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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?

MJD
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Aaryan Dewan
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  • Euclidian said that the point is just a thing without any dimentions. – Mikasa Dec 11 '15 at 16:52
  • @BabakS. Then how can it exist? – Aaryan Dewan Dec 11 '15 at 16:52
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    Points are mathematical abstractions, planck length is something in the physical world. – Eff Dec 11 '15 at 16:53
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    The world is not a rational place. – copper.hat Dec 11 '15 at 16:55
  • Nobody asked him why should we have such this creature! But philosophically it exists – Mikasa Dec 11 '15 at 16:55
  • Mathematics starts with axioms or 'declared truths' and attempts to use various rules (possibly axioms themselves) to derive further truths. Physics tries to find a set of axioms that describes our physical world. In much of geometry, lines and points are abstract objects, and often interchangeable, the character of which is left undefined. (As an aside, Planck’s length is a quantity to which no physical meaning is currently ascribed.) – copper.hat Dec 11 '15 at 17:06
  • See the Wikipedia article on Foundations of Geometry for an introduction to the axiomatic method. As Hilbert once remarked about primitive terms "One must be able to say at all times--instead of points, straight lines, and planes--tables, chairs, and beer mugs" – Bill Dubuque Dec 11 '15 at 18:45

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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.

Stefan Perko
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  • I'd like to upvote you for the first paragraph, but I really can't endorse the second paragraph. The concept of a point is much more general than Cartesian coordinates, and understanding that generality is greatly beneficial to getting past the "mathematics represents reality" misconception. – R.M. Dec 11 '15 at 18:43
  • @R.M. I get your point (pun intented), so I amended my answer. The reason why I wrote the second paragraph that way is because you can understand it with high school mathematics and it easily explains, why there may be infinitely many points between any two points. – Stefan Perko Dec 12 '15 at 07:59
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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.

Deusovi
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  • But $0$ could be ${{}}$ for example so it is something concrete. – Mikasa Dec 11 '15 at 16:58
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    @BabakS. Usually, this set is called $1$. Regardless, I would not say this definition really makes $0$ concrete": Have you ever seen an empty set lying around in nature? – Stefan Perko Dec 11 '15 at 17:08
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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.

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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.

mahbubweb
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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.

Ian Miller
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  • If it does not exist- How can you say “ Sam takes 5 hours to go from one point to another” ? – Aaryan Dewan Dec 11 '15 at 16:54
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    @AaryanDewan: That is a different use of the word - in that sentence, it means "location". – Deusovi Dec 11 '15 at 16:55
  • How can you then define the centre of mass? Its a point off course ? Does that mean- There is no centre of mass of an object? – Aaryan Dewan Dec 11 '15 at 16:57
  • "It's a point off course" would imply there is a direction something should be going and there is a difference between that and the direction currently being followed. If I told you to walk North and you walked North East, then you could be off a few points as a way to describe that difference being something of value that is another interpretation of the term "point." Have I made my point? (Pun intended.) – JB King Dec 11 '15 at 17:06
  • @AaryanDewan: Center of mass is an approximation, just like points themselves. There aren't any perfect shapes in real life and we can't measure any objects with infinite precision. Center of mass makes calculations a lot easier and it works for most practical purposes, but if you're measuring things on the atomic level you'll never be able to find an exact center of mass because one does not exist. – Deusovi Dec 11 '15 at 17:34