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Math Help - A possible new paradox puzzle?

  1. #1
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    A possible new paradox puzzle?

    Start off with an idealized straight line which is one-dimensional.

    Now let's take a ruler to measure this straight line. It appears that the ruler must be two-dimensional to measure the one-dimensional straight line (most rulers are lined which make them two-dimensional - can you think of a one-dimensional ruler which can measure the straight line?)

    What I'm saying is that it might not make sense to say that a straight line is one-dimensional if it can only be measured by a two-dimensional ruler, so again, can you think of a one-dimensional type of ruler?
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    There ain't no such thing as "a line"; we chose to invent markers (like a pencil) and straight edges
    and using those we "draw something we choose to call a line" joining two Cartesian points.
    The length of the line is really not what we measure by "ruler" (that's to give an approximation),
    but is obtained by formula using the coordinates of two end points.
    PROVE to me that your ruler is precise; it can't be, since the "lines" on the ruler have "width"...
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    Quote Originally Posted by wonderboy1953 View Post
    Start off with an idealized straight line which is one-dimensional.

    Now let's take a ruler to measure this straight line. It appears that the ruler must be two-dimensional to measure the one-dimensional straight line (most rulers are lined which make them two-dimensional - can you think of a one-dimensional ruler which can measure the straight line?)

    What I'm saying is that it might not make sense to say that a straight line is one-dimensional if it can only be measured by a two-dimensional ruler, so again, can you think of a one-dimensional type of ruler?
    Here's a one-dimensional ruler of sorts. Say you're a one-dimensional creature, and you have the ability to move forward at constant speed. So you measure distance by moving in a direction and timing yourself from start to finish.
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    Wilmer

    Quote Originally Posted by Wilmer View Post
    There ain't no such thing as "a line"; we chose to invent markers (like a pencil) and straight edges
    and using those we "draw something we choose to call a line" joining two Cartesian points.
    The length of the line is really not what we measure by "ruler" (that's to give an approximation),
    but is obtained by formula using the coordinates of two end points.
    PROVE to me that your ruler is precise; it can't be, since the "lines" on the ruler have "width"...
    "There ain't no such thing as 'a line';" Conceptually there is even if it's undefined (not trying to make a joke here) as that gives the idea of a line its greatest power as that can be applied to many situations.

    "PROVE to me that your ruler is precise;" Why? I never asserted that I'm using a precise ruler. All I'm asserting is that for the one-dimensional concept of a line that you must use a two-dimensional ruler (imprecise that it may be) to measure with, meaning that the one-dimensional line has existence only in two-dimensional (or higher) mathematical space. Can YOU Wilmer, or anybody else, prove otherwise? (that's the puzzle)
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    The creature

    Quote Originally Posted by undefined View Post
    Here's a one-dimensional ruler of sorts. Say you're a one-dimensional creature, and you have the ability to move forward at constant speed. So you measure distance by moving in a direction and timing yourself from start to finish.
    How would this creature measure distance in the first place (e.g. how would it know about a basic unit of distance, say a foot?)
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    Quote Originally Posted by wonderboy1953 View Post
    How would this creature measure distance in the first place (e.g. how would it know about a basic unit of distance, say a foot?)
    The creature can mark points, and can control its speed and direction very well.

    The creature has pondered mathematics and knows about Euclidean norms.
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    Distinguishing the lines

    Quote Originally Posted by undefined View Post
    The creature can mark points, and can control its speed and direction very well.

    The creature has pondered mathematics and knows about Euclidean norms.
    Wouldn't the two lines need to be kept seperate so that the second can measure the first, otherwise how would this creature be able to distinguish the two?
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    Quote Originally Posted by wonderboy1953 View Post
    Wouldn't the two lines need to be kept seperate so that the second can measure the first, otherwise how would this creature be able to distinguish the two?
    What two lines? The creature is only moving along one line. We can even restrict the creature's universe to a single line.

    Alternatively we can suppose the creature has echolocation, and can measure distances that way.
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    Well, if you include echo location, you'll need a reflecting surface which is perpendicular to the plane of motion of the creature I think. So, this introduces the 2nd dimension I think wonderboy1953 is referring to.
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    Quote Originally Posted by Unknown008 View Post
    Well, if you include echo location, you'll need a reflecting surface which is perpendicular to the plane of motion of the creature I think. So, this introduces the 2nd dimension I think wonderboy1953 is referring to.
    You can just have a one-dimensional object that acts as an obstruction.
    Last edited by undefined; September 8th 2010 at 10:45 AM. Reason: "barrier that acts as an obstruction" was redundant
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    Quote Originally Posted by Unknown008 View Post
    Well, if you include echo location, you'll need a reflecting surface which is perpendicular to the plane of motion of the creature I think. So, this introduces the 2nd dimension I think wonderboy1953 is referring to.
    What undefined is saying is that, in a straight line, the creature is sending out a signal that would bounce off of something and return to the creature. But that raises other questions (what is the nature of the thing that the signal is bouncing off of?, how did that reflecting thing get on the line to begin with? can the creature go through the reflecting thing like a ghost? are just some of the questions to consider).
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    Quote Originally Posted by wonderboy1953 View Post
    What undefined is saying is that, in a straight line, the creature is sending out a signal that would bounce off of something and return to the creature. But that raises other questions (what is the nature of the thing that the signal is bouncing off of?, how did that reflecting thing get on the line to begin with? can the creature go through the reflecting thing like a ghost? are just some of the questions to consider).
    Well supposing we restrict the creature's universe to a single straight line, obviously there are no atoms in the creature's universe as there are in ours, so I think in order to decide whether something like that is plausible is merely up to whether we can come up with consistent rules for how things might work, and I see no reason we couldn't come up with such rules, such as rules for whether one object can pass through another, or what forces might be acting on the relevant objects during contact, etc. That universe would only exist in our minds though, that we can would know of. But how would we be able to say such a thing is impossible?
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    Just some comments

    Quote Originally Posted by undefined View Post
    Well supposing we restrict the creature's universe to a single straight line, obviously there are no atoms in the creature's universe as there are in ours, so I think in order to decide whether something like that is plausible is merely up to whether we can come up with consistent rules for how things might work, and I see no reason we couldn't come up with such rules, such as rules for whether one object can pass through another, or what forces might be acting on the relevant objects during contact, etc. That universe would only exist in our minds though, that we can would know of. But how would we be able to say such a thing is impossible?
    When I posted this paradox it was under the assumption that it's within this universe. Matter takes up space making it three-dimensional and waves also take up space making them three-dimensional.

    Echolocation is based on biosonar which uses matter to send signals back and forth. It appears that echolocation wouldn't be feasible in our universe the way that undefined is suggesting (at the one-dimensional level). I also want to mention that at the atomic level, Heisenberg's uncertainty principle is predominant meaning that solid obstructions are relative and can be penetrated by waves, going straight through the solid obstructions (but how would these waves be reflected back?).

    I think you can understand better why I say the concept of one dimensionality leads to a paradox in our universe (think about it please).
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    Quote Originally Posted by wonderboy1953 View Post
    When I posted this paradox it was under the assumption that it's within this universe. Matter takes up space making it three-dimensional and waves also take up space making them three-dimensional.

    Echolocation is based on biosonar which uses matter to send signals back and forth. It appears that echolocation wouldn't be feasible in our universe the way that undefined is suggesting (at the one-dimensional level). I also want to mention that at the atomic level, Heisenberg's uncertainty principle is predominant meaning that solid obstructions are relative and can be penetrated by waves, going straight through the solid obstructions (but how would these waves be reflected back?).

    I think you can understand better why I say the concept of one dimensionality leads to a paradox in our universe (think about it please).
    If you're talking about restrictions to this universe, then I think you never adequately responded to Wilmer's claim that there's no such thing as a line in this universe.

    You responded: "Conceptually there is even if it's undefined (not trying to make a joke here) as that gives the idea of a line its greatest power as that can be applied to many situations."

    Well conceptually there exist all the things I mentioned. You're setting a double standard here.
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    Point by point

    Quote Originally Posted by undefined View Post
    If you're talking about restrictions to this universe, then I think you never adequately responded to Wilmer's claim that there's no such thing as a line in this universe.

    You responded: "Conceptually there is even if it's undefined (not trying to make a joke here) as that gives the idea of a line its greatest power as that can be applied to many situations."

    Well conceptually there exist all the things I mentioned. You're setting a double standard here.
    How can Wilmer claim there's no line in this universe? Wouldn't he first have to know what a line is in the first place to deny it? (btw the concept of a line is taught in schools, straight or otherwise. I can picture Wilmer struggling in class while the teacher is going over lines in geometry).

    In regards to me, no double standard as I'm going by this universe which still leads to a paradox. In regards to what undefined is proposing, it'll work in another universe and here I stand.
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