different kinds of geometries synthetically.

Answer 2

A point is a abstract theoretical mathematical construction. It does not exist in the real world hence it can be smaller than a Planck length.. It has no shape, zero width, zero length, zero height, etc - it is zero dimensional.

From this you should be able to conceptualize how you can have an infinite number of points between any two given points.

Answer 3

That last part is correct. Points do not exist, just like numbers don't exist. You won't find any points anywhere for the same reason you don't see the number 2 in your backyard. Mathematical concepts do not exist in real life. We can use them to get close to real life situations in certain conditions, but all mathematical ideas are just that - ideas.

Answer 4

First of all, Plank length which is commonly referred to as "the smallest length" is not actually the "smallest length", it's something a bit more complicated; but that has nothing to do with your post. It's simple:

Mathematics is not bound by reality.

Answer 5

When in mathematics we talk about geometric point we talk about point on a plane or in three dimension or in higher dimension or in some abstract geometric space. A point is just an element in a set, on which set we define geometry. Just an element. Nothing more.

When you say, between any points on a line there are infinitely many points you consider the line in some $\mathbb{R}^n$. And line is actually $\mathbb{R}$. So, Euclidean property of real number states between any two real numbers there is a rational and irrational number. This is explains that part.