how would you define a mathematical point?

Discussion in 'Math' started by PG1995, Sep 26, 2011.

  1. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    Hello

    How would you define a mathematical point? Here is my 'crude' opinion. A point is simply a supposition from mathematician's point of view that a certain number is there where the point or dot is; although it (the number) is not there. But it serves the purpose. What do you say? Please let me hear. Thanks.
     
  2. Georacer

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    Nov 25, 2009
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    The point in the Euclidean geometry is a primitive notion. That means that we must accept it and "come to terms with it", without understanding it through a certain proof.

    As I see it, a point in space describes one unique spot in that space, a spot that is discrete from any other "nearby" point. That spot can be referred to also uniquely by a set of three numbers, its coordinates.
     
  3. PG1995

    Thread Starter Active Member

    Apr 15, 2011
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    @GeoRacer: Thanks, and my opinion isn't that bad! :) Because as I see it there are some indirect similarities between your description and mine, or is it me only?!
     
  4. Georacer

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    True, but I 'd like to think of a point not as a number but as a place in space that can have a unique label. It helps me with more abstract and complex notions more often that not.
     
  5. studiot

    AAC Fanatic!

    Nov 9, 2007
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    There is a little more to it than this.
    We have a definition as well as just taking it on trust.

    Georacer in particular might be interested in the background.

    Euclid's original definition translates as

    "That which has no parts"

    In geometric terms we take this to mean that a point has zero dimensions.

    However we can also look at it from another point of view.

    Lines are divisible

    A point is defined as "that which is not divisible"

    This definition may be more satisfying.

    In modern usage members of a set are called the points of that set.

    The link to euclidian geometry comes when the set contains an infinite number of members or points, each one infinitesimally small.

    go well
     
  6. Georacer

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    I read the wikipedia article too, but it didn't satisfy me to the extent of referring to it.

    The fact that the point has no dimensions should be pretty much considered common knowledge for anyone with little experience on the topic.

    On the other hand, the fact that the point contains no other points but itself might be useful for understanding its primitive and axiomatic character.

    P.S. Point (or Σημείο/Seemeeo in Greek) translates as the one that has been noted and has the same root with sign.
     
  7. studiot

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    Wikipedia article? I've no idea what you mean. My sources are elsewhere.

    The first definition suggests zero dimensions.

    Later mathematics has problems with stacking up a bunch (even an infinite one) of zeros to make a finite total.

    That is why I posted the second definition (which is also ancient greek) where the line and the point are contrasted.

    This neatly sidesteps the zero problem, as does the point set topology definition I also gave.

    go well
     
    PG1995 likes this.
  8. Georacer

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    Nov 25, 2009
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    Heh, Wikipedia has pretty much the same definitions, that's why I made the assumption of reference. Maybe you actually wrote they Wikipedia article.

    It was an interesting conversation, with profound theoretical background. Now I 'll go back to trying to carve a sheet of metal with a drill and a dremmel. ;P
     
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