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In geometry, what is a point? I have seen Euclid's definition and definitions in some text books. Nowhere have I found a complete notion. And then I made a definition out from everything that I know regarding maths. Now, I need to know what I know is correct or not. One book said, if we make a dot on a paper, it is a model for a point. Another said it has no size. Another said, everybody knows what it is.Another said, if placed one after another makes a straight line.Another said, dimensionless.Another said, can not be seen by any means.

ReekMaths
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    It would help if you provided your definition... – apnorton Jan 25 '13 at 19:18
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    A point is undefined almost always in geometry - it is considered an axiomatic object, like a natural number. You can explain what you want a point to represent, but what it "is" is an abstract concept with no definition other than the properties we assert for it in our axioms. – Thomas Andrews Jan 25 '13 at 19:19
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    @ReeksMaths: What is wrong with MathWorld's definition? Regards – Amzoti Jan 25 '13 at 19:20
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    If he knew the definition, it'd miss the point. – Sniper Clown Jan 25 '13 at 19:36
  • @Mahmud: Don't speak nonsense in this forum.It is not any place to speak nonsense. – ReekMaths Jan 25 '13 at 19:43
  • More complex objects are also considered points, eg ultrafilters in Stone-Cech compactification (these are elusive to describe). – alancalvitti Jan 25 '13 at 19:48
  • The first lecture I remember as an undergraduate the lecturer pointed out that a Venn Diagram was a model for sets using collections of points in the plane with a certain cardinality. It blew my mind at the time, but it got home to me the difference between a model of certain axioms (you have some notion from "somewhere else" of what you mean which guides your understanding) and the system defined by the axioms (which does not depend on any particular model). Of course if you want a complete ordered field all models are isomorphic, but that is a fact to be proved from the axioms. – Mark Bennet Jan 25 '13 at 19:53
  • Another example of more complex point is: a polynomial is a point in a vector space. – alancalvitti Jan 25 '13 at 20:21
  • @ThomasAndrews put that in an answer. – John Alexiou Jan 25 '13 at 21:06
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    ReekMaths: Relax, @Mahmud is allowed to make jokes in the comments. Especially if they are... to the point. –  Jan 26 '13 at 07:20
  • I think it is not a place for jokes.And, don't worry Rahul Narain, I assure you that I was relaxed during the joking. – ReekMaths Jan 26 '13 at 16:33

4 Answers4

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Point, in Euclidean geometry, is an undefined notion.

We do not define what a point is, only what properties points must have, and these properties are completely specified by the axioms. This is certainly a modern view of mathematics, and differs from the approach in Euclid's times. Euclid defines point, but the definition is vague, and it is never used anyway.

The modern view can be seen in Hilbert's book "Foundations of geometry", where a modern treatment of geometry is given, with emphasis on the axiomatic approach. To see the extent to which we do not care about what a point could possibly be, see this question, on a famous quote by Hilbert stating that "One must be able to say at all times -- instead of points, straight lines, and planes -- tables, chairs, and beer mugs".

Of course, we usually want to work with very concrete "models" of the axioms, the most famous being the Cartesian plane. In this model, the plane is just $\mathbb R\times\mathbb R$, and we identify a point with an element of this set, that is, an ordered pair $(x,y)$.

Also, there are several (equivalent) ways of axiomatizing geometry, so details of what "basic properties" we assume as a priori will vary depending on what concrete axiomatization or specific model of the axioms one has in mind or is working with.

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Andrés E. Caicedo
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We can't always define everything or prove all facts. When we define something we are describing it according to other well-known objects, so if we don't accept anything as obvious things, we can not define anything too! This is same for proving arguments and facts, if we don't accept somethings as Axioms like "ZFC" axioms or some else, then we can't speak about proving other facts. About your question, I should say that you want to define "point" according to which objects? If you don't get it obvious you should find other objects you know that can describe "point"!

ReekMaths
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omid saba
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My opinion can be argued, but a point is really just a starting or end place for an object. If we make the assumption that a point is real then it must be an intersection of two lines which in reality can be drawn in space. But two lines have to be created which they themselves begin as points and these collection of points extended out through space until somewhere at some place, these two lines will pass through one another and share just one common point between them. I really believe that it's all just mental construct and a way to connect dots.

Armondo Villaescuza
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A point is a 'mind-erected' entity found in the ‘2 dimensional space’ or the ‘3 dimensional space’ model that is constructed by the human-brain. Any point is also the fundamental 'mind-erected entity' of geometry that is size-less, invisible and indivisible, yet points have the power to generate each of the 2D and 3D objects constructed by the brain.

size-less : Any point is too small to exist physically, we mean this when we say that any point is size-less.

‘2 dimensional space’ model : On seeing a flat wooden slab, we can think of it getting thinner and thinner to such an extent that it becomes thickness-less [As we move forward in future, the flat wooden slab gets thinner and thinner and along the future it has always been existing. At thickness-less, it might seem that the wooden is not existing. Since, there is no obstruction in imagining the existence of an invisible wooden block; therefore we are free to conclude that at thickness-less the wooden exists.]. We take this thickness-less surface directly in front of our mind, letting it to stand vertically and then along the boundary we extend it indefinitely. Let this expansion to finish instantaneously, then what we have got is the ‘2 dimensional space’.

‘3 dimensional space’ model : When residing in a box, if anybody kicks out everything from inside the box including himself, then we get empty space in the box. We then throw the box away and pick up the empty space. We place the space in front of our mind, and enlarge it indefinitely. Let this expansion to finish instantaneously, then what we have got is the ‘3 dimensional space’.

‘constructed by the human-brain’ : constructed in the mind

ReekMaths
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